arXiv Papers of Physics-Informed AI

Abstract:
Physics‑informed neural networks (PINNs) have become a powerful framework for solving PDEs by embedding physical laws into differentiable objectives. Despite their advances, training PINNs remains fragile: recent conflict‑averse optimization schemes alleviate gradient interference between residual and boundary losses, but we show that their effectiveness deteriorates as model capacity increases. In this paper, we identify a capacity‑induced failure mode, where overparameterized networks undergo functional modularity, self‑partitioning into task‑exclusive modules that suppress cross‑objective interaction and hinder convergence toward Pareto‑stationary points. To address this issue, we propose a novel framework, Modular‑Sparsity Synchronization (ModSync), which integrates structural optimization into conflict‑averse training by penalizing task‑exclusive connections while preserving interaction‑promoting pathways. Extensive experiments across diverse PDE benchmarks demonstrate that ModSync consistently prevents capacity‑driven failures, sustains robust cross‑objective coupling, and achieves state‑of‑the‑art accuracy. Codes are available at \urlhttps://github.com/heejokong/ModSync.

Abstract:
We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman‑Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least‑squares QR factorization. The trial space, which we term Linear‑in‑Learnables (LiL), comprises representations whose trainable parameters enter linearly, including random‑feature extreme learning machines, spectral polynomial bases, and trigonometric expansions, each implemented as a physics‑informed neural network. The method thus replaces the nonconvex gradient‑based training that limits standard PINNs with a convex per‑step solve. We establish local Newton‑Kantorovich convergence of the outer iteration to a residual‑limited neighborhood under an explicit smallness condition, with the limiting accuracy governed by the best‑approximation residual of the trial space rather than by an optimization tolerance. The method, denoted LiL‑Q, is assessed on seven benchmarks spanning scalar nonlinear PDEs (Bratu, viscous Burgers, Buckley‑Leverett), coupled systems (plane‑strain elasticity and the incompressible Navier‑Stokes equations in two and three spatial dimensions), and steady‑state Darcy flow with heterogeneous permeability. Across these problems, LiL‑Q converges in single‑digit outer iterations in most cases, even at the coarsest basis sizes and independent of the parameter count. When the exact solution lies in the span of the trial space, the method recovers it to machine precision in a single solve. On the Navier‑Stokes benchmarks, it matches or exceeds published PINN solvers with up to two orders of magnitude fewer trainable parameters, without gradient‑based optimization.

Abstract:
We propose a new weak‑form Physics‑Informed Neural Network approach (named INI‑VPINN). INI‑VPINN naturally incorporates Neumann boundary and interface conditions into the variational formulation. It removes the need for additional loss terms or multiple subdomain networks. This framework employs compact support weighting functions and integration by parts to implicitly impose flux and continuity constraints. In this way, it implicitly ensures physical consistency across material boundaries. The proposed method is tested on Poisson and Laplace problems with sharp interfaces and complex geometries. Results show that, compared with several other Physics Informed Neural Networks‑based formulations, the INI‑VPINN consistently achieves higher accuracy, smoother and faster convergence. The proposed framework provides a general approach for solving multimaterial problems with complex geometries and mixed Neumann‑Dirichlet boundary conditions using neural networks. The implementation is publicly available in a GitHub repository.

Abstract:
Human pose estimation (HPE) utilizing wireless WiFi signals has emerged as a promising technology owing to its device‑free nature, privacy preservation, and robustness against occlusion and poor lighting. However, existing methods often overlook the physical complex phase information of WiFi signals and fail to generalize across diverse environments due to severe domain shifts. In this paper, we present C‑MambaPose, a physics‑informed complex‑valued Mamba‑GraFormer hybrid framework for robust cross‑environment WiFi‑based 3D HPE. Our framework first sanitizes raw WiFi Channel State Information (CSI) phase errors and constructs a phase‑preserving complex‑valued representation. We then employ a Spatiotemporal Complex Mamba encoder with a dynamic selective receptive field to capture fine‑grained phase dynamics. A cross‑attention joint‑query mapper maps the unstructured sequence tokens to human joints, which are decoded by a Graph Convolutional Network (GCN) to predict anatomically coherent 3D coordinates. Extensive evaluations on the MM‑Fi dataset show that C‑MambaPose achieves competitive or superior performance to state‑of‑the‑art baselines across all settings, setting a new state‑of‑the‑art specifically on the challenging cross‑environment split, requiring only 3.78 M parameters‑an 83.1% reduction compared to GraphPose‑Fi~\citechen2026graph and an 85.7% reduction compared to MetaFi++~\citezhou2023metafi++, while maintaining a comparable size to DT‑Pose~\citechen2025towards (which is only 18% smaller) but achieving significantly superior performance without requiring any pretraining. Our code is publicly available at https://github.com/phucngvinuni/cmampose.git.

Abstract:
Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space. However, many existing architectures still struggle to represent nonlinear time‑dependent systems that involve multi‑scale structures, long‑range interactions, and stable long‑time evolution. In this work, we introduce the Hierarchical Adaptive Multi‑scale Neural Operator (HAMNO), a neural‑operator architecture that combines local convolutional representations, global spectral operators, and hierarchical encoder‑decoder processing. The central component of HAMNO is a data‑dependent gating mechanism that adaptively balances local and global information at each spatial location, allowing the model to resolve fine‑scale features while preserving long‑range dependencies. We further develop a physics‑informed extension, PI‑HAMNO, based on a multi‑objective loss strategy that combines data fitting with strong‑ and weak‑form physics constraints. The strong‑form term penalizes the domain‑integrated squared PDE residual in physical coordinates, while the weak‑form term is constructed by multiplying the governing residual by finite‑element test functions and evaluating the resulting element integrals using centroid‑based tetrahedral quadrature. The framework is evaluated on non‑periodic Allen‑Cahn (AC), Cahn‑Hilliard (CH), and Swift‑Hohenberg (SH) equations defined on cubic domains. Across long‑horizon rollout, data‑limited training, out‑of‑distribution initial‑condition shifts, and random‑seed variations, HAMNO improves predictive accuracy over standard neural‑operator baselines, while PI‑HAMNO further enhances stability, physical consistency, and data efficiency. The implementation is publicly available at https://github.com/MBamdad/HAMNO .

Abstract:
Clock Tree Synthesis (CTS) is a computationally expensive stage in the physical design flow, requiring iterative EDA tool invocations to navigate a vast configuration space for optimal power, wirelength, and timing skew. Existing machine learning approaches require computationally expensive retraining or fine‑tuning cycles to adapt to unseen macro architectures and are architecturally mismatched to the millions of evaluations demanded by exhaustive combinatorial search. We present SwiftCTS, a physics‑informed surrogate framework that addresses both limitations simultaneously. By coupling lightweight, physics‑grounded statistical features with gradient‑boosted ensembles, SwiftCTS trains in under five seconds on a CPU and delivers sub‑millisecond inference without GPU support. To handle out‑of‑distribution (OOD) designs without retraining or fine‑tuning, we introduce a K‑shot multiplicative calibration mechanism that anchors predictions to just one or two physical reference runs, reducing power prediction error from 24.5% to 3.3% and wirelength error from 56.6% to under 1% on unseen macros. Integrating this engine with an evolutionary optimizer, SwiftCTS evaluates 100,000 CTS configurations in under ten seconds, yielding Pareto‑optimal frontiers that are physically validated within the OpenROAD flow. Closed‑loop validation confirms prediction errors below 0.5% for power and wirelength, and timing skew predictions within five picoseconds on an OOD benchmark, consistently outperforming default tool heuristics across all target metrics. Code publicly available at: \hrefhttps://anonymous.4open.science/r/SwiftCTS‑7E6Ehttps://github.com/BarsatKhadka/SwiftCTS

Abstract:
Machine learning weather prediction (MLWP) models have achieved impressive forecasting performance at a small fraction of the computational costs required for traditional physics‑based methods. However, they are primarily (1) data‑driven and (2) evaluated using pixel‑wide error metrics (e.g., RMSE), so there are no guarantees that their forecasts are consistent with known physical laws. We introduce PhysMetrics.Weather, an evaluation framework that assesses the physical realism of MLWP models across three types of metrics: conservation, spectral, and dynamical. By quantifying physical realism, this tool guides the development of physics‑informed architectures and helps evaluate whether MLWP models are reliable for operational use. Our framework is available on Github at https://github.com/Emmakast/PhysMetrics.Weather.

Abstract:
Physics‑informed machine learning is often assessed by curve error, although engineering use depends on downstream decisions: ranking candidates, avoiding infeasible designs and limiting regret. We introduce pinn‑gym, an open benchmark for material‑conditioned lattice design that couples a transparent reduced‑order crush‑and‑impact oracle with five printable polymer cards, dimensionless force‑response targets and a protocol spanning curve fidelity, physical admissibility, top‑k retrieval and mass regret. Across per‑material, pooled and cross‑material settings, low nRMSE is frequently insufficient to identify useful design selections. Physics‑informed losses alter trade‑offs rather than monotonically improving all metrics, and dimensionless conditioning improves comparability without making transfer symmetric. The benchmark is not a certified material model; within the released oracle, candidate generator and material cards, pinn‑gym provides a reproducible testbed for evaluating PIML surrogates as decision systems rather than curve predictors alone.

Abstract:
We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three‑phase black‑oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators (FNO) and their physics‑informed variant (PINO). The application focus is the Norne benchmark reservoir, defined on a heterogeneous 46×112×22 grid (N=113,344 cells), with a production history spanning T=30 timesteps covering 3298 days. Our theoretical contributions are organized around four interlocking problems: (1) functional‑analytic formulation in a product‑Sobolev‑space setting, including well‑posedness of the implicit timestep map and sharp local Lipschitz estimates; (2) covariate shift quantification, proving that the Wasserstein‑2 distance grows as W_2 \leq \varepsilon(L^n‑1)/(L‑1), with exponential population‑risk discrepancy for L>1; (3) physics‑constrained spectral stability, showing PINO training with λ_R \geq λ^_R reduces the learned Jacobian spectral radius to ρ_F + Cλ_R^‑1/2, yielding uniform‑in‑time rollout error |δ_n| \leq \varepsilon/(1‑ρ); and (4) K‑step TBPTT gradient analysis, deriving geometric bias decay O(ρ^K), optimal window K^ = O(\log(T/σ^2)), and Adam convergence O(1/\sqrtt) + O(ρ^K^). Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain R^2>0.99 (oil), R^2>0.90 (gas), R^2\approx 0.80 (pressure), and monotonically improving R^2 (water) across the full 3298‑day horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1000‑member ensemble runs in under one minute on a single B200 GPU, giving a ～10^4× wall‑clock speedup over the OPM finite‑volume simulator.

Abstract:
Underwater scene reconstruction is essential for immersive exploration of aquatic environments, yet remains challenging due to complex participating‑media effects such as absorption and scattering, as well as the limited field of view (FoV) of conventional cameras. Although combining panoramic imaging with 3D Gaussian Splatting (3DGS) offers a promising direction for photorealistic underwater rendering, traditional 3DGS struggles with both spherical projection distortion and underwater medium degradation. In this paper, we propose Underwater360, a physics‑informed omnidirectional 3DGS framework for underwater panoramic scene reconstruction. First, we introduce an Omnidirectional Gaussian Splatting module that performs ray casting directly in spherical camera space instead of relying on 2D projection approximations, thereby reducing geometric distortions under 360^\circ FoV. Second, we design a physics‑based appearance‑medium modeling architecture with pose‑conditioned appearance embeddings to explicitly decouple intrinsic scene radiance from depth‑dependent backscatter and attenuation, enabling physically grounded scene appearance restoration. Finally, we establish a new panoramic underwater benchmark dataset containing both synthetic and real‑world scenes. Extensive experiments demonstrate that Underwater360 achieves superior performance in underwater novel view synthesis and scene appearance restoration, delivering improved rendering quality and cross‑view consistency in complex underwater environments. The code and datasets are released at https://github.com/SwcK423/Underwater360

Abstract:
Reconstructing and predicting dynamic 3D scenes from multi‑view videos is a foundational task for robotics, AR/VR, and digital twins. Recent physics‑informed Gaussian Splatting methods achieve impressive future frame extrapolation but lack semantic awareness and suffer from large computational overhead. We introduce R5DGS, a framework that augments a physics‑driven 4D Gaussian representation with compact Identity Encoding vectors, enabling precise Gaussian‑to‑object association. By constructing an offline CLIP‑based object lookup table, we support open‑vocabulary text prompting to retrieve and render object‑specific Gaussians across arbitrary timestamps and viewpoints. Furthermore, we propose a rigid‑body inference constraint that predicts and integrates physical dynamics exclusively for object centroids, propagating motion to associated Gaussians via relative transformations. This optimization yields a 11 FPS speedup during extrapolation without compromising trajectories plausibility.

Abstract:
Efficiently solving Poisson equations on complex, irregular domains remains a fundamental challenge in scientific computing, as classical iterative solvers often suffer from prohibitive runtime due to ill‑conditioned systems. While neural operators offer a fast alternative, they typically rely on large‑scale labeled datasets or struggle with unstable training dynamics when using physics‑informed residual losses. We propose \textscNPSolver, a neural Poisson solver trained without solution labels via iterative physics supervision. Instead of relying on fully converged numerical solutions or raw PDE residuals, \textscNPSolver utilizes a small number of preconditioned conjugate gradient (PCG) steps to refine its own predictions, providing a more stable and well‑scaled training signal. Theoretical analysis confirms that this iterative supervision serves as a well‑conditioned error proxy and that a stop‑gradient design is essential for optimization stability. To better capture boundary‑driven features under mixed boundary conditions, we further introduce the Boundary‑Aware Transolver (\textscBA‑Transolver) architecture that explicitly separates interior and boundary tokenization. Extensive evaluations on 2D and 3D irregular geometries demonstrate that \textscNPSolver outperforms both physics‑informed and data‑driven baselines. Furthermore, a downstream thermal control task highlights the model's capability for conducting efficient and reliable gradient‑based boundary control. We will release our codes and data at https://github.com/intell‑sci‑comput/NPSolver.

Abstract:
While Physics‑Informed Neural Networks (PINNs) are powerful for solving Partial Differential Equations (PDEs), their training is often paralyzed by gradient pathology. The gradients from the PDE residuals and boundary constraints oppose each other, trapping the model in local minima. Current solutions, such as adaptive weighting or hard constraints, either fail to fundamentally resolve this ill‑conditioning or are limited to simple geometries. In this study, we systematically analyze the possible causes of this gradient pathology from the perspectives of loss landscapes and optimization dynamics. Based on the obtained conclusion, we propose Constraint‑Aligned loss with Manifold Lifting (CAML). By reformulating all zeroth‑order terms into aligned constraints, our method effectively mitigates gradient conflicts. In addition, we introduce a delay factor to help the optimizer skip the high‑curvature area. Experiments demonstrate that our CAML significantly enhances numerical stability and efficiency in highly complex PINN problems. Our code is open‑sourced on https://github.com/YichenLuo‑0/CAML.

Abstract:
We introduce EMMA, a physics‑informed multimodal framework that recovers all identifiable dynamical parameters of a system directly from raw video, audio, and image‑based time‑series observations. Unlike prior video‑only approaches that struggle with occluded states, hidden actuation inputs, or assumptions about known initial conditions and coordinate frames, EMMA performs joint inference of explicit parameters, implicit dynamical components, and calibration invariants within a unified continuous‑time model. EMMA leverages a Liquid Time‑Constant (LTC) network to learn latent dynamics from heterogeneous modalities while a physics‑constrained loss enforces consistency with the governing differential equations. A unified feature pipeline enables consistent alignment across video trajectories, acoustic signatures, and chart‑derived measurements, allowing EMMA to estimate parameters under forced, implicit, and multivariate dynamics without requiring segmentation masks, differentiable rendering, or specialized sensors. Across 100+ scenarios including five standard dynamical benchmarks (75 Delfys videos), real‑world rover and quadrotor systems with hidden inputs, and simulation‑chart case studies spanning biological and chaotic systems, EMMA delivers robust multi‑parameter recovery and significantly outperforms existing single‑modality and equation‑discovery baselines. Our results establish EMMA as a general, scalable solution for physics‑consistent model extraction from opportunistic multimodal data. Code and data are available at: https://github.com/ImpactLabASU/EMMA‑CVPR2026

Abstract:
Learning real‑world dynamics from visual observations is crucial for various domains. A common strategy is to calibrate simulators by estimating physical parameters, yet accuracy is ultimately bounded by the underlying physical models, which often assume materials are homogeneous and isotropic. Even if reasonable, real‑world objects typically exhibit mild anisotropy and heterogeneity. After the near‑isotropic backbone is well calibrated, these residual effects become the key bottleneck for further closing the real‑to‑sim gap. Although neural networks can fit dynamics end‑to‑end, such black‑box modeling discards strong physical priors, leading to poor data efficiency and overfitting. Therefore, we propose MoSA, a motion‑constrained stress adaptation framework that targets these residual effects to further improve real‑to‑sim dynamics learning. MoSA uses an isotropic model as a physics prior and learns residual stress operators to capture mild anisotropy and heterogeneity. It progressively adapts stresses via microplane‑constrained redistribution in a physics‑informed cascaded network. We further impose motion constraints by supervising temporal and spatial derivatives of the deformation field. Experimentally, our learned dynamics achieves superior accuracy, generalization, and robustness, while learning physically meaningful residual anisotropy. Finally, we validate MoSA in a robot manipulation setting, showing that better real‑to‑sim dynamics modeling translates into more reliable sim‑to‑real transfer. Project Page is available at https://mercerai.github.io/MoSA/.

Abstract:
Physics‑informed diffusion models typically enforce PDE constraints only on final outputs, leaving intermediate representations unconstrained and prone to shortcut learning under shifted boundary conditions. We introduce REPA‑P, a teacher‑free, architecture‑agnostic framework that aligns intermediate features with physical states using first‑principles residuals. REPA‑P attaches lightweight 1×1 projection heads to selected layers, decodes hidden activations into physical quantities, and applies PDE residual losses during training. These heads are discarded at inference, introducing zero overhead. Across four PDE tasks, including Darcy flow, topology optimization, electrostatic potential, and turbulent channel flow, REPA‑P accelerates convergence by up to 2×, reduces physics residuals by up to 66.4%, and improves out‑of‑distribution robustness by up to 49.3%, with consistent gains on both U‑Net and Diffusion Transformer backbones. Ablations show that supervising a small set of intermediate layers captures most benefits and complements output‑level physics losses. Code is available at [https://github.com/Hxxxz0/REPA‑P](https://github.com/Hxxxz0/REPA‑P).

Abstract:
Multimodal Large Language Models (MLLMs) have made substantial progress in egocentric video understanding, but their ability to reason cooperatively from multiple embodied viewpoints remains largely unexplored. We study this problem through multi‑robot cooperative dynamic spatial reasoning, where a model must answer spatial, temporal, visibility, and coordination questions by integrating synchronized egocentric videos from a team of moving robots. To support this setting, we introduce CoopSR, the first benchmark for this task, together with EgoTeam, a multi‑robot egocentric QA dataset. EgoTeam contains 114,227 QA pairs spanning 19 question types, four difficulty tiers, and three team sizes in Habitat and iGibson, along with a real‑world test set of around 2,326 QAs collected using two quadruped robots. We further propose SP‑CoR (Spectral and Physics‑Informed Cooperative Reasoner), an MLLM framework for fine‑grained cooperative spatial reasoning. SP‑CoR combines dynamics‑aware multi‑robot frame sampling, spectral‑ and physics‑guided view fusion, and physics‑aligned prompt distillation, enabling the model to benefit from privileged robot‑pose supervision during training while requiring only egocentric videos at test time. Across 22 MLLM baselines, SP‑CoR consistently improves cooperative reasoning, outperforming the strongest fine‑tuned baseline by +3.87% on Habitat and +7.12% on iGibson. It also shows stronger generalization to unseen team sizes and real‑world robot tests. Code can be found at https://github.com/KPeng9510/seeing‑together.git.

Abstract:
Time series forecasting under non‑stationarity faces a fundamental tension between capturing stable representations and adapting to distribution shifts. Existing methods implicitly rely on static historical assumptions, leading to a critical failure mode we term Phase Amnesia, where models become blind to the evolving global context. To resolve this, we formalize non‑stationary dynamics through three physical hypotheses: wold decomposition, dynamical phase evolution, and heteroscedastic manifold generation. These principles inspire PULSE, a physics‑informed, plug‑and‑play framework adopting a Disentangle‑‑Evolve‑‑Simulate design philosophy. Specifically, PULSE utilizes phase‑anchored disentanglement to resolve optimization interference caused by dominant trends, employs a Phase Router to actively generate future trajectories, and introduces Statistic‑Aware Mixup (SAM) to ensure robustness against out‑of‑distribution volatility. Empirically, PULSE enables a simple MLP backbone to achieve state‑of‑the‑art or highly competitive performance across 12 real‑world benchmarks. This validates that a correct physics‑informed inductive bias is far more critical than raw architectural complexity for non‑stationary forecasting. The code is available at: https://github.com/Gemost/PULSE.

Abstract:
Computed tomography (CT) images are frequently degraded by acquisition artifacts, including noise, blur, streaking, aliasing, and metal artifacts. Yet CT enhancement is still largely evaluated using image quality metrics with limited perceptual and clinical validity, while existing datasets remain focused on isolated restoration tasks, hindering unified benchmarking across diverse degradation types. We present CT‑DegradBench, a dataset and benchmark for CT degradation detection and severity estimation under controlled single‑ and mixed‑artifact settings. CT‑DegradBench enables systematic evaluation across multiple degradation families and severity levels within a common experimental framework. We further propose SeSpeCT (Semantic‑Spectral CT degradation estimation), a framework that combines semantic priors from medical vision‑language models with complementary frequency‑domain cues for artifact analysis. SeSpeCT constructs a training‑free semantic quality axis in the multimodal embedding space using radiology‑informed text prompts, without task‑specific fine‑tuning, and combines it with spectral features that capture degradation‑specific frequency patterns. The resulting representation enables joint prediction of artifact type and severity. Experimental results show that SeSpeCT consistently outperforms the evaluated baselines under both single‑ and mixed‑degradation settings. The framework is available at https://github.com/yousranb/CT‑DEGRADBENCH.

Abstract:
Physics‑Informed Neural Networks (PINNs) combine deep learning with physical constraints for solving partial differential equations (PDEs), and are widely applied in fluid mechanics, heat transfer, and solid mechanics. However, PINN training still suffers from high‑dimensional non‑convex loss landscapes, imbalanced multiobjective constraints, and ineffective information propagation. Existing curriculum learning and causality‑guided strategies improve training stability, but mainly focus on temporal or parametric progression, lacking explicit treatment of spatial information propagation and inter‑region consistency. Moreover, they are not directly applicable to boundary value problems (BVPs) with strong spatial coupling. To address this issue, we propose a spatially correlated curriculum learning framework for PINNs. To the best of our knowledge, this is the first work to address PINN training difficulties from the perspective of spatial coupling among subregions. First, spatial causal weights guide information from near‑boundary regions inward, reducing optimization failures and spurious convergence. Second, a low‑frequency information bridge enforces pseudo‑label‑based consistency across spatially separated regions, suppressing global low‑frequency drift. Third, a region‑adaptive reweighting strategy adjusts subregion losses to reduce local residuals and recover high‑frequency details. Experiments on PDE benchmarks show that, under comparable computational cost, the proposed method alleviates training failures and improves solution accuracy. The code is available at https://github.com/pigofmomo/CurriculumLearningPINN.

Abstract:
Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics‑informed neural networks (PINNs) for solving high‑dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second‑order‑free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler‑Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high‑order schemes such as Heun can fully eliminate this bias, such schemes re‑introduce second‑order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM‑induced loss bias and propose an unbiased, second‑order‑free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un‑EM‑BSDE.

Abstract:
While modern diffusion models excel at generating diverse single images, extending this to sequential generation reveals a fundamental challenge: balancing narrative dynamism with multi‑character coherence. Existing methods often falter at this trade‑off, leading to artifacts where characters lose their identity or the story stagnates. To resolve this critical tension, we introduce RealDiffusion, a unified framework designed to reconcile robust coherence with narrative dynamism. Heat diffusion serves as a dissipative prior that averages neighboring features along the sequence and removes high‑frequency noise within the subject region. This suppresses attribute drift and stabilizes identity across frames. A region‑aware stochastic process then introduces small perturbations that explore nearby modes and prevent collapse so the story maintains pose change and scene evolution. We thus introduce a lightweight, training‑free Physics‑informed Attention mechanism that injects controllable physical priors into the self‑attention layers during inference. By modeling feature evolution as a configurable physical system, our method regularizes spatio‑temporal relationships without suppressing intentional, prompt‑driven changes. Extensive experiments demonstrate that RealDiffusion achieves substantial gains in character coherence while preserving narrative dynamism, outperforming state‑of‑the‑art approaches. Code is available at https://github.com/ShmilyQi‑CN/RealDiffusion.

Abstract:
Fast surrogate models for expensive simulations are now essential across the sciences, yet they typically operate as black boxes. We present \textttGWAgent, a large language model (LLM)‑based workflow that constructs interpretable analytic surrogates directly from simulation data. Surrogate modeling is well suited to agentic workflows because candidate models can be quantitatively validated against ground‑truth simulations at each iteration. As a demonstration, we build a surrogate for gravitational waveforms from eccentric binary black hole mergers. We show that providing the agent with a physics‑informed domain ansatz substantially improves output model accuracy. The resulting analytic surrogate attains a median Advanced LIGO mismatch of 6.9×10^‑4 together with an ～ 8.4× speedup in waveform evaluation, surpassing both symbolic regression and conventional machine learning baselines. Beyond producing an accurate model, the workflow identifies compact physical structure from the learned representation. As an astrophysical application, we use \textttGWAgent to analyze the eccentricity of GW200129 and infer e_20\mathrmHz=0.099^+0.063_‑0.044. These results show that validation‑constrained agentic workflows can produce accurate, fast, and interpretable surrogates for scientific simulations and inference.

Abstract:
jNO (jax Neural Operators) is a JAX‑native library for neural operators and foundation models with unified support for both data‑driven and physics‑informed training. Its core design is a tracing system in which domains, model calls, residuals, supervised losses, and diagnostics are written in one symbolic language and compiled into one optimization pipeline. This allows users to move between operator regression, mesh‑aware residual evaluation, and PDE‑constrained training without restructuring the surrounding code. jNO also supports multi‑model compositions, fine‑grained control at parameter level (model, optimizer, and learning rate), hyperparameter tuning, and JAX‑native workflows for translated PDE foundation‑model families. The source repository is available at https://github.com/FhG‑IISB/jNO.

Abstract:
Shadows cast by terrain and tall structures remain a major obstacle for high‑resolution satellite image analysis, degrading classification, detection, and 3D reconstruction performance. Public resources offering geometry‑consistent paired shadow/shadow‑free satellite imagery are essentially missing, and most Earth‑observation datasets are designed for shadow detection or 3D modelling rather than removal. Existing deep shadow‑removal datasets either target ground‑level or aerial scenes or rely on unpaired and weakly supervised formulations rather than explicit satellite pairs. We address this gap with deSEO, a geometry‑aware and physics‑informed methodology that, to the best of our knowledge, is the first to derive paired supervision for satellite shadow removal from the S‑EO shadow detection dataset through a fully replicable pipeline. For each tile, deSEO selects a minimally shadowed acquisition as a weak reference and pairs it with shadowed counterparts using temporal and geometric filtering, Jacobian‑based orientation normalisation, and LoFTR‑RANSAC registration. A per‑pixel validity mask restricts learning to reliably aligned regions, enabling supervision despite residual off‑nadir parallax. In addition to this paired dataset, we develop a DSM‑aware deshadowing model that combines residual translation, perceptual objectives, and mask‑constrained adversarial learning. In contrast, a direct adaptation of a UAV‑based SRNet/pix2pix architecture fails to converge under satellite viewpoint variability. Our model consistently reduces the visual impact of cast shadows across diverse illumination and viewing conditions, achieving improved structural and perceptual fidelity on held‑out scenes. deSEO therefore provides the first reproducible, geometry‑aware paired dataset and baseline for shadow removal in satellite Earth observation.

Abstract:
Proteins are inherently multiscale physical systems whose functional properties emerge from coordinated structural organization across multiple spatial resolutions, ranging from atomic interactions to global fold topology. However, existing protein representation learning methods typically operate at a single structural level or treat different sources of structural information as parallel modalities, without explicitly modeling their hierarchical relationships. We introduce PRIME (Protein Representation via Physics‑Informed Multiscale Equivariant Hierarchies), a unified framework that models proteins as a nested family of five physically grounded structural graphs spanning surface, atomic, residue, secondary‑structure, and protein levels. Adjacent levels are connected through deterministic, physics‑informed assignment operators, enabling bidirectional information exchange via bottom‑up aggregation and top‑down contextual refinement. Experiments on standard protein representation learning benchmarks demonstrate strong and competitive performance across diverse tasks, with particularly notable gains on the Fold Classification benchmark, where PRIME outperforms the strongest geometric GNN baseline by margins of 13.80 and 18.30 points on the harder Superfamily and Fold splits, and achieves a state‑of‑the‑art accuracy of 84.10% on Reaction Class prediction, surpassing all baseline methods, including ESM. Ablation studies confirm that each structural level contributes complementary and non‑redundant information, and adaptive cross‑attention analysis reveals that PRIME autonomously identifies the most task‑relevant structural resolutions at prediction time. Our source code is publicly available at https://github.com/HySonLab/PRIME

Abstract:
Operational phase unwrapping is the primary computational bottleneck in InSAR‑based volcanic and seismic monitoring. We challenge the industry trend of adopting high‑complexity computer vision architectures, such as attention mechanisms, without validating their suitability for physics‑constrained geophysical regression. We present the first large‑scale architectural ablation study on a global LiCSAR benchmark (20 frames, 39,724 patches, 651M pixels). Our results reveal a significant "complexity penalty": a vanilla U‑Net (7.76M parameters) achieves R^2=0.834 and RMSE = 1.01 cm, outperforming 11.37M‑parameter attention‑based models by 34% in R^2 and 51% in RMSE. Power Spectral Density (PSD) analysis provides the physical justification: while attention excels at capturing sharp semantic edges in natural images, it injects unphysical high‑frequency artifacts (>0.3 cycles/pixel) into geophysical fields, violating the fundamental smoothness constraints of elastic surface deformation. With a 2.92ms inference latency (a 2.5× speedup), the vanilla U‑Net is the only candidate to comfortably meet the sub‑100ms requirement for operational early‑warning systems. This work bridges the "publication‑to‑practice" gap by proving that convolutional locality outperforms modern complexity for smooth‑field regression, advocating for physics‑informed simplicity in ML4RS. Code available at https://github.com/prabhjotschugh/When‑Less‑is‑More‑InSAR‑Phase‑Unwrapping

Abstract:
Physics‑informed neural networks (PINNs) provide a promising machine learning framework for solving partial differential equations, but their training often breaks down on challenging problems, sometimes converging to physically incorrect solutions despite achieving small residual losses. This failure, we argue, is not merely an optimization difficulty. Rather, it reflects a fundamental weakness of the empirical PDE residual loss, which can admit trivial or spurious solutions during training. From this perspective, we revisit pseudo‑time stepping, a technique that has recently shown strong empirical success in PINNs. We show that its main benefit is not simply to ease optimization; instead, when combined with collocation‑point resampling, it helps reveal and avoid spurious solutions. At the same time, we find that the effectiveness of pseudo‑time stepping depends critically on the choice of step size, which cannot be tuned reliably from the training loss alone. To overcome this limitation, we propose an adaptive pseudo‑time stepping strategy that selects the step size from a finite‑difference surrogate of the local residual Jacobian, yielding the largest step permitted by local stability without per‑problem tuning. Across a diverse set of PDE benchmarks, the proposed method consistently improves both accuracy and robustness. Together, these findings provide a clearer understanding of why PINNs fail and suggest a practical pathway toward more reliable physics‑informed learning. All code and data accompanying this manuscript are available at https://github.com/sifanexisted/jaxpi2.

Abstract:
Retinal laser speckle contrast imaging (LSCI) is a noninvasive optical modality for monitoring retinal blood flow dynamics. However, conventional temporal LSCI (tLSCI) reconstruction relies on sufficiently long speckle sequences to obtain stable temporal statistics, which makes it vulnerable to acquisition disturbances and limits effective temporal resolution. A physically informed reconstruction framework, termed RetinaDiff (Retinal Diffusion Model), is proposed for retinal tLSCI that is robust to motion and requires only a few frames. In RetinaDiff, registration based on phase correlation is first applied to stabilize the raw speckle sequence before contrast computation, reducing interframe misalignment so that fluctuations at each pixel primarily reflect true flow dynamics. This step provides a physics prior corrected for motion and a high quality multiframe tLSCI reference. Next, guided by the physics prior, a conditional diffusion model performs inverse reconstruction by jointly conditioning on the registered speckle sequence and the corrected prior. Experiments on data acquired with a retinal LSCI system developed in house show improved structural continuity and statistical stability compared with direct reconstruction from few frames and representative baselines. The framework also remains effective in a small number of extremely challenging cases, where both the direct 5‑frame input and the conventional multiframe reconstruction are severely degraded. Overall, this work provides a practical and physically grounded route for reliable retinal tLSCI reconstruction from extremely limited frames. The source code and model weights will be publicly available at https://github.com/QianChen113/RetinaDiff.

Abstract:
This paper is a step‑by‑step, self‑contained guide to the complete training cycle of a Physics‑Informed Neural Network (PINN) ‑‑ a topic that existing tutorials and guides typically delegate to automatic differentiation libraries without exposing the underlying algebra. Using a first‑order initial value problem with a known analytical solution as a running example, we walk through every stage of the process: forward propagation of both the network output and its temporal derivative, evaluation of a composite loss function built from the ODE residual and the initial condition, backpropagation of gradients ‑‑ with particular attention to the product rule that arises in hidden layers ‑‑ and a gradient descent parameter update. Every calculation is presented with explicit, verifiable numerical values using a 1‑3‑3‑1 multilayer perceptron with two hidden layers and 22 trainable parameters. From these concrete examples, we derive general recursive formulas ‑‑ expressed as sensitivity propagation relations ‑‑ that extend the gradient computation to networks of arbitrary depth, and we connect these formulas to the automatic differentiation engines used in practice. The trained network is then validated against the exact solution, achieving a relative L^2 error of 4.290 × 10^‑4 using only the physics‑informed loss, without any data from the true solution. A companion Jupyter/PyTorch notebook reproduces every manual calculation and the full training pipeline, providing mutual validation between hand‑derived and machine‑computed gradients.

Abstract:
Accurate Global Horizontal Irradiance (GHI) forecasting is critical for grid stability, particularly in arid regions characterized by rapid aerosol fluctuations. While recent trends favor computationally expensive Transformer‑based architectures, this paper challenges the prevailing "complexity‑first" paradigm. We propose a lightweight, Physics‑Informed Hybrid CNN‑BiLSTM framework that prioritizes domain knowledge over architectural depth. The model integrates a Convolutional Neural Network (CNN) for spatial feature extraction with a Bi‑Directional LSTM for capturing temporal dependencies. Unlike standard data‑driven approaches, our model is explicitly guided by a vector of 15 engineered features including Clear‑Sky indices and Solar Zenith Angle ‑ rather than relying solely on raw historical data. Hyperparameters are rigorously tuned using Bayesian Optimization to ensure global optimality. Experimental validation using NASA POWER data in Sudan demonstrates that our physics‑guided approach achieves a Root Mean Square Error (RMSE) of 19.53 W/m^2, significantly outperforming complex attention‑based baselines (RMSE 30.64 W/m^2). These results confirm a "Complexity Paradox": in high‑noise meteorological tasks, explicit physical constraints offer a more efficient and accurate alternative to self‑attention mechanisms. The findings advocate for a shift towards hybrid, physics‑aware AI for real‑time renewable energy management.

Abstract:
The stable operation of off‑grid photovoltaic systems requires accurate, computationally efficient solar forecasting. Contemporary deep learning models often suffer from massive computational overhead and physical blindness, generating impossible predictions. This paper introduces the Physics‑Informed State Space Model (PISSM) to bridge the gap between efficiency and physical accuracy for edge‑deployed microcontrollers. PISSM utilizes a dynamic Hankel matrix embedding to filter stochastic sensor noise by transforming raw meteorological sequences into a robust state space. A Linear State Space Model replaces heavy attention mechanisms, efficiently modeling temporal dependencies for parallel processing. Crucially, a novel Physics‑Informed Gating mechanism leverages the Solar Zenith Angle and Clearness Index to structurally bound outputs, ensuring predictions strictly obey diurnal cycles and preventing nocturnal errors. Validated on a multi‑year dataset for Omdurman, Sudan, PISSM achieves superior accuracy with fewer than 40,000 parameters, establishing an ultra‑lightweight benchmark for real‑time off‑grid control.

Abstract:
Uncertainty quantification (UQ) is essential for deploying machine learning models in safety‑critical physical systems, yet classical Bayesian approaches incur substantial computational overhead. We establish a formal connection between Born‑rule measurement statistics from variational quantum circuits (VQCs) and Bayesian posterior uncertainty, proving that repeated quantum measurements naturally produce calibrated prediction intervals without requiring explicit Bayesian neural network (BNN) machinery. We demonstrate this framework on physics‑constrained VQCs trained on PDE residuals. Systematic experiments comparing quantum shot‑based UQ against MC Dropout and Deep Ensemble baselines show that quantum UQ achieves coverage probabilities within 1‑3% of target confidence levels at N >= 5000 shots, while MC Dropout systematically over‑covers by 4‑5%. Physics‑constrained circuits reduce the expected calibration error (ECE) by 34‑40% compared to unconstrained counterparts, with interval widths 14‑30% narrower at equivalent coverage. Information‑theoretic analysis reveals that quantum circuits extract ~15% more bits of UQ information per evaluation than MC Dropout and ~42% more than Deep Ensembles (M = 10), owing to the exponential Hilbert space accessible through Born‑rule sampling. These results establish quantum measurement statistics as a principled, computationally efficient framework for uncertainty quantification in physics‑informed learning.

Abstract:
The barren plateau phenomenon; where cost function gradients vanish exponentially with system size; remains a fundamental obstacle to training variational quantum circuits (VQCs) at scale. We demonstrate, both theoretically and numerically, that embedding partial differential equation (PDE) constraints into the VQC loss function provides a natural and effective mitigation mechanism against barren plateaus. We derive analytical gradient variance lower bounds showing that physics‑constrained loss functions composed of local PDE residuals evaluated at spatial collocation points inherit the favorable polynomial scaling of local cost functions, while additionally benefiting from constraint‑induced landscape narrowing that concentrates gradient information. Systematic numerical experiments on the one‑dimensional heat equation, Burgers' equation, and the Saint‑Venant shallow water equations quantify the gradient variance across 4‑8 qubits and 1‑5 layer depths, comparing global cost, local cost, PDE‑constrained, and PDE‑constrained with structured ansatz configurations. We find that PDE‑constrained circuits exhibit favorable gradient variance scaling with system size, with the physics constraints creating a stabilizing effect that resists exponential gradient vanishing. Entanglement entropy analysis reveals that structured ansatze operate in a sub‑maximal entanglement regime consistent with trainability. Convergence experiments confirm that physics‑constrained VQCs achieve lower loss values in fewer epochs. These results establish PDE constraints as a principled, physically motivated strategy for designing trainable variational quantum circuits, with direct implications for quantum physics‑informed neural networks and variational quantum simulation.

Abstract:
We propose a Hybrid Quantum‑Classical Physics‑Informed Neural Network (HQC‑PINN) that integrates parameterized variational quantum circuits into the PINN framework for hydrological PDE‑constrained learning. Our architecture encodes multi‑source remote sensing features into quantum states via trainable angle encoding, processes them through a hardware‑efficient variational ansatz with entangling layers, and constrains the output using the Saint‑Venant shallow water equations and Manning's flow equation as differentiable physics loss terms. The inherent stochasticity of quantum measurement provides a natural mechanism for uncertainty quantification without requiring explicit Bayesian inference machinery. We further introduce a quantum transfer learning protocol that pre‑trains on multi‑hazard disaster data before fine‑tuning on flood‑specific events. Numerical simulations on multi‑modal satellite and meteorological data from the Kalu River basin, Sri Lanka, show that the HQC‑PINN achieves convergence in ~3x fewer training epochs and uses ~44% fewer trainable parameters compared to an equivalent classical PINN, while maintaining competitive classification accuracy. Theoretical analysis indicates that hydrological physics constraints narrow the effective optimization landscape, providing a natural mitigation against barren plateaus in variational quantum circuits. This work establishes the first application of quantum‑enhanced physics‑informed learning to hydrological prediction and demonstrates a viable path toward quantum advantage in environmental science.

Abstract:
Semi‑analytic models (SAMs) have been treating galaxy populations as dynamical systems for \gtrsim50 years, but their evolution equations remain poorly constrained. We introduce sapphire, a modular, automatically differentiable, GPU‑accelerated SAM written from scratch in JAX. For the first time, we compute exact Jacobian matrices of our nonlinear differential equations and show that they have interpretable, non‑random structures, using the Pandya et al. (2023) physical model as an initial example. Both local and global sensitivity analyses reveal that supernova energy loading is a key astrophysical parameter for galaxy evolution. We use gradient descent and Hamiltonian Monte Carlo (HMC) to perform comprehensive mock parameter recovery tests. These indicate that the z=0 stellar‑to‑halo‑mass relation alone does not contain enough information to infer many astrophysical parameters. Using observations of star‑forming galaxies from the MaNGA survey and the Behroozi et al. (2019) empirical model as one baseline, we derive multiple posteriors assuming different combinations of data, including z=0 interstellar medium gas fractions and metallicities. The inferred physical parameters suggest that galaxies self‑regulate their star formation primarily through preventative rather than ejective feedback. Both Fisher and HMC forecasts demonstrate the potential of sapphire to enable precision inference for galaxy formation, but more work is needed to expand its library of models. We discuss how our unique blend of differentiability, massive GPU parallelization, numerical robustness and principled Bayesian methods sets the stage for hybrid physics‑informed, data‑driven discovery of galaxy formation astrophysics and cosmology. We make sapphire publicly available at https://github.com/virajpandya/sapphire.

Abstract:
Topological phases of matter\unicodex2013comprising both insulators and semimetals\unicodex2013offer great potential for quantum applications, but identifying new candidates remains challenging due to expensive first‑principles simulations and labor‑intensive experimental workflows. Here we introduce a physics‑informed chemical rule that integrates compositional, orbital and crystallographic descriptors within an interpretable linear framework. By explicitly encoding electron filling, space‑group symmetry and orbital‑resolved chemical environments, our method overcomes a fundamental limitation of composition‑only heuristics\unicodex2013their inability to distinguish polymorphs with identical stoichiometry but different crystal structures. Using only elemental characteristics, our approach reduces a material's topological propensity to a single, physically interpretable score, enabling rapid and high‑throughput assessment. The model achieves superior predictive performance while maintaining physical transparency, and identifies candidate topological materials where conventional symmetry indicators fail. Consequently, our framework enables rapid and interpretable exploration of complex materials spaces, establishing a scalable paradigm for the intelligent discovery of next‑generation topological and quantum materials.

Abstract:
The conservation of cultural heritage increasingly relies on integrating technological innovation with domain expertise to ensure effective monitoring and predictive maintenance. This paper presents a novel framework to support the preservation of cultural assets, combining Internet of Things (IoT) and Artificial Intelligence (AI) technologies, enhanced with the physical knowledge of phenomena. The framework is structured into four functional layers that permit the analysis of 3D models of cultural assets and elaborate simulations based on the knowledge acquired from data and physics. A central component of the proposed framework consists of Scientific Machine Learning, particularly Physics‑Informed Neural Networks (PINNs), which incorporate physical laws into deep learning models. To enhance computational efficiency, the framework also integrates Reduced Order Methods (ROMs), specifically Proper Orthogonal Decomposition (POD), and is also compatible with classical Finite Element (FE) methods. Additionally, it includes tools to automatically manage and process 3D digital replicas, enabling their direct use in simulations. The proposed approach offers three main contributions: a methodology for processing 3D models of cultural assets for reliable simulation; the application of PINNs to combine data‑driven and physics‑based approaches in cultural heritage conservation; and the integration of PINNs with ROMs to efficiently model degradation processes influenced by environmental and material parameters. The reproducible and open‑access experimental phase exploits simulated scenarios on complex and real‑life geometries to test the efficacy of the proposed framework in each of its key components, allowing the possibility of dealing with both direct and inverse problems. Code availability: https://github.com/valc89/PhysicsInformedCulturalHeritage

Abstract:
Precise knowledge of the point spread function (PSF) underpins many data analysis steps in astronomy, from photometry and astrometry to source de‑blending and deconvolution. In adaptive optics (AO) observations, however, the PSF is highly variable with wavelength, field position, and observing conditions, making it difficult to model. Traditional PSF reconstruction (PSF‑R) requires full AO telemetry and complex infrastructures, limiting its routine use, especially for tomographic systems. We present a practical framework for fast, accurate, and data‑calibrated PSF modeling that captures the spatial and spectral variability of AO‑corrected PSFs without relying on complete AO telemetry. Our approach builds on a Fourier‑based PSF model inspired by astro‑TIPTOP. As inputs, our model uses only a compact set of physically meaningful parameters retrievable from the ESO archive. A lightweight neural network corrects these inputs to achieve the best match with real data. It is trained end to end with the PSF model, allowing it to learn any miscalibrations directly from on‑sky data. The framework achieves high accuracy on on‑sky data. On a test set of MUSE‑NFM standard stars, it yields median errors of 13.5% in the Strehl ratio and 10.9% in the core full width at half maximum (FWHM). In crowded MUSE‑NFM observations of ω Centauri, the method predicts dozens of off‑axis, wavelength‑dependent PSFs with a Strehl error of <5% and a FWHM error of 4.6%, enabling source separation without per‑star PSF extraction. Our compact, physics‑informed, and data‑calibrated model delivers accurate, polychromatic, and field‑varying PSFs without relying on full AO telemetry. While demonstrated on MUSE‑NFM, the method is still transferable to other AO‑assisted instruments.

Abstract:
Achieving real‑time physics‑based animation that generalizes across diverse 3D shapes and discretizations remains a fundamental challenge. We introduce PhysSkin, a physics‑informed framework that addresses this challenge. In the spirit of Linear Blend Skinning, we learn continuous skinning fields as basis functions lifting motion subspace coordinates to full‑space deformation, with subspace defined by handle transformations. To generate mesh‑free, discretization‑agnostic, and physically consistent skinning fields that generalize well across diverse 3D shapes, PhysSkin employs a new neural skinning fields autoencoder which consists of a transformer‑based encoder and a cross‑attention decoder. Furthermore, we also develop a novel physics‑informed self‑supervised learning strategy that incorporates on‑the‑fly skinning‑field normalization and conflict‑aware gradient correction, enabling effective balancing of energy minimization, spatial smoothness, and orthogonality constraints. PhysSkin shows outstanding performance on generalizable neural skinning and enables real‑time physics‑based animation.

Abstract:
Radio maps (RMs) provide spatially continuous propagation characterizations essential for 6G network planning, but high‑fidelity RM construction remains challenging. Rigorous electromagnetic solvers incur prohibitive computational latency, while data‑driven models demand massive labeled datasets and generalize poorly from simplified simulations to complex multipath environments. This paper proposes RadioDiff‑FS, a few‑shot diffusion framework that adapts a pretrained main‑path generator to multipath‑rich target domains with only a small number of high‑fidelity samples. The adaptation is grounded in a theoretical decomposition of the multipath RM into a dominant main‑path component and a directionally sparse residual. This decomposition shows that the cross‑domain shift corresponds to a bounded and geometrically structured feature translation rather than an arbitrary distribution change. A direction‑consistency loss (DCL) is then introduced to constrain diffusion score updates along physically plausible propagation directions, thereby suppressing phase‑inconsistent artifacts that arise in the low‑data regime. Experiments show that RadioDiff‑FS reduces NMSE by 59.5% on static RMs and by 74.0% on dynamic RMs relative to the vanilla diffusion baseline, achieving an SSIM of 0.9752 and a PSNR of 36.37 dB under severely limited supervision. Even in a one‑shot setting with a single target‑domain sample per scene, RadioDiff‑FS outperforms all fully supervised baselines, confirming that the directional constraint provides an effective inductive bias under extreme data scarcity. Code is available at https://github.com/UNIC‑Lab/RadioDiff‑FS.

Abstract:
A comprehensive understanding of heat transport is essential for optimizing various mechanical and engineering applications, including 3D printing. Recent advances in machine learning, combined with physics‑based models, have enabled a powerful fusion of numerical methods and data‑driven algorithms. This progress is driven by the availability of limited sensor data in various engineering and scientific domains, where the cost of data collection and the inaccessibility of certain measurements are high. To this end, we present PiGRAND, a Physics‑informed graph neural diffusion framework. In order to reduce the computational complexity of graph learning, an efficient graph construction procedure was developed. Our approach is inspired by the explicit Euler and implicit Crank‑Nicolson methods for modeling continuous heat transport, leveraging sub‑learning models to secure the accurate diffusion across graph nodes. To enhance computational performance, our approach is combined with efficient transfer learning. We evaluate PiGRAND on thermal images from 3D printing, demonstrating significant improvements in prediction accuracy and computational performance compared to traditional graph neural diffusion (GRAND) and physics‑informed neural networks (PINNs). These enhancements are attributed to the incorporation of physical principles derived from the theoretical study of partial differential equations (PDEs) into the learning model. The PiGRAND code is open‑sourced on GitHub: https://github.com/bu32loxa/PiGRAND

Abstract:
Inverse problems for stiff parabolic partial differential equations (PDEs), such as the inverse heat conduction problem (IHCP), are severely ill‑posed: the forward map rapidly damps high‑frequency interior structure before it reaches the boundary. Soft‑constrained physics‑informed neural networks (PINNs), which embed the PDE as a residual penalty, suffer from gradient pathology in this regime and tend to fit boundary measurements while leaving the interior field essentially untouched. We propose Neural Field Thermal Tomography (NeFTY), a hard‑constrained neural field framework for label‑free three‑dimensional inverse heat conduction. NeFTY represents the unknown diffusivity as a continuous coordinate‑based neural network, and at every optimization step passes the candidate field through a differentiable implicit‑Euler heat solver with harmonic‑mean interface flux, so that the governing PDE holds exactly on the discretization rather than as a soft penalty. Adjoint gradients propagate the surface reconstruction error back to the network weights at solver‑level memory cost, making test‑time inversion tractable on a single GPU. Across synthetic 3D benchmarks, NeFTY substantially outperforms soft‑constrained PINN variants and a voxel‑grid baseline on label‑free volumetric recovery, and it transfers to real thermography data, surpassing classical signal‑processing baselines in both defect segmentation and depth estimation. Additional details at https://cab‑lab‑princeton.github.io/nefty/

Abstract:
Physics‑Informed Neural Networks (PINNs) have shown promise in solving incompressible Navier‑Stokes equations, yet existing approaches are predominantly designed for single‑flow settings. When extended to multi‑flow scenarios, these methods face three key challenges: (1) difficulty in simultaneously capturing both shared physical principles and flow‑specific characteristics, (2) susceptibility to inter‑task negative transfer that degrades prediction accuracy, and (3) unstable training dynamics caused by disparate loss magnitudes across heterogeneous flow regimes. To address these limitations, we propose UniPINN, a unified multi‑flow PINN framework that integrates three complementary components: a shared‑specialized architecture that disentangles universal physical laws from flow‑specific features, a cross‑flow attention mechanism that selectively reinforces relevant patterns while suppressing task‑irrelevant interference, and a dynamic weight allocation strategy that adaptively balances loss contributions to stabilize multi‑objective optimization. Extensive experiments on three canonical flows demonstrate that UniPINN effectively unifies multi‑flow learning, achieving superior prediction accuracy and balanced performance across heterogeneous regimes while successfully mitigating negative transfer. The source code of this paper will be released on https://github.com/Event‑AHU/OpenFusion

Abstract:
Modeling wind‑driven object dynamics from video observations is highly challenging due to the invisibility and spatio‑temporal variability of wind, as well as the complex deformations of objects. We present DiffWind, a physics‑informed differentiable framework that unifies wind‑object interaction modeling, video‑based reconstruction, and forward simulation. Specifically, we represent wind as a grid‑based physical field and objects as particle systems derived from 3D Gaussian Splatting, with their interaction modeled by the Material Point Method (MPM). To recover wind‑driven object dynamics, we introduce a reconstruction framework that jointly optimizes the spatio‑temporal wind force field and object motion through differentiable rendering and simulation. To ensure physical validity, we incorporate the Lattice Boltzmann Method (LBM) as a physics‑informed constraint, enforcing compliance with fluid dynamics laws. Beyond reconstruction, our method naturally supports forward simulation under novel wind conditions and enables new applications such as wind retargeting. We further introduce WD‑Objects, a dataset of synthetic and real‑world wind‑driven scenes. Extensive experiments demonstrate that our method significantly outperforms prior dynamic scene modeling approaches in both reconstruction accuracy and simulation fidelity, opening a new avenue for video‑based wind‑object interaction modeling.

Abstract:
Real‑world optimization problems are often constrained by complex physical laws that limit computational scalability. These constraints are inherently tied to complex regions, and thus learning models that incorporate physical and geometric knowledge, i.e., physics‑informed machine learning (PIML), offer a promising pathway for efficient solution. Here, we introduce PolyFormer, which opens a new direction for PIML in prescriptive optimization tasks, where physical and geometric knowledge is not merely used to regularize learning models, but to simplify the problems themselves. PolyFormer captures geometric structures behind constraints and transforms them into efficient polytopic reformulations, thereby decoupling problem complexity from solution difficulty and enabling off‑the‑shelf optimization solvers to efficiently produce feasible solutions with acceptable optimality loss. Through evaluations across three important problems (large‑scale resource aggregation, network‑constrained optimization, and optimization under uncertainty), PolyFormer achieves computational speedups up to 6,400‑fold and memory reductions up to 99.87%, while maintaining solution quality competitive with or superior to state‑of‑the‑art methods. These results demonstrate that PolyFormer provides an efficient and reliable solution for scalable constrained optimization, expanding the scope of PIML to prescriptive tasks in scientific discovery and engineering applications.

Abstract:
Navigating safely through dense crowds requires collision avoidance that generalizes beyond the densities seen during training. Learning‑based crowd navigation can break under out‑of‑distribution crowd sizes due to density‑sensitive observation normalization and social‑cost scaling, while analytical solvers often remain safe but freeze in tight interactions. We propose a reinforcement learning approach for dense, variable‑density navigation that attains zero‑shot density generalization using a density‑invariant observation encoding with density‑randomized training and physics‑informed proxemic reward shaping with density‑adaptive scaling. The encoding represents the distance‑sorted K nearest pedestrians plus bounded crowd summaries, keeping input statistics stable as crowd size grows. Trained with N\!\in\![11,16] pedestrians in a 3\mathrmm×3\mathrmm arena and evaluated up to N\!=\!21 pedestrians (1.3× denser), our policy reaches the goal in >99% of episodes and achieves 86% collision‑free success in random crowds, with markedly less freezing than analytical methods and a >\!60‑point collision‑free margin over learning‑based benchmark methods. Codes are available at \hrefhttps://github.com/jznmsl/PSS‑Socialhttps://github.com/jznmsl/PSS‑Social.

Abstract:
The upper 6 GHz (U6G) band with XL‑MIMO is a key enabler for sixth‑generation wireless systems, yet intelligent radiomap prediction for such systems remains challenging. Existing datasets support only small‑scale arrays (up to 8x8) with predominantly isotropic antennas, far from the 1024‑element directional arrays envisioned for 6G. Moreover, current methods encode array configurations as scalar parameters, forcing neural networks to extrapolate array‑specific radiation patterns, which fails when predicting radiomaps for configurations absent from training data. To jointly address data scarcity and generalization limitations, this paper advances XL‑MIMO radiomap prediction from three aspects. To overcome data limitations, we construct the first XL‑MIMO radiomap dataset containing 78400 radiomaps across 800 urban scenes, five frequency bands (1.8‑6.7 GHz), and nine array configurations up to 32x32 uniform planar arrays with directional elements. To enable systematic evaluation, we establish a comprehensive benchmark framework covering practical scenarios from coverage estimation without field measurements to generalization across unseen configurations and environments. To enable generalization to arbitrary beam configurations without retraining, we propose the beam map, a physics‑informed spatial feature that analytically computes array‑specific coverage patterns. By decoupling deterministic array radiation from data learned multipath propagation, beam maps shift generalization from neural network extrapolation to physics‑based computation. Integrating beam maps into existing architectures reduces mean absolute error by up to 60.0% when generalizing to unseen configurations and up to 50.5% when transferring to unseen environments. The complete dataset and code are publicly available at https://lxj321.github.io/MulticonfigRadiomapDataset/.

Abstract:
Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics‑informed neural network (PINN) involving challenging optimization landscapes due to higher‑order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk‑on‑Spheres method, we introduce a learning scheme called \emphWalk‑on‑Spheres Neural Operator (WoS‑NO) which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data‑free physics‑informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre‑computed datasets, avoids computing higher‑order derivatives for loss functions that are memory‑intensive and unstable, and demonstrates zero‑shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75× improvement in L_2‑error compared to standard physics‑informed training schemes, up to 6.31× improvement in training speed, and reductions of up to 2.97× in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS‑NO

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a promising mesh‑free paradigm for solving partial differential equations, yet adoption in science and engineering is limited by slow training and modest accuracy relative to modern numerical solvers. We introduce the Sequential Correction Algorithm for Learning Efficient PINN (Scale‑PINN), a learning strategy that bridges modern physics‑informed learning with numerical algorithms. Scale‑PINN incorporates the iterative residual‑correction principle, a cornerstone of numerical solvers, directly into the loss formulation, marking a paradigm shift in how PINN losses can be conceived and constructed. This integration enables Scale‑PINN to achieve unprecedented convergence speed across PDE problems from different physics domain, including reducing training time on a challenging fluid‑dynamics problem for state‑of‑the‑art PINN from hours to sub‑2 minutes while maintaining superior accuracy, and enabling application to representative problems in aerodynamics and urban science. By uniting the rigor of numerical methods with the flexibility of deep learning, Scale‑PINN marks a significant leap toward the practical adoption of PINNs in science and engineering through scalable, physics‑informed learning. Codes are available at https://github.com/chiuph/SCALE‑PINN.

Abstract:
Characterizing two‑dimensional quantum materials from optical microscopy images is challenging due to the subtle layer‑dependent contrast, limited labeled data, and significant variation across laboratories and imaging setups. Existing vision models struggle in this domain since they lack physical priors and cannot generalize to new materials or hardware conditions. This work presents a new physics‑aware multimodal framework that addresses these limitations from both the data and model perspectives. We first present Synthia, a physics‑based synthetic data generator that simulates realistic optical responses of quantum material flakes under thin‑film interference. Synthia produces diverse and high‑quality samples, helping reduce the dependence on expert manual annotation. We introduce QMat‑Instruct, the first large‑scale instruction dataset for quantum materials, comprising multimodal, physics‑informed question‑answer pairs designed to teach Multimodal Large Language Models (MLLMs) to understand the appearance and thickness of flakes. Then, we propose Physics‑Aware Instruction Tuning (QuPAINT), a multimodal architecture that incorporates a Physics‑Informed Attention module to fuse visual embeddings with optical priors, enabling more robust and discriminative flake representations. Finally, we establish QF‑Bench, a comprehensive benchmark spanning multiple materials, substrates, and imaging settings, offering standardized protocols for fair and reproducible evaluation.

Abstract:
Deep generative models such as flow matching and diffusion models have shown great potential in learning complex distributions and dynamical systems, but often act as black‑boxes, neglecting underlying physics. In contrast, physics‑based simulation models described by ODEs/PDEs remain interpretable, but may have missing or unknown terms, unable to fully describe real‑world observations. We bridge this gap with a novel grey‑box method that integrates incomplete physics models directly into generative models. Our approach learns dynamics from observational trajectories alone, without ground‑truth physics parameters, in a simulation‑free manner that avoids scalability and stability issues of Neural ODEs. The core of our method lies in modelling a structured variational distribution within the flow matching framework, by using two latent encodings: one to model the missing stochasticity and multi‑modal velocity, and a second to encode physics parameters as a latent variable with a physics‑informed prior. Furthermore, we present an adaptation of the framework to handle second‑order dynamics. Our experiments on representative ODE/PDE problems and real‑world weather forecasting demonstrate that our method performs on par with or superior to fully data‑driven approaches and previous grey‑box baselines, while preserving the interpretability of the physics model. Our code is available at https://github.com/DMML‑Geneva/VGB‑DM.

Abstract:
Physics‑informed neural networks (PINNs) in energy form, also known as the deep energy method (DEM), offer advantages over strong‑form PINNs such as lower‑order derivatives and fewer hyperparameters, yet dedicated and user‑friendly software for energy‑form PINNs remains scarce. To address this gap, we present LM‑DEM (Large‑Model‑assisted Deep Energy Method), an open‑source, Streamlit‑based platform for solving variational partial differential equations (PDEs) in computational mechanics. LM‑DEM integrates large language models (LLMs) for geometry modeling: users can generate Gmsh‑compatible geometries directly from natural language descriptions or images, significantly reducing the burden of traditional geometry preprocessing. The solution process is driven by the deep energy method, while finite element solutions can be obtained in parallel. The framework supports built‑in problems including Poisson, screened Poisson, linear elasticity, and hyperelasticity in two and three dimensions, as well as user‑defined energy functionals analogous to the \textttUMAT interface in Abaqus. The source code is available at https://github.com/yizheng‑wang/LMDEM, and a web‑based version is accessible at https://ai4m.llmdem.com. LM‑DEM aims to lower the barrier for practitioners and beginners to adopt energy‑form PINNs for variational PDE problems.

Abstract:
High‑fidelity scientific simulations are now producing unprecedented amounts of data, creating a storage and analysis bottleneck. A single simulation can generate tremendous data volumes, often forcing researchers to discard valuable information. A prime example of this is plasma turbulence described by the gyrokinetic equations: nonlinear, multiscale, and 5D in phase space. It constitutes one of the most computationally demanding frontiers of modern science, with runs taking weeks and yielding tens of terabytes of data dumps. The increasing storage demands underscore the importance of compression. However, reconstructed snapshots do not necessarily preserve essential physical quantities. We present a spatiotemporal evaluation pipeline, accounting for structural phenomena and multi‑scale transient fluctuations to assess the degree of physical fidelity. Indeed, we find that various compression techniques lack preservation of both spatial mode structure and temporal turbulence characteristics. Therefore, we explore Physics‑Informed Neural Compression (PINC), which incorporates physics‑informed losses tailored to gyrokinetics and enables extreme compressions ratios of over 70,000x. Entropy coding on top of PINC further pushes it to 120,000x. This direction provides a viable and scalable solution to the prohibitive storage demands of gyrokinetics, enabling post‑hoc analyses that were previously infeasible.

Abstract:
Accurate traffic flow prediction remains a fundamental challenge in intelligent transportation systems, particularly in cross‑domain, data‑scarce scenarios where limited historical data hinders model training and generalisation. The complex spatio‑temporal dependencies and nonlinear dynamics of urban mobility networks further complicate few‑shot learning across different cities. This paper proposes MCPST, a novel Multi‑phase Consensus Spatio‑Temporal framework for few‑shot traffic forecasting that reconceptualises traffic prediction as a multi‑phase consensus learning problem. Our framework introduces three core innovations: (1) a multi‑phase engine that models traffic dynamics through diffusion, synchronisation, and spectral embeddings for comprehensive dynamic characterisation; (2) an adaptive consensus mechanism that dynamically fuses phase‑specific predictions while enforcing consistency; and (3) a structured meta‑learning strategy for rapid adaptation to new cities with minimal data. We establish extensive theoretical guarantees, including representation theorems with bounded approximation errors and generalisation bounds for few‑shot adaptation. Through experiments on four real‑world datasets, MCPST outperforms fourteen state‑of‑the‑art methods in spatio‑temporal graph learning methods, dynamic graph transfer learning methods, prompt‑based spatio‑temporal prediction methods and cross‑domain few‑shot settings, improving prediction accuracy while reducing required training data and providing interpretable insights. The implementation code is available at https://github.com/afofanah/MCPST.

Abstract:
Graph neural networks (GNNs) often struggle in class‑imbalanced settings, where minority classes are under‑represented and predictions are biased toward majorities. We propose PIMPC‑GNN, a physics‑informed multi‑phase consensus framework for imbalanced node classification. Our method integrates three complementary dynamics: (i) thermodynamic diffusion, which spreads minority labels to capture long‑range dependencies, (ii) Kuramoto synchronisation, which aligns minority nodes through oscillatory consensus, and (iii) spectral embedding, which separates classes via structural regularisation. These perspectives are combined through class‑adaptive ensemble weighting and trained with an imbalance‑aware loss that couples balanced cross‑entropy with physics‑based constraints. Across five benchmark datasets and imbalance ratios from 5‑100, PIMPC‑GNN outperforms 16 state‑of‑the‑art baselines, achieving notable gains in minority‑class recall (up to +12.7%) and balanced accuracy (up to +8.3%). Beyond empirical improvements, the framework also provides interpretable insights into consensus dynamics in graph learning. The code is available at \texttthttps://github.com/afofanah/PIMPC‑GNN.

Abstract:
Global ocean modeling is vital for climate science but struggles to balance computational efficiency with accuracy. Traditional numerical solvers are accurate but computationally expensive, while pure deep learning approaches, though fast, often lack physical consistency and long‑term stability. To address this, we introduce HybridOM, a framework integrating a lightweight, differentiable numerical solver as a skeleton to enforce physical laws, with a neural network as the flesh to correct subgrid‑scale dynamics. To enable efficient high‑resolution modeling, we further introduce a physics‑informed regional downscaling mechanism based on flux gating. This design achieves the inference efficiency of AI‑based methods while preserving the accuracy and robustness of physical models. Extensive experiments on the GLORYS12V1 and OceanBench dataset validate HybridOM's performance in two distinct regimes: long‑term subseasonal‑to‑seasonal simulation and short‑term operational forecasting coupled with the FuXi‑2.0 weather model. Results demonstrate that HybridOM achieves state‑of‑the‑art accuracy while strictly maintaining physical consistency, offering a robust solution for next‑generation ocean digital twins. Our source code is available at https://github.com/ChiyodaMomo01/HybridOM.

Abstract:
Deep learning has emerged as a promising paradigm for spatio‑temporal modeling of fluid dynamics. However, existing approaches often suffer from limited generalization to unseen flow conditions and typically require retraining when applied to new scenarios. In this paper, we present LLM4Fluid, a spatio‑temporal prediction framework that leverages Large Language Models (LLMs) as generalizable neural solvers for fluid dynamics. The framework first compresses high‑dimensional flow fields into a compact latent space via reduced‑order modeling enhanced with a physics‑informed disentanglement mechanism, effectively mitigating spatial feature entanglement while preserving essential flow structures. A pretrained LLM then serves as a temporal processor, autoregressively predicting the dynamics of physical sequences with time series prompts. To bridge the modality gap between prompts and physical sequences, which can otherwise degrade prediction accuracy, we propose a dedicated modality alignment strategy that resolves representational mismatch and stabilizes long‑term prediction. Extensive experiments across diverse flow scenarios demonstrate that LLM4Fluid functions as a robust and generalizable neural solver without retraining, achieving state‑of‑the‑art accuracy while exhibiting powerful zero‑shot and in‑context learning capabilities. Code and datasets are publicly available at https://github.com/qisongxiao/LLM4Fluid.

Abstract:
Wireless imaging has become a vital function in future integrated sensing and communication (ISAC) systems. However, traditional model‑based and data‑driven deep learning imaging methods face challenges related to multipath extraction, dataset acquisition, and multi‑scenario adaptation. To overcome these limitations, this study innovatively combines implicit neural representation (INR) with explicit physical models to realize wireless imaging in reconfigurable intelligent surface (RIS)‑aided ISAC systems. INR employs neural networks (NNs) to project physical locations to voxel values, which is indirectly supervised by measurements of channel state information with physics‑informed loss functions. The continuous shape and scattering characteristics of targets are embedded into NN parameters through training, enabling arbitrary image resolutions and off‑grid voxel value prediction. Additionally, three issues related to INR‑based imager are further addressed. First, INR is generalized to enable efficient imaging under multipath interference by jointly learning image and multipath information. Second, the imaging speed and accuracy for dynamic targets are enhanced by embedding prior image information. Third, imaging results are employed to assist in RIS phase design for improved communication performance. Extensive simulations demonstrate that the proposed INR‑based imager significantly outperforms traditional model‑based methods with super‑resolution abilities, and the focal length characteristics of the imaging system is revealed. Moreover, communication performance can benefit from the imaging results. Part of the source code for this paper can be accessed at https://github.com/kiwi1944/INRImager

Abstract:
Modeling the dynamics of open quantum systems on noisy intermediate‑scale quantum (NISQ) devices constitutes a major challenge, as high noise levels and environmental degradations lead to the decay of pure quantum states (decoherence) and energy losses. This situation represents one of the most important problems in the field of quantum information technologies. While existing data‑driven methods struggle to generalize beyond the training data (extrapolation), physics‑informed neural networks (PINNs) require predefined governing equations, which limit their discovery capability when the underlying physics is incomplete or unknown. In this work, we present the ParaQNN (ParaQuantum neural network) architecture, an equation‑free framework for physical discovery. ParaQNN disentangles multi‑scale dynamics without relying on a priori laws by employing a dialetheist logic layer that models coherent signal and decoherent noise as independent yet interacting channels. Through extensive benchmark tests performed on Rabi oscillations, Lindblad dynamics, and particularly complex "mixed regimes" where relaxation and dephasing processes compete, we show that ParaQNN exhibits a consistent performance advantage compared to Random Forest, XGBoost, and PINN models with incomplete physical information. Unlike its competitors, ParaQNN succeeds in maintaining oscillatory and damping dynamics with high accuracy even in extrapolation regions where training data are unavailable, by "discovering" the underlying structural invariants from noisy measurements. These results demonstrate that paraconsistent logic provides a structurally more stable epistemic foundation than classical methods for learning quantum behavior in situations where mathematical equations prove insufficient.

Abstract:
Physics‑informed machine learning holds great promise for solving differential equations, yet existing methods struggle with highly oscillatory, multiscale, or singularly perturbed PDEs due to spectral bias, costly backpropagation, and manually tuned kernel or Fourier frequencies. This work introduces a soft partition‑‑based Kernel‑Adaptive Physics‑Informed Extreme Learning Machine (KAPI‑ELM), a deterministic low‑dimensional parameterization in which smooth partition lengths jointly control collocation centers and Gaussian kernel widths, enabling continuous coarse‑to‑fine resolution without Fourier features, random sampling, or hard domain interfaces. A signed‑distance‑based weighting further stabilizes least‑squares learning on irregular geometries. Across eight benchmarks‑‑including oscillatory ODEs, high‑frequency Poisson equations, irregular‑shaped domains, and stiff singularly perturbed convection‑diffusion problems‑the proposed method matches or exceeds the accuracy of state‑of‑the‑art Physics‑Informed Neural Network (PINN) and Theory of Functional Connections (TFC) variants while using only a single linear solve. Although demonstrated on steady linear PDEs, the results show that soft‑partition kernel adaptation provides a fast, architecture‑free approach for multiscale PDEs with broad potential for future physics‑informed modeling. For reproducibility, the reference codes are available at https://github.com/vikas‑dwivedi‑2022/soft_kapi

Abstract:
We demonstrate a deep learning framework capable of recovering physical parameters from the Nonlinear Schrodinger Equation (NLSE) under severe noise conditions. By integrating Physics‑Informed Neural Networks (PINNs) with automatic differentiation, we achieve reconstruction of the nonlinear coefficient beta with less than 0.2 percent relative error using only 500 sparse, randomly sampled data points corrupted by 20 percent additive Gaussian noise, a regime where traditional finite difference methods typically fail due to noise amplification in numerical derivatives. We validate the method's generalization capabilities across different physical regimes (beta between 0.5 and 2.0) and varying data availability (between 100 and 1000 training points), demonstrating consistent sub‑1 percent accuracy. Statistical analysis over multiple independent runs confirms robustness (standard deviation less than 0.15 percent for beta equals 1.0). The complete pipeline executes in approximately 80 minutes on modest cloud GPU resources (NVIDIA Tesla T4), making the approach accessible for widespread adoption. Our results indicate that physics‑based regularization acts as an effective filter against high measurement uncertainty, positioning PINNs as a viable alternative to traditional optimization methods for inverse problems in spatiotemporal dynamics where experimental data is scarce and noisy. All code is made publicly available to facilitate reproducibility.

Abstract:
Flood models inform strategic disaster management by simulating the spatiotemporal hydrodynamics of flooding. While physics‑based numerical flood models are accurate, their substantial computational cost limits their use in operational settings where rapid predictions are essential. Models designed with graph neural networks (GNNs) provide both speed and accuracy while having the ability to process unstructured spatial domains. Given its flexible input and architecture, GNNs can be leveraged alongside physics‑informed techniques with ease, significantly improving interpretability and generalizability. We introduce a novel flood GNN architecture, DUALFloodGNN, which embeds physical constraints at both global and local scales through explicit loss terms. The model jointly predicts water volume at nodes and flow along edges through a shared message‑passing framework. To improve performance for autoregressive inference, model training is conducted with a multi‑step loss enhanced with dynamic curriculum learning. Compared with standard GNN architectures and state‑of‑the‑art GNN flood models, DUALFloodGNN achieves substantial improvements in predicting multiple hydrologic variables (e.g., water volume, flow, and depth) while maintaining high computational efficiency. The model is open sourced at https://github.com/acostacos/dual_flood_gnn. The dataset is open sourced at https://hdl.handle.net/2123/35293 with the DOI 10.25910/9xav‑0s86.

Abstract:
Physics‑constrained data generation is essential for machine learning in scientific domains where real data are scarce; however, existing approaches often over‑constrain models without identifying which physical components are necessary. We present a systematic ablation study of a physics‑informed grating coupler spectrum generator that maps five geometric parameters to 100‑point spectral responses. By selectively removing explicit energy conservation enforcement, Fabry‑Perot oscillations, bandwidth variation, and noise, we uncover a physics constraint paradox: explicit energy conservation enforcement is mathematically redundant when the underlying equations are physically consistent, with constrained and unconstrained variants achieving identical conservation accuracy (mean error approximately 7 x 10^‑9). In contrast, Fabry‑Perot oscillations dominate threshold‑based bandwidth variability, accounting for a 72 percent reduction in half‑maximum bandwidth spread when removed (with bandwidth spread reduced from 132.3 nm to 37.4 nm). We further identify a subtle pitfall: standard noise‑addition‑plus‑renormalization pipelines introduce 0.5 percent unphysical negative absorption values. The generator operates at 200 samples per second, enabling high‑throughput data generation and remaining orders of magnitude faster than typical full‑wave solvers reported in the literature. Finally, downstream machine learning evaluation reveals a clear physics‑learnability trade‑off: while central wavelength prediction remains unaffected, removing Fabry‑Perot oscillations improves bandwidth prediction accuracy by 31.3 percent in R‑squared and reduces RMSE by 73.8 percent. These findings provide actionable guidance for physics‑informed dataset design and highlight machine learning performance as a diagnostic tool for assessing constraint relevance.

Abstract:
Standard neural network architectures employ fixed activation functions (ReLU, tanh, sigmoid) that are poorly suited for approximating functions with singular or fractional power behavior, a structure that arises ubiquitously in physics, including boundary layers, fracture mechanics, and corner singularities. We introduce Müntz‑Szász Networks (MSN), a novel architecture that replaces fixed smooth activations with learnable fractional power bases grounded in classical approximation theory. Each MSN edge computes ϕ(x) = \sum_k a_k |x|^μ_k + \sum_k b_k \mathrmsign(x)|x|^λ_k, where the exponents \μ_k, λ_k\ are learned alongside the coefficients. We prove that MSN inherits universal approximation from the Müntz‑Szász theorem and establish novel approximation rates: for functions of the form |x|^α, MSN achieves error \mathcalO(|μ‑ α|^2) with a single learned exponent, whereas standard MLPs require \mathcalO(ε^‑1/α) neurons for comparable accuracy. On supervised regression with singular target functions, MSN achieves 5‑8x lower error than MLPs with 10x fewer parameters. Physics‑informed neural networks (PINNs) represent a particularly demanding application for singular function approximation; on PINN benchmarks including a singular ODE and stiff boundary‑layer problems, MSN achieves 3‑6x improvement while learning interpretable exponents that match the known solution structure. Our results demonstrate that theory‑guided architectural design can yield dramatic improvements for scientifically‑motivated function classes.

Authors: Zi Wang, Mingkai Huang, Zhang Shi, Hongjie Hu, Lan Lan, Hui Zhang, Yan Li, Xi Hu, Qing Lu, Zongming Zhu, Qiong Yao, Yuxiang Dai, Fanwen Wang, Yinzhe Wu, Jun Lyu, Qianqian Gao, Guangming Xu, Zhenxuan Zhang, Haosen Zhang, Qing Li, Guangming Wang, Tianxing He, Lizhen Lan, Siyue Li, Le Xue, Mengting Sun, Yuntong Lyu, Junpu Hu, Jiayu Zhu, Rizwan Ahmad, Zhengyu Bu, Xianling Qian, Guanke Cai, Ruiyu Cao, Weirui Cai, Chang Xu, Yuyang Ren, Feidan Yu, Siying Ma, Ziqiang Xu, Xinran Chen, Sha Hua, Daniel Kim, Yajing Zhang, Chen Ouyang, Wenjia Bai, Jing Qin, Yucheng Yang, Daniel Rueckert, He Wang, Qian Tao, Claudia Prieto, Michael Markl, Alistair Young, Lianming Wu, Shuo Wang, Chen Qin, Mengsu Zeng, Xihong Hu, Haibo Xu, Xiaobo Qu, Hao Li, Guang Yang, Chengyan Wang

Abstract:
Multimodal cardiovascular magnetic resonance (CMR) imaging provides comprehensive and non‑invasive insights into cardiovascular disease (CVD) diagnosis and underlying mechanisms. Despite decades of advancements, its widespread clinical adoption remains constrained by prolonged scan times, inconsistent image quality, and heterogeneity across medical environments. This underscores the urgent need for a generalist reconstruction foundation model for ultra‑fast CMR imaging, one formulated for physics‑constrained inverse problems in the sensor (k‑space) domain, capable of adapting across diverse imaging scenarios and serving as the essential substrate for all downstream analyses. To enable this goal, we curate MMCMR‑427K, the largest and most comprehensive multimodal CMR k‑space database to date, comprising 427,465 multi‑coil k‑space data paired with structured metadata across 13 international centers, 12 CMR modalities, 15 scanners spanning four field strengths, and 17 CVD categories in populations across three continents. Building on this unprecedented resource, we introduce CardioMM, a generalist reconstruction foundation model capable of dynamically adapting to heterogeneous fast CMR imaging scenarios. CardioMM unifies semantic contextual understanding with physics‑informed data consistency to deliver robust reconstructions across varied scanners, protocols, and patient presentations. Comprehensive evaluations demonstrate that CardioMM achieves state‑of‑the‑art performance across internal centers and exhibits strong zero‑shot generalization to unseen external settings. Importantly, CardioMM supports acceleration up to 24x, providing the first evidence that such extreme acquisition speed can preserve key cardiac phenotypes, quantitative myocardial biomarkers, and diagnostic image quality without compromising clinical integrity.

Abstract:
Physics‑aware driving world model is essential for drive planning, out‑of‑distribution data synthesis, and closed‑loop evaluation. However, existing methods often rely on a single diffusion model to directly map driving actions to videos, which makes learning difficult and leads to physically inconsistent outputs. To overcome these challenges, we propose GenieDrive, a novel framework designed for physics‑aware driving video generation. Our approach starts by generating 4D occupancy, which serves as a physics‑informed foundation for subsequent video generation. 4D occupancy contains rich physical information, including high‑resolution 3D structures and dynamics. To facilitate effective compression of such high‑resolution occupancy, we propose a VAE that encodes occupancy into a latent tri‑plane representation, reducing the latent size to only 58% of that used in previous methods. We further introduce Mutual Control Attention (MCA) to accurately model the influence of control on occupancy evolution, and we jointly train the VAE and the subsequent prediction module in an end‑to‑end manner to maximize forecasting accuracy. Together, these designs yield a 7.2% improvement in forecasting mIoU at an inference speed of 41 FPS, while using only 3.47 M parameters. Additionally, a Normalized Multi‑View Attention is introduced in the video generation model to generate multi‑view driving videos with guidance from our 4D occupancy, significantly improving video quality with a 20.7% reduction in FVD. Experiments demonstrate that GenieDrive enables highly controllable, multi‑view consistent, and physics‑aware driving video generation.

Abstract:
We present DeepVekua, a hybrid architecture that unifies geometric deep learning with spectral analysis to solve partial differential equations (PDEs) in sparse data regimes. By learning a diffeomorphic coordinate transformation that maps complex geometries to a latent harmonic space, our method outperforms state‑of‑the‑art implicit representations on advection‑diffusion systems. Unlike standard coordinate‑based networks which struggle with spectral bias, DeepVekua separates the learning of geometry from the learning of physics, solving for optimal spectral weights in closed form. We demonstrate a 100x improvement over spectral baselines. The code is available at https://github.com/VladimerKhasia/vekuanet.

Abstract:
Coordinate‑based neural networks have emerged as a powerful tool for representing continuous physical fields, yet they face two fundamental pathologies: spectral bias, which hinders the learning of high‑frequency dynamics, and the curse of dimensionality, which causes parameter explosion in discrete feature grids. We propose the Adaptive Vekua Cascade (AVC), a hybrid architecture that bridges deep learning and classical approximation theory. AVC decouples manifold learning from function approximation by using a deep network to learn a diffeomorphic warping of the physical domain, projecting complex spatiotemporal dynamics onto a latent manifold where the solution is represented by a basis of generalized analytic functions. Crucially, we replace the standard gradient‑descent output layer with a differentiable linear solver, allowing the network to optimally resolve spectral coefficients in a closed form during the forward pass. We evaluate AVC on a suite of five rigorous physics benchmarks, including high‑frequency Helmholtz wave propagation, sparse medical reconstruction, and unsteady 3D Navier‑Stokes turbulence. Our results demonstrate that AVC achieves state‑of‑the‑art accuracy while reducing parameter counts by orders of magnitude (e.g., 840 parameters vs. 4.2 million for 3D grids) and converging 2‑3x faster than implicit neural representations. This work establishes a new paradigm for memory‑efficient, spectrally accurate scientific machine learning. The code is available at https://github.com/VladimerKhasia/vecua.

Abstract:
Representing and rendering dynamic scenes with complex motions remains challenging in computer vision and graphics. Recent dynamic view synthesis methods achieve high‑quality rendering but often produce physically implausible motions. We introduce NeHaD, a neural deformation field for dynamic Gaussian Splatting governed by Hamiltonian mechanics. Our key observation is that existing methods using MLPs to predict deformation fields introduce inevitable biases, resulting in unnatural dynamics. By incorporating physics priors, we achieve robust and realistic dynamic scene rendering. Hamiltonian mechanics provides an ideal framework for modeling Gaussian deformation fields due to their shared phase‑space structure, where primitives evolve along energy‑conserving trajectories. We employ Hamiltonian neural networks to implicitly learn underlying physical laws governing deformation. Meanwhile, we introduce Boltzmann equilibrium decomposition, an energy‑aware mechanism that adaptively separates static and dynamic Gaussians based on their spatial‑temporal energy states for flexible rendering. To handle real‑world dissipation, we employ second‑order symplectic integration and local rigidity regularization as physics‑informed constraints for robust dynamics modeling. Additionally, we extend NeHaD to adaptive streaming through scale‑aware mipmapping and progressive optimization. Extensive experiments demonstrate that NeHaD achieves physically plausible results with a rendering quality‑efficiency trade‑off. To our knowledge, this is the first exploration leveraging Hamiltonian mechanics for neural Gaussian deformation, enabling physically realistic dynamic scene rendering with streaming capabilities.

Abstract:
Light scattering by spherical‑shaped particles of sizes comparable to the wavelength is foundational in many areas of science, from chemistry to atmospheric science, photonics and nanotechnology. With the new capabilities offered by machine learning, there is a great interest in end‑to‑end differentiable frameworks for scattering calculations. Here we introduce PyMieDiff, a fully differentiable, GPU‑compatible implementation of Mie scattering for layered, spherical particles in PyTorch. The library provides native, autograd‑compatible spherical Bessel and Hankel functions, vectorized evaluation of Mie coefficients, and APIs for computing efficiencies, angular scattering, and near‑fields. All inputs ‑ geometry, material dispersion, wavelengths, and observation angles and positions ‑ are represented as tensors, enabling seamless integration with gradient‑based optimisation or physics‑informed neural networks. The toolkit can also be combined with "TorchGDM" for end‑to‑end differentiable multi‑particle scattering simulations. PyMieDiff is available under an open source licence at https://github.com/UoS‑Integrated‑Nanophotonics‑group/MieDiff.

Abstract:
Traditional physics‑informed neural networks (PINNs) do not always satisfy physics based constraints, especially when the constraints include differential operators. Rather, they minimize the constraint violations in a soft way. Strict satisfaction of differential‑algebraic equations (DAEs) to embed domain knowledge and first‑principles in data‑driven models is generally challenging. This is because data‑driven models consider the original functions to be black‑box whose derivatives can only be obtained after evaluating the functions. We introduce DAE‑HardNet, a physics‑constrained (rather than simply physics‑informed) neural network that learns both the functions and their derivatives simultaneously, while enforcing algebraic as well as differential constraints. This is done by projecting model predictions onto the constraint manifold using a differentiable projection layer. We apply DAE‑HardNet to several systems and test problems governed by DAEs, including the dynamic Lotka‑Volterra predator‑prey system and transient heat conduction. We also show the ability of DAE‑HardNet to estimate unknown parameters through a parameter estimation problem. Compared to multilayer perceptrons (MLPs) and PINNs, DAE‑HardNet achieves orders of magnitude reduction in the physics loss while maintaining the prediction accuracy. It has the added benefits of learning the derivatives which improves the constrained learning of the backbone neural network prior to the projection layer. For specific problems, this suggests that the projection layer can be bypassed for faster inference. The current implementation and codes are available at https://github.com/SOULS‑TAMU/DAE‑HardNet.

Abstract:
Optical aberrations significantly degrade image quality in microscopy, particularly when imaging deeper into samples. These aberrations arise from distortions in the optical wavefront and can be mathematically represented using Zernike polynomials. Existing methods often address only mild aberrations on limited sample types and modalities, typically treating the problem as a black‑box mapping without leveraging the underlying optical physics of wavefront distortions. We propose ZRNet, a physics‑informed framework that jointly performs Zernike coefficient prediction and optical image Restoration. We contribute a Zernike Graph module that explicitly models physical relationships between Zernike polynomials based on their azimuthal degrees‑ensuring that learned corrections align with fundamental optical principles. To further enforce physical consistency between image restoration and Zernike prediction, we introduce a Frequency‑Aware Alignment (FAA) loss, which better aligns Zernike coefficient prediction and image features in the Fourier domain. Extensive experiments on CytoImageNet demonstrates that our approach achieves state‑of‑the‑art performance in both image restoration and Zernike coefficient prediction across diverse microscopy modalities and biological samples with complex, large‑amplitude aberrations. We further validate on experimental PSF data from a physical microscope and demonstrate robustness to realistic sensor noise, confirming generalisation beyond simulated conditions. Code is available at https://github.com/janetkok/ZRNet.

Abstract:
Channel state information (CSI) is essential for adaptive beamforming and maintaining robust links in wireless communication systems. However, acquiring CSI incurs significant overhead, consuming up to 25% of spectrum resources in 5G networks due to frequent pilot transmissions at millisecond‑scale intervals. Recent approaches aim to reduce this burden by reconstructing CSI from spatiotemporal RF measurements, such as signal strength and direction‑of‑arrival. While effective in offline settings, these methods often suffer from inference latencies in the 5‑100 ms range, making them impractical for real‑time systems. We present GSpaRC: Gaussian Splatting for Real‑time Reconstruction of RF Channels, a method that achieves accurate channel reconstruction with latency in the low‑millisecond regime or below. GSpaRC represents the RF environment using a compact set of 3D Gaussian primitives, each parameterized by a lightweight neural model augmented with physics‑informed features such as distance‑based attenuation. Unlike traditional vision‑based splatting pipelines, GSpaRC is tailored for RF reception: it employs an equirectangular projection onto a hemispherical surface centered at the receiver to reflect omnidirectional antenna behavior. A custom CUDA pipeline enables fully parallelized directional sorting, splatting, and rendering across frequency and spatial dimensions. Evaluated on multiple RF datasets, GSpaRC achieves similar CSI reconstruction fidelity to recent state‑of‑the‑art methods while reducing training and inference time by over an order of magnitude. These results illustrate that modest GPU computation can substantially reduce pilot overhead, making GSpaRC a scalable low‑latency approach for channel estimation in 5G and future wireless systems.

Abstract:
Accurate extraction and segmentation of the cerebral arteries from digital subtraction angiography (DSA) sequences is essential for developing reliable clinical management models of complex cerebrovascular diseases. Conventional loss functions often rely solely on pixel‑wise overlap, overlooking the geometric and physical consistency of vascular boundaries, which can lead to fragmented or unstable vessel predictions. To overcome this limitation, we propose a novel Physics‑Informed Loss (PIL) that models the interaction between the predicted and ground‑truth boundaries as an elastic process inspired by dislocation theory in materials physics. This formulation introduces a physics‑based regularization term that enforces smooth contour evolution and structural consistency, allowing the network to better capture fine vascular geometry. The proposed loss is integrated into several segmentation architectures, including U‑Net, U‑Net++, SegFormer, and MedFormer, and evaluated on two public benchmarks: DIAS and DSCA. Experimental results demonstrate that PIL consistently outperforms conventional loss functions such as Cross‑Entropy, Dice, Active Contour, and Surface losses, achieving superior sensitivity, F1 score, and boundary coherence. These findings confirm that the incorporation of physics‑based boundary interactions into deep neural networks improves both the precision and robustness of vascular segmentation in dynamic angiographic imaging. The implementation of the proposed method is publicly available at https://github.com/irfantahir301/Physicsis_loss.

Abstract:
Radio maps (RMs) serve as environment‑aware electromagnetic (EM) representations that connect scenario geometry and material properties to the spatial distribution of signal strength, enabling localization without costly in‑situ measurements. However, constructing high‑fidelity indoor RMs remains challenging due to the prohibitive latency of EM solvers and the limitations of learning‑based methods, which often rely on sparse measurements or assumptions of homogeneous material, which are misaligned with the heterogeneous and multipath‑rich nature of indoor environments. To overcome these challenges, we propose iRadioDiff, a sampling‑free diffusion‑based framework for indoor RM construction. iRadioDiff is conditioned on access point (AP) positions, and physics‑informed prompt encoded by material reflection and transmission coefficients. It further incorporates multipath‑critical priors, including diffraction points, strong transmission boundaries, and line‑of‑sight (LoS) contours, to guide the generative process via conditional channels and boundary‑weighted objectives. This design enables accurate modeling of nonstationary field discontinuities and efficient construction of physically consistent RMs. Experiments demonstrate that iRadioDiff achieves state‑of‑the‑art performance in indoor RM construction and received signal strength based indoor localization, which offers effective generalization across layouts and material configurations. Code is available at https://github.com/UNIC‑Lab/iRadioDiff.

Abstract:
In multi‑objective optimization, multiple loss terms are weighted and added together to form a single objective. These weights are chosen to properly balance the competing losses according to some meta‑goal. For example, in physics‑informed neural networks (PINNs), these weights are often adaptively chosen to improve the network's generalization error. A popular choice of adaptive weights is based on the neural tangent kernel (NTK) of the PINN, which describes the evolution of the network in predictor space during training. The convergence of such an adaptive weighting algorithm is not clear a priori. Moreover, these NTK‑based weights would be updated frequently during training, further increasing the computational burden of the learning process. In this paper, we prove that under appropriate conditions, gradient descent enhanced with adaptive NTK‑based weights is convergent in a suitable sense. We then address the problem of computational efficiency by developing a randomized algorithm inspired by a predictor‑corrector approach and matrix sketching, which produces unbiased estimates of the NTK up to an arbitrarily small discretization error. Finally, we provide numerical experiments to support our theoretical findings and to show the efficacy of our randomized algorithm. Code Availability: https://github.com/maxhirsch/Efficient‑NTK

Abstract:
Flood prediction is critical for emergency planning and response to mitigate human and economic losses. Traditional physics‑based hydrodynamic models generate high‑resolution flood maps using numerical methods requiring fine‑grid discretization; which are computationally intensive and impractical for real‑time large‑scale applications. While recent studies have applied convolutional neural networks for flood map super‑resolution with good accuracy and speed, they suffer from limited generalizability to unseen areas. In this paper, we propose a novel approach that leverages latent diffusion models to perform super‑resolution on coarse‑grid flood maps, with the objective of achieving the accuracy of fine‑grid flood maps while significantly reducing inference time. Experimental results demonstrate that latent diffusion models substantially decrease the computational time required to produce high‑fidelity flood maps without compromising on accuracy, enabling their use in real‑time flood risk management. Moreover, diffusion models exhibit superior generalizability across different physical locations, with transfer learning further accelerating adaptation to new geographic regions. Our approach also incorporates physics‑informed inputs, addressing the common limitation of black‑box behavior in machine learning, thereby enhancing interpretability. Code is available at https://github.com/neosunhan/flood‑diff.

Abstract:
Estimating the geometric and volumetric properties of transparent deformable liquids is challenging due to optical complexities and dynamic surface deformations induced by container movements. Autonomous robots performing precise liquid manipulation tasks, such as dispensing, aspiration, and mixing, must handle containers in ways that inevitably induce these deformations, complicating accurate liquid state assessment. Current datasets lack comprehensive physics‑informed simulation data representing realistic liquid behaviors under diverse dynamic scenarios. To bridge this gap, we introduce Phys‑Liquid, a physics‑informed dataset comprising 97,200 simulation images and corresponding 3D meshes, capturing liquid dynamics across multiple laboratory scenes, lighting conditions, liquid colors, and container rotations. To validate the realism and effectiveness of Phys‑Liquid, we propose a four‑stage reconstruction and estimation pipeline involving liquid segmentation, multi‑view mask generation, 3D mesh reconstruction, and real‑world scaling. Experimental results demonstrate improved accuracy and consistency in reconstructing liquid geometry and volume, outperforming existing benchmarks. The dataset and associated validation methods facilitate future advancements in transparent liquid perception tasks. The dataset and code are available at https://dualtransparency.github.io/Phys‑Liquid/.

Abstract:
4D flow magnetic resonance imaging (MRI) is a reliable, non‑invasive approach for estimating blood flow velocities, vital for cardiovascular diagnostics. Unlike conventional MRI focused on anatomical structures, 4D flow MRI requires high spatiotemporal resolution for early detection of critical conditions such as stenosis or aneurysms. However, achieving such resolution typically results in prolonged scan times, creating a trade‑off between acquisition speed and prediction accuracy. Recent studies have leveraged physics‑informed neural networks (PINNs) for super‑resolution of MRI data, but their practical applicability is limited as the prohibitively slow training process must be performed for each patient. To overcome this limitation, we propose PINGS‑X, a novel framework modeling high‑resolution flow velocities using axes‑aligned spatiotemporal Gaussian representations. Inspired by the effectiveness of 3D Gaussian splatting (3DGS) in novel view synthesis, PINGS‑X extends this concept through several non‑trivial novel innovations: (i) normalized Gaussian splatting with a formal convergence guarantee, (ii) axes‑aligned Gaussians that simplify training for high‑dimensional data while preserving accuracy and the convergence guarantee, and (iii) a Gaussian merging procedure to prevent degenerate solutions and boost computational efficiency. Experimental results on computational fluid dynamics (CFD) and real 4D flow MRI datasets demonstrate that PINGS‑X substantially reduces training time while achieving superior super‑resolution accuracy. Our code and datasets are available at https://github.com/SpatialAILab/PINGS‑X.

Abstract:
Learning dexterous and agile policy for humanoid and dexterous hand control requires large‑scale demonstrations, but collecting robot‑specific data is prohibitively expensive. In contrast, abundant human motion data is readily available from motion capture, videos, and virtual reality, which could help address the data scarcity problem. However, due to the embodiment gap and missing dynamic information like force and torque, these demonstrations cannot be directly executed on robots. To bridge this gap, we propose Scalable Physics‑Informed DExterous Retargeting (SPIDER), a physics‑based retargeting framework to transform and augment kinematic‑only human demonstrations to dynamically feasible robot trajectories at scale. Our key insight is that human demonstrations should provide global task structure and objective, while large‑scale physics‑based sampling with curriculum‑style virtual contact guidance should refine trajectories to ensure dynamical feasibility and correct contact sequences. SPIDER scales across diverse 9 humanoid/dexterous hand embodiments and 6 datasets, improving success rates by 18% compared to standard sampling, while being 10X faster than reinforcement learning (RL) baselines, and enabling the generation of a 2.4M frames dynamic‑feasible robot dataset for policy learning. As a universal physics‑based retargeting method, SPIDER can work with diverse quality data and generate diverse and high‑quality data to enable efficient policy learning with methods like RL.

Abstract:
Accurately simulating systems governed by PDEs, such as voltage fields in cardiac electrophysiology (EP) modelling, remains a significant modelling challenge. Traditional numerical solvers are computationally expensive and sensitive to discretisation, while canonical deep learning methods are data‑hungry and struggle with chaotic dynamics and long‑term predictions. Physics‑Informed Neural Networks (PINNs) mitigate some of these issues by incorporating physical constraints in the learning process, yet they remain limited by mesh resolution and long‑term predictive stability. In this work, we propose a Physics‑Informed Neural Operator (PINO) approach to solve PDE problems in cardiac EP. Unlike PINNs, PINO models learn mappings between function spaces, allowing them to generalise to multiple mesh resolutions and initial conditions. Our results show that PINO models can accurately reproduce cardiac EP dynamics over extended time horizons and across multiple propagation scenarios, including zero‑shot evaluations on scenarios unseen during training. Additionally, our PINO models maintain high predictive quality in long roll‑outs (where predictions are recursively fed back as inputs), and can scale their predictive resolution by up to 10x the training resolution. These advantages come with a significant reduction in simulation time compared to numerical PDE solvers, highlighting the potential of PINO‑based approaches for efficient and scalable cardiac EP simulations.

Abstract:
Eigenvalue problems have a distinctive forward‑inverse structure and are fundamental to characterizing a system's thermal response, stability, and natural modes. Physics‑Informed Neural Networks (PINNs) offer a mesh‑free alternative for solving such problems but are often orders of magnitude slower than classical numerical schemes. In this paper, we introduce a reformulated PINN approach that casts the search for eigenpairs as a biconvex optimization problem, enabling fast and provably convergent alternating convex search (ACS) over eigenvalues and eigenfunctions using analytically optimal updates. Numerical experiments show that PINN‑ACS attains high accuracy with convergence speeds up to 500× faster than gradient‑based PINN training. We release our codes at https://github.com/NeurIPS‑ML4PS‑2025/PINN_ACS_CODES.

Abstract:
Physics Informed Neural Networks (PINNs) are gaining attention for their ability to embed physical laws into deep learning models, which is particularly useful in structural engineering tasks with limited data. This paper aims to explore the use of PINNs to predict the weight of small scale spaghetti bridges, a task relevant to understanding load limits and potential failure modes in simplified structural models. Our proposed framework incorporates physics‑based constraints to the prediction model for improved performance. In addition to standard PINNs, we introduce a novel architecture named Physics Informed Kolmogorov Arnold Network (PIKAN), which blends universal function approximation theory with physical insights. The structural parameters provided as input to the model are collected either manually or through computer vision methods. Our dataset includes 15 real bridges, augmented to 100 samples, and our best model achieves an R^2 score of 0.9603 and a mean absolute error (MAE) of 10.50 units. From applied perspective, we also provide a web based interface for parameter entry and prediction. These results show that PINNs can offer reliable estimates of structural weight, even with limited data, and may help inform early stage failure analysis in lightweight bridge designs. The complete data and code are available at https://github.com/OmerJauhar/PINNS‑For‑Spaghetti‑Bridges.

Abstract:
Although Finite Element Analysis (FEA) is an integral part of the product design lifecycle, the analysis is computationally expensive, making it unsuitable for many design optimization problems. The deep learning models can be a great solution. However, selecting the architecture that emulates the FEA with great accuracy is a challenge. This paper presents a comprehensive evaluation of graph neural networks (GNNs) and 3D U‑Nets as surrogates for FEA of parametric I‑beams. We introduce a Physics‑Informed Neural Network (PINN) framework, governed by the Navier Cauchy equations, to enforce physical laws. Crucially, we demonstrate that a curriculum learning strategy, pretraining on data followed by physics informed fine tuning, is essential for stabilizing training. Our results show that GNNs fundamentally outperform the U‑Net. Even the worst performer among GNNs, the GCN framework, achieved a relative L2 error of 8.7% while the best framework among U Net, U Net with attention mechanism trained on high resolution data, achieved 13.0% score. Among the graph‑based architectures, the Message Passing Neural Networks (MPNN) and Graph Transformers achieved the highest accuracy, achieving a relative L2 score of 3.5% and 2.6% respectively. The inclusion of physics fundamental laws (PINN) significantly improved the generalization, reducing error by up to 11.3% on high‑signal tasks. While the Graph Transformer is the most accurate model, it is more 37.5% slower during inference when compared to second best model, MPNN PINN. The PINN enhanced MPNN (MPNN PINN) provides the most practical solution. It offers a good compromise between predictive performance, model size, and inference speed.

Abstract:
We propose Physics‑informed Attention‑enhanced Fourier Neural Operator (PIANO) to solve the Nonlinear Force‑Free Field (NLFFF) problem in solar physics. Unlike conventional approaches that rely on iterative numerical methods, our proposed PIANO directly learns the 3D magnetic field structure from 2D boundary conditions. Specifically, PIANO integrates Efficient Channel Attention (ECA) mechanisms with Dilated Convolutions (DC), which enhances the model's ability to capture multimodal input by prioritizing critical channels relevant to the magnetic field's variations. Furthermore, we apply physics‑informed loss by enforcing the force‑free and divergence‑free conditions in the training process so that our prediction is consistent with underlying physics with high accuracy. Experimental results on the ISEE NLFFF dataset show that our PIANO not only outperforms state‑of‑the‑art neural operators in terms of accuracy but also shows strong consistency with the physical characteristics of NLFFF data across magnetic fields reconstructed from various solar active regions. The GitHub of this project is available https://github.com/Autumnstar‑cjh/PIANO

Abstract:
The 3D incompressible Navier‑Stokes equations model essential fluid phenomena, including turbulence and aerodynamics, but are challenging to solve due to nonlinearity and limited solution regularity. Despite extensive research, the full mathematical understanding of the 3D incompressible Navier‑Stokes equations continues to elude scientists, highlighting the depth and difficulty of the problem. Classical solvers are costly, and neural network‑based methods typically assume strong solutions, limiting their use in underresolved regimes. We introduce WAN3DNS, a weak‑form neural solver that recasts the equations as a minimax optimization problem, allowing learning directly from weak solutions. Using the weak formulation, WAN3DNS circumvents the stringent differentiability requirements of classical physics‑informed neural networks (PINNs) and accommodates scenarios where weak solutions exist, but strong solutions may not. We evaluated WAN3DNS's accuracy and effectiveness in three benchmark cases: the 2D Kovasznay, 3D Beltrami, and 3D lid‑driven cavity flows. Furthermore, using Galerkin's theory, we conduct a rigorous error analysis and show that the L^2 training error is controllably bounded by the architectural parameters of the network and the norm of residues. This implies that for neural networks with small loss, the corresponding L^2 error will also be small. This work bridges the gap between weak solution theory and deep learning, offering a robust alternative for complex fluid flow simulations with reduced regularity constraints. Code: https://github.com/Wenran‑Li/WAN3DNS

Abstract:
Remote photoplethysmography (rPPG) measurement enables non‑contact physiological monitoring but suffers from accuracy degradation under head motion and illumination changes. Existing deep learning methods are mostly heuristic and lack theoretical grounding, limiting robustness and interpretability. In this work, we propose a physics‑informed rPPG paradigm derived from the Navier‑Stokes equations of hemodynamics, showing that the pulse signal follows a second‑order dynamical system whose discrete solution naturally leads to a causal convolution, justifying the use of a Temporal Convolutional Network (TCN). Based on this principle, we design the PHASE‑Net, a lightweight model with three key components: 1) Zero‑FLOPs Axial Swapper module to swap or transpose a few spatial channels to mix distant facial regions, boosting cross‑region feature interaction without changing temporal order; 2) Adaptive Spatial Filter to learn a soft spatial mask per frame to highlight signal‑rich areas and suppress noise for cleaner feature maps; and 3) Gated TCN, a causal dilated TCN with gating that models long‑range temporal dynamics for accurate pulse recovery. Extensive experiments demonstrate that PHASE‑Net achieves state‑of‑the‑art performance and strong efficiency, offering a theoretically grounded and deployment‑ready rPPG solution. The source code is available at https://github.com/Alex036225/PhaseNet.

Abstract:
Physics‑informed graph neural networks (PIGNNs) have emerged as fast AC power‑flow solvers that can replace the classic NewtonRaphson (NR) solvers, especially when thousands of scenarios must be evaluated. However, current PIGNNs still need accuracy improvements at parity speed; in particular, the soft constraint on the physics loss is inoperative at inference, which can deter operational adoption. We address this with PIGNN‑Attn‑LS, combining an edge‑aware attention mechanism that explicitly encodes line physics via per‑edge biases to form a fully differentiable knownoperator layer inside the computation graph, with a backtracking line‑search‑based globalized correction operator that restores an operative decrease criterion at inference. Training and testing use a realistic High‑/Medium‑Voltage scenario generator, with NR used only to construct reference states. On held‑out HV cases consisting of 4‑32‑bus grids, PIGNN‑Attn‑LS achieves a test RMSE of 0.00033 p.u. in voltage and 0.08 deg in angle, outperforming the PIGNN‑MLP baseline by 99.5% and 87.1%, respectively. With streaming micro‑batches, it delivers 2‑5x faster batched inference than NR on 4‑1024‑bus grids.

Abstract:
We present the fundamental theory and implementation guidelines underlying Evidential Physics‑Informed Neural Network (E‑PINN) ‑‑ a novel class of uncertainty‑aware PINN. It leverages the marginal distribution loss function of evidential deep learning for estimating uncertainty of outputs, and infers unknown parameters of the PDE via a learned posterior distribution. Validating our model on two illustrative case studies ‑‑ the 1D Poisson equation with a Gaussian source and the 2D Fisher‑KPP equation, we found that E‑PINN generated empirical coverage probabilities that were calibrated significantly better than Bayesian PINN and Deep Ensemble methods. To demonstrate real‑world applicability, we also present a brief case study on applying E‑PINN to analyze clinical glucose‑insulin datasets that have featured in medical research on diabetes pathophysiology.

Abstract:
Graph neural networks have recently shown strong promise for event reconstruction tasks in Liquid Argon Time Projection Chambers, yet their performance remains limited for underrepresented classes of particles, such as Michel electrons. In this work, we investigate physics‑informed strategies to improve semantic segmentation within the NuGraph2 architecture. We explore three complementary approaches: (i) enriching the input representation with context‑aware features derived from detector geometry and track continuity, (ii) introducing auxiliary decoders to capture class‑level correlations, and (iii) incorporating energy‑based regularization terms motivated by Michel electron energy distributions. Experiments on MicroBooNE public datasets show that physics‑inspired feature augmentation yields the largest gains, particularly boosting Michel electron precision and recall by disentangling overlapping latent space regions. In contrast, auxiliary decoders and energy‑regularization terms provided limited improvements, partly due to the hit‑level nature of NuGraph2, which lacks explicit particle‑ or event‑level representations. Our findings highlight that embedding physics context directly into node‑level inputs is more effective than imposing task‑specific auxiliary losses, and suggest that future hierarchical architectures such as NuGraph3, with explicit particle‑ and event‑level reasoning, will provide a more natural setting for advanced decoders and physics‑based regularization. The code for this work is publicly available on Github at https://github.com/vitorgrizzi/nugraph_phys/tree/main_phys.

Abstract:
We present HyPINO, a multi‑physics neural operator designed for zero‑shot generalization across a broad class of PDEs without requiring task‑specific fine‑tuning. Our approach combines a Swin Transformer‑based hypernetwork with mixed supervision: (i) labeled data from analytical solutions generated via the Method of Manufactured Solutions (MMS), and (ii) unlabeled samples optimized using physics‑informed objectives. The model maps PDE parameterizations to target Physics‑Informed Neural Networks (PINNs) and can handle linear elliptic, hyperbolic, and parabolic equations in two dimensions with varying source terms, geometries, and mixed Dirichlet/Neumann boundary conditions, including interior boundaries. HyPINO achieves strong zero‑shot accuracy on seven benchmark problems from PINN literature, outperforming U‑Nets, Poseidon, and Physics‑Informed Neural Operators (PINO). Further, we introduce an iterative refinement procedure that treats the residual of the generated PINN as "delta PDE" and performs another forward pass to generate a corrective PINN. Summing their contributions and repeating this process forms an ensemble whose combined solution progressively reduces the error on six benchmarks and achieves a >100x lower L_2 loss in the best case, while retaining forward‑only inference. Additionally, we evaluate the fine‑tuning behavior of PINNs initialized by HyPINO and show that they converge faster and to lower final error than both randomly initialized and Reptile‑meta‑learned PINNs on five benchmarks, performing on par on the remaining two. Our results highlight the potential of this scalable approach as a foundation for extending neural operators toward solving increasingly complex, nonlinear, and high‑dimensional PDE problems. The code and model weights are publicly available at https://github.com/rbischof/hypino.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs). However, standard MLP‑based PINNs often fail to converge when dealing with complex initial value problems, leading to solutions that violate causality and suffer from a spectral bias towards low‑frequency components. To address these issues, we introduce NeuSA (Neuro‑Spectral Architectures), a novel class of PINNs inspired by classical spectral methods, designed to solve linear and nonlinear PDEs with variable coefficients. NeuSA learns a projection of the underlying PDE onto a spectral basis, leading to a finite‑dimensional representation of the dynamics which is then integrated with an adapted Neural ODE (NODE). This allows us to overcome spectral bias, by leveraging the high‑frequency components enabled by the spectral representation; to enforce causality, by inheriting the causal structure of NODEs, and to start training near the target solution, by means of an initialization scheme based on classical methods. We validate NeuSA on canonical benchmarks for linear and nonlinear wave equations, demonstrating strong performance as compared to other architectures, with faster convergence, improved temporal consistency and superior predictive accuracy. Code and pretrained models are available in https://github.com/arthur‑bizzi/neusa.

Abstract:
Optical microrobots actuated by optical tweezers (OT) offer great potential for biomedical applications such as cell manipulation and microscale assembly. These tasks demand accurate three‑dimensional perception to ensure precise control in complex and dynamic biological environments. However, the transparent nature of microrobots and low‑contrast microscopic imaging challenge conventional deep learning methods, which also require large annotated datasets that are costly to obtain. To address these challenges, we propose a physics‑informed, data‑efficient framework for depth estimation of optical microrobots. Our method augments convolutional feature extraction with physics‑based focus metrics, such as entropy, Laplacian of Gaussian, and gradient sharpness, calculated using an adaptive grid strategy. This approach allocates finer grids over microrobot regions and coarser grids over background areas, enhancing depth sensitivity while reducing computational complexity. We evaluate our framework on multiple microrobot types and demonstrate significant improvements over baseline models. Specifically, our approach reduces mean squared error (MSE) by over 60% and improves the coefficient of determination (R^2) across all test cases. Notably, even when trained on only 20% of the available data, our model outperforms ResNet50 trained on the full dataset, highlighting its robustness under limited data conditions. Our code is available at: https://github.com/LannWei/CBS2025.

Abstract:
We present a hybrid framework that couples finite element methods (FEM) with physics‑informed DeepONet to model fluid transport in porous media from sharp, localized Gaussian sources. The governing system consists of a steady‑state Darcy flow equation and a time‑dependent convection‑diffusion equation. Our approach solves the Darcy system using FEM and transfers the resulting velocity field to a physics‑informed DeepONet, which learns the mapping from source functions to solute concentration profiles. This modular strategy preserves FEM‑level accuracy in the flow field while enabling fast inference for transport dynamics. To handle steep gradients induced by sharp sources, we introduce an adaptive sampling strategy for trunk collocation points. Numerical experiments demonstrate that our method is in good agreement with the reference solutions while offering orders of magnitude speedups over traditional solvers, making it suitable for practical applications in relevant scenarios. Implementation of our proposed method is available at https://github.com/erkara/fem‑pi‑deeponet.

Abstract:
This paper presents a novel Energy‑Equidistributed adaptive sampling framework for multi‑dimensional conservative PDEs, introducing both location‑based and velocity‑based formulations of Energy‑Equidistributed moving mesh PDEs (EMMPDEs). The framework utilizes the energy density function as the monitor function, ensuring that mesh adaptation dynamically tracks energy evolution during temporal integration. These theoretical developments are integrated with deep neural networks to establish the Energy‑Equidistributed Moving Sampling Physics‑Informed Neural Networks (EEMS‑PINNs), which integrate physics‑informed learning with energy‑adaptive mesh optimization. Extensive numerical experiments demonstrate that EEMS‑PINNs effectively maintain solution accuracy in long‑time simulations while preserving conserved energy. The framework's robustness is further evidenced by its stable performance in non‑conservative systems. The code for this paper can be found at https://github.com/sufe‑Ran‑Zhang/EMMPDE.

Abstract:
We present a new efficient hybrid parameter estimation method based on the idea, that if nonlinear dynamic models are stated in terms of a system of equations that is linear in terms of the parameters, then regularized ordinary least squares can be used to estimate these parameters from time series data. We introduce the term "Physics‑Informed Regression" (PIR) to describe the proposed data‑driven hybrid technique as a way to bridge theory and data by use of ordinary least squares to efficiently perform parameter estimation of the model coefficients of different parameter‑linear models; providing examples of models based on nonlinear ordinary equations (ODE) and partial differential equations (PDE). The focus is on parameter estimation on a selection of ODE and PDE models, each illustrating performance in different model characteristics. For two relevant epidemic models of different complexity and number of parameters, PIR is tested and compared against the related technique, physics‑informed neural networks (PINN), both on synthetic data generated from known target parameters and on real public Danish time series data collected during the COVID‑19 pandemic in Denmark. Both methods were able to estimate the target parameters, while PIR showed to perform noticeably better, especially on a compartment model with higher complexity. Given the difference in computational speed, it is concluded that the PIR method is superior to PINN for the models considered. It is also demonstrated how PIR can be applied to estimate the time‑varying parameters of a compartment model that is fitted using real Danish data from the COVID‑19 pandemic obtained during a period from 2020 to 2021. The study shows how data‑driven and physics‑informed techniques may support reliable and fast ‑‑ possibly real‑time ‑‑ parameter estimation in parameter‑linear nonlinear dynamic models.

Abstract:
This paper investigates the generalisability of Koopman‑based representations for chaotic dynamical systems, focusing on their transferability across prediction and control tasks. Using the Lorenz system as a testbed, we propose a three‑stage methodology: learning Koopman embeddings through autoencoding, pre‑training a transformer on next‑state prediction, and fine‑tuning for safety‑critical control. Our results show that Koopman embeddings outperform both standard and physics‑informed PCA baselines, achieving accurate and data‑efficient performance. Notably, fixing the pre‑trained transformer weights during fine‑tuning leads to no performance degradation, indicating that the learned representations capture reusable dynamical structure rather than task‑specific patterns. These findings support the use of Koopman embeddings as a foundation for multi‑task learning in physics‑informed machine learning. A project page is available at https://kikisprdx.github.io/.

Abstract:
This paper proposes a Physics‑Informed Kolmogorov‑Arnold Network for analyzing elasticity problems in multi‑material electronic packaging structures. The method replaces traditional Multi‑Layer Perceptrons with Kolmogorov‑Arnold Networks within an energy‑based Physics‑Informed Neural Network framework. By constructing admissible displacement fields satisfying essential boundary conditions and optimizing network parameters through numerical integration, the proposed method effectively handles material property discontinuities. Unlike traditional methods that require domain decomposition and interface constraints for multi‑material problems, Kolmogorov‑Arnold Networks' trainable B‑spline activation functions provide inherent piecewise characteristics. This capability stems from B‑splines' local support, which enables effective approximation of discontinuities despite their individual smoothness. Consequently, this approach enables accurate approximation across the entire domain using a single network and simplifying the computational framework. Numerical experiments demonstrate that the proposed method achieves excellent accuracy and robustness in multi‑material elasticity problems, validating its practical potential for electronic packaging analysis. Source codes are available at https://github.com/yanpeng‑gong/PIKAN‑MultiMaterial.

Abstract:
As a critical component of Wearable AI, IMU‑based Human Activity Recognition (HAR) has attracted increasing attention from both academia and industry in recent years. Although HAR performance has improved considerably in specific scenarios, its generalization capability remains a key barrier to widespread real‑world adoption. For example, domain shifts caused by variations in users, sensor positions, or environments can significantly decrease the performance in practice. As a result, in this survey, we explore the rapidly evolving field of IMU‑based generalizable HAR, reviewing 229 research papers alongside 25 publicly available datasets to provide a broad and insightful overview. We first present the background and overall framework of IMU‑based HAR tasks, as well as the generalization‑oriented training settings. Then, we categorize representative methodologies from two perspectives: (i) model‑centric approaches, including pre‑training method, end‑to‑end method, and large language model (LLM)‑based learning method; and (ii) data‑centric approaches, including multi‑modal learning and data augmentation techniques. In addition, we summarize widely used datasets in this field, as well as relevant tools and benchmarks. Building on these methodological advances, the broad applicability of IMU‑based HAR is also reviewed and discussed. Finally, we discuss persistent challenges (e.g., data scarcity, efficient training, and reliable evaluation) and also outline future directions for HAR, including the adoption of foundation and large language models, physics‑informed and context‑aware reasoning, generative modeling, and resource‑efficient training and inference. The complete list of this survey is available at https://github.com/rh20624/Awesome‑IMU‑Sensing, which will be updated continuously.

Abstract:
In this paper, we aim to model 3D scene geometry, appearance, and physical information just from dynamic multi‑view videos in the absence of any human labels. By leveraging physics‑informed losses as soft constraints or integrating simple physics models into neural nets, existing works often fail to learn complex motion physics, or doing so requires additional labels such as object types or masks. We propose a new framework named TRACE to model the motion physics of complex dynamic 3D scenes. The key novelty of our method is that, by formulating each 3D point as a rigid particle with size and orientation in space, we directly learn a translation rotation dynamics system for each particle, explicitly estimating a complete set of physical parameters to govern the particle's motion over time. Extensive experiments on three existing dynamic datasets and one newly created challenging synthetic datasets demonstrate the extraordinary performance of our method over baselines in the task of future frame extrapolation. A nice property of our framework is that multiple objects or parts can be easily segmented just by clustering the learned physical parameters.

Abstract:
Estimating heat flux in the nuclear fusion device EAST is a critically important task. Traditional scientific computing methods typically model this process using the Finite Element Method (FEM). However, FEM relies on grid‑based sampling for computation, which is computationally inefficient and hard to perform real‑time simulations during actual experiments. Inspired by artificial intelligence‑powered scientific computing, this paper proposes a novel Physics‑Informed Neural Network (PINN) to address this challenge, significantly accelerating the heat conduction estimation process while maintaining high accuracy. Specifically, given inputs of different materials, we first feed spatial coordinates and time stamps into the neural network, and compute boundary loss, initial condition loss, and physical loss based on the heat conduction equation. Additionally, we sample a small number of data points in a data‑driven manner to better fit the specific heat conduction scenario, further enhancing the model's predictive capability. We conduct experiments under both uniform and non‑uniform heating conditions on the top surface. Experimental results show that the proposed thermal conduction physics‑informed neural network achieves accuracy comparable to the finite element method, while achieving ×40 times acceleration in computational efficiency. The dataset and source code will be released on https://github.com/Event‑AHU/OpenFusion.

Abstract:
Solving high‑frequency oscillatory partial differential equations (PDEs) is a critical challenge in scientific computing, with applications in fluid mechanics, quantum mechanics, and electromagnetic wave propagation. Traditional physics‑informed neural networks (PINNs) suffer from spectral bias, limiting their ability to capture high‑frequency solution components. We introduce Separated‑Variable Spectral Neural Networks (SV‑SNN), a novel framework that addresses these limitations by integrating separation of variables with adaptive spectral methods. Our approach features three key innovations: (1) decomposition of multivariate functions into univariate function products, enabling independent spatial and temporal networks; (2) adaptive Fourier spectral features with learnable frequency parameters for high‑frequency capture; and (3) theoretical framework based on singular value decomposition to quantify spectral bias. Comprehensive evaluation on benchmark problems including Heat equation, Helmholtz equation, Poisson equations and Navier‑Stokes equations demonstrates that SV‑SNN achieves 1‑3 orders of magnitude improvement in accuracy while reducing parameter count by over 90% and training time by 60%. These results establish SV‑SNN as an effective solution to the spectral bias problem in neural PDE solving. The implementation will be made publicly available upon acceptance at https://github.com/xgxgnpu/SV‑SNN.

Abstract:
Nuclear reactors are often modeled as inflexible baseload generators with fixed downtimes and restrictive ramping constraints. In practice, however, a reactor's operational flexibility is closely tied to its fuel cycle and associated reactivity margin. A key physical constraint for power maneuverability is xenon poisoning, caused from the transient buildup of neutron‑absorbing xenon following a power reduction. This transient can delay or prevent subsequent power ramp‑up due to suppressed core reactivity. Additionally, if a reactor is shutdown during periods of low reactivity, restart times can vary significantly, leading to prolonged downtimes. This work introduces a physics‑informed modeling framework that embeds fuel cycle dynamics within a unit commitment (UC) formulation. The framework tracks reactivity margin, dynamically enforces xenon induced constraints, and endogenously schedules refueling outages based on core conditions. By capturing intracycle reactivity evolution, the model enables operation dependent nuclear dispatch that reflects both techno‑economic requirements and irreducible nuclear physics limits. Application to a representative reactor fleet shows that flexible operation can slow reactivity degradation and extend fuel cycles. Results further demonstrate that different operational modes substantially affect VRE utilization, curtailment, and nuclear fleet capacity factors. These findings highlight the importance of fuel cycle aware flexibility modeling for accurate reactor scheduling and integration of nuclear power into energy system models.

Abstract:
Neural operators offer a powerful paradigm for solving partial differential equations (PDEs) that cannot be solved analytically by learning mappings between function spaces. However, there are two main bottlenecks in training neural operators: they require a significant amount of training data to learn these mappings, and this data needs to be labeled, which can only be accessed via expensive simulations with numerical solvers. To alleviate both of these issues simultaneously, we propose PICore, an unsupervised coreset selection framework that identifies the most informative training samples without requiring access to ground‑truth PDE solutions. PICore leverages a physics‑informed loss to select unlabeled inputs by their potential contribution to operator learning. After selecting a compact subset of inputs, only those samples are simulated using numerical solvers to generate labels, reducing annotation costs. We then train the neural operator on the reduced labeled dataset, significantly decreasing training time as well. Across four diverse PDE benchmarks and multiple coreset selection strategies, PICore achieves up to 78% average increase in training efficiency relative to supervised coreset selection methods with minimal changes in accuracy. We provide code at https://github.com/Asatheesh6561/PICore.

Abstract:
Physics‑Informed Neural Networks (PINNs) are deep learning models that incorporate the governing physical laws of a system into the learning process, making them well‑suited for solving complex scientific and engineering problems. Recently, PINNs have gained widespread attention as a powerful framework for combining physical principles with data‑driven modeling to improve prediction accuracy. Despite their successes, however, PINNs often exhibit poor extrapolation performance outside the training domain and are highly sensitive to the choice of activation functions (AFs). In this paper, we introduce a transfer learning (TL) method to improve the extrapolation capability of PINNs. Our approach applies transfer learning (TL) within an extended training domain, using only a small number of carefully selected collocation points. Additionally, we propose an adaptive AF that takes the form of a linear combination of standard AFs, which improves both the robustness and accuracy of the model. Through a series of experiments, we demonstrate that our method achieves an average of 40% reduction in relative L2 error and an average of 50% reduction in mean absolute error in the extrapolation domain, all without a significant increase in computational cost. The code is available at https://github.com/LiuzLab/PINN‑extrapolation .

Abstract:
Accurate representation of wells is essential for reliable reservoir characterization and simulation of operational scenarios in subsurface flow models. Physics‑informed neural networks (PINNs) have recently emerged as a promising method for reservoir modeling, offering seamless integration of monitoring data and governing physical equations. However, existing PINN‑based studies face major challenges in capturing fluid pressure near wells, particularly during the early stage after injection begins. To address this, we propose WellPINN, a modeling workflow that combines the outputs of multiple sequentially trained PINN models to accurately represent wells. This workflow iteratively approximates the radius of the equivalent well to match the actual well dimensions by decomposing the domain into stepwise shrinking subdomains with a simultaneously reducing equivalent well radius. Our results demonstrate that sequential training of superimposing networks around the pumping well is the first workflow that focuses on accurate inference of fluid pressure from pumping rates throughout the entire injection period, significantly advancing the potential of PINNs for inverse modeling and operational scenario simulations. All data and code for this paper will be made openly available at https://github.com/linuswalter/WellPINN.

Abstract:
We present a nonparametric Bayesian framework to infer radial distribution functions from experimental scattering measurements with uncertainty quantification using non‑stationary Gaussian processes. The Gaussian process prior mean and kernel functions are designed to mitigate well‑known numerical challenges with the Fourier transform, including discrete measurement binning and detector windowing, while encoding fundamental yet minimal physical knowledge of liquid structure. We demonstrate uncertainty propagation of the Gaussian process posterior to unmeasured quantities of interest. Experimental radial distribution functions of liquid argon and water with uncertainty quantification are provided as both a proof of principle for the method and a benchmark for molecular models. The full implementation is available on GitHub at: https://github.com/hoepfnergroup/LiquidStructureGP‑Sullivan.

Abstract:
Body dynamics are crucial information for the analysis of human motions in important research fields, ranging from biomechanics, sports science to computer vision and graphics. Modern approaches collect the body dynamics, external reactive force specifically, via force plates, synchronizing with human motion capture data, and learn to estimate the dynamics from a black‑box deep learning model. Being specialized devices, force plates can only be installed in laboratory setups, imposing a significant limitation on the learning of human dynamics. To this end, we propose a novel method for estimating human ground reaction dynamics directly from the more reliable motion capture data with physics laws and computational simulation as constrains. We introduce a highly accurate and robust method for computing ground reaction forces from motion capture data using Euler's integration scheme and PD algorithm. The physics‑based reactive forces are used to inform the learning model about the physics‑informed motion dynamics thus improving the estimation accuracy. The proposed approach was tested on the GroundLink dataset, outperforming the baseline model on: 1) the ground reaction force estimation accuracy compared to the force plates measurement; and 2) our simulated root trajectory precision. The implementation code is available at https://github.com/cuongle1206/Phys‑GRD

Abstract:
World models have become indispensable tools for embodied intelligence, serving as powerful simulators capable of generating realistic robotic videos while addressing critical data scarcity challenges. However, current embodied world models exhibit limited physical awareness, particularly in modeling 3D geometry and motion dynamics, resulting in unrealistic video generation for contact‑rich robotic scenarios. In this paper, we present RoboScape, a unified physics‑informed world model that jointly learns RGB video generation and physics knowledge within an integrated framework. We introduce two key physics‑informed joint training tasks: temporal depth prediction that enhances 3D geometric consistency in video rendering, and keypoint dynamics learning that implicitly encodes physical properties (e.g., object shape and material characteristics) while improving complex motion modeling. Extensive experiments demonstrate that RoboScape generates videos with superior visual fidelity and physical plausibility across diverse robotic scenarios. We further validate its practical utility through downstream applications including robotic policy training with generated data and policy evaluation. Our work provides new insights for building efficient physics‑informed world models to advance embodied intelligence research. The code is available at: https://github.com/tsinghua‑fib‑lab/RoboScape.

Abstract:
Musculoskeletal humanoids are robots that closely mimic the human musculoskeletal system, offering various advantages such as variable stiffness control, redundancy, and flexibility. However, their body structure is complex, and muscle paths often significantly deviate from geometric models. To address this, numerous studies have been conducted to learn body schema, particularly the relationships among joint angles, muscle tension, and muscle length. These studies typically rely solely on data collected from the actual robot, but this data collection process is labor‑intensive, and learning becomes difficult when the amount of data is limited. Therefore, in this study, we propose a method that applies the concept of Physics‑Informed Neural Networks (PINNs) to the learning of body schema in musculoskeletal humanoids, enabling high‑accuracy learning even with a small amount of data. By utilizing not only data obtained from the actual robot but also the physical laws governing the relationship between torque and muscle tension under the assumption of correct joint structure, more efficient learning becomes possible. We apply the proposed method to both simulation and an actual musculoskeletal humanoid and discuss its effectiveness and characteristics.

Abstract:
Recent advances in generative modeling ‑‑ particularly diffusion models and flow matching ‑‑ have achieved remarkable success in synthesizing discrete data such as images and videos. However, adapting these models to physical applications remains challenging, as the quantities of interest are continuous functions governed by complex physical laws. Here, we introduce FunDiff, a novel framework for generative modeling in function spaces. FunDiff combines a latent diffusion process with a function autoencoder architecture to handle input functions with varying discretizations, generate continuous functions evaluable at arbitrary locations, and seamlessly incorporate physical priors. These priors are enforced through architectural constraints or physics‑informed loss functions, ensuring that generated samples satisfy fundamental physical laws. We theoretically establish minimax optimality guarantees for density estimation in function spaces, showing that diffusion‑based estimators achieve optimal convergence rates under suitable regularity conditions. We demonstrate the practical effectiveness of FunDiff across diverse applications in fluid dynamics and solid mechanics. Empirical results show that our method generates physically consistent samples with high fidelity to the target distribution and exhibits robustness to noisy and low‑resolution data. Code and datasets are publicly available at https://github.com/sifanexisted/fundiff.

Abstract:
Given trajectory data, a domain‑specific study area, and a user‑defined threshold, we aim to find anomalous trajectories indicative of possible GPS spoofing (e.g., fake trajectory). The problem is societally important to curb illegal activities in international waters, such as unauthorized fishing and illicit oil transfers. The problem is challenging due to advances in AI generated in deep fakes generation (e.g., additive noise, fake trajectories) and lack of adequate amount of labeled samples for ground‑truth verification. Recent literature shows promising results for anomalous trajectory detection using generative models despite data sparsity. However, they do not consider fine‑scale spatiotemporal dependencies and prior physical knowledge, resulting in higher false‑positive rates. To address these limitations, we propose a physics‑informed diffusion model that integrates kinematic constraints to identify trajectories that do not adhere to physical laws. Experimental results on real‑world datasets in the maritime and urban domains show that the proposed framework results in higher prediction accuracy and lower estimation error rate for anomaly detection and trajectory generation methods, respectively. Our implementation is available at https://github.com/arunshar/Physics‑Informed‑Diffusion‑Probabilistic‑Model.

Abstract:
Neural rendering has advanced significantly in 3D reconstruction and novel view synthesis, and integrating physics into these frameworks opens new applications such as physically accurate digital twins for robotics and XR. However, the inverse problem of estimating physical parameters from visual observations remains challenging. Existing physics‑aware neural rendering methods typically require dense multi‑view videos, making them impractical for scalable, real‑world deployment. Under sparse‑view settings, the sequential optimization strategies employed by current approaches suffer from severe error accumulation: inaccuracies in initial 3D reconstruction propagate to subsequent stages, degrading physical state and material parameter estimates. On the other hand, simultaneous optimization of all parameters fails due to the highly non‑convex and often non‑differentiable nature of the problem. We propose ProJo4D, a progressive joint optimization framework that gradually expands the set of jointly optimized parameters. This design enables physics‑informed gradients to refine geometry while avoiding the instability of direct joint optimization over all parameters. Evaluations on synthetic and real‑world datasets demonstrate that ProJo4D substantially outperforms prior work in 4D future state prediction and physical parameter estimation, achieving up to 10x improvement in geometric accuracy while maintaining computational efficiency. Please visit the project webpage: https://daniel03c1.github.io/ProJo4D/

Abstract:
We present a novel dual‑stream architecture that achieves state‑of‑the‑art underwater image enhancement by explicitly integrating the Jaffe‑McGlamery physical model with capsule clustering‑based feature representation learning. Our method simultaneously estimates transmission maps and spatially‑varying background light through a dedicated physics estimator while extracting entity‑level features via capsule clustering in a parallel stream. This physics‑guided approach enables parameter‑free enhancement that respects underwater formation constraints while preserving semantic structures and fine‑grained details. Our approach also features a novel optimization objective ensuring both physical adherence and perceptual quality across multiple spatial frequencies. To validate our approach, we conducted extensive experiments across six challenging benchmarks. Results demonstrate consistent improvements of +0.5dB PSNR over the best existing methods while requiring only one‑third of their computational complexity (FLOPs), or alternatively, more than +1dB PSNR improvement when compared to methods with similar computational budgets. Code and data will be available at https://github.com/iN1k1/.

Abstract:
Large Language Models (LLMs) demonstrate strong reasoning and task planning capabilities but remain fundamentally limited in physical interaction modeling. Existing approaches integrate perception via Vision‑Language Models (VLMs) or adaptive decision‑making through Reinforcement Learning (RL), but they fail to capture dynamic object interactions or require task‑specific training, limiting their real‑world applicability. We introduce APEX (Anticipatory Physics‑Enhanced Execution), a framework that equips LLMs with physics‑driven foresight for real‑time task planning. APEX constructs structured graphs to identify and model the most relevant dynamic interactions in the environment, providing LLMs with explicit physical state updates. Simultaneously, APEX provides low‑latency forward simulations of physically feasible actions, allowing LLMs to select optimal strategies based on predictive outcomes rather than static observations. We evaluate APEX on three benchmarks designed to assess perception, prediction, and decision‑making: (1) Physics Reasoning Benchmark, testing causal inference and object motion prediction; (2) Tetris, evaluating whether physics‑informed prediction enhances decision‑making performance in long‑horizon planning tasks; (3) Dynamic Obstacle Avoidance, assessing the immediate integration of perception and action feasibility analysis. APEX significantly outperforms standard LLMs and VLM‑based models, demonstrating the necessity of explicit physics reasoning for bridging the gap between language‑based intelligence and real‑world task execution. The source code and experiment setup are publicly available at https://github.com/hwj20/APEX_EXP .

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a new learning paradigm for solving partial differential equations (PDEs) by enforcing the constraints of physical equations, boundary conditions (BCs), and initial conditions (ICs) into the loss function. Despite their successes, vanilla PINNs still suffer from poor accuracy and slow convergence due to the intractable multi‑objective optimization issue. In this paper, we propose a novel Dual‑Balanced PINN (DB‑PINN), which dynamically adjusts loss weights by integrating inter‑balancing and intra‑balancing to alleviate two imbalance issues in PINNs. Inter‑balancing aims to mitigate the gradient imbalance between PDE residual loss and condition‑fitting losses by determining an aggregated weight that offsets their gradient distribution discrepancies. Intra‑balancing acts on condition‑fitting losses to tackle the imbalance in fitting difficulty across diverse conditions. By evaluating the fitting difficulty based on the loss records, intra‑balancing can allocate the aggregated weight proportionally to each condition loss according to its fitting difficulty level. We further introduce a robust weight update strategy to prevent abrupt spikes and arithmetic overflow in instantaneous weight values caused by large loss variances, enabling smooth weight updating and stable training. Extensive experiments demonstrate that DB‑PINN achieves significantly superior performance than those popular gradient‑based weighting methods in terms of convergence speed and prediction accuracy. Our code and supplementary material are available at https://github.com/chenhong‑zhou/DualBalanced‑PINNs.

Abstract:
Auto‑regressive partial differential equation (PDE) foundation models have shown great potential in handling time‑dependent data. However, these models suffer from the shortcut problem deeply rooted in auto‑regressive prediction, causing error accumulation. The challenge becomes particularly evident for out‑of‑distribution data, as the pretraining performance may approach random model initialization for downstream tasks with long‑term dynamics. To deal with this problem, we propose physics‑informed temporal alignment (PITA), a self‑supervised learning framework inspired by inverse problem solving. Specifically, PITA aligns the physical dynamics discovered at different time steps on each given PDE trajectory by integrating physics‑informed constraints into the self‑supervision signal. The alignment is derived from observation data without relying on known physics priors, indicating strong generalization ability to the out‑of‑distribution data. Extensive experiments show that PITA significantly enhances the accuracy and robustness of existing foundation models on diverse time‑dependent PDE data. The code is available at https://github.com/SCAILab‑USTC/PITA.

Abstract:
Parametric differential equations of the form du/dt = f(u, x, t, p) are fundamental in science and engineering. While deep learning frameworks such as the Fourier Neural Operator (FNO) can efficiently approximate solutions, they struggle with inverse problems, sensitivity estimation (du/dp), and concept drift. We address these limitations by introducing a sensitivity‑based regularization strategy, called Sensitivity‑Constrained Fourier Neural Operators (SC‑FNO). SC‑FNO achieves high accuracy in predicting solution paths and consistently outperforms standard FNO and FNO with physics‑informed regularization. It improves performance in parameter inversion tasks, scales to high‑dimensional parameter spaces (tested with up to 82 parameters), and reduces both data and training requirements. These gains are achieved with a modest increase in training time (30% to 130% per epoch) and generalize across various types of differential equations and neural operators. Code and selected experiments are available at: https://github.com/AMBehroozi/SC_Neural_Operators

Abstract:
Velocity, pressure, and temperature are the key variables for understanding thermal convection, and measuring them all is a complex task. In this paper, we demonstrate a method to reconstruct temperature and pressure fields based on given Lagrangian velocity data. A physics‑informed neural network (PINN) based on a multilayer perceptron architecture and a periodic sine activation function is used to reconstruct both the temperature and the pressure for two cases of turbulent Rayleigh‑Bénard convection (Pr = 6.9, Ra = 10^9). The first dataset is generated with DNS and it includes Lagrangian velocity data of 150000 tracer particles. The second contains a PTV experiment with the same system parameters in a water‑filled cubic cell, and we observed about 50000 active particle tracks per time step with the open‑source framework proPTV. A realistic temperature and pressure field could be reconstructed in both cases, which underlines the importance of PINNs also in the context of experimental data. In the case of the DNS, the reconstructed temperature and pressure fields show a 90% correlation over all particles when directly validated against the ground truth. Thus, the proposed method, in combination with particle tracking velocimetry, is able to provide velocity, temperature, and pressure fields in convective flows even in the hard turbulence regime. The PINN used in this paper is compatible with proPTV and is part of an open source project. It is available on request at https://github.com/DLR‑AS‑BOA.

Abstract:
Learning‑based methods, such as imitation learning (IL) and reinforcement learning (RL), can produce excel control policies over challenging agile robot tasks, such as sports robot. However, no existing work has harmonized learning‑based policy with model‑based methods to reduce training complexity and ensure the safety and stability for agile badminton robot control. In this paper, we introduce Hamlet, a novel hybrid control system for agile badminton robots. Specifically, we propose a model‑based strategy for chassis locomotion which provides a base for arm policy. We introduce a physics‑informed "IL+RL" training framework for learning‑based arm policy. In this train framework, a model‑based strategy with privileged information is used to guide arm policy training during both IL and RL phases. In addition, we train the critic model during IL phase to alleviate the performance drop issue when transitioning from IL to RL. We present results on our self‑engineered badminton robot, achieving 94.5% success rate against the serving machine and 90.7% success rate against human players. Our system can be easily generalized to other agile mobile manipulation tasks such as agile catching and table tennis. Our project website: https://dreamstarring.github.io/HAMLET/.

Abstract:
Solving high‑dimensional partial differential equations (PDEs) is a critical challenge where modern data‑driven solvers often lack reliability and rigorous error guarantees. We introduce Simulation‑Calibrated Scientific Machine Learning (SCaSML), a framework that systematically improves pre‑trained PDE solvers at inference time without any retraining. Our core idea is to use defect correction method that derive a new PDE, termed Structural‑preserving Law of Defect, that precisely describes the error of a given surrogate model. Since it retains the structure of the original problem, we can solve it efficiently with traditional stochastic simulators and correct the initial machine‑learned solution. We prove that SCaSML achieves a faster convergence rate, with a final error bounded by the product of the surrogate and simulation errors. On challenging PDEs up to 160 dimensions, SCaSML reduces the error of various surrogate models, including PINNs and Gaussian Processes, by 20‑80%. Code of SCaSML is available at https://github.com/Francis‑Fan‑create/SCaSML.

Abstract:
A physics‑informed neural network (PINN) models the dynamics of a system by integrating the governing physical laws into the architecture of a neural network. By enforcing physical laws as constraints, PINN overcomes challenges with data scarsity and potentially high dimensionality. Existing PINN frameworks rely on fully observed time‑course data, the acquisition of which could be prohibitive for many systems. In this study, we developed a new PINN learning paradigm, namely Constrained Learning, that enables the approximation of first‑order derivatives or motions using non‑time course or partially observed data. Computational principles and a general mathematical formulation of Constrained Learning were developed. We further introduced MPOCtrL (Message Passing Optimization‑based Constrained Learning) an optimization approach tailored for the Constrained Learning framework that strives to balance the fitting of physical models and observed data. Its code is available at github link: https://github.com/ptdang1001/MPOCtrL Experiments on synthetic and real‑world data demonstrated that MPOCtrL can effectively detect the nonlinear dependency between observed data and the underlying physical properties of the system. In particular, on the task of metabolic flux analysis, MPOCtrL outperforms all existing data‑driven flux estimators.

Abstract:
Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many‑query scenarios, such as uncertainty quantification, remeshing applications, and topology optimization. This limitation has motivated the development of data‑driven surrogate models, where microscale computations are substituted by black‑box mappings between macroscale quantities. While these approaches offer significant speedups, they typically struggle to incorporate microscale physical constraints, such as the balance of linear momentum. In this contribution, we propose the Equilibrium Neural Operator (EquiNO), a physics‑informed PDE surrogate in which equilibrium is hard‑enforced by construction. EquiNO achieves this by projecting the solution onto a set of divergence‑free basis functions obtained via proper orthogonal decomposition (POD), thereby ensuring satisfaction of equilibrium without relying on penalty terms or multi‑objective loss functions. We compare EquiNO with variational physics‑informed neural and operator networks that enforce physical constraints only weakly through the loss function, as well as with purely data‑driven operator‑learning baselines. Our framework, applicable to multiscale FE^\,2 computations, introduces a finite element‑operator learning (FE‑OL) approach that integrates the finite element (FE) method with operator learning (OL). We apply the proposed methodology to quasi‑static problems in solid mechanics and demonstrate that FE‑OL yields accurate solutions even when trained on restricted datasets. The results show that EquiNO achieves speedup factors exceeding 8000‑fold compared to traditional methods and offers a robust and physically consistent alternative to existing data‑driven surrogate models.

Abstract:
Human Activity Recognition (HAR) primarily relied on traditional RGB cameras to achieve high‑performance activity recognition. However, the challenging factors in real‑world scenarios, such as insufficient lighting and rapid movements, inevitably degrade the performance of RGB cameras. To address these challenges, biologically inspired event cameras offer a promising solution to overcome the limitations of traditional RGB cameras. In this work, we rethink human activity recognition by combining the RGB and event cameras. The first contribution is the proposed large‑scale multi‑modal RGB‑Event human activity recognition benchmark dataset, termed HARDVS 2.0, which bridges the dataset gaps. It contains 300 categories of everyday real‑world actions with a total of 107,646 paired videos covering various challenging scenarios. Inspired by the physics‑informed heat conduction model, we propose a novel multi‑modal heat conduction operation framework for effective activity recognition, termed MMHCO‑HAR. More in detail, given the RGB frames and event streams, we first extract the feature embeddings using a stem network. Then, multi‑modal Heat Conduction blocks are designed to fuse the dual features, the key module of which is the multi‑modal Heat Conduction Operation layer. We integrate RGB and event embeddings through a multi‑modal DCT‑IDCT layer while adaptively incorporating the thermal conductivity coefficient via FVEs into this module. After that, we propose an adaptive fusion module based on a policy routing strategy for high‑performance classification. Comprehensive experiments demonstrate that our method consistently performs well, validating its effectiveness and robustness. The source code and benchmark dataset will be released on https://github.com/Event‑AHU/HARDVS/tree/HARDVSv2

Abstract:
Thermal analysis is crucial in 3D‑IC design due to increased power density and complex heat dissipation paths. Although operator learning frameworks such as DeepOHeat~\citeliu2023deepoheat have demonstrated promising preliminary results in accelerating thermal simulation, they face critical limitations in prediction capability for multi‑scale thermal patterns, training efficiency, and trustworthiness of results during design optimization. This paper presents DeepOHeat‑v1, an enhanced physics‑informed operator learning framework that addresses these challenges through three key innovations. First, we integrate Kolmogorov‑Arnold Networks with learnable activation functions as trunk networks, enabling an adaptive representation of multi‑scale thermal patterns. This approach achieves a 1.25x and 6.29x reduction in error in two representative test cases. Second, we introduce a separable training method that decomposes the basis function along the coordinate axes, achieving 62x training speedup and 31x GPU memory reduction in our baseline case, and enabling thermal analysis at resolutions previously infeasible due to GPU memory constraints. Third, we propose a confidence score to evaluate the trustworthiness of the predicted results, and further develop a hybrid optimization workflow that combines operator learning with finite difference (FD) using Generalized Minimal Residual (GMRES) method for incremental solution refinement, enabling efficient and trustworthy thermal optimization. Experimental results demonstrate that DeepOHeat‑v1 achieves accuracy comparable to optimization using high‑fidelity finite difference solvers, while speeding up the entire optimization process by 70.6× in our test cases, effectively minimizing the peak temperature through optimal placement of heat‑generating components. Open source code is available at https://github.com/xlyu0127/DeepOHeat‑v1.

Abstract:
Graph Neural Networks (GNNs) and differential equations (DEs) are two rapidly advancing areas of research that have shown remarkable synergy in recent years. GNNs have emerged as powerful tools for learning on graph‑structured data, while differential equations provide a principled framework for modeling continuous dynamics across time and space. The intersection of these fields has led to innovative approaches that leverage the strengths of both, enabling applications in physics‑informed learning, spatiotemporal modeling, and scientific computing. This survey aims to provide a comprehensive overview of the burgeoning research at the intersection of GNNs and DEs. We will categorize existing methods, discuss their underlying principles, and highlight their applications across domains such as molecular modeling, traffic prediction, and epidemic spreading. Furthermore, we identify open challenges and outline future research directions to advance this interdisciplinary field. A comprehensive paper list is provided at https://github.com/Emory‑Melody/Awesome‑Graph‑NDEs. This survey serves as a resource for researchers and practitioners seeking to understand and contribute to the fusion of GNNs and DEs

Abstract:
Creating a physical digital twin of a real‑world object has immense potential in robotics, content creation, and XR. In this paper, we present PhysTwin, a novel framework that uses sparse videos of dynamic objects under interaction to produce a photo‑ and physically realistic, real‑time interactive virtual replica. Our approach centers on two key components: (1) a physics‑informed representation that combines spring‑mass models for realistic physical simulation, generative shape models for geometry, and Gaussian splats for rendering; and (2) a novel multi‑stage, optimization‑based inverse modeling framework that reconstructs complete geometry, infers dense physical properties, and replicates realistic appearance from videos. Our method integrates an inverse physics framework with visual perception cues, enabling high‑fidelity reconstruction even from partial, occluded, and limited viewpoints. PhysTwin supports modeling various deformable objects, including ropes, stuffed animals, cloth, and delivery packages. Experiments show that PhysTwin outperforms competing methods in reconstruction, rendering, future prediction, and simulation under novel interactions. We further demonstrate its applications in interactive real‑time simulation and model‑based robotic motion planning.

Abstract:
Accurate and efficient 3D mapping of large‑scale outdoor environments from LiDAR measurements is a fundamental challenge in robotics, particularly towards ensuring smooth and artifact‑free surface reconstructions. Although the state‑of‑the‑art methods focus on memory‑efficient neural representations for high‑fidelity surface generation, they often fail to produce artifact‑free manifolds, with artifacts arising due to noisy and sparse inputs. To address this issue, we frame surface mapping as a physics‑informed energy optimization problem, enforcing surface smoothness by optimizing an energy functional that penalizes sharp surface ridges. Specifically, we propose a deep learning based approach that learns the signed distance field (SDF) of the surface manifold from raw LiDAR point clouds using a physics‑informed loss function that optimizes the L_2‑Hessian energy of the surface. Our learning framework includes a hierarchical octree based input feature encoding and a multi‑scale neural network to iteratively refine the signed distance field at different scales of resolution. Lastly, we introduce a test‑time refinement strategy to correct topological inconsistencies and edge distortions that can arise in the generated mesh. We propose a \textttCUDA‑accelerated least‑squares optimization that locally adjusts vertex positions to enforce feature‑preserving smoothing. We evaluate our approach on large‑scale outdoor datasets and demonstrate that our approach outperforms current state‑of‑the‑art methods in terms of improved accuracy and smoothness. Our code is available at \hrefhttps://github.com/HrishikeshVish/HessianForge/https://github.com/HrishikeshVish/HessianForge/

Abstract:
We introduce AG‑VPReID, a new large‑scale dataset for aerial‑ground video‑based person re‑identification (ReID) that comprises 6,632 subjects, 32,321 tracklets and over 9.6 million frames captured by drones (altitudes ranging from 15‑120m), CCTV, and wearable cameras. This dataset offers a real‑world benchmark for evaluating the robustness to significant viewpoint changes, scale variations, and resolution differences in cross‑platform aerial‑ground settings. In addition, to address these challenges, we propose AG‑VPReID‑Net, an end‑to‑end framework composed of three complementary streams: (1) an Adapted Temporal‑Spatial Stream addressing motion pattern inconsistencies and facilitating temporal feature learning, (2) a Normalized Appearance Stream leveraging physics‑informed techniques to tackle resolution and appearance changes, and (3) a Multi‑Scale Attention Stream handling scale variations across drone altitudes. We integrate visual‑semantic cues from all streams to form a robust, viewpoint‑invariant whole‑body representation. Extensive experiments demonstrate that AG‑VPReID‑Net outperforms state‑of‑the‑art approaches on both our new dataset and existing video‑based ReID benchmarks, showcasing its effectiveness and generalizability. Nevertheless, the performance gap observed on AG‑VPReID across all methods underscores the dataset's challenging nature. The dataset, code and trained models are available at https://github.com/agvpreid25/AG‑VPReID‑Net.

Abstract:
Time‑dependent partial differential equations are a significant class of equations that describe the evolution of various physical phenomena over time. One of the open problems in scientific computing is predicting the behaviour of the solution outside the given temporal region. Most traditional numerical methods are applied to a given time‑space region and can only accurately approximate the solution of the given region. To address this problem, many deep learning‑based methods, basically data‑driven and data‑free approaches, have been developed to solve these problems. However, most data‑driven methods require a large amount of data, which consumes significant computational resources and fails to utilize all the necessary information embedded underlying the partial differential equations (PDEs). Moreover, data‑free approaches such as Physics‑Informed Neural Networks (PINNs) may not be that ideal in practice, as traditional PINNs, which primarily rely on multilayer perceptrons (MLPs) and convolutional neural networks (CNNs), tend to overlook the crucial temporal dependencies inherent in real‑world physical systems. We propose a method denoted as PhysicsSolver that merges the strengths of two approaches: data‑free methods that can learn the intrinsic properties of physical systems without using data, and data‑driven methods, which are effective at making predictions. Extensive numerical experiments have demonstrated the efficiency and robustness of our proposed method. We provide the code at \hrefhttps://github.com/PhysicsSolver/PhysicsSolverhttps://github.com/PhysicsSolver.

Abstract:
This paper develops a novel deep learning approach for solving evolutionary equations, which integrates sequential learning strategies with an enhanced hard constraint strategy featuring trainable parameters, addressing the low computational accuracy of standard Physics‑Informed Neural Networks (PINNs) in large temporal domains.Sequential learning strategies divide a large temporal domain into multiple subintervals and solve them one by one in a chronological order, which naturally respects the principle of causality and improves the stability of the PINN solution. The improved hard constraint strategy strictly ensures the continuity and smoothness of the PINN solution at time interval nodes, and at the same time passes the information from the previous interval to the next interval, which avoids the incorrect/trivial solution at the position far from the initial time. Furthermore, by investigating the requirements of different types of equations on hard constraints, we design a novel influence function with trainable parameters for hard constraints, which provides theoretical and technical support for the effective implementations of hard constraint strategies, and significantly improves the universality and computational accuracy of our method. In addition, an adaptive time‑domain partitioning algorithm is proposed, which plays an important role in the application of the proposed method as well as in the improvement of computational efficiency and accuracy. Numerical experiments verify the performance of the method. The data and code accompanying this paper are available at https://github.com/zhizhi4452/HCS.

Abstract:
Background: MRI is crucial for brain imaging but is highly susceptible to motion artifacts due to long acquisition times. This study introduces PI‑MoCoNet, a physics‑informed motion correction network that integrates spatial and k‑space information to remove motion artifacts without explicit motion parameter estimation, enhancing image fidelity and diagnostic reliability. Materials and Methods: PI‑MoCoNet consists of a motion detection network (U‑net with spatial averaging) to identify corrupted k‑space lines and a motion correction network (U‑net with Swin Transformer blocks) to reconstruct motion‑free images. The correction is guided by three loss functions: reconstruction (L1), perceptual (LPIPS), and data consistency (Ldc). Motion artifacts were simulated via rigid phase encoding perturbations and evaluated on IXI and MR‑ART datasets against Pix2Pix, CycleGAN, and U‑net using PSNR, SSIM, and NMSE. Results: PI‑MoCoNet significantly improved image quality. On IXI, for minor artifacts, PSNR increased from 34.15 dB to 45.95 dB, SSIM from 0.87 to 1.00, and NMSE reduced from 0.55% to 0.04%. For moderate artifacts, PSNR improved from 30.23 dB to 42.16 dB, SSIM from 0.80 to 0.99, and NMSE from 1.32% to 0.09%. For heavy artifacts, PSNR rose from 27.99 dB to 36.01 dB, SSIM from 0.75 to 0.97, and NMSE decreased from 2.21% to 0.36%. On MR‑ART, PI‑MoCoNet achieved PSNR gains of ~10 dB and SSIM improvements of up to 0.20, with NMSE reductions of ~6%. Ablation studies confirmed the importance of data consistency and perceptual losses, yielding a 1 dB PSNR gain and 0.17% NMSE reduction. Conclusions: PI‑MoCoNet effectively mitigates motion artifacts in brain MRI, outperforming existing methods. Its ability to integrate spatial and k‑space information makes it a promising tool for clinical use in motion‑prone settings. Code: https://github.com/mosaf/PI‑MoCoNet.git.

Abstract:
This paper introduces PINT (Physics‑Informed Neural Time Series Models), a framework that integrates physical constraints into neural time series models to improve their ability to capture complex dynamics. We apply PINT to the ERA5 WeatherBench dataset, focusing on long‑term forecasting of 2m‑temperature data. PINT incorporates the Simple Harmonic Oscillator Equation as a physics‑informed prior, embedding its periodic dynamics into RNN, LSTM, and GRU architectures. This equation's analytical solutions (sine and cosine functions) facilitate rigorous evaluation of the benefits of incorporating physics‑informed constraints. By benchmarking against a linear regression baseline derived from its exact solutions, we quantify the impact of embedding physical principles in data‑driven models. Unlike traditional time series models that rely on future observations, PINT is designed for practical forecasting. Using only the first 90 days of observed data, it iteratively predicts the next two years, addressing challenges posed by limited real‑time updates. Experiments on the WeatherBench dataset demonstrate PINT's ability to generalize, capture periodic trends, and align with physical principles. This study highlights the potential of physics‑informed neural models in bridging machine learning and interpretable climate applications. Our models and datasets are publicly available on GitHub: https://github.com/KV‑Park.

Abstract:
Physics‑Informed Neural Networks (PINNs) are a kind of deep‑learning‑based numerical solvers for partial differential equations (PDEs). Existing PINNs often suffer from failure modes of being unable to propagate patterns of initial conditions. We discover that these failure modes are caused by the simplicity bias of neural networks and the mismatch between PDE's continuity and PINN's discrete sampling. We reveal that the State Space Model (SSM) can be a continuous‑discrete articulation allowing initial condition propagation, and that simplicity bias can be eliminated by aligning a sequence of moderate granularity. Accordingly, we propose PINNMamba, a novel framework that introduces sub‑sequence modeling with SSM. Experimental results show that PINNMamba can reduce errors by up to 86.3% compared with state‑of‑the‑art architecture. Our code is available at https://github.com/miniHuiHui/PINNMamba.

Abstract:
Deep learning methods ‑ usually consisting of a class of deep neural networks (DNNs) trained by a stochastic gradient descent (SGD) optimization method ‑ are nowadays omnipresent in data‑driven learning problems as well as in scientific computing tasks such as optimal control (OC) and partial differential equation (PDE) problems. In practically relevant learning tasks, often not the plain‑vanilla standard SGD optimization method is employed to train the considered class of DNNs but instead more sophisticated adaptive and accelerated variants of the standard SGD method such as the popular Adam optimizer are used. Inspired by the classical Polyak‑Ruppert averaging approach, in this work we apply averaged variants of the Adam optimizer to train DNNs to approximately solve exemplary scientific computing problems in the form of PDEs and OC problems. We test the averaged variants of Adam in a series of learning problems including physics‑informed neural network (PINN), deep backward stochastic differential equation (deep BSDE), and deep Kolmogorov approximations for PDEs (such as heat, Black‑Scholes, Burgers, and Allen‑Cahn PDEs), including DNN approximations for OC problems, and including DNN approximations for image classification problems (ResNet for CIFAR‑10). In each of the numerical examples the employed averaged variants of Adam outperform the standard Adam and the standard SGD optimizers, particularly, in the situation of the scientific machine learning problems. The Python source codes for the numerical experiments associated to this work can be found on GitHub at https://github.com/deeplearningmethods/averaged‑adam.

Abstract:
Surface wave dispersion curve inversion is crucial for estimating subsurface shear‑wave velocity (vs), yet traditional methods often face challenges related to computational cost, non‑uniqueness, and sensitivity to initial models. While deep learning approaches show promise, many require large labeled datasets and struggle with real‑world datasets, which often exhibit varying period ranges, missing values, and low signal‑to‑noise ratios. To address these limitations, this study introduces DispFormer, a transformer‑based neural network for v_s profile inversion from Rayleigh‑wave phase and group dispersion curves. DispFormer processes dispersion data independently at each period, allowing it to handle varying lengths without requiring network modifications or strict alignment between training and testing datasets. A depth‑aware training strategy is also introduced, incorporating physical constraints derived from the depth sensitivity of dispersion data. DispFormer is pre‑trained on a global synthetic dataset and evaluated on two regional synthetic datasets using zero‑shot and few‑shot strategies. Results show that even without labeled data, the zero‑shot DispFormer generates inversion profiles that outperform the interpolated reference model used as the pretraining target, providing a deployable initial model generator to assist traditional workflows. When partial labeled data available, the few‑shot trained DispFormer surpasses traditional global search methods. Real‑world tests further confirm that DispFormer generalizes well to dispersion data with varying lengths and achieves lower data residuals than reference models. These findings underscore the potential of DispFormer as a foundation model for dispersion curve inversion and demonstrate the advantages of integrating physics‑informed deep learning into geophysical applications.

Abstract:
Electrical Impedance Tomography (EIT) is a promising noninvasive imaging technique that reconstructs the spatial conductivity distribution from boundary voltage measurements. However, it poses a highly nonlinear and ill‑posed inverse problem. Traditional regularization‑based methods are sensitive to noise and often produce significant artifacts. Physics‑Embedded learning frameworks, particularly Physics‑Informed Neural Networks (PINNs), have shown success in solving such inverse problems under ideal conditions with abundant internal data. Yet in practical EIT applications, only sparse and noisy boundary measurements are available. Moreover, changing boundary excitations require the simultaneous training of multiple forward networks and one inverse network, which significantly increases computational complexity and hampers convergence. To overcome these limitations, we propose a Physics‑Embedded Dual‑Learning Imaging Framework for EIT. The dual‑learning strategy is composed of a supervised CNN‑based forward network, which learns to predict a discrete internal potential distribution under fixed Neumann‑to‑Dirichlet boundary conditions, and an unsupervised PINN‑based inverse network, which reconstructs the conductivity by enforcing the governing PDE through discrete numerical differentiation of the predicted potentials. This decoupled architecture removes the need for smooth conductivity assumptions, reduces the number of forward networks required from K to 1, and improves reconstruction robustness and efficiency under realistic measurement constraints.(https://github.com/XuanxuanYang/CNN‑PINNframework.git)

Abstract:
The use of neural networks for solving differential equations is practically difficult due to the exponentially increasing runtime of autodifferentiation when computing high‑order derivatives. We propose n‑TangentProp, the natural extension of the TangentProp formalism \citesimard1991tangent to arbitrarily many derivatives. n‑TangentProp computes the exact derivative d^n/dx^n f(x) in quasilinear, instead of exponential time, for a densely connected, feed‑forward neural network f with a smooth, parameter‑free activation function. We validate our algorithm empirically across a range of depths, widths, and number of derivatives. We demonstrate that our method is particularly beneficial in the context of physics‑informed neural networks where \ntp allows for significantly faster training times than previous methods and has favorable scaling with respect to both model size and loss‑function complexity as measured by the number of required derivatives. The code for this paper can be found at https://github.com/kyrochi/n\_tangentprop.

Abstract:
Initial boundary value problems arise commonly in applications with engineering and natural systems governed by nonlinear partial differential equations (PDEs). Operator learning is an emerging field for solving these equations by using a neural network to learn a map between infinite dimensional input and output function spaces. These neural operators are trained using a combination of data (observations or simulations) and PDE‑residuals (physics‑loss). A major drawback of existing neural approaches is the requirement to retrain with new initial/boundary conditions, and the necessity for a large amount of simulation data for training. We develop a physics‑informed transformer neural operator (named PINTO) that efficiently generalizes to unseen initial and boundary conditions, trained in a simulation‑free setting using only physics loss. The main innovation lies in our new iterative kernel integral operator units, implemented using cross‑attention, to transform the PDE solution's domain points into an initial/boundary condition‑aware representation vector, enabling efficient learning of the solution function for new scenarios. The PINTO architecture is applied to simulate the solutions of important equations used in engineering applications: advection, Burgers, and steady and unsteady Navier‑Stokes equations (three flow scenarios). For these five test cases, we show that the relative errors during testing under challenging conditions of unseen initial/boundary conditions are only one‑fifth to one‑third of other leading physics informed operator learning methods. Moreover, our PINTO model is able to accurately solve the advection and Burgers equations at time steps that are not included in the training collocation points. The code is available at https://github.com/quest‑lab‑iisc/PINTO

Abstract:
The numerical approximation of partial differential equations (PDEs) using neural networks has seen significant advancements through Physics‑Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi‑Layer Perceptrons (MLPs), which struggle to effectively learn high‑frequency and nonlinear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive bias of MLPs. However, they usually require high‑resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting. In addition, the fixed positions of the mesh parameters restrict their flexibility, making accurate approximation of complex PDEs challenging. To overcome these limitations, we propose Physics‑Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics‑Informed‑Gaussians/

Abstract:
Time series forecasting often demands a trade‑off between accuracy and efficiency. While recent Transformer models have improved forecasting capabilities, they come with high computational costs. Linear‑based models have shown better accuracy than Transformers but still fall short of ideal performance. We propose PIAD‑SRNN, a physics‑informed adaptive decomposition state‑space RNN, that separates seasonal and trend components and embeds domain equations in a recurrent framework. We evaluate PIAD‑SRNN's performance on indoor air quality datasets, focusing on CO2 concentration prediction across various forecasting horizons, and results demonstrate that it consistently outperforms SoTA models in both long‑term and short‑term time series forecasting, including transformer‑based architectures, in terms of both MSE and MAE. Besides proposing PIAD‑SRNN which balances accuracy with efficiency, this paper also provides four curated datasets. Code and data: https://github.com/ahmad‑shirazi/DSSRNN

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a tool for approximating the solution of Partial Differential Equations (PDEs) in both forward and inverse problems. PINNs minimize a loss function which includes the PDE residual determined for a set of collocation points. Previous work has shown that the number and distribution of these collocation points have a significant influence on the accuracy of the PINN solution. Therefore, the effective placement of these collocation points is an active area of research. Specifically, available adaptive collocation point sampling methods have been reported to scale poorly in terms of computational cost when applied to high‑dimensional problems. In this work, we address this issue and present the Point Adaptive Collocation Method for Artificial Neural Networks (PACMANN). PACMANN incrementally moves collocation points toward regions of higher residuals using gradient‑based optimization algorithms guided by the gradient of the PINN loss function, that is, the squared PDE residual. We apply PACMANN for forward and inverse problems, and demonstrate that this method matches the performance of state‑of‑the‑art methods in terms of the accuracy/efficiency tradeoff for the low‑dimensional problems, while outperforming available approaches for high‑dimensional problems. Key features of the method include its low computational cost and simplicity of integration into existing physics‑informed neural network pipelines. The code is available at https://github.com/CoenVisser/PACMANN.

Abstract:
Diffusion models have recently emerged as a potent tool in generative modeling. However, their inherent iterative nature often results in sluggish image generation due to the requirement for multiple model evaluations. Recent progress has unveiled the intrinsic link between diffusion models and Probability Flow Ordinary Differential Equations (ODEs), thus enabling us to conceptualize diffusion models as ODE systems. Simultaneously, Physics Informed Neural Networks (PINNs) have substantiated their effectiveness in solving intricate differential equations through implicit modeling of their solutions. Building upon these foundational insights, we introduce Physics Informed Distillation (PID), which employs a student model to represent the solution of the ODE system corresponding to the teacher diffusion model, akin to the principles employed in PINNs. Through experiments on CIFAR 10 and ImageNet 64x64, we observe that PID achieves performance comparable to recent distillation methods. Notably, it demonstrates predictable trends concerning method‑specific hyperparameters and eliminates the need for synthetic dataset generation during the distillation process. Both of which contribute to its easy‑to‑use nature as a distillation approach for Diffusion Models. Our code and pre‑trained checkpoint are publicly available at: https://github.com/pantheon5100/pid_diffusion.git.

Abstract:
Prediction of Kelvin‑Helmholtz instability (KHI) is crucial across various fields, requiring extensive high‑fidelity data. However, experimental data are often sparse and noisy, while simulated data may lack credibility due to discrepancies with real‑world configurations and parameters. This underscores the need for field reconstruction and parameter inference from sparse, noisy data, which constitutes inverse problems. Based on the physics‑informed neural networks (PINNs), the KH‑PINN framework is established in this work to solve the inverse problems of KHI flows. By incorporating the governing physical equations, KH‑PINN reconstructs continuous flow fields and infer unknown transport parameters from sparse, noisy observed data. The 2D unsteady incompressible flows with both constant and variable densities are studied. To our knowledge, this is the first application of PINNs to unsteady incompressible flows with variable densities. To address the spatiotemporal multiscale issue and enhance the reconstruction accuracy of small‑scale structures, the multiscale embedding (ME) strategy is adopted. To address the magnitude multiscale issue and enhance the reconstruction accuracy of small‑magnitude velocities, which are critical for KHI problems, the small‑velocity amplification (SVA) strategy is proposed. The results demonstrate that KH‑PINN can accurately reconstruct the fields with complex, evolving vortices and infer unknown parameters across a broad range of Reynolds numbers. Additionally, the energy‑decaying and entropy‑increasing curves are accurately obtained. The effectiveness of ME and SVA is validated through comparative studies, and the anti‑noise and few‑shot learning capabilities of KH‑PINN are also validated. The code for this work is available at https://github.com/CAME‑THU/KH‑PINN.

Abstract:
Surrogate modeling of non‑linear oscillator networks remains challenging due to discrepancies between simplified analytical models and real‑world complexity. To bridge this gap, we investigate hybrid reservoir computing, combining reservoir computing with "expert" analytical models. Simulating the absence of an exact model, we first test the surrogate models with parameter errors in their expert model. Second, in a residual physics task, we assess their performance when their expert model lacks key non‑linear coupling terms present in an extended ground‑truth model. We focus on short‑term forecasting across diverse dynamical regimes, evaluating the use of these surrogates for control applications. We show that hybrid reservoir computers generally outperform standard reservoir computers and exhibit greater robustness to parameter tuning. This advantage is less pronounced in the residual physics task. Notably, unlike standard reservoir computers, the performance of the hybrid does not degrade when crossing an observed spectral radius threshold. Furthermore, there is good performance for dynamical regimes not accessible to the expert model, demonstrating the contribution of the reservoir.

Abstract:
Numerical methods for contact mechanics are of great importance in engineering applications, enabling the prediction and analysis of complex surface interactions under various conditions. In this work, we propose an energy‑based physics‑informed neural network (PINNs) framework for solving frictionless contact problems under large deformation. Inspired by microscopic Lennard‑Jones potential, a surface contact energy is used to describe the contact phenomena. To ensure the robustness of the proposed PINN framework, relaxation, gradual loading and output scaling techniques are introduced. In the numerical examples, the well‑known Hertz contact benchmark problem is conducted, demonstrating the effectiveness and robustness of the proposed PINNs framework. Moreover, challenging contact problems with the consideration of geometrical and material nonlinearities are tested. It has been shown that the proposed PINNs framework provides a reliable and powerful tool for nonlinear contact mechanics. More importantly, the proposed PINNs framework exhibits competitive computational efficiency to the commercial FEM software when dealing with those complex contact problems. The codes used in this manuscript are available at https://github.com/JinshuaiBai/energy_PINN_Contact.(The code will be available after acceptance)

Abstract:
Arterial spin labeling (ASL) magnetic resonance imaging (MRI) enables cerebral perfusion measurement, which is crucial in detecting and managing neurological issues in infants born prematurely or after perinatal complications. However, cerebral blood flow (CBF) estimation in infants using ASL remains challenging due to the complex interplay of network physiology, involving dynamic interactions between cardiac output and cerebral perfusion, as well as issues with parameter uncertainty and data noise. We propose a new spatial uncertainty‑based physics‑informed neural network (PINN), SUPINN, to estimate CBF and other parameters from infant ASL data. SUPINN employs a multi‑branch architecture to concurrently estimate regional and global model parameters across multiple voxels. It computes regional spatial uncertainties to weigh the signal. SUPINN can reliably estimate CBF (relative error ‑0.3 \pm 71.7), bolus arrival time (AT) (30.5 \pm 257.8), and blood longitudinal relaxation time (T_1b) (‑4.4 \pm 28.9), surpassing parameter estimates performed using least squares or standard PINNs. Furthermore, SUPINN produces physiologically plausible spatially smooth CBF and AT maps. Our study demonstrates the successful modification of PINNs for accurate multi‑parameter perfusion estimation from noisy and limited ASL data in infants. Frameworks like SUPINN have the potential to advance our understanding of the complex cardio‑brain network physiology, aiding in the detection and management of diseases. Source code is provided at: https://github.com/cgalaz01/supinn.

Abstract:
We propose the first learning scheme for functional differential equations (FDEs). FDEs play a fundamental role in physics, mathematics, and optimal control. However, the numerical analysis of FDEs has faced challenges due to its unrealistic computational costs and has been a long standing problem over decades. Thus, numerical approximations of FDEs have been developed, but they often oversimplify the solutions. To tackle these two issues, we propose a hybrid approach combining physics‑informed neural networks (PINNs) with the cylindrical approximation. The cylindrical approximation expands functions and functional derivatives with an orthonormal basis and transforms FDEs into high‑dimensional PDEs. To validate the reliability of the cylindrical approximation for FDE applications, we prove the convergence theorems of approximated functional derivatives and solutions. Then, the derived high‑dimensional PDEs are numerically solved with PINNs. Through the capabilities of PINNs, our approach can handle a broader class of functional derivatives more efficiently than conventional discretization‑based methods, improving the scalability of the cylindrical approximation. As a proof of concept, we conduct experiments on two FDEs and demonstrate that our model can successfully achieve typical L^1 relative error orders of PINNs ～ 10^‑3. Overall, our work provides a strong backbone for physicists, mathematicians, and machine learning experts to analyze previously challenging FDEs, thereby democratizing their numerical analysis, which has received limited attention. Code is available at \urlhttps://github.com/TaikiMiyagawa/FunctionalPINN.

Abstract:
Recent advancements have integrated camera pose as a user‑friendly and physics‑informed condition in video diffusion models, enabling precise camera control. In this paper, we identify one of the key challenges as effectively modeling noisy cross‑frame interactions to enhance geometry consistency and camera controllability. We innovatively associate the quality of a condition with its ability to reduce uncertainty and interpret noisy cross‑frame features as a form of noisy condition. Recognizing that noisy conditions provide deterministic information while also introducing randomness and potential misguidance due to added noise, we propose applying epipolar attention to only aggregate features along corresponding epipolar lines, thereby accessing an optimal amount of noisy conditions. Additionally, we address scenarios where epipolar lines disappear, commonly caused by rapid camera movements, dynamic objects, or occlusions, ensuring robust performance in diverse environments. Furthermore, we develop a more robust and reproducible evaluation pipeline to address the inaccuracies and instabilities of existing camera control metrics. Our method achieves a 25.64% improvement in camera controllability on the RealEstate10K dataset without compromising dynamics or generation quality and demonstrates strong generalization to out‑of‑domain images. Training and inference require only 24GB and 12GB of memory, respectively, for 16‑frame sequences at 256x256 resolution. We will release all checkpoints, along with training and evaluation code. Dynamic videos are best viewed at https://zgctroy.github.io/CamI2V.

Abstract:
Motion Planning (MP) is a critical challenge in robotics, especially pertinent with the burgeoning interest in embodied artificial intelligence. Traditional MP methods often struggle with high‑dimensional complexities. Recently neural motion planners, particularly physics‑informed neural planners based on the Eikonal equation, have been proposed to overcome the curse of dimensionality. However, these methods perform poorly in complex scenarios with shaped robots due to multiple solutions inherent in the Eikonal equation. To address these issues, this paper presents PC‑Planner, a novel physics‑constrained self‑supervised learning framework for robot motion planning with various shapes in complex environments. To this end, we propose several physical constraints, including monotonic and optimal constraints, to stabilize the training process of the neural network with the Eikonal equation. Additionally, we introduce a novel shape‑aware distance field that considers the robot's shape for efficient collision checking and Ground Truth (GT) speed computation. This field reduces the computational intensity, and facilitates adaptive motion planning at test time. Experiments in diverse scenarios with different robots demonstrate the superiority of the proposed method in efficiency and robustness for robot motion planning, particularly in complex environments.

Abstract:
Mapping and motion planning are two essential elements of robot intelligence that are interdependent in generating environment maps and navigating around obstacles. The existing mapping methods create maps that require computationally expensive motion planning tools to find a path solution. In this paper, we propose a new mapping feature called arrival time fields, which is a solution to the Eikonal equation. The arrival time fields can directly guide the robot in navigating the given environments. Therefore, this paper introduces a new approach called Active Neural Time Fields (Active NTFields), which is a physics‑informed neural framework that actively explores the unknown environment and maps its arrival time field on the fly for robot motion planning. Our method does not require any expert data for learning and uses neural networks to directly solve the Eikonal equation for arrival time field mapping and motion planning. We benchmark our approach against state‑of‑the‑art mapping and motion planning methods and demonstrate its superior performance in both simulated and real‑world environments with a differential drive robot and a 6 degrees‑of‑freedom (DOF) robot manipulator. The supplementary videos can be found at https://youtu.be/qTPL5a6pRKk, and the implementation code repository is available at https://github.com/Rtlyc/antfields‑demo.

Abstract:
Physics‑informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill‑conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics‑informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics‑aware models. Furthermore, while traditional physics‑informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test‑time optimization performance. The code is available at https://github.com/2ailesB/neural‑parametric‑solver.

Abstract:
Recent breakthroughs in machine learning and artificial intelligence, fueled by scientific data, are revolutionizing the discovery of new materials. Despite the wealth of existing scientific literature, the availability of both structured experimental data and chemical domain knowledge that can be easily integrated into data‑driven workflows is limited. The motivation to integrate this information, as well as additional context from first‑principle calculations and physics‑informed deep learning surrogate models, is to enable efficient exploration of the relevant chemical space and to predict structure‑property relationships of new materials a priori. Ultimately, such a framework could replicate the expertise of human subject‑matter experts. In this work, we present dZiner, a chemist AI agent, powered by large language models (LLMs), that discovers new compounds with desired properties via inverse design (property‑to‑structure). In specific, the agent leverages domain‑specific insights from foundational scientific literature to propose new materials with enhanced chemical properties, iteratively evaluating them using relevant surrogate models in a rational design process, while accounting for design constraints. The model supports both closed‑loop and human‑in‑the‑loop feedback cycles, enabling human‑AI collaboration in molecular design with real‑time property inference, and uncertainty and chemical feasibility assessment. We demonstrate the flexibility of this agent by applying it to various materials target properties, including surfactants, ligand and drug candidates, and metal‑organic frameworks. Our approach holds promise to both accelerate the discovery of new materials and enable the targeted design of materials with desired functionalities. The methodology is available as an open‑source software on https://github.com/mehradans92/dZiner.

Abstract:
As a neural network's depth increases, it can improve generalization performance. However, training deep networks is challenging due to gradient and signal propagation issues. To address these challenges, extensive theoretical research and various methods have been introduced. Despite these advances, effective weight initialization methods for tanh neural networks remain insufficiently investigated. This paper presents a novel weight initialization method for neural networks with tanh activation function. Based on an analysis of the fixed points of the function \tanh(ax), the proposed method aims to determine values of a that mitigate activation saturation. A series of experiments on various classification datasets and physics‑informed neural networks demonstrates that the proposed method outperforms Xavier initialization methods~(with or without normalization) in terms of robustness across different network sizes, data efficiency, and convergence speed. Code is available at https://github.com/1HyunwooLee/Tanh‑Init

Abstract:
Finding solutions to partial differential equations (PDEs) is an important and essential component in many scientific and engineering discoveries. One of the common approaches empowered by deep learning is Physics‑informed Neural Networks (PINNs). Recently, a new type of fundamental neural network model, Kolmogorov‑Arnold Networks (KANs), has been proposed as a substitute of Multilayer Perceptions (MLPs), and possesses trainable activation functions. To enhance KANs in fitting accuracy, a modification of KANs, so called ReLU‑KANs, using "square of ReLU" as the basis of its activation functions, has been suggested. In this work, we propose another basis of activation functions, namely, Higherorder‑ReLU (HR), which is simpler than the basis of activation functions used in KANs, namely, Bsplines; allows efficient KAN matrix operations; and possesses smooth and non‑zero higher‑order derivatives, essential to physicsinformed neural networks. We name such KANs with Higher‑order‑ReLU (HR) as their activations, HRKANs. Our detailed experiments on two famous and representative PDEs, namely, the linear Poisson equation and nonlinear Burgers' equation with viscosity, reveal that our proposed Higher‑order‑ReLU‑KANs (HRKANs) achieve the highest fitting accuracy and training robustness and lowest training time significantly among KANs, ReLU‑KANs and HRKANs. The codes to replicate our experiments are available at https://github.com/kelvinhkcs/HRKAN.

Abstract:
Recent advances in generative models for medical imaging have shown promise in representing multiple modalities. However, the variability in modality availability across datasets limits the general applicability of the synthetic data they produce. To address this, we present a novel physics‑informed generative model capable of synthesizing a variable number of brain MRI modalities, including those not present in the original dataset. Our approach utilizes latent diffusion models and a two‑step generative process: first, unobserved physical tissue property maps are synthesized using a latent diffusion model, and then these maps are combined with a physical signal model to generate the final MRI scan. Our experiments demonstrate the efficacy of this approach in generating unseen MR contrasts and preserving physical plausibility. Furthermore, we validate the distributions of generated tissue properties by comparing them to those measured in real brain tissue.

Abstract:
Physics‑informed machine learning (PIML) has emerged as a promising alternative to classical methods for predicting dynamical systems, offering faster and more generalizable solutions. However, existing models, including recurrent neural networks (RNNs), transformers, and neural operators, face challenges such as long‑time integration, long‑range dependencies, chaotic dynamics, and extrapolation, to name a few. To this end, this paper introduces state‑space models implemented in Mamba for accurate and efficient dynamical system operator learning. Mamba addresses the limitations of existing architectures by dynamically capturing long‑range dependencies and enhancing computational efficiency through reparameterization techniques. To extensively test Mamba and compare against another 11 baselines, we introduce several strict extrapolation testbeds that go beyond the standard interpolation benchmarks. We demonstrate Mamba's superior performance in both interpolation and challenging extrapolation tasks. Mamba consistently ranks among the top models while maintaining the lowest computational cost and exceptional extrapolation capabilities. Moreover, we demonstrate the good performance of Mamba for a real‑world application in quantitative systems pharmacology for assessing the efficacy of drugs in tumor growth under limited data scenarios. Taken together, our findings highlight Mamba's potential as a powerful tool for advancing scientific machine learning in dynamical systems modeling. (The code will be available at https://github.com/zheyuanhu01/State_Space_Model_Neural_Operator upon acceptance.)

Abstract:
This paper introduces an efficient tensor‑vector product technique for the rapid and accurate approximation of integral operators within physics‑informed deep learning frameworks. Our approach leverages neural network architectures to evaluate problem dynamics at specific points, while employing Gaussian quadrature formulas to approximate the integral components, even in the presence of infinite domains or singularities. We demonstrate the applicability of this method to both Fredholm and Volterra integral operators, as well as to optimal control problems involving continuous time. Additionally, we outline how this approach can be extended to approximate fractional derivatives and integrals and propose a fast matrix‑vector product algorithm for efficiently computing the fractional Caputo derivative. In the numerical section, we conduct comprehensive experiments on forward and inverse problems. For forward problems, we evaluate the performance of our method on over 50 diverse mathematical problems, including multi‑dimensional integral equations, systems of integral equations, partial and fractional integro‑differential equations, and various optimal control problems in delay, fractional, multi‑dimensional, and nonlinear configurations. For inverse problems, we test our approach on several integral equations and fractional integro‑differential problems. Finally, we introduce the pinnies Python package to facilitate the implementation and usability of the proposed method.

Abstract:
Accurate 3D seismic velocity modeling through seismic travel‑time tomography using both active‑ and passive‑source data provides critical underpinning models for seismicity monitoring and hazard assessment. Because travel‑time tomography is an inherently ill‑posed inverse problem, UQ of the estimated models using Bayesian methods is also important for reliable downstream interpretations and analyses. However, Bayesian inference for 3D tomography based on conventional grid‑based representations faces the ``curse of dimensionality'' and severe computational bottlenecks. Consequently, rigorous Bayesian UQ for margin‑wide 3D travel‑time tomography has remained largely unexplored. In this study, we propose a meshless 3D Bayesian travel‑time tomography method that combines PINNs with a neural representation of the velocity structure, enabling tractable and data‑efficient Bayesian inference through function‑space particle‑based variational inference. To efficiently integrate passive‑source data into the Bayesian estimation of the velocity structure, we conduct analytical marginalization treating uncertain source parameters as nuisance parameters, with passive‑source relocation carried out in post‑processing. We validated the capability of our approach for 3D problems through synthetic experiments. Furthermore, we applied the method to a real‑world dataset from marine active‑source surveys and natural earthquakes off the Kii Peninsula, Nankai Trough. Our probabilistic 3D ensemble successfully resolves key geological features and provides data‑consistent uncertainty maps. The posterior mean hypocenters shifted mainly in the vertical direction by 10‑15 km, consistent with a previous relocation result. Finally, the neural representation drastically reduces storage requirements for the entire ensemble velocity model, highlighting the scalability and data efficiency of the proposed framework.

Abstract:
The Morse index of a free boundary minimal surface is encoded in its Jacobi‑Steklov spectrum, and we test how faithfully a physics‑informed neural network (PINN) reproduces that spectrum on a problem whose answer is already known in closed form. The benchmark is the critical catenoid in the unit ball \mathbbB^3, where it is well known that the Morse index equals 4 and the nullity equals 2. Separating the angular variable reduces the eigenvalue problem to a family of one‑dimensional Robin problems on [‑T,T], one for each Fourier mode. A network that enforces the parity of each mode by construction, and carries the eigenvalue as a trainable parameter, returns the three eigenvalues below the stability threshold to within 10^‑6 to 10^‑4 of their exact values, with PDE residuals of order 10^‑4; assembling them recovers the index 4 and the nullity 2. We then track the spectrum along a one‑parameter homotopy joining a flat reference operator to the catenoid Jacobi operator and identify the crossings at which the index changes. Since the critical catenoid is rigid, a fact we prove, this homotopy deforms operators rather than surfaces. We close by explaining how the same pipeline, with its one‑dimensional solver replaced by a two‑dimensional one, is poised to address genuinely geometric families in ellipsoidal balls, where the boundary curvature is no longer constant, and the Morse index is not yet known.

Abstract:
Deep learning in computational microscopy is severely constrained by the scarcity of densely annotated datasets. While synthetic data generation has bridged this gap in macroscopic computer vision, traditional graphics engines rely on geometric ray‑tracing, failing to capture the micro‑optical phenomena required for microscopy. Conversely, while wave‑optics formulations exist, rendering them computationally tractable at the scale required for deep learning remains a massive systems challenge. To address this, we introduce the Optical Synthetic Object Generator (OSOG), a high‑performance, fully differentiable forward‑modeling engine. Drawing on established physical models of diffraction and phase retardation, OSOG maps continuous Optical Path Difference (OPD) calculations into a highly optimized, PyTorch‑native Structure‑of‑Arrays (SoA) architecture. We validate this computational framework across three axes: First, object detection models (YOLOv11‑OBB) trained purely on OSOG‑generated data achieve robust zero‑shot transfer to real‑world highly occluded Lysozyme micrographs. Second, we introduce DiffOSOG, demonstrating that the engine's end‑to‑end differentiability allows for the exact recovery of continuous optical parameters via curriculum‑guided inverse rendering. Finally, OSOG bypasses the \mathcalO(N) bottlenecks of sequential ray‑tracing, demonstrating sub‑linear scaling by synthesizing 40,000 complex wave‑optic particles in under 50 milliseconds (\>20 FPS). By providing a fast, scalable, and physically grounded tensor pipeline, OSOG enables true real‑time, on‑the‑fly dataset generation.

Abstract:
Purpose: Physics‑Informed Neural Networks (PINNs) struggle with stiff, regime‑changing transport equations due to instability, loss imbalance, and violations of physical consistency. This paper investigates these failures through the Marshak wave equations ‑ a canonical benchmark from radiative transport ‑ where initial and boundary conditions differ by up to 12 orders of magnitude, and proposes targeted modifications to the standard PINN framework to overcome them. Design/methodology/approach: Three modifications are introduced: (1) a ScaledSigmoid final activation enforcing physical bounds and positivity of the unknowns; (2) a logarithmic MSE loss replacing the standard quadratic loss for initial and boundary conditions, enabling training across extreme scale disparities; and (3) explicit enforcement of global conservation laws derived from the governing equations as an additional physics loss term. Monte Carlo sampling with exponential time weighting is used throughout. Findings: The proposed framework successfully recovers the Marshak wave dynamics ‑ including the hot, cold, and wave‑front regions ‑ in agreement with a reference Implicit Monte Carlo solution, with run times under 30 minutes. Ablation studies confirm that each ingredient is essential: linear activation, absence of the logarithmic loss, or removal of the PDE term each independently cause the method to fail qualitatively. Originality/value: This work identifies and resolves three concrete failure modes of standard PINNs on stiff hyperbolic systems with nonlinear coupling. The combination of bounded activations, scale‑aware loss functions, and conservation law enforcement constitutes a novel and practically validated framework, with applicability to radiative transport and other coupled stiff PDE systems in engineering.

Abstract:
In robust variational physics‑informed neural networks (RVPINNs), the loss function is formulated in terms of the Riesz representative of the variational residual within a discrete test space. This approach guarantees that the loss function is robust with respect to the true error in the energy norm up to a remainder term that depends on both the neural network approximation and the discrete space configuration. However, in problems with localized singularities, steep gradients, or interface layers, a fixed coarse test space may fail to resolve the continuous Riesz representative of the residual during training. Although this can be avoided by using a sufficiently fine test space from the start, doing so may be computationally inefficient. We therefore propose an adaptive algorithm that enriches the test space only where the error between the discrete and continuous Riesz representatives is pronounced. We establish theoretical adaptive strategies within the RVPINN framework and derive their error bounds. Furthermore, we propose a computable refinement indicator and prove that, under the saturation assumption, it serves as a reliable and efficient error estimator for the non‑computable discrepancy between the discrete and continuous Riesz representatives. Finally, we propose a practical adaptive algorithm and demonstrate its effectiveness through numerical experiments on elliptic Dirichlet problems.

Abstract:
Physics‑informed and data‑driven surrogates are increasingly used to approximate mechanics‑governed flow fields, but the target quantities assigned to such models are not always identifiable from the input variables available at prediction time. We introduce an input‑schema identifiability certificate for computational surrogates. Starting from a reduced physical model, the certificate decomposes a target field into components that are measurable from geometry, components that require boundary‑condition information, and components identifiable only up to a symmetry quotient. This yields a pre‑training audit: it predicts which oracle‑channel interventions should reduce error, which should fail, and which ambiguity cannot be removed by changing the architecture, loss, optimizer, or sample size. We instantiate the framework for incompressible tubular flow using a Cosserat‑rod reduction, where lumen velocity separates into a mesh‑measurable tangent direction, a boundary‑condition‑dependent magnitude, and a signed‑orientation ambiguity. Controlled experiments on patient‑specific aortic CFD geometries, analytic Womersley flows, and an advection‑diffusion transfer problem confirm the predicted pattern: supplying signed direction collapses angular error to the oracle regime, whereas supplying magnitude without orientation leaves the predicted sign ambiguity and yields 16‑33 percent per‑node sign flips. The results provide a mechanics‑based diagnostic for deciding whether a surrogate modelling task is physically identifiable before training, and expose failure modes that aggregate error metrics can hide.

Abstract:
To address the challenges of interoperability in computational science, we present the latest updates to the software package MaRDI Open Interfaces. This software package aims to decrease the time and coding/testing efforts spent by computational scientists on tasks such as writing bindings to numerical solvers and adapting experiment codes to the varying interfaces of solvers for the same problem type (e.g., for benchmarking, which solver is better). By streamlining these tasks, this software package helps researchers focus on the actual essence of their computational projects. Here, we demonstrate a recently developed interface for nonlinear optimization and illustrate how it can be applied for computational experiments with optimization problems. As an example of such problem, we consider training of physics‑informed neural networks to predict the solutions of viscous Burgers' equation.

Abstract:
Physics‑Informed Neural Networks (PINNs) solve Partial Differential Equations (PDEs) by embedding physical laws into neural network training. However, their performance suffers from unstable convergence, training plateaus, and strong sensitivity to architectural and optimization hyperparameters due to the highly non‑convex and multi‑term structure of the physics‑informed loss. In this setting, the outer‑loop hyperparameter search is a noisy and black‑box optimization problem over heterogeneous parameters, where classical local or gradient‑based strategies are easily trapped in suboptimal regions. Evolutionary algorithms, with their population‑based exploration and ability to handle mixed, non‑differentiable search spaces, provide a more robust mechanism for discovering promising configurations. We propose and investigate a two‑stage approach based on evolutionary algorithms that combines exploration and exploitation parts of PINNs training to improve solution accuracy and robustness under fixed computational budgets. In the first stage, we perform low‑fidelity training runs with truncated epochs to rapidly screen candidate configurations, treating hyperparameter selection as a black‑box outer‑loop problem. In the second stage, only the most promising candidates are fully trained with standard gradient‑based optimizers to refine the solution. Evaluated on three popular problems, namely Advection, Klein‑Gordon and Helmholtz equations, our method consistently outperforms standard training and achieves significantly lower mean error within constrained computational resources.

Abstract:
We develop QCPIKAN, the first quantum‑classical physics‑informed Kolmogorov‑Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev‑polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high‑frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single‑phase flow, component transport and two‑phase flow. Compared with existing quantum‑classical physics‑informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.

Abstract:
Deep learning has managed to evade numerous intuitions from classical statistics to achieve unprecedented performance on a number of real‑world tasks. In this article, we investigate the key features and surprises of deep learning from a physics‑informed perspective, taking care to point out and justify where possible the many choices inherent in constructing a deep learning model. In particular, we review the phenomenon of neural scaling laws and discuss their interplay with the constraints and inductive biases which may be present when applying machine learning to problems in physics.

Abstract:
We introduce SEA‑PINN, a novel architecture that incorporates a Squeeze‑Excitation‑like attention mechanism into physics‑informed neural networks to dynamically recalibrate the importance of neurons across layers. A key feature of SEA‑PINN is its highly stable initialization. On 17 out of 20 benchmark problems, SEA‑PINN exhibit nearly negligible variance and significantly reduced initial loss, establishing a quasi‑deterministic and favorable starting point for optimization. Notably, without employing Fourier feature embeddings or periodic activation functions, SEA‑PINN attained competitive accuracy (83% vs. 90% improvement relative to FNN‑PINN on the high‑frequency case 7) as compared with TSA‑PINN‑a model specifically engineered for high‑frequency problems via learnable frequencies in sinusoidal activations. Furthermore, integrating SEA‑PINN into TSA‑PINN boosted performance by 42.49%. These results underscore SEA‑PINN as a lightweight plug‑in module that enhances nonlinear representation power, promotes more robust and efficient convergence, and strengthens the overall reliability of physics‑informed learning.

Abstract:
Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems. While traditional numerical solvers are robust, they often incur prohibitive computational costs due to mesh dependencies, whereas recent Physics‑Informed Neural Networks (PINNs) offer a mesh‑free alternative but frequently suffer from slow convergence and optimization instability. To bridge this gap, this article proposes the Physics‑Informed Broad Learning System (PIBLS), a novel backpropagation‑free framework that reformulates PDE solving as a direct least‑squares optimization. We improved an algorithm within this framework to handle nonlinear PDEs efficiently and provide a rigorous mathematical proof establishing the universal approximation property of PIBLS for these equations. Experiments on linear and nonlinear PDEs demonstrate that PIBLS is one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy. This framework provides a computationally efficient paradigm for scientific machine learning, offering a practical, high‑speed alternative for real‑time simulation and design optimization tasks.

Abstract:
This chapter reviews recent advances in Scientific Machine Learning (SciML) for modeling coupled fluid flow and transport phenomena governed by the incompressible Navier‑Stokes and scalar transport equations. Such systems, found in applications like turbidity currents and thermal convection, feature strong nonlinear coupling and multiscale behavior that make high‑fidelity simulations computationally expensive. To address this, the chapter surveys state‑of‑the‑art SciML methods for building efficient surrogate models, including linear reduced‑order techniques based on Singular Value Decomposition (such as Dynamic Mode Decomposition) and nonlinear neural network approaches like Physics‑Informed Neural Networks (PINNs) and β‑Variational Autoencoders (β‑VAEs). It first covers the authors' work combining these models with High Performance Computing strategies, including Adaptive Mesh Refinement/Coarsening (AMR/C) and scientific floating‑point data compression. It then presents two new contributions: surrogate modeling of turbidity currents via PINNs, and the extraction of disentangled nonlinear modes from thermal flows using β‑VAEs. Governing equations and representative benchmarks, including lock‑exchange flows and Rayleigh‑Bénard convection, illustrate these methodologies. The chapter is intentionally long, covering both the mathematical and physical foundations of coupled fluid flow and the computational aspects of state‑of‑the‑art modeling. Overall, it demonstrates how SciML enables fast, accurate approximations of complex coupled systems within the specific data regimes and modeling assumptions considered, while substantially reducing computational cost relative to full‑order simulations. Broader capabilities such as real‑time prediction and uncertainty quantification remain active research directions whose feasibility depends strongly on the problem at hand.

Abstract:
Scientific machine learning (SciML) has emerged as a promising approach for accelerating simulations of complex physical systems, yet achieving physically consistent and generalizable predictions for nonlinear, history‑dependent problems remains a central challenge. In this study, we propose a hybrid GNN‑‑FEM framework for efficient and generalizable phase‑field fracture modeling. While phase‑field approaches provide a robust variational framework for simulating complex crack evolution, their high computational cost limits practical applications because they require solving coupled, nonlinear, and history‑dependent systems within an incremental finite element procedure. To address this challenge, a graph neural network surrogate is integrated into the conventional staggered scheme, replacing the phase‑field update at each load increment while retaining the FEM‑based displacement solver to enforce mechanical equilibrium and boundary conditions. By preserving the incremental solution structure, the framework remains consistent with history‑dependent fracture evolution without requiring the surrogate to approximate the full solution trajectory. This selective surrogate strategy emphasizes the identification of a physically meaningful and incrementally structured learning target, rather than relying on brute‑force data generation to learn the full fracture process. The proposed framework achieves strong generalization across varying geometries, loading conditions, material properties, and discretizations through dimensionless feature design, a graph‑based formulation on mesh‑based domains, and a physics‑informed loss derived from the governing phase‑field equation. Numerical experiments demonstrate that the hybrid approach reduces computational cost while maintaining accuracy compared with conventional FEM, and exhibits robust predictive performance across diverse problem settings.

Abstract:
Identifying anisotropic yield functions remains challenging since yielding is not directly observed in full‑field mechanical measurements, directional calibration can require many loading directions, and selecting an appropriate analytical form is nontrivial. This study proposes a physics‑informed framework for discovering yield functions from full‑field displacement data and reaction force data, without stress observations, plastic strain measurements, direct yield surface data, or a prescribed parametric yield function. The framework identifies the yield function as a mechanically constrained constitutive component inside elastoplastic stress integration, rather than through direct stress‑space supervision. The yield function is represented by a convex neural network that enforces convexity and positive homogeneity of degree one while imposing the assumed tension‑compression symmetry, and this neural yield function is trained with a differentiable stress update and a physics‑informed force equilibrium loss across multiple loading cases. The proposed framework is validated using finite element (FE) benchmark studies with von Mises, Hill 1948, and Yld2000‑2d yield functions, assessing yield contour agreement, displacement‑noise sensitivity, identifiability through plastically active stress states, epistemic uncertainty, and polynomial‑surrogate deployment. This study provides a mechanics‑constrained pathway for discovering anisotropic yield functions from displacement and force data while keeping the identified component within the structure of elastoplastic stress integration.

Abstract:
In this work we investigate the role of neural architectures as implicit functional priors in control problems governed by ordinary differential equations. Rather than focusing on highly complex problems, our objective is to investigate architecture‑dependent effects in controlled dynamical systems within the simplest physically interpretable settings possible. In particular, we study a controlled linear RLC electrical circuit and a nonlinear Duffing‑type dynamical system. Both systems are analyzed first through classical optimal‑control formulations and later through PINN‑based approaches. We compare different combinations of multilayer perceptrons (MLPs) and Fourier‑based KAN‑like architectures, and analyze their influence on the resulting controls. The numerical experiments suggest that different architectural choices systematically generate qualitatively distinct controls, even under identical governing equations, loss functionals, initial and target states, training parameters and physical constraints. Significant differences appear in the spectral structure, smoothness, energy distribution, and phase‑space behavior of the learned solutions. A central observation of this work is the emergence of a functional specialization phenomenon when the neural architectures are allowed sufficient freedom to shape the structure of the learned controls. More specifically, in the systems considered here, Fourier‑based architectures tend to produce trajectories with richer oscillatory content, whereas smoother low‑frequency‑biased architectures tend to generate more regular and energetically efficient controls. This suggests that different functional components of the control problem may be handled more efficiently by different neural architectures, leading to an implicit specialization between state representation and control generation.

Abstract:
High‑fidelity simulation of spatiotemporal dynamics is computationally prohibitive, necessitating efficient super‑resolution techniques to reconstruct high‑resolution data from coarse‑grained inputs. Traditional data‑driven methods often lack physical constraints, and simple physics‑informed learning struggles with irregular spatial geometries and intricately evolving temporal dynamics. To tackle these challenges, we propose a Physics‑augmented Koopman‑enhanced Graph Convolutional Network (P‑K‑GCN) for spatiotemporal super‑resolution on irregular geometries. Specifically, a continuous spline‑based GCN is first designed to extract spatial dependencies directly from coarse graph, and Koopman operator theory is incorporated to project the nonlinear dynamics into a compact latent space where temporal progression is linearized. Second, we augment the optimization objective with a physics‑based loss to force the data‑driven reconstructions to adhere to physical laws for improving predictive fidelity and robustness. Finally, we provide a rigorous theoretical analysis, establishing that the physics augmentation and Koopman regularization mathematically guarantees a reduction in super‑resolution error by diminishing Rademacher complexity and tightening generalization bounds. We evaluate our framework on reconstructing spatially high‑resolution cardiac electrodynamics across a 3D heart geometry from sparse low‑resolution measurements. Numerical experiments demonstrate that our method achieves superior accuracy compared to baseline models.

Abstract:
AI systems can increasingly automate scientific workflows, but the reasoning that links prior evidence, generated ideas, experiments and final claims often remains implicit inside model inference. Here we introduce Xcientist, a research harness that externalizes research synthesis and experimental validation into inspectable, contract‑governed processes. Xcientist organizes literature evidence, idea states, implementation plans, ablation records and repair traces as persistent research artifacts, so that generated mechanisms can be grounded, executed, tested and revised without losing their evidential basis. We identify claim drift as a failure mode of automated research, where runnable artifacts no longer support the mechanism originally claimed. Across training‑free memory systems, graph‑structured traffic forecasting and multi‑scale physics‑informed neural networks, Xcientist preserves traceable trajectories from problem formulation to mechanism design, validation and bounded revision. These results suggest that AI scientists should be evaluated not only by their final artifacts, but by whether their synthesis and validation processes remain attributable, inspectable and scientifically accountable.

Abstract:
The concept of geodesic‑like curves was introduced by Chen in 2010 as a method for estimating shortest paths (geodesics) on parametric surfaces, with its convergence established theoretically. However, an efficient numerical computational framework has not yet been developed. In this paper, we propose an elegant and efficient approach for computing geodesic‑like curves by leveraging deep learning and Physics‑Informed Neural Networks (PINNs). Under the proposed framework, not only can single parametric surfaces be handled efficiently, but a broad class of complex parametric surfaces including multi‑surface systems with C^0 or higher continuity and surfaces of revolution can also be robustly addressed.

Abstract:
Photonic quantum machine learning offers a route to trainable physical representations built from phase, interference and measurement. However, its role in scientific machine learning remains largely unexplored. Physics‑informed neural fields provide a natural setting, because differential equations require trial spaces that preserve phase, frequency and derivative structure. Here we introduce a photonic quantum neural field in which coordinates become trainable optical phases, are mixed by multi‑photon Fock‑space interference and are decoded from photon‑number measurements. The photonic circuit is optimized as the neural‑field representation itself, not as a fixed feature map or hardware accelerator. Photonic measurement is therefore a trainable representation on which the physics‑informed residual is minimized. Across seven elliptic, wave, nonlinear dispersive and inverse PDE benchmarks, we observe a phase‑complexity transition: classical coordinate and Fourier‑feature networks suffice in smooth regimes, whereas the photonic field is most accurate when residual derivatives amplify phase mismatch. In the hardest regimes it gives the lowest errors, with margins reaching an order of magnitude and about one quarter of the trainable parameters of classical baselines. Frozen and shuffled controls, together with noise stress tests, attribute this gain to learned interference and stable Fock‑probability readout under compound perturbations. These results identify photonic quantum measurement as a representation‑learning principle for scientific machine learning.

Abstract:
Time‑dependent partial differential equations (PDEs) arise in many engineering systems, including thermo‑fluid applications. Classical numerical simulations of such systems can become computationally expensive for long‑time dynamics because they typically require sequential time integration with time steps constrained by stability, accuracy, or nonlinear solvers. Although scientific machine learning provides an alternative for approximating PDE solutions, standard neural network approximations often degrade when extrapolated beyond the training time interval. In this work, we propose a steady‑state‑informed neural network representation for dissipative PDE systems whose solutions relax toward a stationary equilibrium. The proposed ansatz decomposes the solution into a steady‑state component and a transient correction modulated by a time‑dependent decay profile. When the decay profile vanishes at long time and the transient correction remains bounded, the representation embeds convergence to the prescribed steady state directly into the architecture, rather than enforcing it through an additional penalty term. This allows the network to learn the transient dynamics while preserving the correct asymptotic behavior. We implement the approach within a physics‑informed neural network (PINN) framework and train the resulting model using the SOAP optimizer. The method is evaluated on a sequence of problems of increasing physical and geometric complexity, ranging from the one‑dimensional heat equation to incompressible Navier‑Stokes flow in a lid‑driven cavity, natural convection in a square cavity, and a full three‑dimensional conjugate heat transfer problem. The numerical results show that the steady‑state‑informed architecture substantially improves temporal extrapolation beyond the training interval compared with architectures that do not explicitly enforce the asymptotic condition.

Abstract:
Safe coordination in networked cyber‑physical systems forces learning algorithms to simultaneously handle hybrid discrete‑continuous actions, hard training‑time safety constraints, and physics‑governed dynamics. We show that these three features form a directed cycle of biases that defeats any naive composition of off‑the‑shelf modules, and formalize this as a three‑way coupling lemma. We then introduce TRIDENT, the first MARL framework whose three components are co‑designed to cancel each leak: a Richardson‑Romberg gradient correction reducing Gumbel‑Softmax bias from O(tau) to O(tau^2), a Lyapunov‑constrained sequential trust‑region update enforcing per‑iterate feasibility, and a physics‑informed residual critic that decomposes value rather than reward. We prove an O~(1/sqrt(K)) convergence rate to a constrained Nash equilibrium and an O(sqrt(K)) cumulative‑violation bound. On multi‑UAV mobile‑edge computing, autonomous intersection management, and a hybrid SMAC variant, TRIDENT cuts training‑time violations by 95.5% over MADDPG and 76.3% over MACPO, while improving reward by 13.5% over the strongest unconstrained baseline.

Abstract:
Learned dynamics models often answer global physical questions, such as fault severity or impact stiffness, by pooling a per‑step feature sequence into one readout vector. This sequence‑to‑global interface creates an under‑studied temporal credit problem: with only trajectory‑level supervision, a model can predict accurately in training conditions while reading from abundant smooth correlates rather than the brief physical events that determine the target. We call this failure temporal credit dilution. It is not exposed by the training loss and is not removed by standard physics‑informed residuals, because the error lies in where the global readout assigns functional credit. We introduce Credit‑in‑Event, an interface‑level probe for measuring how much pooled credit lands on event steps, and prove in closed form that a pooled linear reader routes credit to a spurious background channel as the event fraction shrinks. We then propose CREST, a training‑free and label‑free readout that estimates a transient event core from learned features and re‑anchors the pooled representation through event‑versus‑rest contrast. Across simulated gear and impact systems, recurrent and attention encoders, and public bearing vibration data, CREST reduces out‑of‑distribution error while restoring event credit. Ablations show that stable‑step selection and receptive‑field shrinking fail, confirming that the gain comes from event‑core credit re‑anchoring rather than a generic locality or stability prior.

Abstract:
Machine Learning (ML) models for grain growth prediction are typically trained on idealized synthetic data, yet practical applications require generalization to conditions outside the training distribution. This study evaluated the Out‑Of‑Distribution (OOD) generalization capability of the trained model from our previous study across three test cases, including experimental microstructures, microstructures characterized by a bimodal grain size distribution, and abnormal grain growth. To further probe whether physics‑informed architectural design could improve robustness under these different conditions, a boundary‑masked attention mechanism was proposed specifically for grain growth, constraining attention to grain boundary pixels. Both the baseline and the proposed physics‑informed attention model were evaluated without retraining or fine‑tuning on the OOD data. Both models successfully generalized to all three test cases, yet the boundary‑masked attention mechanism provided substantial improvements, with the most notable gains for microstructures characterized by a bimodal grain size distribution, where Structural Similarity Index Measure (SSIM) improved from \num0.6221 to \num0.7609 and mean grain size (\overlineR) error decreased from \SI8.75\percent to \SI3.57\percent. The attention heatmap analysis revealed that the boundary‑masked attention model learned to concentrate attention on large grain boundaries in a manner consistent with curvature‑driven grain growth physics, emerging from training without being explicitly encoded into the architecture. These results indicate that models trained on synthetic data can generalize to diverse OOD conditions without retraining, and that physics‑informed attention may improve accuracy when the boundary morphology matches the training domain.

Abstract:
Learning‑based single‑shot fringe projection profilometry (FPP) has been studied mostly at close range. The long‑range regime (standoff beyond 1 m) remains largely unaddressed: inverse‑square intensity falloff lowers fringe signal‑to‑noise ratio and degrades physical ground truth, the single‑shot problem is ill‑posed because fringe‑order information is absent from one image, and these architectures have not been studied mechanistically. We present a diagnose‑repair‑verify study using mechanistic interpretability (MI) and conformal uncertainty quantification (UQ) as convergent diagnostics: they agree on one physical failure locus, driving and verifying an architectural repair. On a photorealistic synthetic benchmark (15,600 fringe images, 50 objects at 1.5‑2.1 m), a best UNet baseline reaches 14.54 mm object mean absolute error (MAE). Three probes (linear probing, Grad‑CAM, flat‑plane out‑of‑distribution test) converge: the baseline solves the task via object‑boundary shape priors rather than fringe‑phase decoding. We repair this with PhiCalNet, which outputs wrapped phase rather than depth and applies a fixed differentiable calibration layer mapping phase to depth, removing the shape‑prior solution from the hypothesis space architecturally rather than by a loss penalty. A physics‑informed loss that enforces the same physics as a soft penalty on a depth‑regressing network yields no measurable gain, isolating the architecture as the operative factor. PhiCalNet reduces object MAE 3.3x to 4.46 mm; the residual is carried by 0.103% of pixels at the +/‑pi wrap discontinuity. Pixel‑wise conformal UQ confirms the diagnosis: rejecting the top 5% of object pixels by snapshot disagreement cuts PhiCalNet RMSE by 64% (20.6‑>7.4 mm) versus 3.5% for the baseline. MI and UQ converge on the same failure locus.

Abstract:
Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high‑frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNN, a reparameterized neural network model with activation ReLU or tanh designed for high‑frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one‑ and four‑dimensional function approximations, forward and inverse PDE problems in combination with physics‑informed neural networks (PINNs), and operator learning for an earthquake problem using real data, demonstrate that RepNN improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features with slightly additional computational cost. These results indicate that RepNN provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.

Abstract:
Multivariate forecasting in physical systems requires models that predict coupled temporal variables while preserving meaningful state evolution. Deep forecasters can fit temporal correlations, and physics‑informed models can regularize predictions with scientific constraints, but these directions are often connected only at the decoded‑output level. As a result, the hidden predictive state that generates future trajectories may remain statistically useful but physically unstructured. We introduce Phys‑JEPA, a physics‑informed joint‑embedding predictive architecture for multivariate time‑series forecasting. Phys‑JEPA learns a latent world model in which predictive states are decomposed into physical and residual components, and physical consistency is imposed directly on latent states and latent transitions rather than only on decoded forecasts. This formulation uses known physical variables to organize the representation space while retaining residual capacity for unresolved dynamics. On Jena Climate 2009‑‑2016, Phys‑JEPA reduces aggregate MSE from 0.12482 to 0.12273 and temperature MSE from 0.01892 to 0.01831 at H=24. On Traffic, full Phys‑JEPA improves aggregate MSE over the supervised baseline across all tested horizons, reducing H=192 MSE from 0.800784 to 0.773873. On Electricity, the best variant depends on horizon: static latent consistency is strongest at H=24 and H=48, while full Phys‑JEPA gives the best aggregate and target‑variable MSE at H=192. These initial results suggest that moving physics‑informed learning from output space to latent predictive state space is a promising direction for interpretable temporal world models.

Abstract:
Accurate prediction of hydrodynamic performance is central to ship design, yet high‑fidelity computational fluid dynamics remains prohibitively expensive for large‑scale parametric exploration. This motivates the development of data‑driven surrogate models that provide rapid approximations to hydrodynamic predictions at substantially reduced cost. We present ShipNet, a geometric deep‑learning surrogate that predicts both hull‑surface pressure distributions and far‑field free‑surface wave patterns directly from hull geometry and speed. The network employs a regularized dynamic graph convolutional backbone on hull point clouds, with a multi‑head decoder for simultaneous near‑body pressure and free‑surface elevation outputs. Training data consist of 420 inviscid free‑surface simulations generated using a potential‑flow panel method for two parent yacht hulls, each parameterized into 70 variants and evaluated at three speeds. ShipNet predicts per‑point pressure coefficient and two‑dimensional wave elevation map using a composite loss that combines point‑wise regression and image‑structure terms. On a geometry‑held‑out test set, ShipNet achieves R^2=0.98 for hull pressure and R^2=0.91 for wave fields. Inference requires approximately 0.15s per case, yielding over a 550x speedup relative to the potential‑flow solver on conventional hardware. Limitations include the restricted geometry and speed ranges and the inviscid training data, while future work will extend the model to high‑fidelity viscous simulations with physics‑informed regularization.

Abstract:
Greenland iceberg discharge exhibits complex nonlinear dynamics with limited observability, challenging traditional predictive models. We present a Hybrid NARX‑LLM framework that combines a nonlinear autoregressive model with exogenous inputs (NARX) and a large language model (LLM) for residual correction. We further propose a Physics‑Informed Prompt (PIP) method that transforms unstructured physical knowledge into structured prompts for zero‑shot in‑context reasoning. The primary objective is to explore the corrective potential of this framework for modeling Greenland iceberg discharge, rather than merely optimizing predictive accuracy. The NARX component captures intrinsic temporal dependencies, while the LLM, guided by PIP, encodes glacier dynamics and environmental drivers and perceives key trend patterns to correct systematic prediction errors. This integration allows the model to reason about unmodeled factors and produce interpretable residuals, enhancing overall predictive accuracy. Applied to Greenland iceberg discharge time series, our approach addresses extreme events that are difficult to predict due to rare variations and nonstationary trends, a limitation often overlooked by traditional methods. By fusing structured time‑series modeling with knowledge‑driven foundation AI, the framework offers a scalable and interpretable pathway to bridge data‑limited climate forecasting with physics‑informed LLM reasoning. The code is available.

Abstract:
This work presents a transparent and reproducible benchmark study of a direct dual‑network Physics‑Informed Neural Network (PINN) formulation for the optimal control of a mass‑spring‑damper system. The classical linear‑quadratic optimal control problem is solved by two independent classical methods ‑‑ Pontryagin's Minimum Principle with single shooting, and direct transcription through trapezoidal collocation ‑‑ and recast as a constrained optimization problem solved by two feedforward neural networks: a state network whose boundary conditions are enforced exactly through a composite cubic‑and‑mask ansatz, and an unconstrained control network. The composite loss combines the physics residual at the collocation points with a trapezoidal approximation of the cost functional, weighted by a single scalar hyperparameter. On the benchmark considered, the PINN reproduces the classical optimal cost to four significant digits, satisfies the terminal state constraints exactly by construction, and produces pointwise state and control errors that fall within the spread of the two classical references. Training is approximately two orders of magnitude slower than classical shooting on this benchmark, which is honestly reported. The contribution is methodological clarity rather than methodological novelty: the formulation and the accompanying Google Colab implementation are intended to lower the barrier to entry for practitioners exploring PINN‑based optimal control without prior exposure to adjoint methods or two‑point boundary value problems.

Abstract:
Accurate and adaptive modeling of vehicle dynamics is paramount for the safety of autonomous driving systems, particularly under extreme maneuvers and time‑varying parameters. While Deep Koopman operator theory offers a promising global linearization framework, its online application faces a theoretical bottleneck: the high‑dimensional lifted state space inherently induces a rank‑deficient problem, rendering traditional recursive least squares based updates numerically unstable. To address this, we propose a novel tire‑force‑driven modeling framework with guaranteed online stability. First, an offline Deep Koopman model is constructed by embedding 7DOF dynamic equilibrium constraints into the learning objective, ensuring the structural fidelity and physical interpretability of the lifted manifold. Second, we theoretically reformulate the operator update in the rank‑deficient space as a minimum‑norm solution problem. A Physics‑Informed Variable Step‑Size Normalized Least Mean Squares (PI‑VSS‑NLMS) algorithm is proposed, which leverages the projection property of NLMS to act as a stable pseudo‑inverse solver while incorporating an anchoring mechanism to suppress parameter drift. Extensive simulations on CarSim and Hardware‑in‑the‑Loop validation on dSPACE MicroAutobox III confirm the superiority of the proposed algorithm. It achieves robust prediction accuracy under unseen excitations while guaranteeing real‑time feasibility with an average execution time of 0.421 ms, thus bridging the gap between theoretical models and practical deployment.

Abstract:
Turbulent combustion simulations are crucial for many scientific and engineering systems. However, the high cost to fully resolve the complex multiscale and multiphysics behavior makes direct simulation typically infeasible. The equivalent reactor network (ERN) approach attempts to improve computational efficiency by replacing a multidimensional turbulent simulation with a series of much cheaper 0‑D and 1‑D chemical reactors, providing a surrogate model that retains detailed chemistry at the cost of simplified flow physics. However, their development remains a challenge, often requiring either expert analysis, or automated approaches that sacrifice accuracy. In this work, we develop an automated machine‑learning‑assisted framework for constructing ERNs of the Sandia‑D turbulent methane/air flame. Principal component analysis is first used to reduce high‑dimensional thermochemical computational fluid dynamics (CFD) data to a low‑dimensional latent space, where k‑means clustering identifies physically interpretable flame regions used to initialize a reactor‑network graph. This initialization is then refined using finite‑difference gradient descent wrapped around non‑differentiable Cantera reactor simulations. Across 30 RANS simulations spanning a range of pilot temperatures and inlet methane compositions, the optimized 7‑reactor ERN achieves a maximum‑temperature R^2 score of 0.7945 while preserving a ～6000× speedup over the CFD solver. Outlet CO prediction remains more challenging, with a final R^2 score of ‑0.4183, but improves substantially from the unoptimized clustering initialization. These results show that unsupervised thermochemical feature extraction can provide effective physics‑informed initializations for ERN construction, while gradient‑based refinement can significantly improve predictive accuracy without manual reactor‑network design.

Abstract:
Data‑driven machine learning approaches have become increasingly attractive for nonlinear system identification, but standard models often fail to preserve the underlying physical structure and remain difficult to interpret, especially when no analytical model is available. In this context, port‑Hamiltonian (pH) models provide a natural physics‑informed representation. However, when these models are parameterized with standard multilayer perceptrons (MLPs), the learned constitutive components often remain poorly interpretable. In this paper, we propose a structure‑preserving identification framework for nonlinear port‑Hamiltonian systems based on Kolmogorov‑Arnold Networks (KANs). The proposed PH‑KAN model parameterizes the interconnection matrix, dissipation matrix, Hamiltonian, and input mapping using dedicated KAN blocks, while enforcing the port‑Hamiltonian constraints by construction. This yields constitutive representations in which the nonlinear functions defining the identified pH components can be explicitly inspected, leading to a more interpretable model than with standard MLP‑based parameterizations.

Abstract:
Accurate prediction of protein‑ligand binding affinity is essential for structure‑based drug discovery. Recent geometric deep learning methods have achieved promising performance by representing protein‑ligand complexes as three‑dimensional graphs. However, most existing approaches mainly rely on static interaction geometry from a single bound conformation, while neglecting molecular flexibility and binding‑induced conformational changes. To address this limitation, we propose a curvature‑informed potential energy surface (CPES) graph neural network for protein‑ligand binding affinity prediction, which incorporates physics‑informed curvature representations to model conformational flexibility. CPES first derives curvature spectral descriptors from the Hessian of the potential energy surface evaluated at equilibrium configurations, whose eigenvalues define the local principal curvatures of the potential energy surface. It then uses spectral cross‑attention to compare the unbound ligand and protein with the bound complex, thereby capturing binding‑induced changes in conformational dynamics. In parallel, hierarchical protein‑ligand interaction representations are learned from static structural features through geometry‑aware message passing, soft clustering, and bidirectional cross‑attention. Finally, CPES fuses the curvature‑informed dynamic representations with static interaction representations for affinity regression. Extensive evaluations on multiple benchmark datasets demonstrate that CPES achieves improved predictive performance and offers physical interpretability.

Abstract:
Physics‑informed neural networks (PINNs) are meshless and carry moving geometry and topology change through resampling of collocation points; the finite‑element method (FEM) is the workhorse for boundary‑fitted discretisations. Coupling the two across a shared interface promises the best of both, yet existing PINN‑FEM schemes are validated only empirically. We put the coupling on a domain‑decomposition footing: viewing each solver as a Steklov‑Poincaré (trace‑to‑flux) operator, we transfer the classical Dirichlet‑Neumann (DN) divergence diagnosis and its Robin‑Neumann (RN) cure, including a closed‑form, sweep‑free interface impedance, and prove a PINN‑specific contraction theorem: a trained network realises only a perturbed Steklov operator with a per‑step training residual, and RN still contracts, with no shared‑eigenbasis hypothesis, to a floor set by the achieved training loss. Because a PINN has no stiffness matrix, we introduce a Fourier‑mode interface probe that recovers the network's resolvable Steklov eigenvalues to within 0.5% and doubles as a diagnostic of the network's spectral cap. The theory predicts measured PINN‑FEM contraction rates to within 7% on 1D and 2D Poisson couplings, and a two‑slab analogue of the large‑added‑mass regime shows RN's per‑mode impedance matching winning decisively where tuned scalar relaxation saturates. We demonstrate the framework on a Stokes/rigid‑disc problem with Alart‑Curnier contact: the meshless PINN fluid absorbs the topology change at contact by collocation exclusion alone, no remeshing and no cut cells, and the static‑equilibrium contact reaction matches the submerged weight to 0.4% under mesh refinement. We quantify remaining limitations: the warm‑started PINN drifts off the Stokes manifold over long horizons, and matched FEM‑FEM benchmarks attribute pre‑impact squeeze‑film signatures to PINN under‑resolution.

Abstract:
Mineral prospectivity modelling (MPM) underpins exploration economics, yet most operational pipelines reduce to data‑driven classifiers trained on shallow surface proxies. Such models are blind to the subsurface physics that actually localises ore: heat advection, fluid flow, and lithology‑dependent precipitation. We present Korzhinskii‑Net, a 2‑D radial physics‑informed neural network (PINN) that couples Darcy flow, advective‑diffusive heat transport, and a softplus‑saturated reaction rate into a single differentiable forward model, weakly supervised by surface and remote‑sensing proxies. The network is named after Dmitri S. Korzhinskii (1899‑1985), whose theory of infiltration metasomatism provides the physical scaffold. We evaluate Korzhinskii‑Net on five ore provinces spanning four commodity classes ‑‑ Norilsk (Ni‑Cu‑PGE), Pechenga (Ni‑Cu sulphide), Udokan (sandstone‑hosted Cu), Sukhoi Log (orogenic Au), and Mirny (kimberlitic diamond) ‑‑ under a fair, leakage‑controlled 5‑fold cross‑validation protocol with hard ring‑shaped negatives. Korzhinskii‑Net attains a mean PR‑AUC of 0.885 versus 0.281 for the strongest classical baseline (gradient boosting), and a mean fractional rank of 0.019 versus 0.413. The improvement is consistent across all five provinces and four commodity systems, suggesting that physics‑informed differentiable simulators, even when constrained only by global open‑data proxies, can recover localisation patterns that pure feature‑based learners systematically miss. We release the full pipeline and evaluation harness as open source.

Abstract:
Wave parameters in the nearshore are crucial for coastal engineering, shoreline protection, marine hazard assessment, and coastal management for climate resilience. Traditional monitoring systems like buoys and radar platforms offer accurate monitoring but can have high installation and maintenance expenses and limited spatial coverage. Passive ocean monitoring using video has been achieved by leveraging deep learning, however, many methods are not physically interpretable, feasible, and validated for oceanography. In thiswork, a Physics‑Guided Deep Spatiotemporal Learning Framework for direct estimation of nearshore wave peak periods from passive coastal video stream is proposed. The framework combines automated temporal‑variance based region‑of‑interest detection, multi‑stage Sim‑to‑Real transfer learning, and physics‑informed regularization to enhance the predictive accuracy and physical consistency. A variety of spatiotemporal architectures were assessed, such as transformer‑based and recurrent‑convolutional ones, alongside synthetic pretraining,silver‑label adaptation, and expert fine‑tuning. The results show that transformer‑based architectures outperformed in terms of the accuracy of the instantaneous prediction, while lightweight recurrent‑convolutional architectures achieved higher temporal stability and operational oceanographic skill. Ablation studies also demonstrated the benefits of physics‑guided regularization in terms of trend‑following consistency, and physically implausible predictions. Explainability auditing also helped to focus attention in hydrodynamically active surf‑zone regions and showed good agreement with the physically derived wave propagation behavior. In general, the proposed framework shows the promise of physics‑guided video‑based deep learning systems for long‑term coastal wave monitoring that are cost‑efficient and operationally feasible.

Abstract:
AI systems are being deployed across medical imaging faster than their failure modes are understood. At this point in time, the failure of greatest clinical concern is hallucination: clinically plausible but factually incorrect outputs, including fabricated anatomical structures, missed findings, incorrect laterality, and invented measurements in generated reports, with direct consequences, for example, for biopsy decisions, staging, and treatment planning. This structured narrative synthesizes peer‑reviewed studies, benchmark datasets, and FDA regulatory guidance across five imaging modalities to produce a cross‑modality analysis of hallucination taxonomy, etiology, detection, and mitigation. Specifically, we address three questions in this study: (1) how can existing taxonomies be unified across modalities?, (2) how do medical‑specialized foundation models hallucinate less than general‑purpose ones?, and (3) which mitigation strategies are effective and compatible with FDA lifecycle oversight? We note that three taxonomic frameworks together cover the imaging pipeline in a way no single framework does alone. We also highlight that general‑purpose foundation models outperform medical‑specialized models on hallucination‑specific benchmarks, indicating that narrow domain fine‑tuning can introduce overfitting‑induced confabulation. At the same time, the oversight of radiologists remains essential; for instance, a very high percentage of of AI‑generated flags required expert correction before clinical use. Physics‑informed architectural constraints, Chain‑of‑Thought prompting, and human‑in‑the‑loop safeguards each address different failure modes and is effective when combined. All findings are mapped to the FDA's Total Product Lifecycle and Predetermined Change Control Plan frameworks, which treat hallucination management as a lifecycle obligation rather than a pre‑deployment checklist.

Abstract:
Underactuated spacecraft faces controllability limitations and heightened sensitivity to environmental disturbances, complicating attitude maneuvering and stabilization. Due to the lack of control authority along the underactuated axis, conventional controllers cannot directly stabilize all attitude components and therefore require reference planning strategies. Furthermore, MPC approaches remain sensitive to inertia uncertainty and unmodeled dynamic couplings, resulting in degraded tracking performance under mismatch. To address these issues, we consider a hierarchical architecture integrating three layers: (i) a nonlinear model predictive controller (NMPC) for constraint and underactuation‑aware maneuver planning and nominal closed‑loop stability under actuator limits; (ii) a physics‑informed neural network (PINN) trained offline on simulation data to estimate residual disturbance torques, with loss terms that enforce consistency with rigid‑body rotational dynamics; (iii) a Lyapunov‑based supervisory safety mechanism that evaluates the learned correction online and bounds or suppresses its influence to preserve the stability properties of the baseline controller. The architecture is evaluated in a high‑fidelity simulation environment modelling reaction wheel dynamics, actuator saturation, and environmental disturbances. Monte Carlo studies show statistically significant reductions in steady‑state attitude error relative to standalone NMPC while maintaining robust behavior under uncertainty. The supervisory layer ensures graceful degradation to purely model‑based control when the learning‑based augmentation is unreliable.

Abstract:
Physics‑Informed Neural Networks (PINNs) are a machine learning method for solving forward and inverse Partial Differential Equations (PDEs). When applied to PDEs with Dirac delta functions in the forcing terms, boundary conditions, or initial conditions, PINNs require approximating them with smooth surrogate functions, a practice that can introduce significant modeling errors. In this work, we exploit the interpretation of PINNs as Residual Least Squares (RLS) methods and show that this perspective enables direct treatment of Dirac delta terms by integrating the weak‑form equation. Among RLS formulations other than PINN, we focus on the Radial Basis Function (RBF) expansion (also known as a single‑layer RBF Network). We show that while integrating out the Dirac delta in PINNs causes residuals to fail to converge to zero, RBF‑RLS consistently provides good forward and inverse solutions to transport problems. We explain this finding using the Neural Tangent Kernel (NTK) theory. We test both approaches on linear PDEs that represent groundwater flow and transport in porous media and rivers. We solve inverse problems to fit synthetic data, noisy synthetic data, and real‑world measurements.

Abstract:
Physics‑Informed Neural Networks (PINNs) are an attractive tool for partial‑observation problems in biology, where the governing dynamics are known but some compartments cannot be measured. Chemotherapy pharmacokinetics (PK) is a clean instance: drug concentration in plasma is routinely measured, but concentration in tissue ‑‑ which determines tumour kill and off‑target toxicity ‑‑ is not. We benchmark a PINN against the standard clinical baseline (nonlinear least‑squares on the analytical biexponential plasma solution, hereafter NLS) and a physics‑agnostic neural baseline (a data‑only MLP) on two PK problems. On the linear two‑compartment problem, NLS is near‑optimal; the PINN matches it to within a small constant factor while also producing the tissue curve in a single training pass, whereas the data‑only MLP fails on tissue by roughly 10x. On a Michaelis‑Menten extension (saturable elimination), the biexponential closed form no longer exists, so NLS is mis‑specified and silently returns meaningless rate constants. The PINN instead exposes a deeper fact: the Michaelis‑Menten two‑compartment model is non‑identifiable from plasma alone, and the PINN reports this honestly by converging to a basin with k12 ‑> 0. Adding two sparse tissue observations largely resolves identifiability: across five seeds the PINN recovers k21 to within 1% of truth and Vmax, Km to within one standard‑deviation bar, while k12 moves in the correct direction (0.02 ‑> 0.82) but remains ~2 sigma below truth ‑‑ a recovery the closed‑form NLS estimator cannot attempt at all, because its biexponential ansatz describes only plasma. Our claim is not that PINNs beat NLS. It is that PINNs offer a uniform recipe that ties the textbook estimator on the textbook problem, exposes structural identifiability that the textbook estimator hides, and absorbs heterogeneous measurements within a single loss.

Abstract:
This paper presents an improved physics‑informed neural network for simulating the spatio‑temporal solution profile of the multidimensional coupled Burgers' equations with high Reynolds numbers. As time evolves, the sharp shock fronts emerge in the solution, creating significant computational challenges for the conventional mesh‑based numerical methods. In particular, numerical methods such as finite differences and finite elements suffer from poor stability and strong mesh dependency when resolving the steep solution gradients. To address these challenges, the proposed framework employs a layerwise self‑adaptive weighting strategy that dynamically adjusts the penalty weights for the physics residual, initial conditions, and boundary conditions throughout training. Moreover, the framework uses a dual‑phase optimization strategy to achieve more stable convergence. To check the effectiveness and accuracy of the proposed framework, a set of numerical experiments is conducted to compare it with the standard Physics‑Informed Neural Network (PINN) with and without Limited‑memory Broyden‑Fletcher‑Goldfarb‑Shanno (L‑BFGS) optimization. Numerical results exhibit that the proposed framework achieves higher accuracy in terms of relative L_2‑ error norm than the standard PINN and is able to capture the development of sharp shock fronts as time evolves in the solution.

Abstract:
Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint‑based optimization, while physics‑informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE‑constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three‑dimensional Allen‑‑Cahn reaction identification, and unsteady Navier‑‑Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid‑based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time‑dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN‑warm‑started adjoint strategy then recovers adjoint‑level accuracy at substantially reduced cost.

Abstract:
Physics‑informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations. While existing results provide rigorous \empha posteriori upper bounds for PINN prediction errors, complete certification also requires complementary lower information in order to obtain computable two‑sided error enclosures. In this paper, we derive computable \empha posteriori lower bounds for PINN errors in ordinary differential equations on suitable certified state‑space domains under a localized strong monotonicity condition. We combine these estimates with complementary localized upper bounds under a one‑sided Lipschitz condition, which is weaker than the global Lipschitz assumption used in previous work and can yield sharper upper error bands. The resulting bounds depend only on the neural‑network approximation, the ODE residual, and local monotonicity and growth constants, and therefore do not require access to the exact solution. For linear time‑invariant and time‑varying systems, we further derive explicit formulas in terms of the minimal and maximal eigenvalues of the symmetric part of the system matrix. We also discuss the distinction between soft and hard enforcement of initial conditions in PINNs and explain why exact enforcement can make the scalar lower certificate uninformative. To recover nontrivial lower information in the linear setting, we use a signed‑residual finite‑probe certificate based on coordinate unit vectors. We also formulate a certificate‑informed training strategy in which the propagated upper certificate is used as an auxiliary regularizer, while lower certificates remain post‑training diagnostics. Altogether, the proposed framework provides rigorous and practically computable error certificates for PINN approximations of ODEs, while making explicit the domains and model classes for which the assumptions can be verified.

Abstract:
Reliable extrapolation remains a central challenge for generative models in computational physics, because models trained over finite ranges of time, parameters, or geometries may produce physically inconsistent predictions outside the training distribution. We introduce a least‑action‑principle‑guided diffusion, LAPG, a framework that promotes physical consistency during inference rather than relying solely on constraints imposed during training. The method combines a conditional score‑based diffusion model with an action‑derived physical guidance score. In the first stage, the learned score model generates an in‑distribution proposal; in the second, an action‑based variational prior refines this proposal toward the target out‑of‑distribution condition. This formulation turns the principle of least action into a differentiable inference‑time correction mechanism and provides an alternative to pointwise residual penalties that often require empirical loss balancing. We evaluate LAPG on representative ordinary‑ and partial‑differential‑equation systems, including free fall, conservative and dissipative spring‑mass dynamics, interacting point vortices, and potential flow over parameterized airfoils. In temporal, parameter, and geometric extrapolation tests, LAPG reduces phase drift, preserves dissipative decay, captures vortex motion, and improves the lift response of airfoil flows compared with training‑time physics‑informed baselines.

Abstract:
Rendezvous is a critical task for multi‑agent systems, requiring agents to coordinate to meet at an unspecified location. However, achieving this in fluid environments presents a challenge, as it remains unclear how agents can exploit underlying fluid kinematics to facilitate convergence. In this study, we adopt a multi‑agent reinforcement learning (MARL) approach to develop physics‑informed rendezvous strategies in vortical flows. Compared to a naive strategy, where agents navigate toward their counterparts, MARL strategies significantly improve the rendezvous rate. MARL strategies also show transferability across varying vortex intensities, vortex scales, and swarm sizes. By breaking the symmetry of the state‑action map, MARL strategy leverages a non‑intuitive mechanism that prevents agents from becoming trapped in separate vortices, thereby enhancing rendezvous success. Additionally, a heuristic strategy is extracted from the learned strategy and also outperforms the naive strategy. Furthermore, a theoretical analysis demonstrates that fluid deformation impedes the rendezvous process. Large finite‑time Lyapunov exponents identify where fluid effects separate adjacent agents, suggesting that targets should be planned in weak‑deformation regions. Our findings reveal the important role that agent‑fluid interactions play in multi‑agent tasks and highlight the MARL capability to explore swarm intelligence in complex flow environments.

Abstract:
Gradient‑based inversion of reaction‑diffusion systems is typically approached via surrogate models or physics‑informed neural networks (PINNs), while the most direct route, backpropagation through the PDE's structure itself, has largely been avoided. We pursue this direct route as a diagnostic probe, backpropagating a steady‑state loss through unrolled Gray‑Scott simulation to recover its parameters, with no surrogate or neural‑network augmentation. Optimization fails to converge, and plotting the landscape directly locates the failure in its geometry ‑‑ flat plateaus with no gradient signal, bounded by sharp cliffs that align with bifurcation boundaries ‑‑ a structure that recurs across loss functions and is inherited however the gradients are routed to parameters. Reading this minimal setup as an ablation of PINN, we disentangle each component's role: with the neural network fixed, the residual loss is quadratic in the PDE parameters and yields a smooth landscape, so it alone already avoids the pathology, by implicitly encoding the full PDE dynamics across all initial conditions. The neural network, for its part, cannot repair an ill‑posed parameter subspace, and so serves only to complete the observed data ‑‑ a division of labor not previously made explicit. These findings carry concrete design implications for PINN‑type methods and a broader heuristic on when added dimensions actually help.

Abstract:
Generative models are increasingly used to propose designs, data, and control actions for physical systems, yet many such systems are governed by hard physical constraints rather than by perceptual plausibility. Semiconductor manufacturing provides a demanding test case: generated masks, layouts, synthetic defect data, and process recipes must obey lithography, transport, reaction, and device‑physics constraints, because physically invalid samples are not merely low quality but unusable. This Perspective argues that semiconductor manufacturing exposes a broader computational‑science challenge, namely that generative AI for constrained physical domains must be physics‑informed by construction, not corrected only through post‑hoc filtering. We survey the emerging architectural toolkit, including physics‑informed diffusion, PDE‑constrained variational models, neural‑operator priors, and conservation‑law‑respecting generative networks, and show how it connects to differentiable lithography, TCAD, process simulation, and autonomous experimentation. We identify four integration patterns between generative models and physics‑based simulators, and we propose a research agenda centered on physics‑fidelity benchmarks, differentiable simulator infrastructure, and multimodal foundation models for physical design and manufacturing. The central claim is analytical rather than rhetorical: where physical validity is the binding criterion of success, architectures that enforce it by construction should be expected to outperform those that filter for it after the fact, and the fab is the setting where this distinction is sharpest.

Abstract:
Physics‑informed neural particle flow (PINPF) learns a deterministic transport field that moves particles from a prior distribution toward a Bayesian posterior while enforcing the governing probability‑evolution equation. However, the standard PINPF velocity model processes particles independently and therefore does not explicitly condition its transport decisions on the empirical particle population. This paper introduces population‑aware PINPF (PA‑PINPF), which augments each particle update with a permutation‑invariant Deep Sets representation of the full particle set. We investigate two population encoders. PA‑PINPF‑State summarizes the particle states, whereas PA‑PINPF‑Feature summarizes the complete local physics‑informed feature vectors, including particle position, pseudo‑time, measurement information, likelihood values, and score information. The latter allows the population context to represent not only particle‑cloud geometry, but also the population‑level Bayesian transport geometry. The methods retain the original unsupervised physics‑informed residual objective and require no ground‑truth posterior samples during training. Experiments on range‑measurement tasks and nonlinear time‑difference‑of‑arrival posterior transport demonstrate that both population‑aware variants improve over particle‑wise PINPF, while feature‑population encoding provides the strongest performance. These results show that population‑level physics features provide useful global information for learned Bayesian particle transport.

Abstract:
Reconstructing local stress fields in heterogeneous microstructures under non‑linear, history‑dependent loading remains a major computational bottleneck in multi‑scale simulations. We propose a coupled LSTM‑GNN framework that links the temporal and spatial aspects of local stress field reconstruction. A Long Short‑Term Memory network encodes macroscopic stress‑strain sequences into a compact hidden state that captures the path‑dependent constitutive response, while a physics‑informed Graph Neural Network reconstructs the spatially‑resolved stress field at each time step. We introduce a relative weighting strategy with linear warm‑up to balance the data‑driven reconstruction loss and a discrete divergence‑based equilibrium penalty. This resolves the scale mismatch that prevents fixed‑weight formulations from converging in the elasto‑plastic regime. The model is trained on 10,000 non‑proportional loading paths applied to a periodic plate‑with‑a‑hole microstructure and von Mises elasto‑plasticity. The model achieves three orders of magnitude speedup over finite element simulations and generalizes to loading sequences twice the training length, with 1.9% cumulative error. Because the graph relies on mesh connectivity instead of the specific element type, one trained surrogate can be applied directly without retraining to meshes with different element types and to both coarser and finer resolutions, while in all cases reproducing the high‑fidelity quad‑element FE field used during training. Indeed, the message passing characteristics inherent to GNN and MeshGraphNet architecture render the model mesh‑agnostic. Analysis of the LSTM hidden states suggests a low‑dimensional structure related to the internal state variables of the constitutive model.

Abstract:
The pulsar magnetosphere has only recently been addressed using Physics‑Informed Neural Networks (PINNs), by deploying a domain‑decomposition approach and treating the separatrix and equatorial current sheet as infinitesimally thin discontinuities. However, this baseline requires extensive manual hyperparameter tuning, achieves limited final accuracy and demands several hours of training. We refine this framework by introducing domain‑specific neural architectures based on Kolmogorov‑Arnold networks, an automated adaptive training pipeline and a physics‑based convergence criterion that eliminate the need for manual calibration. The proposed methodology delivers self‑consistent axisymmetric magnetosphere solutions with mean squared errors of the PDE residuals at O(1e‑6) in double precision ‑ an improvement of two orders of magnitude over the baseline ‑ while achieving convergence in under 20 minutes in single precision. Importantly, the method reliably resolves stellar radii reduced by up to 80% compared to the baseline, overcoming the severe spatial scale disparities that also challenge traditional solvers. Furthermore, by varying the flux that opens to infinity, we provide a correction to the equation that connects it to the equatorial T‑point's position. The complete framework is released as the open‑source library PulsarX.

Abstract:
While physics‑informed neural networks (PINNs) have shown strong potential for process modeling, physical equations are only enforced as soft constraints during training, and thus, they do not guarantee constraint satisfaction at inference. We propose a framework, called piecewise‑linear Karush‑‑Kuhn‑‑Tucker hard‑constrained PINNs (PL‑KKT‑hPINNs), that strictly enforces nonlinear equality constraints through piecewise‑linear projection. This extends the KKT‑hPINN framewor, which exactly enforces linear equalities through the Karush‑‑Kuhn‑‑Tucker (KKT) conditions associated with orthogonally projecting neural network outputs onto the constraint feasible region. The method is demonstrated on a continuous stirred‑tank reactor (CSTR) case study for both one and two inputs. Results show that PL‑KKT‑hPINN preserves predictive accuracy comparable to that of a standard neural network while achieving substantially lower constraint violations. In addition, the proposed model shows improved robustness in low‑data regimes, yielding lower RMSE than the unconstrained neural network for limited training sample sizes. These results demonstrate that PL‑KKT‑hPINN provides a computationally efficient and physically consistent framework for surrogate modeling of nonlinear chemical engineering systems.

Abstract:
We propose a new optimization method, the Nyström‑enhanced relaxed scalar auxiliary variable method (N‑RSAV), which incorporates curvature information into the RSAV framework to accelerate convergence while preserving an unconditional modified energy dissipation law. Existing RSAV‑based methods rely solely on first‑order information and often suffer from slow convergence, particularly for ill‑conditioned problems such as those arising in physics‑informed neural networks (PINNs). To address this limitation, we design the linear operator in the RSAV scheme using approximate Hessian information obtained from a randomized low‑rank Nyström approximation. To preserve the dissipation structure, we enforce positive semidefiniteness through eigenvalue truncation. Furthermore, we introduce an adaptive strategy that reuses the approximate Hessian based on the deviation between the original and modified energies, significantly reducing computational cost. We also provide a convergence analysis of the RSAV scheme with a general positive semidefinite operator under the Polyak‑Lojasiewicz (PL) condition and establish corresponding convergence guarantees for N‑RSAV under the PL condition and an additional convexity assumption. Numerical experiments on ill‑conditioned problems with effectively low‑rank structure, including convex quadratic problems and training of PINNs, demonstrate that the proposed methods achieve substantially faster convergence than conventional RSAV‑based approaches.

Abstract:
Continuous approximations of flow data are useful for downstream analysis, differentiation, and visualization, but purely data‑driven reconstructions do not, in general, preserve the governing physics. This limitation becomes particularly important when input data are physically inconsistent, whether due to low‑fidelity discretizations or unmodeled discrepancies. In such cases, reconstructed fields may exhibit inaccurate PDE residuals, violated balance laws, or unreliable derived quantities. To address this, we propose a physics‑informed B‑spline framework that embeds physical constraints directly into the reconstruction process. The method constructs compact, continuously differentiable representations of discrete fields using tensor‑product B‑splines and determines spline control points by solving an optimization problem balancing data fidelity with residuals of the governing PDEs, alongside initial and boundary conditions. Leveraging exact analytical derivatives of the B‑spline basis enables efficient and accurate evaluation of physical residuals without storing full‑resolution fields. We refer to this approach as physics‑informed multivariate functional approximation (PI‑MFA). Numerical studies on the 1D convection‑diffusion, 2D coupled Burgers, and 2D incompressible Navier‑Stokes equations show PI‑MFA reduces PDE residuals and improves global balance‑law consistency. Compared with standard and regularized MFA, PI‑MFA produces more physically faithful reconstructions and, for physically inconsistent data, lower approximation errors, while offering computational advantages over tested physics‑informed neural networks. Overall, PI‑MFA preserves the compactness, local support, and exact differentiability of classical spline spaces while producing reliable continuous flow fields for scientific analysis and visualization.

Abstract:
Finite element (FE) methods are the benchmark for solid mechanics simulations, yet their computational cost becomes prohibitive for problems with localised nonlinearities, fine‑scale features, or long‑time dynamic evolution. In our earlier FE‑neural operator (FE‑NO) hybrid framework [1], physics‑informed deep operator networks were coupled with FE solvers through overlapping domain decomposition with Dirichlet‑Dirichlet interface exchange, accelerating intensive subdomains while preserving FE fidelity elsewhere. Two limitations remained: the overlapping formulation required redundant interface computations that increased inner Schwarz iteration counts, and the convolutional feature extractor restricted the NO subdomain to structured grids, precluding irregular geometries. A non‑overlapping Schwarz alternating method with Neumann‑Dirichlet interface exchange replaces it, transmitting traction from the NO to FE rather than displacement. This eliminates the overlap layer and reduces inner Schwarz iterations while maintaining bounded error accumulation across all tested time horizons. For arbitrarily shaped subdomains, a Point‑DeepONet operates on unstructured FE point clouds without interpolation, extending it to non‑convex and irregular geometries. Strain and stress operators are derived analytically from the displacement operators via kinematic equations, rather than as independent networks, reducing trainable parameter sets while enforcing mechanical consistency by construction. The framework is validated on three benchmarks: static linear elasticity, quasi‑static hyperelasticity, and elastodynamics with regular and irregular geometries. These results establish a non‑overlapping FE‑NO coupling paradigm that is geometry‑flexible, parameter‑efficient, and convergence‑stable, providing a pathway for hybrid physics‑based and operator‑learning solvers in large‑scale dynamic solid mechanics.

Abstract:
While global data‑driven models excel at predicting continuous atmospheric variables, three‑dimensional hydrometeor forecasting remains challenging due to the zero‑inflated, long‑tailed distributions of these variables. Standard deep learning optimization often yields overly smooth forecasts, attenuating extreme events and spatial textures. We propose PredHydro‑Net, a physics‑guided dual‑decoding framework that mitigates this smoothing. To resolve multi‑variable optimization conflicts, it employs a decoupled architecture where macroscopic thermodynamic and dynamic fields unidirectionally modulate hydrometeor generation. By integrating wavelet‑based frequency decoupling, spectral amplitude matching, and adversarial training, the model achieves a favorable trade‑off between quantitative accuracy and spatial fidelity. In a 72‑h global evaluation, PredHydro‑Net outperforms both spatiotemporal deep learning baselines (Earthformer and PredRNNv2) and the operational Global Forecast System (GFS) in extreme‑event detection and spectral representation. Furthermore, it demonstrates strong climatological consistency with Global Precipitation Measurement (GPM) satellite retrievals. The model reasonably reproduces the three‑dimensional cloud structures in extreme weather events, such as Hurricane Ian. Feature attribution confirms its dependence on physical precursors such as relative humidity and wind convergence, offering a robust, physics‑informed approach to long‑tailed atmospheric prediction.

Abstract:
High‑dimensional transient heat diffusion under noisy boundary conditions exposes a fundamental limitation of classical numerical methods: accuracy degrades catastrophically where physical noise is unavoidable. This paper presents a Physics‑Informed Neural Network (PINN) framework as a systematic solution to this problem across one, two, and three spatial dimensions, establishing clear operational regimes that redefine solver selection in noisy thermal systems. Under 20% boundary noise in 3D, PINN sustains approximately 91% accuracy while Finite Difference Method (FDM) collapses to 36%, a clear decisive advantage. This is further confirmed in a physical copper thermal system, where PINN reduces boundary reconstruction error by 3.3 times under realistic noise conditions. This noise resilience is accompanied by a dimensionality‑driven efficiency crossover: PINN requires fewer spacetime nodes than FDM in 3D while achieving superior accuracy, exposing the true cost of classical discretization at scale. These findings reframe solver selection: the decisive axis is not accuracy alone, but noise exposure and dimensionality jointly. When noise and dimensionality are both high, the classical solver paradigm is insufficient; this work provides the foundation to justify PINN as the operational standard in such regimes.

Abstract:
Progress in AI‑driven crystal materials science has so far been carried by narrow architectures purpose‑built for individual tasks ‑‑ graph neural networks for property prediction, diffusion and flow‑matching models for crystal generation ‑‑ each excelling within its niche yet unable to act as a shared backbone across the full spectrum of materials problems. Generative large language models offer a fundamentally different paradigm, in which structural representation, quantitative prediction, and structure‑activity reasoning can be unified within one model, but the materials community has yet to see this paradigm realized at a level competitive with established narrow specialists. Here we present MatMind, a generative foundation model purpose‑built for crystal materials science under this paradigm, developed through the coordinated activation of structure‑activity knowledge and physics‑informed feedback within a progressive training framework ‑‑ combining structure‑activity knowledge injection, a dual‑head architecture that jointly trains language reasoning and numerical regression in a shared representation space, and multi‑objective physics‑informed reinforcement learning over stability, novelty, and structural diversity. Across three task families, MatMind attains the lowest mean absolute error on energy above hull, bulk modulus, and band gap ‑‑ surpassing graph neural network predictors purpose‑built for these tasks ‑‑ reaches an S.U.N. rate of 65.3% on unconditional crystal generation, and achieves a comparable multiplicative improvement on magnetization‑density‑conditioned generation, where only 21 positive samples exist within over 600000 training entries. By matching or surpassing narrow specialists on their own ground while operating within a single unified model, MatMind shows that the LLM‑based paradigm can serve as a viable backbone for crystal materials science going forward.

Abstract:
Physics‑Informed Neural Network (PINN) is a way of including knowledge in the form of equations in Machine Learning methods. Beyond equations, knowledge exists in other forms, such as text and network structure. While existing PINN‑based approaches discover equation parameters from data, they rely solely on experimental measurements. We propose a new PINN framework that enriches parameter discovery by incorporating auxiliary knowledge sources. We instantiate our framework for microbiology, where generalised Lotka‑Volterra (gLV) serves as a biological foundation for modelling microbial communities. We demonstrate that incorporating knowledge improves microbial community modelling. Our framework enriches the gLV parameters using peer‑reviewed metagenomics literature, as text provides biological context on external influences that gLV alone cannot capture. We combine this knowledge with experimental measurements of microbial abundance using a data‑driven integration approach. We integrate network‑based structural knowledge by explicitly modelling microbial interactions. Our knowledge‑inclusive framework infers microbial networks, revealing ecological insights. We validate these findings against ecological roles documented in the literature. We evaluate on real and simulated datasets spanning human‑ and plant‑associated microbial communities. Our framework improves over the state‑of‑the‑art by up to 53%, even without knowledge. Knowledge addition yields gains of up to 23% in Bray‑Curtis Dissimilarity‑based accuracy and 47% in \mathrmR^2.

Abstract:
At commissioning time, Photovoltaic (PV) operators must forecast production before target‑site observations are available, limiting the direct use of standard supervised forecasters. This cold‑start setting is addressed with a zero‑shot pipeline that generates a synthetic production history from plant metadata and meteorological covariates, enabling time‑series foundation models (TSFMs) to forecast through inference‑time conditioning. Five TSFMs are benchmarked against classical baselines under strict Cold‑Start Baseline, Real Feedback, and Self‑Forecast Feedback strategies. The evaluation spans 440 PV sites across four datasets and diverse climate regimes. Covariate‑aware foundation models outperform baselines by approximately 1.7‑2×: TabPFN‑TS achieves the lowest error under Real Feedback (MAE 0.514, RMSE 0.721 kWh kWp^‑1 d^‑1), while Chronos‑2 is most robust under Self‑Forecast Feedback. Performance is largely insensitive to the synthetic‑history source, indicating that accuracy is driven more by the availability of plausible temporal context than by the specific generator.

Abstract:
This work presents a differentiable framework for the parametrization and shape optimization of industrial CAD geometries represented by multi‑patch NURBS surfaces. The method enables the deformation of complex CAD models through a physics‑informed geometric parametrization, allowing direct morphing driven by physical constraints without the need to prescribe a predefined deformation strategy. A neural displacement field, implemented as a multi‑layer perceptron acting on the NURBS control points, provides a compact parametrization of the admissible design space while preserving patch connectivity. Global geometric quantities relevant to hydrostatic design, including displaced volume, wetted surface area and buoyancy centroid, are formulated as differentiable integral operators evaluated on the parametric domain. These quantities are computed through Gauss‑Legendre quadrature combined with analytical B‑spline derivatives for surface metric evaluation, allowing gradient propagation to the deformation parameters while limiting the computational overhead of automatic differentiation. The proposed framework operates directly on CAD representations without intermediate mesh generation. Numerical experiments on a modified KVLCC2 hull demonstrate the capability of the method to satisfy competing hydrostatic constraints while producing smooth CAD‑compatible geometries and showing stable convergence across multiple random initializations.

Abstract:
Physics‑informed learning is increasingly used for partial differential equation (PDE)‑governed inverse problems, but its reliability remains difficult to certify. This paper develops a no‑harm certification‑and‑selection framework for physics‑informed inverse learning. A learned reconstruction is accepted only when its residual‑calibrated radius is no worse than the baseline radius, namely when R_\mathrmlearn\le R_\mathrmbase+\varepsilon_\mathrmsafe;otherwise, the method returns the baseline. The certificate combines data, physics, boundary or initial‑condition, and optimization residuals. Under a conditional stability estimate, these residuals yield an a posteriori reconstruction‑error bound and a deterministic uncertainty radius. A high‑probability certificate is also derived for physics residuals estimated from independent random collocation points. Numerical tests on Poisson source recovery, inverse heat reconstruction, limited‑angle tomography, elliptic coefficient identification, and stochastic residual validation show that the selector accepts certified improvements, rejects shifted, hallucinated, or unfinished candidates, and becomes conservative in strongly ill‑posed regimes. The framework is therefore a certification‑and‑selection layer, not another reconstruction architecture.

Abstract:
Accurate and scalable flood mapping remains challenging due to limited ground observations, heterogeneous terrain conditions, and the difficulty of enforcing hydrodynamic consistency within data‑driven models. This work introduces a physics‑guided deep learning framework that integrates multi‑modal remote sensing (Sentinel‑1 SAR, Sentinel‑2 optical imagery, and DEM‑derived terrain features) with constraints from the depth‑averaged shallow water equations (SWE). The proposed hybrid architecture combines a UNet to capture fine‑scale spatial details with a Fourier Neural Operator (FNO) to model basin‑scale hydraulic interactions, while physics‑informed residual losses ensure mass and momentum consistency. Evaluated across diverse floodplain settings, the hybrid model achieves an Intersection over Union of 0.82 and an F1 score of 0.90 for flood extent prediction, outperforming UNet‑only and FNO‑only baselines. Using hydrodynamic simulations as reference data, the model achieves an RMSE of 0.21 m for water depth and 0.15 m/s for flow velocity. Physics consistency is maintained, with low residuals and mass imbalance below 2.1%. Ablation studies confirm that removing physicsbased regularization significantly degrades performance, underscoring the value of physical constraints for stability and generalization. These results demonstrate that embedding hydrodynamic principles into deep learning yields more accurate, reliable, and physically coherent flood predictions, offering strong potential for operational monitoring and large‑scale deployment.

Abstract:
Urban green‑space extraction from ultra‑high‑resolution (UHR) imagery is commonly performed patch by patch, which limits semantic reuse among spatially separated but visually similar vegetation patterns. Directly injecting the Normalized Difference Vegetation Index (NDVI) into red‑green‑blue (RGB) backbones can also blur the roles of visual appearance learning and physical vegetation confidence. We propose GMBFormer, a SegFormer‑based framework that replaces adjacency‑driven feature propagation with selective, similarity‑driven prototype retrieval. Only RGB channels enter the backbone and decoder, while NDVI is decoupled as a physics‑informed gate that admits high‑confidence vegetation descriptors into a compact global memory bank through momentum updates. During training and inference, the current patch queries stored prototypes through memory‑mediated cross‑attention, and the retrieved response is integrated with bounded overhead. Experiments use a self‑constructed Chengdu UHR dataset with 7,700 labeled 512 x 512 patches and two reduced‑label settings derived from the public International Society for Photogrammetry and Remote Sensing (ISPRS) Potsdam dataset. Under the same training and evaluation protocol, GMBFormer obtains mean intersection over union (mIoU)/mean Dice (mDice) scores of 89.25%/94.31%, 92.17%/95.92%, and 83.72%/90.86%, respectively, improving the controlled SegFormer‑B4 baseline in each setting. Ablation studies indicate that decoupled NDVI admission, memory retrieval, capacity, and momentum jointly shape the final performance.

Abstract:
Time‑dependent high‑dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics‑informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high‑dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time‑dependent setting by treating space and time as a unified domain without any explicit time marching. A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn. Unlike conventional adaptive strategies that require explicit time stepping or moving meshes, high‑residual regions are automatically identified and tracked across both space and time, driven purely by the PDE residual distribution. The effectiveness of the proposed strategy is assessed on a range of benchmark problems, from sharp and moving features in two spatial dimensions to localised structures in up to eight spatial dimensions.

Abstract:
Wall shear stress (WSS) governs near‑wall transport dynamics and is a key hemodynamic indicator in cardiovascular flows, yet remains difficult to infer accurately due to the need for precise computation of near‑wall velocity gradients. Passive scalar fields, such as concentration or temperature, are advected by the same underlying velocity field and have the potential to uncover hidden flow physics metrics such as WSS. In this work, we demonstrate such reconstruction from spatially limited passive scalar observations using two fundamentally different inverse frameworks: a differentiable physics framework based on discrete adjoint, PDE‑constrained optimization, which enforces the governing equations as hard constraints, and physics‑informed neural networks (PINNs), which treat them as soft constraints. Benchmark problems include a 2D canonical backward‑facing step (2D‑BFS) and a 3D patient‑specific stenotic coronary artery. For the 2D‑BFS case, evaluated under three measurement scenarios (near‑wall, far‑field, and combined), PINN achieves high accuracy when near‑wall data are available but fails when restricted to far‑field measurements, whereas the differentiable physics approach recovers accurate WSS across all scenarios. In the 3D patient‑specific case, the differentiable physics framework outperforms PINNs, yielding accurate WSS reconstruction. These results establish that measurement location and inverse formulation jointly determine reconstruction fidelity in scalar‑based near‑wall flow inference. The proposed framework opens a path toward estimation of near‑wall hemodynamics from scalar transport data, with broader applicability to fluid flow problems where passive scalars can be observed.

Abstract:
Physics‑informed neural networks (PINNs) have been widely applied to solve Navier‑Stokes equations by enforcing outputs and gradients of deep models to satisfy target equations. However, conventional PINNs only constrain the boundary terms by means of the L^2‑norm when addressing the equations with non‑homogeneous boundary conditions. This single constraint strategy may cause inaccurate boundary simulation, further resulting in the decline of prediction accuracy. To resolve this critical issue, this paper proposes an improved physics‑informed neural network by correcting the error of the boundary value, which is called Tr‑PINNs. Based on the results of nonhomogeneous Stokes problem, the algorithm error analysis of Tr‑PINNs is established. The efficacy of the Tr‑PINNs algorithm is demonstrated via numerical experiments, which further demonstrate that the Tr‑PINNs algorithm achieves a remarkable improvement in computational accuracy.

Abstract:
Physics‑Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom (d_eff) in a physics‑constrained model. Unlike the classical d_eff which measures how many parameter directions are informed by data against a statistical prior, our d_eff measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite‑dimensional kernel, we show that d_eff converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite‑dimensional kernel, d_eff instead measures the network's finite‑dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, d_eff also serves as an a priori structural diagnostic. Driving d_eff of a well‑posed problem to zero certifies that the physics and boundary constraints have absorbed the network's free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre‑trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient‑based fine‑tuning can match or exceed this but needs more wall‑clock time and tuning, whereas subspace projection delivers near‑equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.

Abstract:
Physics‑informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input‑output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross‑instance generalization of neural operators with the data efficiency of physics‑informed learning. Despite this promise, how to train PINOs efficiently and robustly remains less well‑understood than the training of either data‑driven neural operators or physics‑informed neural networks (PINNs). To bridge this gap, we examine key components of the PINO training pipeline, including architecture design, optimizer choice, loss balancing, and collocation‑point sampling strategy. We study three representative operator backbones, Deep Operator Network (DeepONet), Fourier Neural Operator (FNO), and Continuous Vision Transformer (CViT), across five diverse parametric PDE systems. Our results show that CViT provides consistently strong and stable performance across the considered benchmarks. Beyond architecture, we find that several optimization pathologies previously identified in PINN training naturally arise in PINOs, including gradient conflicts and causal violation. We also find that mitigation algorithms developed for PINNs remain effective in the PINO setting. We further compare physics‑informed and data‑driven training under different data regimes, revealing that a carefully designed physics‑informed training pipeline can match, and in some cases, outperform purely data‑driven neural operators. Taken together, these findings provide a systematic empirical understanding of the optimization challenges in PINO training and inform a practical pipeline for efficient and robust physics‑informed operator learning. Code and data are available at https://github.com/NanxiiChen/PI‑CViT.

Abstract:
We introduce a microlocal phase‑space‑filtered physics‑informed neural network (PINN‑‑TDPSF or Microlocal PINNFilter) framework for wave propagation on unbounded domains. The method combines a slabwise neural residual approximation of the interior evolution with a time‑dependent phase‑space filter applied in a buffer surrounding the physical computational domain. The central idea is to replace local artificial‑boundary penalties by a phase‑space radiation mechanism: a component is removed only when it is localized near the artificial boundary and its group velocity points outward. The proposed method is not intended to replace FFT, spectral, or split‑step solvers for known‑coefficient forward problems where such methods are available and highly accurate. Instead, it embeds the time‑dependent phase‑space filter into a residual‑based neural framework. This coupling is useful when open‑domain wave propagation must be combined with nonlinear residuals, sparse or off‑grid observations, unknown coefficients, variable interior media, or other non‑FFT‑diagonalizable physics. Numerical experiments for linear Schrödinger propagation, potential scattering, anisotropic Schrödinger dynamics, nonlinear Schrödinger wave packets, soliton stress tests, linearized Euler waves, and sparse‑data recovery of a localized acoustic defect show that the method reduces artificial reflection and wraparound, uses group velocity correctly in anisotropic media, preserves physically incoming branch components, and provides diagnostics when the assumptions behind outgoing‑packet filtering are violated.

Abstract:
Autonomous spacecraft require rapid, lightweight, and reliable onboard detection of cyber‑RF threats. Using the SPARTA attack model, we analyze the latency‑accuracy trade‑offs of TinyML‑compatible classical models ‑‑ Random Forest, Logistic Regression, SVM, and MLP ‑‑ for detecting uplink jamming, Fake‑NR spoofing, payload manipulation, ground‑segment compromise, and unauthorized command injection. We present a physics‑informed theoretical analysis of each model's computational complexity, VC dimension, Lipschitz continuity, and latency scaling, supported by empirical measurements on adversarial RF spectrograms generated via BandErasure, FakeNR, and NoiseBurst corruption modes. Results show that Logistic Regression achieves microsecond‑level inference with only a 1% accuracy drop relative to Random Forest, making it an effective TinyML baseline for onboard autonomy. The study also identifies opportunities for advancing spacecraft cybersecurity through richer feature encoders and multi‑timescale learning architectures, building on recent progress in edge intelligence and trustworthy AI.

Abstract:
In power systems, alternating current optimal power flow (AC‑OPF) has been a challenging problem for decades due to its nonconvexity, but fast and efficient solutions are even more needed because of high penetration of large scale renewable generation and load growth. Recently, neural networks (NN) have gained attention in solving AC‑OPF, but it is still in an early stage to be applicable for real and large‑scale power system operation with topology‑changing characteristics. To end this, we propose a novel framework called GraphOPF that considers topology‑adaptability, scalability, NN training time, self‑supervision, and feasibility altogether. Extensive experiments show that the proposed framework against the baselines is up to 200 times faster in NN training and up to 66 times faster in solving AC‑OPF for large‑scale power systems including the real Korean power system, while achieving more than 99% feasibility.

Abstract:
Large‑scale video generation models have made remarkable progress in semantic consistency and visual quality, producing videos that are increasingly coherent and visually convincing. Nevertheless, the dynamics induced by pixel‑level fitting do not naturally accommodate the regularities that govern real‑world motion and interaction, resulting in persistent shortcomings in physical plausibility. To address this limitation, we propose PILA (Physics‑Informed Latent Alignment), a framework that injects physics‑structured latent guidance into the frozen flow‑matching dynamics of pretrained video models. Specifically, PILA first employs anchored field estimation to map frozen‑generator latents into an operational physical attribute bank organized by field‑proxy slots, using observable motion as a kinematic anchor for constructing less directly observed proxies. To handle the heterogeneity of real‑world dynamics, PILA adopts a mixture‑of‑experts design over physical categories. Label‑prior masked expert routing selects category‑specific operator experts, whose refinements are regularized by operational residuals abstracted from physical relations. Finally, the refined proxies are fused into the physical attribute bank and decoded into a correction to the flow‑matching vector field, injecting physics‑aware guidance while preserving the visual prior of the pretrained backbone. With staged adapter training on Wan 2.1‑1.3B and direct transfer of the learned adapter to Wan 2.2‑14B, PILA achieves state‑of‑the‑art results on VBench‑2.0, VideoPhy‑2, and PhyGenBench in both visual quality and benchmark‑measured physical plausibility.

Abstract:
Physics‑informed neural networks (PINNs) have become a promising framework for simulating partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, recent studies show that PINN optimisation is sensitive to numerical precision. Existing implementations commonly use either single precision (FP32), which is computationally efficient but prone to failure modes, or double precision (FP64), which is robust but substantially expensive. This creates a trade‑off between computational efficiency and numerical accuracy. To reduce the computational cost of double‑precision training while retaining prediction accuracy, we propose a curvature‑aware precision controller that adapts numerical precision during training rather than treating it as a fixed implementation choice. The proposed method reuses curvature information derived from the limited‑memory BFGS (L‑BFGS) optimiser to construct a precision controller, retaining FP32 when lower precision is sufficient and promoting computation to FP64 when the training dynamics indicate numerical sensitivity or precision‑limited stagnation. We evaluate the proposed approach on four canonical PINN failure‑mode benchmarks and an irradiance‑driven ordinary differential equation example. We further test the proposed approach across different neural network architectures. The method consistently matches or even slightly exceeds full FP64 solution accuracy while reducing training time relative to full double‑precision training on all benchmark equations. The obtained results indicate that precision sensitivity in PINN optimisation is phase‑dependent, and that selectively applying higher precision only during numerically critical stages can lower computational cost without sacrificing predictive accuracy.

Abstract:
Physics‑informed neural networks (PINNs) approximate solutions of ODEs and PDEs by minimising a weighted combination of residual, boundary, initial, and data losses. Their performance is often dominated by the choice of loss weights: a poor weighting can drive training to a degenerate solution in which one physical constraint is satisfied while another is ignored. Existing methods select or adapt a single good set of weights. We take a different view: instead of tuning one weight vector, we explore the entire weight space during training. We introduce LC‑PINN, which adapts the loss‑conditional training of Dosovitskiy and Djolonga (2020) to the PDE‑residual setting: the conditioning vector (either the loss weights or a scalar physical coefficient) is treated as a network input and sampled from a simple prior at every optimisation step. This turns PINN training into learning a continuous family of solutions indexed by that vector, with no solver‑generated paired data. LC‑PINN thus lies between classical PINNs and operator learning: it stays fully physics‑informed but amortises training over a parametric family. Our contribution is not the loss‑conditional construction itself, but its extension to PINNs, the unification of the loss‑weight and parametric‑coefficient regimes under one architecture (concatenation for loss weights, FiLM for coefficients), and a fixed‑quadrature L‑BFGS finishing protocol that makes the parametric‑coefficient regime trainable. We give a lambda‑invariance result for the conditional optimum and study LC‑PINN on parametric Helmholtz, Schrodinger, viscous Burgers, and Buckley‑Leverett equations. A single LC‑PINN matches or improves retrained per‑weight PINN baselines while parameterising the full family in one model, at a total cost that amortises favourably against per‑instance retraining.

Abstract:
This study develops a two‑domain physics‑informed neural network framework for contaminant transport through a GCL/SL composite liner system, in which the thin GCL layer is treated using a steady‑state advection‑dispersion‑biodegradation formulation and the underlying soil liner is modeled as a transient transport domain. Two formulations are evaluated against analytical and finite‑element reference solutions under different leachate‑head conditions: a standard PINN with soft constraint enforcement (Std‑PINN) and a hard‑constrained PINN (H‑PINN), in which selected boundary and initial conditions are embedded directly into the trial solutions. The Std‑PINN captures the overall breakthrough behavior but shows larger errors during the early transport stage, particularly under higher leachate heads where advective transport becomes more pronounced. The H‑PINN reduces the optimization burden associated with penalty‑based constraint enforcement and provides more accurate and stable concentration predictions, lowering the MAE from approximately 0.058‑0.067 for the Std‑PINN to about 0.011‑0.023 for the H‑PINN, while reducing the MRE from approximately 9.10%‑19.16% to about 2.08%‑3.14%. Parametric analyses confirm that the H‑PINN with the tanh activation function and an optimized network structure provides the best predictive accuracy. The H‑PINN is further extended to inverse modeling for identifying the SL degradation half‑life from limited concentration observations, showing reliable convergence toward prescribed values and acceptable robustness under low‑to‑moderate observation noise.

Abstract:
Accurate flood forecasting is essential for mitigating disaster risks and protecting communities. However, purely data‑driven machine learning models often struggle in data‑scarce environments and may violate fundamental hydrological principles. Standard Long Short‑Term Memory (LSTM) networks can generate physically inconsistent predictions, particularly when extrapolating to extreme weather conditions. To address these limitations, we propose a Physics‑Informed Machine Learning (PIML) framework that incorporates hydrological knowledge directly into the loss function of an LSTM model. Specifically, a Trend Alignment constraint penalizes directional inconsistencies between precipitation and discharge trends, improving model robustness without requiring complex hydrodynamic equations. This regularization encourages the model to learn physically plausible hydrograph behavior, even with limited training data, while enhancing reliability during peak flood events. Experimental results show that the proposed physics‑informed model outperforms a standard LSTM baseline in data‑scarce settings, increasing the Nash‑Sutcliffe Efficiency (NSE) from 0.20 to 0.23 when trained on only 5% of the available data. Additional stress tests under simulated extreme climate scenarios demonstrate that the baseline model exhibits unstable behavior, whereas the physics‑informed model maintains directional consistency and physical plausibility. Although accurately predicting extreme peak magnitudes remains challenging with limited data, the proposed approach substantially reduces unphysical fluctuations common in purely data‑driven models. These findings demonstrate that simple physical constraints can significantly improve the reliability of deep learning models for real‑time flood forecasting, offering a practical solution for ungauged basins and evolving climate conditions.

Abstract:
We present a probabilistic modeling framework for incorporating small‑scale spatial heterogeneity into macroscopic descriptions of material behavior for polycrystalline metallic materials. Spatially heterogeneous material state fields are represented using probability density functions (PDFs), providing a principled statistical description of microstructural variability and state evolution across different computational polycrystalline realizations. The framework is built on the inverse identification of a probabilistic transport model, formulated as a Liouville equation with an unknown drift term. To enable accurate, stable, and interpretable inference of this drift field in high‑dimensional, transport‑dominated settings, we develop a Separable Probability Learning Technique via Physics‑Informed Neural Networks (SPLIT‑PINN). This method incorporates a marginal‑correction drift decomposition, orthogonality constraints, and residual‑based adaptive training to enhance well‑posedness, numerical stability, and physical consistency without imposing restrictive parametric assumptions. Using SPLIT‑PINN, the drift field governing the temporal evolution of joint state PDFs is inferred directly from data. After benchmark validation, the framework is applied to physical computational datasets describing the evolution of polycrystalline microstructural states, including von Mises stress, dislocation density, and equivalent plastic strain rate. The learned Liouville model, trained on a single dataset, is subsequently used in forward predictions of the temporal evolution of joint and marginal PDFs for multiple unseen polycrystal realizations. Quantitative comparisons with reference PDFs demonstrate that the proposed framework yields accurate and robust probabilistic predictions and generalizes effectively across datasets.

Abstract:
Reconstructing interactive, simulation‑ready 3D scenes from a single image is a critical bottleneck for robotic manipulation. While recent single‑image lifters recover plausible per‑object shapes, composing them yields scenes that collapse under physical simulation due to interpenetrating, hovering, or sinking objects. Existing physics‑aware methods address this strictly as a post‑hoc layout correction, leaving the underlying geometric errors unresolved. To address this, we introduce SimuScene, a compositional 3D reconstruction pipeline that puts physics in the loop of shape and layout estimation. Rather than using physics merely for layout cleanup, we utilize the physics engine as a diagnostic measurement tool during the generative process itself. By diagnostically simulating reconstructed objects under gravity, we convert penetration and support failures into quantitative correction signals that drive gravity‑axis stretching and amodal shape resampling. This physics‑informed feedback loop mitigates accumulated reconstruction errors and produces a stable, simulation‑ready compositional 3D scene. Extensive experiments demonstrate state‑of‑the‑art performance on physical stability and geometric alignment benchmarks. We further highlight SimuScene's utility by deploying reconstructed environments in humanoid control and robot‑arm manipulation tasks.

Abstract:
Structural Health Monitoring (SHM) aims at the real‑time monitoring of the integrity of engineering structures, with Guided‑waves (GWs) providing high sensitivity to damage presence and to ageing effects for thin‑walled components. In conventional GW‑based SHM, a bonded piezoelectric transducer (PZT) emits a short tone burst that produces an Initial Wave Packet (IWP) propagating through the structure. As this packet interacts with boundaries and potential damages, additional scattered wave packets are produced. A major limitation of such approaches lies in the simultaneous excitation of multiple dispersive GW modes by a single PZT, which significantly complicates signal interpretation and damage monitoring. In this context, this work proposes the Physics‑Informed Single Atom Matching Pursuit (PISAMP) method, a signal decomposition method grounded in the physical principles governing wave propagation. In contrast with purely data‑driven or numerically intensive techniques, the proposed approach embeds strong physical constraints into a low‑dimensional and computationally efficient signal representation. This formulation enables the direct identification of key physically meaningful features, including modal wavenumber functions and propagation distances between actuator, damage and sensors. These extracted features, especially source‑damage‑sensor distances, allows to subsequently perform damage location using well established Elliptical Localization techniques. The principal novelty of this study lies in integrating wave propagation physics into a compact signal decomposition framework and developing an interpretable damage localization methodology for GW‑SHM applications.

Abstract:
Multi‑objective optimization (MOO) underlies many machine learning problems, yet MOO solvers across the loss‑balancing, gradient‑balancing, and Pareto‑based families almost universally hand their reconciled directions to Adam~\citekingma2015adam. We show this coupling introduces two systematic gaps between the solver's intent and the optimizer's execution. The first is a \emphweighting mismatch: Adam's second‑moment denominator entangles the time‑varying preference vector with gradient statistics, marginalizing the preference into a history average and collapsing distinct Pareto trade‑offs toward a near‑uniform mixture. The second is a \emphgeometric mismatch: Adam's adaptive metric distorts the Euclidean geometry MOO solvers assume, turning aligned objectives into apparent conflicts. To resolve both jointly, we introduce MAdam (Metric‑Aware Multi‑Objective Adam), a drop‑in wrapper that leaves both solver and optimizer unchanged. MAdam preconditions the reconciled direction by the preference‑conditioned curvature of the scalarized objective; on this whitened input, Adam's second moment collapses to identity, so the realized update is governed by the preference‑conditioned metric. Across multi‑task learning, Pareto‑front recovery, physics‑informed neural networks, and medical imaging, MAdam consistently improves over Adam for every solver family.

Abstract:
Learning‑based and physics‑informed methods are increasingly used for inverse estimation in controlled nonlinear dynamical systems. However, in many such approaches, the theoretic requirements that make unknown‑input reconstruction meaningful, namely well‑posedness in the sense of Hadamard, are often disregarded or weakly addressed through generic regularization terms with no explicit guarantees. In this work, we adopt a complementary viewpoint in which these control‑theoretic and structural conditions inform the estimator design and constrain its training. We thus develop a physics‑informed input‑state neural estimator for joint unknown‑input and state estimation in nonlinear controlled systems with partial measurements. In the present work, this general framework is instantiated on a model of autonomic cardiac regulation, provides a concrete study case. The estimator is formulated as an inverse neural map conditioned on time and measured outputs, and is trained under data fidelity and dynamical consistency constraints. To ensure it complies with the same structural requirements imposed in robust estimation, we derive left‑invertibility conditions by differential‑algebraic elimination and embed the resulting constraints directly into the training objective. We further analyze a priori the stability of the inverse mapping to output perturbations and derive a conservative Lipschitz bound that guides the tuning of cost functional hyper‑parameters. The framework is evaluated on simulated data, where ground truth data is available, and on two distinct datasets of real cardiovascular recordings. The results show that incorporating control‑theoretic solvability constraints into physics‑informed learning improves the reliability of inverse inference beyond forward consistency alone.

Abstract:
Physics‑informed operator learning provides an efficient framework for approximating solution operators of partial differential equations by combining observational data with governing physical laws. However, most existing methods implicitly assume that the prescribed governing equation is accurate. This assumption may fail in practical applications, where model simplifications, missing physical effects, parameter drift, or incomplete constitutive relations can lead to model misspecification. In this work, we propose a physics‑guided operator correction framework for learning solution operators under misspecified governing equations. At the operator level, the target mapping is decomposed into a prior operator induced by an approximate physical model and a learnable correction operator that accounts for the remaining discrepancy. Although the formulation is architecture independent, we realize it using a serial DeepONet architecture, where the first DeepONet provides a prior prediction and the second DeepONet learns an additive correction conditioned on both the input function and the prior prediction. The learned correction is incorporated into the physics residual and trained together with data‑consistency constraints, allowing the model to retain useful physical structure while adapting to inaccurate governing equations. Numerical experiments on diffusion‑reaction, Burgers, cavity flow, and hyperelastic problems show that the proposed method substantially reduces errors induced by misspecified physics. Additional tests under sparse and noisy observations further demonstrate the robustness of the framework and its ability to provide informative uncertainty estimates through deep ensembles.

Abstract:
Physics models are inherently imperfect due to misspecified or missing mechanisms, resulting in systematic discrepancies between model predictions and real‑world observations. The Kennedy‑O'Hagan (KOH) framework addresses this issue through explicit discrepancy modeling. However, its non‑amortized, per‑instance formulation limits scalability across families of related systems. We introduce Amortized Physics‑Informed Calibration (APIC), a population‑level extension of KOH that leverages Neural Processes to perform scalable Bayesian inference across realizations. Our framework employs a two‑branch latent architecture to disentangle instance‑specific physical parameters from shared, state‑dependent structural discrepancies. By integrating differentiable physics into an amortized inference backbone, APIC enables rapid calibration of unseen realizations from sparse observations while quantifying uncertainty. Experiments on the damped spring oscillator, the Lotka‑Volterra system, and the advection‑diffusion PDE with misspecified physics demonstrate improved parameter recovery and consistent identification of the systemic discrepancy structure compared to other calibration approaches.

Abstract:
In AI for Science, physics‑informed losses are increasingly used to train learned compressors for scientific data, but their rate‑distortion implications remain poorly understood. At fixed bitrate, these objectives often improve preservation of a target physical observable while degrading standard reconstruction fidelity. We develop a local geometric theory showing that this tradeoff is governed by the interaction of latent‑space sensitivities induced by the entropy model, the physical observable, and the distortion metric. At each operating point, these induce preferred directions along which compression noise should be suppressed, yielding an anisotropic error‑allocation mechanism. When these directions are misaligned, improving the observable at fixed rate necessarily worsens standard distortion, establishing a fundamental limit on simultaneous preservation. We formalise this through a local tangent‑space rate‑distortion law and introduce a practical alignment diagnostic based on dominant eigenspace overlap. Experiments across scientific domains test the theory and validate that the alignment diagnostic correlates with observed data‑ and physics‑space trade‑offs.

Abstract:
Automatic‑differentiation‑based inverse analysis methods, including physics‑informed neural networks (PINNs) and differentiable programming, have recently shown great promise due to their ability to compute accurate gradients and convergence efficiency. However, their applicability to falling weight deflectometer (FWD) backcalculation remains unexplored. This study critically evaluates PINN‑based inverse analysis for a multilayer pavement system and investigates differentiable finite element method (DiffFEM) as an alternative based on a synthetic benchmark. The standard PINN does not recover layer moduli because of the sharp domain discontinuities inherent to layered pavement systems. Although we use an extended PINN with domain decomposition (XPINN), which shows better performance on discontinuous domains, its performance remains highly sensitive to loss weighting and network architecture, and degrades under measurement noise. By contrast, DiffFEM consistently achieves more accurate, stable, and computationally efficient inversion results. These results indicate that DiffFEM, which enforces the governing physics as a hard constraint, yields better accuracy, robustness, and computational efficiency than PINN‑based approaches, in which the governing physics is imposed as a soft constraint through the loss function. More broadly, the findings suggest that the choice between PINN‑ and DiffFEM‑based inverse analysis needs careful consideration, with DiffFEM offering practical advantages when an efficient and robust differentiable forward solver is available.

Abstract:
This paper examines three approaches for modeling the dynamics of a flexible‑link 2‑DoF robotic arm to address unmodeled dynamics not captured by rigid‑body models. Two physics informed models combine rigid‑body dynamics (RBD) formulations with a Gaussian Mixture Model (GMM) to capture residual model errors and linkage flexibility. A kinematics‑based regression model serves as a purely data‑driven baseline. Using an open‑source dataset, torque predictions are first estimated using Ridge regression on kinematic features, while the physicsbased baseline is constructed from published specifications, and ordinary least‑squares regression is subsequently used to estimate the same parameter set directly from data. Results show that the physics‑based parameters yield the poorest accuracy, while regularized and least‑squares estimators align more closely with measured torques. Residual analysis and error metrics highlight the limitations of purely parametric models for flexible‑link systems and underscore the value of regularization and data‑driven identification, supporting developments of semi‑parametric residual learning methods.

Abstract:
Solving time‑dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics‑informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence‑model‑based approaches parameterize time evolution using general‑purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large‑scale or high‑dimensional settings. This work introduces a PINN approach that incorporates oscillatory state‑space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear‑oscillator‑based temporal evolution, together with a PDE‑aware spectral basis in space. This design enables closed‑form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high‑dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence‑model‑based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics‑aligned and computationally efficient PINN architectures.

Abstract:
This work investigates a nonlinear two‑point boundary value problem arising in diffusion‑reaction processes in catalyst slabs with power‑law kinetics and fractional reaction order. The model exhibits a free‑boundary structure, where an unknown interface separates a dead‑core region with vanishing concentration from an active region with positive concentration. We propose a Physics‑Informed Neural Network (PINN) framework that incorporates a structured, hard‑constrained trial solution embedding the asymptotic behavior near the interface. The dead‑core location is treated as a trainable parameter, enabling the simultaneous approximation of the concentration profile and identification of the free boundary without explicit interface tracking. The method is validated against analytical solutions and high‑precision numerical shooting. Numerical experiments demonstrate that the approach accurately captures both the solution profile and the free‑boundary location while maintaining a computationally manageable training cost.

Abstract:
We propose a localized physics‑informed kernel function neural network (LPIKFNN), which is an improved physics‑informed neural network (PINN) based on physics‑informed kernel function (PIKF). In the LPIKFNN framework, the localized collocation scheme discretizes the physical quantities within the local domain, where the physical field is represented as a linear combination of PIKFs. Based on this representation, the multilayer perceptron is trained to iteratively learn the physical quantities. To overcome the computational challenges of conventional PINN in higher‑order derivative and high wavenumber problems, the LPIKFNN constructs the loss function using the PIKF and a localized collocation scheme rather than relying on automatic differentiation. As a result, the costly derivative evaluations required to enforce governing equations during iterative training are eliminated, leading to significantly improved computational efficiency and training performance. Moreover, incorporating PIKFs into the loss function enables the proposed LPIKFNN to significantly improve computational accuracy in high‑wavenumber problems characterized by highly oscillatory physical fields. To overcome the computational bottleneck of the physics‑informed kernel function neural network (PIKFNN) in heterogeneous problems, the LPIKFNN introduces a localized collocation scheme that removes reliance on global PIKFs, enabling accurate predictions where global PIKFs are unavailable. The feasibility and accuracy of the proposed LPIKFNN are demonstrated through a series of benchmark studies, including high wavenumber problems, higher‑order derivative problems, nonlinear problems, heterogeneous problems, and potential‑based inverse electromyography. The numerical predictions obtained by LPIKFNN show excellent agreement with available analytical solutions and experimental measurements.

Abstract:
Classical finite‑difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint‑sensitive regions. This paper studies a hybrid strategy in which a physics‑informed neural network (PINN) is used not as the final solver, but as an off‑grid residual probe for adaptive mesh refinement. The PINN residual is sampled over the domain, converted into cellwise indicators, and used to guide refinement before the final approximation is computed by a finite‑difference solver. The method is evaluated on three benchmarks. The main full‑solver validation uses the one‑dimensional viscous Burgers equation with a nonuniform finite‑difference solve on the adapted meshes. PINN‑threshold refinement attains final relative L^2 error 0.021067 with 60 degrees of freedom, compared with 0.022617 for uniform refinement with 192 degrees of freedom. At matched mesh size, PINN‑threshold reduces the error by about 67.5%. PINN‑Dörfler refinement gives similar performance, with error 0.021264 using 58 degrees of freedom. A gradient indicator remains slightly more accurate, so the result supports usefulness rather than universal superiority. Manufactured 2D and 3D proxy tests, based on a nonlinear Schrödinger equation and an incompressible Navier‑‑Stokes system, show that PINN residuals can organise structured refinement and improve over random refinement, although they do not consistently outperform gradient or uniform baselines. The results support PINN‑guided AMR as a residual‑indicator strategy for transferring physics‑informed diagnostic information into finite‑difference mesh adaptation while preserving the classical solver as the final approximation engine.

Abstract:
Entropy production governs irreversibility and uncertainty in both physical and information‑theoretic systems. While Physics‑Informed Neural Networks (PINNs) successfully solve differential equations, current architectures remain inherently domain‑specific. The extraction of domain‑invariant entropy representations across fundamentally different physical laws remains unexplored. This paper introduces a unified Physics‑Informed Deep Learning (PIDL) framework that simultaneously enforces differential equation residuals and information‑theoretic bounds within a single neural architecture. We demonstrate this framework via two canonical studies: (i) a thermodynamic continuous stirred‑tank reactor (CSTR) model solving governing ODEs, where a Softplus constraint strictly enforces the Second Law of Thermodynamics; and (ii) an information‑theoretic financial market model solving the inverse Fokker‑Planck PDE to infer latent drift and diffusion coefficients, guaranteeing diffusion positivity via a Softplus constraint while naturally inducing Shannon entropy. Three model variants are evaluated: two domain‑specific baselines and one shared‑encoder architecture. The PIDL framework guarantees absolute thermodynamic admissibility with zero Second‑Law violations and exhibits exceptional data efficiency, retaining >90% predictive accuracy using merely 30% of available training data. Furthermore, a post‑hoc Ruppeiner Riemannian geometric analysis of the learned entropy surface successfully identifies thermodynamic phase instabilities. This methodology provides a robust, domain‑agnostic architecture for physics‑constrained entropy modeling, advancing applications in sustainable process design and quantitative financial risk assessment.

Abstract:
Full waveform inversion (FWI) reconstructs heterogeneous material properties from receiver data but remains computationally demanding. Physics‑informed neural networks (PINNs) and their domain‑decomposed variants (FBPINNs) offer a mesh‑free alternative but face convergence challenges when representing complex velocity fields. We present a hybrid quantum‑classical FBPINN for acoustic FWI, bringing together quantum computing and classical machine learning, in which the decomposed wavefield network and the global velocity network are implemented as classical‑to‑quantum pipelines terminating in parameterized quantum circuits (PQCs). The PQCs are realized as differentiable JAX statevector simulators, enabling end‑to‑end automatic differentiation through the classical PINN, the quantum circuit, and the physics‑informed loss. On a geophysical anomaly benchmark, the quantum hybrid reaches a lower L1 velocity error than the primary classical FBPINN baseline in approximately 8x fewer training iterations, despite using approximately 33% fewer trainable parameters, and it outperforms all 15 classical hyperparameter variants tested. A second benchmark (checkerboard) demonstrates the generality of the inversion pipeline, confirming that the quantum hybrid architecture can recover structured spatial variations beyond the localized anomaly benchmark. Our framework is broadly applicable to wave‑based inverse problems beyond geophysics, including medical ultrasound tomography and non‑destructive evaluation.

Abstract:
Medical treatment recommendation poses several challenges to reinforcement learning (RL): patient physiology evolves in continuous time, measurements and interventions are performed at irregular intervals, and treatment effects vary substantially across individuals. Existing RL formulations and simulated environments, however, are based on discrete‑time MDP or POMDP abstractions with fixed or pre‑specified decision intervals. Thus, it remains difficult to evaluate whether RL methods can handle time‑interval‑dependent disease progression, personalized treatment response, and safety between consecutive measurement points. To address this gap, we introduce MedGym, a benchmark environment for dynamic treatment recommendation. MedGym models longitudinal patient evolution in a continuous‑time framework and constructs a configurable medical RL benchmark from clinical data by using Physics‑Informed Neural Networks. The resulting benchmark supports both offline and online RL, and enables direct comparison between discrete‑time and continuous‑time methods under irregular treatment timing and patient‑specific dynamics. Besides, MedGym supports evaluation from clinically important perspectives, including personalization, trajectory‑level safety, and the performance gap between model‑based offline learning and online deployment. By providing a standardized and configurable benchmark for continuous‑time dynamic treatment, MedGym aims to facilitate more realistic and informative evaluation of medical RL methods.

Abstract:
Symbolic regression (SR) offers a route to scientific discovery by converting observations into interpretable governing equations. However, despite its promise, its reliability degrades sharply when spatiotemporal measurements are sparse, noisy, or physically incomplete, as commonly occurring in practice. Data enrichment (DE) has been shown to be able to mitigate this limitation, yet additional samples can mislead equation discovery unless they preserve the physical structure of the target system. Such implication of DE requires narrow domain expertise as well as technical fluidity, highly limiting its practical usefulness. In this study, we introduce a physics‑guided latent diffusion framework for DE for down the line SR models. The proposed framework combines a variational autoencoder, a conditional latent diffusion model, and a physics‑informed residual corrector to complete sparse observations with synthetic fields constrained by governing relations. We evaluate the approach on heat conduction, incompressible Navier‑Stokes flow, and a moving single‑mass Newtonian gravitational potential, using GPLearn, DEAP, and PySR as downstream SR backends. Our results reveal that physics‑corrected enrichment consistently improves recovery in sparse regimes across physical dynamics and SR models. These results show that generative enrichment can strengthen equation discovery without additional domain expertise.

Abstract:
Inference and control in engineered physical systems pay a heavy physics cost at deployment: state estimators, inverse‑problem solvers, model‑predictive controllers, schedulers, and observers are often not closed‑form and must re‑solve a numerical optimization per instance, with the operator re‑supplied each time. Physics‑informed learning moves this cost to training, but uses a single encoder pathway whose latent geometry de‑learns under fine‑tuning and admits no quantitative transfer guarantee. We propose an asymmetric two‑pathway architecture that resolves both issues. A teacher encoder consumes privileged dense states from a high‑fidelity simulator and represents the system through operator‑polynomial features stable under spectral perturbation; a student encoder learns the same latent geometry from sparse field data and operator descriptors. At deployment the teacher is discarded, and the frozen student runs in a single forward pass with a transfer certificate. The design connects to privileged‑information learning, knowledge distillation, and cross‑modal distillation, but targets cross‑instance transfer rather than fixed‑instance prediction: topology and operator may change, while the latent task does not. We establish sufficient and near‑necessary transfer conditions via Wasserstein proximity between latent laws, yielding a zero‑shot error bound, and develop a finite‑sample certification protocol with active expansion when coverage is incomplete. The framework applies wherever a system admits an operator with reportable spectrum. On power‑system estimation, it achieves zero‑shot transfer to 100 unseen topologies, a 95% certificate pass rate, accuracy competitive with topology‑aware Newton‑‑Raphson, and sub‑millisecond inference. These results suggest asymmetric pathways plus operator‑anchored latent geometry provide a foundation for certified zero‑shot inference and control.

Abstract:
Physics‑Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill‑conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data‑fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator‑level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman‑Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call ``FK‑PINNs", and using the excess risk decomposition approach, we derive non‑asymptotic L^2(Ω)‑error bounds for FK‑PINNs with \tanh activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo‑dimension bounds for first‑ and second‑order derivatives of \tanh neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK‑PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

Abstract:
Understanding the human brain requires access to its microscopic tissue architecture. Diffusion magnetic resonance imaging (MRI) provides the only noninvasive window into whole‑brain microstructure in vivo, yet reliable quantitative mapping remains confined to specialized research settings requiring dense sampling and optimized acquisition protocols. To address this gap, we present a physics‑informed generative microstructure network (PIGMENT) that learns a universal generative prior of human brain microstructure and adapts it zero‑shot to each participant's measured data to recover subject‑specific maps. Trained on 11375 scans spanning multiple sites, vendors, and field strengths, PIGMENT enabled reliable quantitative mapping for tensor, kurtosis, and NODDI models across external datasets from five independent centers. It remains effective where conventional fitting becomes unreliable, recovering meaningful maps from extremely sparse acquisitions while supporting downstream tractography and structural connectivity mapping. PIGMENT estimates demonstrated strong biological validity, preserving submillimeter cortical microarchitectural patterns and early‑childhood white matter developmental trajectories from 10‑fold accelerated scans. Furthermore, PIGMENT enables reliable quantitative tensor mapping on cost‑efficient low‑field systems and the extraction of tumor‑related biomarkers using ultra‑fast clinical protocols. Together, these results establish PIGMENT as a physics‑informed foundation model that extends quantitative diffusion MRI into regimes traditionally too sparse, heterogeneous, or clinically constrained for reliable analysis.

Abstract:
World models, internal simulators that learn the structure and dynamics of an environment, have emerged as a central paradigm in the pursuit of artificial general intelligence, enabling agents to predict, plan, and reason within learned representations. Despite rapid progress across reinforcement learning, robotics, autonomous driving, and video generation, the field lacks a unified framework integrating its diverse architectural choices, training methods, reasoning mechanisms, and application settings. This survey addresses that gap with a multi‑axis taxonomy organized along four dimensions: (i) architecture, encompassing representation format, dynamics formulation, input modality, learning paradigm, and downstream application; (ii) methodological family, including state‑space and recurrent approaches, transformer‑based models, diffusion‑based generators, physics‑informed networks, and language‑augmented multimodal systems; (iii) reasoning strategy, covering imagination‑based planning, latent policy learning, counterfactual reasoning, and planning under uncertainty; and (iv) application domain, spanning robotics, autonomous driving, video prediction, multimodal agents, reinforcement learning, scientific modeling, medical imaging, educational measurement, and business and finance. Tracing the field from early cognitive‑science foundations to milestone systems such as PlaNet, the Dreamer family, MuZero, Sora, Cosmos, and Genie, we examine how these dimensions interact and highlight the recent convergence of chain‑of‑thought reasoning with world‑model imagination. We review evaluation protocols and benchmarks, identify persistent challenges such as compounding prediction errors, sim‑to‑real transfer, and fragmented evaluation, and outline future directions toward unified multimodal world models, foundation‑scale interactive simulators, and safe deployment in safety‑critical domains.

Abstract:
Robotic manipulation depends on the ability to anticipate how actions reshape objects, contacts, and scene geometry before execution. Learned world models provide this capability by predicting task‑relevant future evolution under robot intervention, yet the term now spans latent dynamics models, action‑conditioned video generators, three‑ and four‑dimensional scene predictors, physics‑informed simulators, and predictive modules inside vision‑language‑action systems. This breadth has fragmented the literature and obscured the design choices that matter for manipulation. We survey world models for robotic manipulation through three questions: what future representation is predicted, how prediction is connected to action, and when prediction is used in the robot‑learning pipeline. We operationally define a world model as an action‑conditioned predictive system and distinguish it from perception modules, inverse models, policies, rewards, and value functions. We then organize existing work into five representation families, develop a functional taxonomy that separates integrated prediction‑action models from explicit predictive planners, and characterize infrastructure roles including synthetic experience generation, candidate filtering, search‑based evaluation, learned environments, and outcome verification. We further map these roles across pretraining, post‑training, and inference adaptation, review 34 manipulation datasets, and synthesize evaluation protocols for predictive fidelity, task performance, and simulator reliability. This survey shows that world models are evolving from task‑specific dynamics predictors into predictive infrastructure for robot learning, while exposing open challenges in contact modeling, hallucination control, action alignment, and benchmarking under closed‑loop use.

Abstract:
This study develops a dimensionless multi‑domain physics‑informed neural network (PINN) framework for electro‑osmotic radial consolidation considering smear effects and combined vacuum and surcharge loading. Three PINN‑based models are investigated: a standard soft‑constrained PINN (Std‑PINN), a modified gated PINN (Mod‑PINN), and a modified gated PINN with hard‑constraint boundary encoding (Mod‑HC‑PINN). The models are evaluated against FEM reference solutions under four loading cases, including constant vacuum, exponential vacuum, exponential vacuum with ramp surcharge, and exponential vacuum with cyclic haversine surcharge. The results indicate that the gated architecture applied in Mod‑PINN improves the resolution of steep pressure gradients near the cathode and smear‑zone interface under constant vacuum loading. Under time‑dependent loading, the soft‑constrained Mod‑PINN shows reduced accuracy because it must learn multiple competing objectives simultaneously. The Mod‑HC‑PINN mitigates this issue by embedding the cathode boundary and initial conditions into the output structure, thereby reducing the optimization burden and improving physical consistency. The Mod‑HC‑PINN achieves MAE values of 0.43, 0.41, and 0.27 kPa for the exponential vacuum, ramp surcharge, and cyclic surcharge cases, respectively. Sensitivity analyses further demonstrate that the proposed framework remains robust across practical ranges of network architecture, collocation density, and permeability contrast.

Abstract:
We present a neural‑network‑based framework for the solution of three‑dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through functions that are holomorphic with respect to a suitable complex variable. These functions are subsequently approximated using holomorphic neural networks, which guaranty fulfillment of the holomorphicity requirement. A key feature of the proposed formulation is that the governing partial differential equations (PDEs) are satisfied exactly by construction. Therefore, in contrast to standard physics‑informed neural networks, no residual minimization of PDEs is required in the interior of the domain, and training is based exclusively on boundary collocation points. The method is validated against three‑dimensional Laplace and linear elasticity problems, where, in the latter case, displacement and stress fields are expressed via the Papkovich‑Neuber potentials. The numerical results show an accurate approximation of both scalar and vector fields, with errors remaining controlled throughout the domain. Overall, the work demonstrates that the incorporation of analytical structures into neural network architectures provides a natural and effective framework for the meshless approximation of three‑dimensional boundary value problems while preserving the underlying properties of the governing equations.

Abstract:
Riemannian diffusion models generalize score‑based generative modeling to manifold‑supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat kernel, which is rarely available in closed form beyond a few highly symmetric manifolds. We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics‑informed neural network (PINN). Given an explicit manifold specification, we choose a coordinate system, derive the corresponding heat (Fokker‑‑Planck) equation and a short‑time asymptotic approximation, and then train a PINN to learn the log heat kernel. The resulting surrogate enables both forward noising (heat‑kernel sampling) and conditional‑score evaluation for denoising score matching. We demonstrate the method on diverse manifolds including S^2, SO(3), \mathrmSPD(n), and permutation‑quotiented point clouds.

Abstract:
We propose a novel neural network architecture, termed Multi‑Scale Separable Fourier Neural Networks (MS‑SFNN), for the accurate and efficient solution of linear and nonlinear high‑frequency partial differential equations (PDEs). MS‑SFNN exploits a separable representation: given a d‑dimensional input, it employs d independent subnetworks ‑‑ each acting on a single coordinate ‑‑ and constructs basis functions via element‑wise multiplication of their outputs. The PDE solution is approximated as a linear combination of these basis functions, with coefficients determined by least squares. Critically, all network weights and biases are randomly initialized once, from a uniform distribution with unit variance, and remain fixed thereafter. To enhance expressivity, a tunable scaling factor is introduced in each subnetwork to modulate the frequency content of the resulting basis functions. Fourier features are explicitly embedded through cosine activations, endowing the method with strong spectral approximation capabilities. To mitigate the memory bottleneck associated with dense collocation in high‑frequency or three‑dimensional problems, we replace automatic differentiation with analytically derived basis function derivatives and develop a memory‑efficient batched QR decomposition algorithm for solving large‑scale least‑squares systems. Numerical experiments demonstrate that MS‑SFNN achieves unprecedented accuracy across a range of challenging PDEs, significantly outperforming state‑of‑the‑art methods such as Physics‑Informed Neural Networks (PINN) and Separated‑Variable Spectral Neural Networks (SV‑SNN).

Abstract:
Learning‑based surrogates for partial differential equations have recently matched the accuracy of classical solvers while achieving orders‑of‑magnitude speedups, predominantly in fluid settings and structured geometries. In contrast, robust surrogates for deformable solids remain underexplored, despite the presence of nonlinear elasticity, plasticity, and transient behavior that challenge standard architectures. We introduce a multigrid graph neural network for solid mechanics that couples an encoder‑processor‑decoder backbone with a physics‑informed coarsening strategy. Instead of downsampling via geometric heuristics, our method scores nodes using a residual‑based measure of local physical activity and preferentially retains regions of high strain or stress concentration, allocating multiscale capacity where it is most needed. This preserves long‑range interactions through hierarchical message passing while improving stability over long rollouts. We evaluate on multiple datasets covering linear, nonlinear, and transient regimes, and observe consistent gains in accuracy and rollout stability compared to standard sampling baselines. Our results highlight the importance of physics‑informed coarsening for scalable surrogate modeling in solid mechanics.

Abstract:
Physics‑Informed Neural Networks (PINNs) are a common class of machine learning‑based partial differential equation (PDE) solvers which train a network to represent a solution by minimizing a residual loss that encodes the PDE. Despite their successes, they are known to fail on certain simple equations, converging to an incorrect solution despite low loss. These failure modes have garnered significant attention in the literature over the past several years, motivating both architectural and optimization based solutions. By directly visualizing the residual, we show that failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Finally, we extend double backpropagation over the full set of residuals, and use it to achieve state‑of‑the‑art performance on four standard failure mode equations with up to 23× fewer collocation points and a vanilla architecture.

Abstract:
Learning to reach arbitrary goals from sparse feedback requires agents to infer a rich notion of reachability across state‑‑goal pairs. Goal‑conditioned reinforcement learning (GCRL) tackles this challenge by learning policies that generalize across goals, but this generalization becomes increasingly difficult as the underlying dynamics become high‑dimensional, hybrid, or contact‑dependent. To address this issue, physics‑informed GCRL (Pi‑GCRL) introduces optimal‑control‑inspired inductive biases into goal‑conditioned value learning. While Pi‑GCRL methods have proven effective in navigation and object‑free goal‑reaching domains, their reliability in contact‑rich tasks remains unclear, where contact interactions induce hybrid dynamics, mode‑dependent controllability, and nonsmooth value landscapes. In this work, we show that these structural properties can cause existing Pi‑GCRL methods to degrade when applied naively to contact‑rich manipulation. Motivated by this analysis, we introduce contact‑aware and hierarchical formulations that apply physics‑informed inductive biases selectively across the manipulation problem. Our results provide a principled step toward extending Pi‑GCRL to contact‑rich manipulation.

Abstract:
Radio‑frequency (RF) fingerprinting identifies wire‑less transmitters using hardware‑induced imperfections present in baseband I/Q signals. However, deep learning models often degrade under receiver and channel distribution shifts, particularly as transmitter populations grow. This work proposes the Hamiltonian Transformer, a physics‑informed attention architecture that enforces norm preserving value dynamics within each attention head using a learned skew‑symmetric generator and a Störmer‑Verlet leapfrog integration step. An additional phase‑increment embedding exposes oscillator dynamics at the input layer. All experiments use non‑equalized raw I/Q signals from the WiSig dataset under four protocols: same‑day classification, cross‑receiver generalisation, cross‑day generalisation, and transmitter scaling up to 150 devices. The Hamiltonian Transformer achieves 99.12% accuracy under same‑day conditions and 61.64% at 150 transmitters, consistently outperforming CNN and Transformer baselines across all scale points. A controlled ablation study identifies norm‑preservation in the value update as the primary inductive bias driving the scaling advantage, with the phase increment embedding providing the single largest per‑component improvement. These results indicate that embedding physics‑informed structural priors into attention mechanisms is an effective approach to large‑scale transmitter identification on raw wireless signals.

Abstract:
Physics‑informed neural networks (PINNs) formulate the solution of partial differential equations as residual minimization problems over neural network parameterizations. Although highly flexible, optimization of PINNs using modern variants of Stochastic Gradient Descent algorithms is expensive. On the other hand, iterative computation of PINN parameterization using the Gauss‑Newton method suffers from convergence difficulties, dense Jacobian structures, and poor conditioning that limit the effectiveness of second‑order optimization methods. In this work, we introduce IGA‑ODIL, a spline‑based residual minimization framework combining ideas from Optimizing DIscrete Loss (ODIL), robust variational residual minimization, and Isogeometric Analysis (IGA). Instead of neural‑network parameterizations of PINNs, the unknown solution is represented by smooth B‑spline basis functions, leading to sparse structured Jacobians and efficient Gauss‑‑Newton optimization. We also derive robust residual formulations based on weighted Gram operators, making the loss function related with the true error. The resulting systems inherit locality, sparsity, and approximation‑theoretic properties of classical finite element and isogeometric methods while preserving the residual‑learning philosophy of scientific machine learning. The proposed methodology is evaluated on several benchmark problems, including Poisson equations, convection‑dominated advection‑‑diffusion equations, Helmholtz problems with highly oscillatory solutions, nonlinear Allen‑‑Cahn equations, and inverse Helmholtz parameter identification. Numerical experiments demonstrate orders‑of‑magnitude speedups compared with PINNs and CRVPINNs while maintaining high accuracy and robustness.

Abstract:
We introduce Cosmo‑PINN, a Physics‑Informed Neural Network for reconstruction of the cosmological theory. In this work we demonstrate the application of the Cosmo‑PINN in the reconstruction of the dark energy equation of state parameter w_DE\left( z\right) directly from late‑time cosmological observations. This framework overcomes the main limitation shared by Gaussian Process and Artificial Neural Network reconstruction approaches, where the recovered solution is driven by the data and it is not necessarily true that it is physically consistent, by embedding the cosmological constraints directly into the loss function as hard constraints, ensuring that the reconstructed quantities satisfy the physical laws at every point during the training. For the training of the network, we employed background data, and specifically the Baryon Acoustic Oscillation from DESI DR2, the Cosmic Chronometers and three different Supernova compilations, while we simultaneously introduce the cosmological parameters H_0,~Ω_m0 and r_\mathrmdrag as trained parameters. The reconstruction shows that the trained w_DE\left( z\right) crosses the phantom divide within the redshift range z=0.27‑0.42 in agreement with the value obtained by the Chevallier‑Polarski‑Linder model. In the quintessence scenario, for large redshifts the dark energy Ω_DE\left( z\right) provides a pressureless nonzero contribution to the cosmological fluid suggesting a unified scenario. Finally, we demonstrate the significance of imposing the physical constraints within the loss function by comparing the Cosmo‑PINN reconstruction against a purely data‑driven neural network with the same architecture.

Abstract:
This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least‑squares solve, thereby bypassing traditional gradient‑based training. The key methodological contribution include: (1) an efficient tensor‑product scheme that generates multi‑dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time‑marching strategy to improve computational efficiency in long‑time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least‑squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics‑Informed Neural Networks (PINNs).

Abstract:
Closure‑level accuracy in neural kinetic shock solvers is not guaranteed by accurate density, velocity and temperature profiles, because the relevant observables are velocity‑weighted projections of the nonequilibrium distribution. We study this observability problem for one‑dimensional Bhatnagar‑‑Gross‑‑Krook (BGK) shock waves using a positive macro‑‑micro physics‑informed neural network (PINN) in which the distribution is represented as a local Maxwellian multiplied by a bounded exponential correction. Independent discrete‑velocity method (DVM) references are used for validation. Shock‑tube tests show that sparse joint anchoring of heat flux and normal stress stabilises the primary nonequilibrium layer, whereas residual‑only, macro‑only and single‑moment variants fail in distinct ways. In a stationary Mach‑2 normal shock, a flux‑locked compact model recovers ρ, u_x, T, q_x, σ_xx and m_xxx^cl, but leaves R_xx^cl with order‑unity error. DVM diagnostics show that R_xx^cl is controlled by a sign‑changing, tail‑weighted cancellation weakly observed by lower moments. A shock‑local closure correction aligned with this missing projection reduces the relative R_xx^cl error to 1.12×10^‑1 while preserving the lower moments. A common‑initialisation ablation shows that optional distribution‑function probe losses are diagnostic rather than constitutive. A supplementary DVM‑‑PINN comparison for the scalar fourth‑order excess Δ shows that the obstruction is anisotropic, sign‑changing tail weighting rather than fourth‑order polynomial degree alone.

Abstract:
Neural networks trained under different hyperparameter settings can fall into distinct training "regimes," with consistent behavior within regimes and qualitative differences across regimes. In this paper, we study such multi‑regime behavior in scientific machine learning (SciML) models through a regime‑aware diagnostic framework that jointly analyzes performance, training dynamics, and loss‑landscape geometry. We identify three key findings: (i) a consistent three‑regime structure emerges across many standard SciML models, different constraint enforcements, and various optimizer designs; (ii) optimization effectiveness is regime‑specific, with no single method performing well across all regimes; and (iii) SciML models can exhibit fine‑grained failure modes that can challenge conventional interpretations of standard loss‑landscape metrics. Our results provide an approach to establish a unified, task‑oblivious perspective on failure modes in SciML and to inform regime‑aware guidance for improving robustness. We validate these findings across widely‑used SciML models, including physics‑informed neural networks, neural operators, and neural ordinary differential equations, on benchmarks spanning representative ordinary and partial differential equations.

Abstract:
This work presents an end‑to‑end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user‑friendly methodology applicable to a wide range of problems, from data assimilation to closure modeling. Our approach combines a baseline differentiable PDE solver, which predicts the state w from the nonlinear system R(w) = 0, with a generic additive, parametrized, and differentiable correction f_ϕ(w), with trainable parameters ϕ. We show how to optimize phi within a fully differentiable Python workflow by reformulating the PDE as an implicit layer, enabling its integration into arbitrary objective functions, while leveraging PyTorch's automatic differentiation graph. The method is demonstrated on the Reynolds‑Averaged Navier‑Stokes equations for compressible flows, where the closure term, or a portion of it, is modeled using trainable parameters or a Neural Network. The first application considers the 2D NASA Wall‑Mounted Hump test case, where a production‑term parameter is optimized against time‑averaged LES data. A second application is carried out on the VKI LS‑59 turbine blade, where the Spalart‑Allmaras eddy viscosity field is reconstructed through the optimization of a trainable spatial field. A dataset is generated starting from the VKI LS‑59 turbine blade geometry using the differentiable BROADCAST solver with the Spalart‑Allmaras turbulence model. The results highlight the flexibility of the framework, showing its applicability beyond turbulence modeling to a broader class of physics‑informed PDE‑constrained problems with data‑driven components.

Abstract:
Occupant comfort and grid‑aware energy efficiency are competing objectives whose joint optimization depends critically on how reward functions are specified in deep reinforcement learning (DRL) controllers for buildings. Yet reward design remains largely ad hoc: comfort terms are either hand‑tuned heuristics or simple temperature‑deviation proxies without explicit grounding in thermal‑comfort physics. We present PIRS (Physics‑Informed Reward Shaping), which replaces these ad‑hoc comfort proxies with the ISO 7730 Predicted Mean Vote (PMV) formulation inside a weighted multi‑objective reward for Soft Actor‑Critic (SAC). By anchoring the comfort signal in the ISO 7730 PMV formulation, PIRS improves reward interpretability and provides a standards‑grounded comfort proxy without changing any other component of the learning pipeline. We evaluate PIRS in CityLearn v2.1.2 (challenge 2022 phase 1) with a central SAC agent trained for 50k steps over five random seeds, and compare against a rule‑based controller (RBC), a manually engineered reward (E2), an energy‑only reward (E3), and a naive temperature‑deviation comfort reward (E4). District‑level key performance indicators (KPIs), reported as ratios versus RBC, show that PIRS attains cost, carbon, and electricity metrics on par with the manual baseline while substantially outperforming non‑physics‑grounded designs ‑‑ particularly on load ramping (1.78x vs. ~2.4x RBC) and daily peak demand. All DRL policies remain above RBC at this training budget; we interpret this gap honestly and position PIRS as an interpretable, standards‑aligned foundation for reward design rather than a claim of dominance over classical control at limited compute.

Abstract:
Evolutionary computation offers a variety of tools to solve complex real‑world optimization problems. However, research often focuses on smaller, simplified problems and optimization algorithms that sometimes miss expectations in real‑world scenarios. Additionally, trust in the applied algorithm and the solutions it provides is often essential in such settings, but requires an understanding of the search process itself. This leads to evolutionary computation often not being seriously considered by practitioners in many application contexts, among them physics‑based modeling. In this article, techniques from evolutionary computation are detailed that can alleviate these problems. First, five real‑world physics‑based optimization problems are introduced and described by domain experts. For each of these, the requirements for the evolutionary algorithm regarding performance and explainability to increase trust and usability are presented. We found that all domain experts expect fast convergence to a good solution and want some explanations for how the results were formed, while other requirements strongly depend on the respective problem. Finally, we present existing approaches that can be leveraged to improve those aspects of evolutionary algorithms but have to our knowledge never been employed in complex real‑world scenarios. This implies a gap between both domains that needs to be closed to exploit the full potential of evolutionary computation.

Abstract:
Autonomous space systems operating in extreme thermal environments require accurate and efficient thermal modeling to support both pre‑mission system design and onboard autonomy. For lunar rovers, large temperature gradients, radiative heat transfer, and variable surface conditions make reliable thermal prediction especially challenging. High‑fidelity physics‑based simulations provide accurate results but are computationally expensive, while simplified models and lookup‑table approach often lack sufficient accuracy. Physics‑informed machine learning (PIML) offers a promising alternative by combining data‑driven models with embedded physical knowledge. This paper presents a PIML framework for thermal analysis of a simplified lunar rover with internal heat sources, where machine learning enables environment‑adaptive coarse meshing. The proposed architecture integrates a transfer neural network (TNN) that adaptively determines 3D finite‑difference nodalization based on thermal loads and initial conditions, enabling more accurate coarse‑mesh calculations. A differentiable finite‑difference thermal simulator is embedded within the framework to enforce physical consistency and support efficient training, while an upscaling layer reconstructs high‑resolution temperature fields from the coarse‑grid solution. The proposed PIML approach is evaluated against high‑fidelity fine‑mesh simulations, low‑fidelity fixed coarse‑mesh models, and a purely data‑driven artificial neural network (ANN). Results show that the PIML framework improves prediction accuracy by 50% and 39% relative to the coarse‑mesh physics model and ANN model, respectively, while maintaining physically consistent thermal distributions. Computationally, the framework is also 3x faster than high‑fidelity simulations, demonstrating an effective balance between accuracy and efficiency for thermal modeling of lunar rover systems.

Abstract:
We propose the Lowner‑‑John deform‑cloud scheme as a Lagrangian diagnostic for incompressible stochastic flows with an inertial range. A volume‑filled particle cloud is released at the ultraviolet scale and summarized at each time by two objects: the inertia tensor of its minimum‑volume enclosing ellipsoid and the velocity gradient coarse‑grained over that ellipsoid. We test the scheme on a two‑dimensional isotropic incompressible Gaussian‑‑Holder finite‑time‑correlated velocity field with Kolmogorov exponent, generated spectrally with Ornstein‑‑Uhlenbeck Fourier modes. The resulting empirical train shows a broadly fluctuating but statistically saturated ellipsoid aspect ratio, a clear scale dependence of the perceived gradient, and an approximately ordinary tensor‑level strain‑‑vorticity balance. We then formulate reduced modeling of the train as physics‑informed generator identification. In intrinsic variables describing scale, aspect ratio, strain amplitude, vorticity, and strain‑‑ellipsoid alignment, the aspect‑ratio dynamics separates into an aligned‑strain source and a Lowner‑‑John residual. The final open‑box closure models strain and vorticity as scale‑dependent stochastic drivers, represents alignment by a stationary von‑‑Mises bias, and closes the residual by a scale‑dependent affine feedback. Thus the observed aspect‑ratio saturation is not merely fitted; it is explained as a balance between persistent strain alignment and geometric relaxation of the enclosing ellipsoid. The construction provides a portable route from particle‑cloud data to interpretable finite‑dimensional stochastic dynamics for future turbulent‑flow applications.

Abstract:
Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete geometric elements (usually triangles). Recently, physics‑informed neural networks (PINNs) have emerged as a continuous, mesh‑free alternative that does not suffer from FEM's sensitivity to mesh quality or geometric discretization errors. We present PINNSur, a simple framework for using PINNs on curved surfaces: we train a neural field to approximate the surface's normals, and then we express surface differential operators using their projection from \mathbbR^3 onto the surface. Since every orientable manifold has well‑defined normals, our method is suitable for all such surfaces, regardless of curvature or topology, enabling many geometry processing applications. Moreover, despite their empirical success in solving PDEs in flat Euclidean domains, PINNs lack convergence guarantees to the true solution of the underlying PDE, and there is limited systematic experimental evidence demonstrating such convergence. This gap restricts their adoption as reliable solvers compared to established methods like FEM, where convergence to the true solution is well understood and theoretically grounded. These surface PDEs are particularly challenging to solve convergently, as one must not only deal with the convergence of the function approximation, but also with the convergence of the geometric approximation of the surface itself. In this work, we empirically investigate the convergence behavior of PINNs for solving surface PDEs by introducing a simple empirical convergence test.

Abstract:
We train a pair of autoregressive models to construct zero‑mean control variates to mitigate the sign problem in quantum Monte Carlo simulations. The two autoregressive networks are confined to the positive‑ and negative‑sign sectors with strictly disjoint support, and each is exactly normalized over its sector. Their difference is therefore structurally zero‑mean, providing an unbiased auxiliary observable whose correlation with the sign estimator controls the variance reduction. We implement the method within the stochastic series expansion framework, which we extend to frustrated lattices by developing an incremental loop‑topology update. Sign‑ergodic sampling is achieved through a twist channel, which is the unique sign‑changing mechanism on non‑bipartite lattices. We implement the control variates as autoregressive transformers with an end‑of‑sequence parity mask that enforces exact sign‑sector resolution, while the incremental loop‑count change and cumulative frustration parity are incorporated as topological features. On the triangular‑lattice Heisenberg antiferromagnet, we benchmark the method in the small‑N limit. The control variate reduces the standard error of the average sign by up to an order of magnitude and that of the energy estimator by a factor of three to five, remaining effective even when the average sign drops below 10^‑3. This work lays out the framework and provides a proof‑of‑principle demonstration that autoregressive control variates can effectively mitigate the sign problem. Scaling to larger systems with physics‑informed architectures is the subject of future work.

Abstract:
Time‑dependent partial differential equations (PDEs) often develop sharp fronts, localized peaks, and other moving structures that occupy only a small portion of the space‑‑time domain but dominate the approximation error. This makes fixed or uniformly sampled collocation strategies inefficient for physics‑informed neural networks (PINNs), especially in high dimensions and over long‑time prediction intervals. We propose the predictive moving sample method (PMSM), which builds on the moving sample method (MSM) in \citexu2026moving by replacing its full time domain iterative training with a progressive time‑stepping strategy and simplifying the velocity‑field loss to further reduce the per‑step cost. To improve practicality for long‑time prediction, we further introduce the windowed‑reset predictive moving sample method (WR‑PMSM), which restricts extension training to an active time window and periodically resets the reference state, thereby reducing the growth of optimization cost while preserving global consistency through a final refinement stage. Across four representative benchmarks, PMSM consistently outperforms both standard PINNs and the original MSM under matched collocation budgets. These results suggest that transporting samples according to residual dynamics provides an effective and practical route to neural network solvers for time‑dependent PDEs.

Abstract:
Reconstructing continuous state trajectories of chaotic dynamical systems from sparse, noisy observations remains a fundamental open problem in nonlinear science. We introduce the Physics‑Informed Diffusion Model with Dormand‑Prince Integration (PIDM‑DP), which embeds a fully differentiable 5th‑order Dormand‑Prince (DP‑RK45) ODE integrator directly into the reverse sampling loop of a Denoising Diffusion Probabilistic Model (DDPM). At each denoising step, physics residuals are back‑propagated via automatic differentiation, constraining every generated trajectory to satisfy the system's governing equations to 5th‑order accuracy. A linear‑scheduled guidance mechanism that ramps the physics weight from zero at high noise levels to its full value near the clean‑data limit prevents the gradient explosions that cause naive physics‑informed approaches to fail on stiff systems with Jacobian eigenvalues of order O(10^3). Evaluated across five benchmark systems of increasing complexity 3D Lorenz, 3D Rössler, 5D Hyperchaotic, 20D Lorenz‑96, and the stiff 3D Rabinovich‑Fabrikant at 10% observation density with additive Gaussian noise (σ=0.05), PIDM‑DP achieves reconstruction RMSE improvements of up to 15.4× over an unconstrained diffusion baseline and decisively outperforms the Ensemble Kalman Filter on stiff systems where ensemble covariance collapses. On the Rabinovich‑Fabrikant out‑of‑distribution benchmark, PIDM‑DP attains RMSE 0.1097 \pm 0.0269 versus 0.9443 \pm 0.5288 (unconstrained diffusion, 8.6× worse) and 0.3561 \pm 0.3040 (EnKF, 3.2× worse), with p<0.001 in paired Wilcoxon tests (N = 30). Topological validation via the Rosenstein Lyapunov estimator confirms that PIDM‑DP preserves the chaotic invariant measure.

Abstract:
Turbulence is ubiquitous in engineering and science, yet direct simulation is prohibitively expensive. The Reynolds‑averaged Navier‑Stokes (RANS) equations provide savings exceeding ten orders of magnitude but introduce unclosed terms (the closure problem). Offline‑trained machine‑learning (ML) closures suffer distribution shift in predictive simulations, while ML methods that bypass the governing equations struggle to generalise from scarce high‑fidelity data. We develop a physics‑derived deep learning closure model for RANS, the Deep Algebraic Reynolds Stress Model (DARSM), which can be trained on small datasets and accurately generalise across Reynolds numbers, to unseen geometries, and to different flow regimes. A neural network maps flow invariants to empirical parameters in an implicit algebraic Reynolds stress equation, derived from the Reynolds stress transport equations under the weak‑equilibrium assumption, imposing physics‑based structure on the ML closure. End‑to‑end optimisation through the governing PDEs and the coupled implicit closure eliminates distribution shift, but both unrolled and implicit automatic differentiation fail on the stiff coupled solver. We derive adjoint equations that exploit the solver's implicit‑explicit structure for efficient optimisation. On canonical square‑duct and periodic‑hill benchmarks, DARSM reduces average test velocity error over baseline RANS by 2‑4× across Reynolds number, geometries, and flow regimes, with peak case‑level reductions of 12×. The model trained on attached, anisotropy‑dominated flows (square duct) accurately generalises without retraining to separated flows (periodic hills), a regime change in the underlying physics. DARSM also outperforms five established ML methods: offline training, tensor‑basis neural networks, field‑inversion machine learning, DeepONets, and physics‑informed neural networks.

Abstract:
Physics‑informed machine learning (PIML) integrates mechanistic knowledge, typically in the form of partial differential equations (PDE), into data‑driven models. Despite strong empirical performance, its statistical generalisation properties remain poorly understood, particularly in the regression setting with unbounded losses. Existing analyses rely on approximation or stability arguments and do not fully capture how physical structure influences generalisation from finite data. In this work, we develop a PAC‑Bayesian framework for PIML that provides high‑probability generalisation guarantees in the presence of unbounded losses. We adopt a multi‑task perspective that jointly treats data fidelity, PDE residuals, initial and boundary conditions, avoiding the looseness induced by standard union‑bound approaches. Our analysis leverages the structure of physics‑informed objectives to derive novel bounds where the complexity scales with input‑gradient norms of the losses, revealing a direct link between physical regularity and generalisation. We instantiate this framework under Sobolev and Poincaré‑type assumptions, yielding two classes of bounds that trade off statistical complexity and smoothness in different regimes. Building on these results, we propose a self‑bounding‑aware learning algorithm that directly optimises tractable surrogates of the derived bounds, along with a practical procedure to estimate the associated constants in realistic settings. Empirical evaluations on standard PDE benchmarks demonstrate that our bounds are non‑vacuous, significantly tighter than union‑bound baselines, and can be effectively minimised during training. Overall, our results provide a principled statistical foundation for the generalisation of physics‑informed models.

Abstract:
A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot K in the 3‑sphere S^3, and the signed count of minimal surfaces in hyperbolic 4‑space \mathrmH^4 meeting the sphere at infinity at K, with prescribed genus and self‑intersection number. In this paper, we develop a novel machine learning framework based on Physics‑Informed Neural Networks (PINNs) to solve the minimal surface equation in hyperbolic space. We utilise this framework to test Fine's Conjecture by constructing near‑minimal surfaces bounding various families of knots in S^3. Furthermore, we develop an algorithmic method to find self‑intersections and compute their sign. For every knot analysed, the computationally discovered minimal surfaces and their self‑intersection numbers perfectly align with the predictions of Fine's Conjecture, providing empirical evidence for it.

Abstract:
Quantum control is essential for quantum information science and technology, yet designing high‑fidelity control protocols remains challenging due to complex optimization landscapes, hardware noise, and long pulse sequences. Existing numerical solvers often require problem‑specific engineering and produce opaque control amplitudes, while naive large language models (LLMs) lack the physical consistency and long‑horizon precision for reliable quantum control synthesis. Here we introduce VF‑QCTRL, a physics‑informed large language model framework for general quantum control that combines symbolic reasoning with optimization to propose analytic control ansätze and coherently refine their parameters through feedback. To systematically evaluate LLM‑driven quantum control, we develop QCTRL‑BENCH, a benchmark spanning sixteen tasks across single‑ and multi‑qubit systems, closed and open quantum dynamics, noiseless and noisy settings, and both analytic and numerical protocols. Across the benchmark, VF‑QCTRL demonstrates strong universality, accuracy, efficiency, and interpretability: it applies to generic quantum control systems without task‑specific training, achieves performance competitive with or exceeding state‑of‑the‑art conventional solvers in both noiseless and noisy regimes with query efficiency, exhibits favorable inference‑time scaling and pulse resolution scaling, and derives physically interpretable analytical protocols directly from prompts. Our results establish physics‑informed LLM‑based quantum control as a promising paradigm for accurate, efficient, interpretable, and training‑free quantum control protocol design across a broad range of quantum systems.

Abstract:
Cardiac function evaluation necessitates continuous, non‑invasive monitoring, a capability limited in MRI. Millimeter‑wave (mmWave) radar and its Synthetic Aperture Radar (SAR) mode offer a privacy‑preserving and portable point‑of‑care clinical applications. However, reconstructing high‑fidelity 3D cardiac geometry from SAR remains an open challenge. Traditional radar methods generate sparse point clouds that lack continuous surface topology. Meanwhile, direct application of optical reconstruction networks performs poorly due to the severe speckle noise and ambiguous boundaries inherent in SAR images. To bridge this gap, we propose SAR2Mesh, a novel framework that reformulates the task as a coarse‑to‑fine mesh deformation process. By initializing with a topological template, our approach explicitly preserves anatomical connectivity through progressive mesh deformation.We introduce a geometry‑aware feature projection module to extract multi‑view features via 3D‑to‑2D sampling, and a physics‑informed radar loss to enforce consistency between the predicted geometry and raw radar echoes. Furthermore, we present Cardiac Mesh‑SAR, the first large‑scale paired SAR‑mesh dataset. Extensive experiments demonstrate that SAR2Mesh significantly outperforms existing image‑based baselines, achieving accurate and physically consistent cardiac reconstructions.

Abstract:
Accurate reconstruction of magnetic fields in inaccessible regions is vital for many high‑precision experiments in physics. Traditional methods, such as spherical harmonic expansion, often suffer from truncation errors that limit their precision. This study proposes an advanced Physics‑Informed Neural Network (PINN) framework for high‑precision 3D magnetic field mapping. Unlike conventional data‑driven models, the proposed PINN integrates Maxwell's equations directly into the loss function, enforcing divergence‑free and curl‑free conditions across the entire domain. A key innovation is the inclusion of explicit physics‑residual losses at measurement locations, ensuring rigorous physical consistency beyond random collocation sampling. Validation using simulated data achieves a reconstruction accuracy of 10^‑4, a tenfold improvement over existing PINN benchmarks. Furthermore, experimental validation using a custom coil assembly demonstrates robust reconstruction with sub‑percent relative accuracy, reaching the 10^‑3 level under ambient conditions. This AI‑driven methodology provides a robust, high‑precision solution for field monitoring and measurement in complex experimental environments where direct sensor placement is restricted.

Abstract:
An extensive diffusion analysis is presented for binary Ni‑X and ternary Ni‑Al‑X (X = Cr, Mo, Ta, W, Re) systems, which play a crucial role in microstructural evolution and phase stability in Ni‑Al‑based superalloys. Specifically, we highlight changes in the diffusion coefficients of X in the presence of Al and compare diffusional interactions across systems considered. First‑principles calculations, combined with activation energies derived from temperature‑dependent experiments, reveal consistent trends in Ni‑X systems, with variations in activation energies largely attributed to differences in migration energies. In ternary systems, diffusion coefficients estimated from intersecting diffusion profiles show that the main interdiffusion coefficient of X is comparable to its binary counterpart, with similar activation energies. However, cross‑diffusion coefficients are shown to significantly influence fluxes, either enhancing or reducing diffusion lengths depending on the relative directions of diffusing elements. For Ni‑Al‑Re, a single‑profile method is employed to overcome uncertainties in estimating composition gradients at the near‑end‑member intersecting composition. The diffusion coefficients obtained correlate well with the nature of diffusion paths when represented on Gibbs triangles. To extend these findings, a physics‑informed neural network (PINN) optimization method is applied to extract composition‑dependent diffusion coefficients across the full composition range. The analysis demonstrates the necessity of incorporating experimentally estimated diffusion coefficients as equality constraints, without which optimization reliability is compromised. Overall, the results establish a robust framework for diffusion studies in Ni‑Al‑X systems, highlighting the critical role of cross‑diffusion effects and constraint‑enhanced numerical methods.

Abstract:
Neural networks have emerged as effective tools for solving ill‑posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on synthetic data generated from the forward model. This requires specifying unknown parameters in the forward model and solving the model to generate synthetic observations. Typically, the unknown parameters are sampled from a prescribed probability distribution. Here, we show that this sampling strategy is not a neutral preprocessing step, but instead defines an implicit regularization operator. This result follows from the fact that the learned inverse operator minimizes empirical risk together with the classical result that conditional expectation minimizes mean‑square error. We present theoretical results for the implicit regularization operator in both infinite‑ and finite‑data settings, including Physics Informed Neural Networks (PINNs). These results are demonstrated numerically on three inverse problems of increasing complexity: a 1D linear Fredholm integral equation, a 1D nonlinear subsurface interface inversion, and a 2D nonlinear cross‑well seismic traveltime tomography problem. Across all three problems, three distinct sources of regularization are identified in the learned operator: prior sampling, architectural, and physics‑informed regularization. A mismatched sampling distribution is shown to degrade reconstruction quality in ways that neither more expressive architectures nor augmented physics residuals can fully correct. The results demonstrate that the sampling distribution should be chosen with the same care as a classical regularization functional and provide a practical framework for implementing more sophisticated regularization operators using neural networks.

Abstract:
Reliable forecasting of renewable energy generation is a foundational requirement for grid stability energy trading battery scheduling and carbon aware operational planning Solar and wind resources are inherently intermittent their output fluctuates with cloud cover wind speed atmospheric turbulence seasonal patterns and local terrain The proliferation of IoT and edge devices spanning smart meters inverters anemometers pyranometers weather stations and grid interface sensors has created an unprecedented volume of real time operational data that conventional forecasting pipelines are ill equipped to exploit fully This review investigates how large language model LLM agents can enhance renewable energy forecasting by integrating heterogeneous sensor streams weather API data historical generation records grid constraints and contextual reasoning into unified decision support workflows We survey classical forecasting methods statistical time series models deep learning architectures physics hybrid approaches and emerging LLM agent frameworks for explanation uncertainty communication and operator guidance A six layer taxonomy is proposed covering data acquisition preprocessing feature engineering model inference uncertainty estimation and natural language reporting The review identifies twelve open challenges spanning real time deployment model drift under distribution shift uncertainty quantification hallucination control in LLM agents interoperability of edge hardware and integration with energy management systems The paper concludes by recommending a research agenda centred on open benchmarks physics informed LLM grounding and federated forecasting architectures

Abstract:
Ultrasound computed tomography (UCT) via full waveform inversion (FWI) enables high‑resolution quantitative imaging for tissue characterization and disease diagnosis. However, UCT suffers from large computational burden and severe convergence issues due to highly nonlinear optimization. Deep learning can accelerate UCT reconstruction, but supervised training requires large‑scale labeled datasets difficult to obtain in vivo. To address these limitations, we propose SDA‑UCT, a two‑stage self‑supervised domain‑adaptive framework for rapid and accurate UCT imaging of musculoskeletal tissues. SDA‑UCT employs an attention‑enhanced network (AttUCT) pre‑trained on simulation datasets and transfers to in‑vivo data via physics‑informed self‑supervised learning, effectively bridging the simulation‑to‑real domain gap. A Low‑Rank Adaptation (LoRA) mechanism is integrated to enable efficient adaptation across diverse clinical scenarios. Results showed that AttUCT achieved high‑quality SOS reconstruction for simulated human forearm with a PSNR of 29.23 dB and SSIM of 0.928, outperforming conventional FWI and existing deep learning methods. Validated on in‑vivo data, SDA‑UCT successfully reconstructed SOS images revealing complex anatomical structures (skin, fat, muscle, tendon, bone and bone marrow) for human forearm, in high concordance with MRI references. The LoRA mechanism adjusting only 3% of parameters achieved comparable performance to full fine‑tuning. The rapid reconstruction (5 ms per frame) enables real‑time 3D visualization, achieving five‑orders‑of‑magnitude improvement over traditional FWI. This work represents the first self‑supervised domain‑adaptive deep learning for rapid, high‑resolution in‑vivo UCT imaging, showing potential for musculoskeletal disease diagnosis.

Abstract:
Advanced driver assistance systems (ADAS) play an important role in modern automotive intelligence, significantly enhancing vehicle safety and stability. The performance of ADAS critically relies on accurate and reliable vehicle state estimation, particularly from vehicle dynamic sensors. Among these signals, wheel load is a key variable for chassis control and safety‑critical functions, yet it remains difficult to estimate robustly due to complex suspension geometry, nonlinear dynamics, and measurement noise. To address this issue, we propose DBPnet, a Bayesian physics‑informed neural network (PINN) with a physics‑aware embedding module inspired by damper characteristics. First, this paper presents a suspension linkage‑level modeling (SLLM) approach that constructs a nonlinear instantaneous dynamic model by explicitly considering the complex geometric structure of the suspension. Building upon SLLM, Bayesian inference is integrated into the PINN to effectively cope with noise and uncertainty in the vehicle chassis system, thereby improving the model's robustness. Then, a physics‑informed loss function is employed to ensure consistency with fundamental physical principles, while the damper characteristics‑inspired embedding module extracts temporal variation features of input signals and incorporates them into each layer of the PINN, ensuring that physical observations guide the neural network without being constrained by fixed physical models. Extensive evaluations on high‑fidelity simulations and real‑world experiments demonstrate that our DBPnet consistently achieves lower RMSE and MaxError than baseline methods. These results highlight the potential of our DBPnet to advance wheel load estimation and contribute to the development of more reliable ADAS actuator functions.

Abstract:
This work presents a high‑fidelity computational fluid dynamics (CFD) and data‑driven modeling framework for assembly‑level flow characterization in a four‑loop pressurized water reactor (PWR). A full lower‑plenum and core‑inlet domain was constructed using publicly available geometry and operating conditions, enabling transient simulations with pump‑induced swirl boundary conditions. The results show that cold‑leg swirl and lower‑plenum transport generate strongly heterogeneous assembly‑wise inlet flow distributions, particularly near the lower core region, while axial resistance and mixing progressively homogenize the flow at higher elevations. These physics‑informed datasets were subsequently used to evaluate machine learning (ML) applications for partial field reconstruction and short‑term autoregressive prediction. A 3D convolutional‑based inpainting model successfully recon‑structed missing assembly‑level mass flow rates from partial observations, with errors concentrated in the highly turbulent base (bottom) layer and diminishing significantly in upper layers. Comparative analysis across multiple ML models demon‑strates that spatially aware architectures, particularly ConvLSTM, significantly outperform sequence‑based (LSTM) and operator‑learning (DeepONet) approaches by effectively capturing coupled spatio‑temporal dynamics. The study also high‑lights key challenges, including the sensitivity of inlet flow predictions to turbulence and mesh resolution, as well as the absence of full‑scale experimental validation data. Despite these limitations, the results remain consistent with expected physical behavior. Overall, this work establishes high‑fidelity CFD as a critical foundation for developing data‑driven surrogates, sparse sensing strategies, and future multiphysics coupling frameworks.

Abstract:
This letter introduces a physics‑informed self‑supervised framework for sonar image despeckling that reformulates despeckling as residual consistency in the homomorphic log domain. By constraining the log‑ratio residual to obey multiplicative speckle statistics, the proposed method eliminates the need for clean supervision while preventing degenerate identity solutions. A variance‑targeted statistical loss combined with edge‑aware structural regularization and median‑guided curriculum stabilization enables effective speckle suppression with preserved structural fidelity. This formulation along with a lightweight neural network achieves state‑of‑the‑art performance across multiple real sonar datasets and demonstrates excellent cross‑dataset robustness, while remaining suitable for real‑time deployment.

Abstract:
We propose a Weak‑form Physics‑Informed Neural Operator (WINO), a data‑free framework that combines the efficiency of neural operators with the geometric flexibility of the φ‑finite element method (φ‑FEM). φ‑FEM is an unfitted method that accommodates geometric variations without body‑fitted meshes, where the domain geometry is represented by the level‑set function φ. To impose the boundary conditions, Dirichlet problems adopt the φ‑FEM lifting so only the homogeneous displacement contribution is learned, whereas traction‑driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak‑form residuals aligned with φ‑FEM together with squared penalties on the cut‑cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear φ‑FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold‑started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50‑‑80% compared with purely data‑driven methods.

Abstract:
Determining crystal structures from powder X‑ray diffraction (PXRD) has been a significant challenge in materials science, particularly when experimental data contain noise or the target structure has a high complexity. While recent AI generative models show promise for rapid structure generation, they predominantly employ data‑driven approaches to learn direct mappings between PXRD patterns and crystal structures, often failing on complex or out‑of‑distribution cases. In this work, we present a hybrid ab‑initio approach that decomposes structure determination into a two‑stage optimization problem: (1) discrete selection of space group symmetry, unit cell parameters, and Wyckoff site combinations; and (2) continuous optimization of atomic coordinates within the selected Wyckoff positions. By integrating AI‑based techniques for peak profile analysis, density estimation and energy minimization with physics‑informed constraints, our method systematically overcomes limitations of purely data‑driven PXRD solvers. We demonstrate that this hierarchical optimization framework enables robust structure determination even for challenging cases with high structural complexity or limited experimental data quality. Our approach provides a principled pathway for incorporating crystallographic knowledge into AI models for more reliable and generalizable crystal structure determination.

Abstract:
We present an improved neural field architecture for solving partial differential equations (PDEs). Current physics‑informed neural networks (PINNs) provide a flexible framework for solving PDEs, but they struggle to achieve highly accurate solutions and require computation that scales poorly with parameter count. Our model, which we call beignet (Bandlimited Embedding with Interpolated Grid Network), replaces the random Fourier feature embedding used by existing PINN models with a trainable multi‑resolution Fourier feature pyramid. To query beignet at a continuous coordinate, we use Fourier interpolation at each level of the pyramid to return features at the input coordinate, and then decode this vector with a fully‑connected neural network trunk. Our model provides multiple benefits: 1) Spatial derivatives can be computed efficiently by using the chain rule to compose derivatives of the neural network computed with automatic differentiation with derivatives of the feature grid computed spectrally by the Fast Fourier transform (FFT). 2) beignet can achieve higher accuracy in a compute‑efficient manner by scaling the parameter count of this Fourier feature pyramid, instead of the less‑efficient strategy of scaling the neural network architecture. 3) beignet can directly control the representation bandlimit, resulting in more stable optimization for difficult PDEs. We demonstrate that beignet finds significantly more accurate solutions on PDE benchmarks using fewer parameters than state‑of‑the‑art PINN methods. We further evaluate beignet on the self‑similar inviscid Burgers blowup problem and show that it can minimize residuals to near machine precision using Adam, an accuracy regime previously attained only by using computationally expensive higher‑order optimizers.

Abstract:
Automated detection and masking of individual methane plumes from satellite imagery is important for operational emission attribution and quantification. We present a machine learning framework for plume detection from MethaneSAT retrieved column‑averaged dry‑air mole fractions of methane. We address two core challenges: the scarcity of labeled MethaneSAT data and the need for inference reliability across diverse atmospheric and surface conditions. We first demonstrate that Mask R‑CNN with a ResNet‑50 backbone outperforms U‑Net semantic segmentation on both MethaneAIR (an airborne version of MethaneSAT) and MethaneSAT data, with pixel‑level F1 score gains of 10.49 and 5.48 respectively. To address MethaneSAT data scarcity, we evaluate three cross‑sensor transfer strategies leveraging MethaneAIR flights and synthetic plumes. Mask R‑CNN with ResNet‑50 fine‑tuned from MethaneAIR pre‑trained weights is the most effective strategy, achieving instance‑level precision of 0.60 and a near‑perfect recall of 0.98 at the baseline operating point. A physics‑informed post‑processing pipeline converts detections into two operationally distinct modes. The first is a high‑sensitivity mode that applies morphological filtering and proximity‑based merging for comprehensive emission screening, achieving precision of 0.71 and recall of 0.94. The second is a high‑precision mode that additionally applies a distribution‑based classifier for confident source attribution, achieving precision of 0.92 and recall of 0.70. Manual review of detections classified as false positives against our wavelet‑based ground truth labels reveals that a meaningful fraction of cases correspond to real methane enhancements excluded by conservative labeling criteria, indicating that precision values reported are lower bounds on true detection performance... Our data and code are available at: https://doi.org/10.7910/DVN/FR959H

Abstract:
Frequency‑modulated continuous‑wave radar sensing often relies on labeled measurements that are costly, restricted, or difficult to collect at scale. This work evaluates physics‑informed digital twins as controlled testbeds for early‑stage quantum‑classical radar learning. Two synthetic radar benchmarks are considered: unmanned aerial vehicle classification from range‑Doppler maps and human fall detection from Doppler‑time spectrograms. For both tasks, inputs are standardized, reduced using principal component analysis, and classified using either a radial basis function support vector classifier or a quantum support vector classifier. All quantum‑kernel results are obtained using noiseless classical simulation; no quantum hardware is used, and no quantum‑advantage claim is made. Across five random seeds, the quantum support vector classifier improves the UAV benchmark from four principal components onward, reaching an accuracy of 0.941 +/‑ 0.012 at eight components, compared with 0.880 +/‑ 0.029 for the classical baseline. On the fall‑detection benchmark, both classifiers perform similarly, with a small quantum‑kernel improvement at higher feature dimensions. A Gaussian‑noise robustness study shows limited performance degradation across the tested noise levels, while preserving the UAV quantum‑kernel gain. These results support digital twins as useful, controlled environments for radar‑QML benchmarking prior to measured‑data validation and hardware execution.

Abstract:
The growing complexity and scale of scientific workflows in high performance computing (HPC) environments have led to significant challenges in managing energy consumption without compromising computational performance. Traditional scheduling strategies often fail to account for the complex interplay between thermal dynamics, workload diversity, and system scalability, leading to inefficient and unsustainable energy usage. This paper introduces a novel, scalable, and AI‑assisted scheduling framework for optimizing energy consumption in HPC environments without compromising performance. Central to our approach is the integration of Computational Fluid Dynamics (CFD) with a Physics‑Informed Variational Autoencoder (PIVAE), enabling the generation of physically realistic synthetic workload data that bridges the gap between thermodynamic behavior and scheduler decision‑making in complex, multi‑scale HPC environments. By categorizing workflows based on resource utilization profiles, we evaluate multiple scheduling strategies such as Locality Aware and Speculative Aware Scheduling. These workflows, ranging from event reconstruction to anomaly detection, represent diverse computational intensities. Our results show that modest reductions in CPU performance (e.g., to 15%) can yield substantial energy savings (up to 10%) with only minor turnaround time increases (approximately 5‑6%), identifying an optimal operational sweet spot. This work demonstrates how physics‑informed generative modeling can enable adaptive, sustainable, and data‑efficient scheduling for next‑generation HPC infrastructures.

Abstract:
When learning dynamical systems from data, embedding physical structure can constrain the solution space and improve generalization, but many physics‑informed models assume access to the full system state. This limits their use in partially observed settings, where some state variables are completely unobserved and must be inferred without direct supervision. Here, we present neural Hamiltonian ordinary differential equations (NHODE), a framework that combines Hamiltonian neural networks (HNNs) with neural ordinary differential equations (neural ODEs) to learn partially observed dynamical systems from data. The Hamiltonian structure enforces energy conservation by construction, while the neural ODE framework enables a flexible training procedure that allows the loss to be defined only on observed variables. We also incorporate additional physical constraints through symmetry‑aware coordinate transformations and separable energy formulations. The framework is evaluated on systems of increasing complexity, from linear and nonlinear mass‑spring systems to the chaotic three‑body problem. Across all examples, increasing the amount of embedded physical structure improves the accuracy and long‑horizon stability of the predictions. Even in the most challenging regimes, the NHODE framework captures both observed and latent dynamics, whereas purely data‑driven baselines become unstable.

Abstract:
Physics‑informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter‑equation coupling strengthens. We provide a theoretical explanation for this phenomenon through neural tangent kernel (NTK) analysis: for linearly coupled systems, we prove that the standard NTK's spectral radius grows as Ω(γ^2) with coupling strength γ, shrinking the stable learning rate, while block‑diagonal Gauss‑‑Newton (GN) preconditioning yields a preconditioned NTK K_P = J H^+ J^\top (where H is the block‑diagonal GN Hessian) whose spectral radius is bounded by S (S = number of networks), independent of γ. We verify the Ω(γ^2) growth numerically across symmetric, asymmetric, and nonlinear coupled PDE systems, and confirm λ_\max(K_P) = S with equality in all cases. Combining the Kronecker‑preconditioned optimizer SOAP with inverse‑gradient‑norm loss balancing (SOAP+GN) yields coupling‑robust accuracy: across 234 experiments spanning three 1D systems of increasing nonlinearity and a 2D electroosmotic flow benchmark, SOAP+GN maintains final‑epoch L_2 degradation \leq 1.1× (ratio of strong‑ to weak‑coupling error) even as coupling parameters vary over one to two orders of magnitude, compared with > 10^2× for Adam+GN. SOAP+GN further scales to a 2D, 6‑PDE electroosmotic flow system at EDL‑resolved conditions ‑‑ a regime that all prior PINN electrokinetics studies have avoided through simplified physics ‑‑ where Adam+GN fails entirely (L_2 > 0.9).

Abstract:
Autonomous aerial vehicles necessitate control strategies that balance computational efficiency with robust performance in dynamic operational environments. This paper proposes a model predictive control (MPC) framework for aerial platforms that leverages physics‑informed machine learning (PIML) to achieve an optimal balance between computational tractability and robust performance. At the core of the proposed approach lies a sparse, control‑affine model identified via the PIML method, which provides a parsimonious yet interpretable representation of the system dynamics by embedding first‑principles knowledge and learning residual uncertainties from operational data. This model is incorporated within a robust MPC scheme that adopts a high‑order Runge‑Kutta discretization to ensure prediction accuracy and an adaptive tube‑based mechanism to guarantee constraint satisfaction under uncertainty. The online adaptation of the tube, directly informed by the residual error of the PIML model, ensures robust stability without introducing excessive conservatism. Rigorous theoretical proofs are provided to establish recursive feasibility and stability. Numerical simulations and experiments on a quadrotor demonstrate that our method significantly reduces computational load compared to nonlinear MPC and robust MPC using a high‑fidelity model, while outperforming PID, nonlinear MPC, neural‑network‑based MPC, and fixed‑tube robust MPC in tracking performance and robustness, showcasing the practical efficiency of the proposed PIML‑based control synthesis for resource‑constrained aerial systems.

Abstract:
Forces govern how fluids deform biological tissues, regulate cardiovascular function, and determine the performance and failure of soft materials. Recent advances in flow birefringence, including the use of suspended anisotropic nanomaterials to optically encode stress in fluids, have made direct stress measurement experimentally accessible in projection. However, direct experimental access to all six components of the three‑dimensional (3D) fluid stress tensor has remained unattainable because optical measurements provide only path‑integrated observables. Recovering local 3D stresses from such data constitutes an intrinsically underdetermined tensor tomography problem, where two optical observables must determine six independent stress components. Here we introduce U‑FlowPET, an unsupervised physics‑informed framework that integrates photoelastic tomography with the governing equations of fluid mechanics to reconstruct the full 3D stress tensor without relying on constitutive assumptions, geometric symmetry, or labeled training data. Rather than learning from labeled reference stress fields, the method identifies physically admissible stress fields that satisfy momentum balance and continuity while remaining consistent with measured optical projections. We validate the approach using analytical, numerical, and experimental datasets. In axisymmetric pipe flow with an analytical solution, all six stress components are reconstructed with normalized mean absolute errors below 4%. Robust reconstruction is further demonstrated in curved‑pipe flow without symmetry assumptions and in experimental pipe‑flow data despite measurement noise. By enabling direct 3D stress‑field reconstruction from optical data alone, U‑FlowPET extends fluid analysis from observing motion to quantifying force and establishes a new framework for stress‑based diagnostics in biological flows and functional materials.

Abstract:
Active traffic management (ATM) is frequently hindered by traditional macroscopic models and rigid empirical thresholds that fail to capture metastable phase precursors, resulting in delayed, reactive interventions. To address this, we propose SpinFlow, a physics‑informed spin‑field framework unifying Kerner's three‑phase theory with statistical physics for continuous macroscopic traffic phase inference. Inspired by the Heisenberg model, SpinFlow parametrizes spatially varying phase weights via a latent spin vector and a competitive‑equilibrium mapping, allowing synchronized flow to emerge naturally. A physics‑regularized Expectation‑Maximization algorithm inverts this latent structure from high‑resolution trajectories, jointly optimizing the spin field while softly enforcing mass conservation and spatial smoothness. We introduce the Phase Equilibrium Degree (PED) to quantify structural alignment and topologically localize phase‑transition points. Across four real‑world trajectory datasets, SpinFlow achieves R_q^2 up to 0.940, PED drops of 94.9‑100%, and interpretable phase maps that outperform three heterogeneous baselines on forward accuracy, physics consistency, and bottleneck localization. SpinFlow pinpoints congestion nucleation without prior network topology, yielding a data‑driven, physics‑consistent trigger for ATM.

Abstract:
The discovery of stable, lead‑free halide perovskites for optoelectronic applications is constrained by vast compositional space and limited interpretability of conventional screening approaches. We present a genome‑guided, physics‑informed framework that decodes thermodynamic stability and optoelectronic behavior through four physically interpretable descriptor families: packing, bonding, polarization, and electronic identity. Trained on 1,221 DFT‑calculated A2BB'X6 compounds, machine‑learning surrogates achieve robust predictive performance, with a recall‑optimized stability classifier (ROC‑AUC = 0.92) and an XGBoost regressor for band‑gap prediction (R2 = 0.93 on held‑out data). Applying a staged inverse‑design constraint stack to 13,088 charge‑balanced, lead‑free compositions reduces the search space to five DFT‑validated, phase‑stable semiconductors: Rb2SnMnBr6, Cs2CdSnBr6, Cs2CdSnI6, Cs2KGaI6, and Cs2AgAlBr6. These candidates lie on the convex hull (E_hull <= 0 meV/atom), preserve ordered double‑perovskite structures, and exhibit strong optical absorption (alpha peak ~1e5 cm^‑1). Genotype‑phenotype coupling analysis reveals a hierarchical control mechanism: packing genes define structural formability, bonding genes govern near‑edge optical transitions and conductivity, and optoelectronic response genes regulate dielectric response and exciton screening (epsilon0 = 4.6‑8.2). This work establishes a generalizable paradigm for interpretable inverse design, linking descriptor‑level genomics to experimentally relevant optoelectronic phenotypes and providing design rules for discovering stable, lead‑free double perovskites for photovoltaics, sensing, and transparent electronic applications.

Abstract:
Reconstructing continuous physical fields from sparse measurements is a central inverse problem, but data‑driven generative models can produce states that violate governing dynamics. We introduce a physics‑informed generative solver that separates stable prior learning from inference‑time enforcement of conservation laws. Martingale‑Regularized Score Matching regularizes score pretraining with a Score Fokker‑Planck constraint, yielding a dynamically stable prior. Physics‑Informed Implicit Score Sampling then guides denoising trajectories by gradients of physical residuals, projecting samples toward admissible manifolds without retraining. In acoustics, the method co‑generates pressure and particle velocity from sparse sensors, enabling dense virtual arrays that suppress spatial aliasing. The same framework generalizes to real‑world ERA5 meteorological fields under extreme sparsity. Together, this work establishes a rigorous and generalizable paradigm for solving high‑dimensional inverse problems, bridging the gap between generative artificial intelligence and first‑principles science.

Abstract:
Complex dynamical systems governed by holomorphic maps such as z^2 + c exhibit fractal boundaries with extreme sensitivity to initial conditions. Accurately modelling these structures from data requires methods that respect the underlying complex‑analytic geometry, yet Multi‑Layer Perceptrons (MLPs) within Neural Ordinary Differential Equations (Neural ODEs) lack complex‑analytic priors, violate the Cauchy‑‑Riemann conditions, and function as opaque approximators incapable of yielding governing equations. We introduce Holomorphic KAN‑ODE, a framework that replaces the MLP with a Kolmogorov‑Arnold Network (KAN) whose learnable B‑spline activations reside on network edges, and incorporates Cauchy‑‑Riemann equations as a differentiable regularization to preserve holomorphic structure. We evaluate on six families of complex dynamical systems spanning polynomial and transcendental classes. With only 280 parameters (16× fewer than the MLP baseline), the network achieves velocity‑field R^2 > 0.95 on all six systems, correctly identifies all six governing symbolic families through automatic spline‑to‑formula fitting, and reconstructs Julia set fractal boundaries with up to 98.0% agreement. Crucially, the model exhibits only 4% MSE degradation under 10% observation noise versus 15.2× for MLPs, and achieves 90.4% improvement in transfer learning from quadratic to cubic dynamics. While the MLP attains lower pointwise reconstruction error due to its larger capacity, the KAN uniquely provides interpretable symbolic equations, enforced holomorphic structure, and superior noise resilience, capabilities that are entirely absent in black‑box architectures. These results establish KANs as a parameter‑efficient, interpretable alternative to MLPs for physics‑informed discovery of holomorphic dynamics.

Abstract:
We present a four‑dimensional equation of state for strongly interacting matter at finite temperature and conserved charge densities, constructed using a deep neural network. It is designed for direct use in hybrid models of relativistic heavy‑ion collisions: it reproduces hadron resonance gas thermodynamics at typical particlization scales, is consistent with lattice QCD at low baryon chemical potential, and extrapolates into the high‑density region inaccessible to either approach, which is precisely the regime targeted by RHIC BES, FAIR, HADES, and CBM. Thermodynamic consistency throughout the full phase space is enforced via a physics‑informed loss function. We demonstrate the developed equation of state by implementing it at zero net strangeness and fixed electric‑to‑baryon charge ratio within the integrated hydrokinetic model.

Abstract:
The Monge‑Ampère equation is a fundamental fully nonlinear elliptic partial differential equation that finds extensive applications across multiple disciplines. This study proposes a novel physics‑informed neural network integrated with attention feature expansion (PINN‑AFE) for its numerical solution. A multi‑head attention enhanced feature pool is constructed to enable adaptive nonlinear feature representation, and input convex neural networks are adopted to impose strict convexity of solutions with rigorous theoretical guarantees. Meanwhile, a dynamically weighted loss function combined with hybrid optimization is formulated to accelerate training convergence. Comprehensive numerical experiments validate the accuracy and computational efficiency of the developed framework. The PINN‑AFE paradigm is further extended to image processing tasks, delivering high‑quality and physically consistent results in both image enhancement and medical image registration scenarios.

Abstract:
Accurate modeling of aerodynamic loads is essential for understanding and predicting the responses of complex structural systems. However, these models often rely on simplifications of the true physical forces, introducing assumptions that can limit their accuracy. Validating such models becomes particularly challenging in the presence of noisy or incomplete data. To address this, we introduce a probabilistic physics‑informed machine learning approach designed to reconstruct the underlying aerodynamic loads from noisy measurements of structural dynamic responses. The model avoids overfitting, eliminates the need for regularization schemes, and allows for the use of heterogeneous and multi‑fidelity data during the training process. The efficacy of the approach is demonstrated through the reconstruction of aerodynamic loads on the Great Belt East Bridge, simulated under a linear unsteady assumption. Results show a strong agreement between true and predicted loads, particularly related to root mean squared errors, magnitude, phase angle and peak values of the signals. The method for load reconstructing holds broad applicability, such as modeling validation, future load estimation, and structural damage prognosis.

Abstract:
Computational imaging enables compact infrared systems, but deep‑learning pipelines that combine image reconstruction and object detection often introduce substantial inference latency. Most existing acceleration strategies compress the reconstruction network while overlooking physical priors from the optical path, leaving a trade‑off between accuracy and speed. We present Physics‑aware Dual‑Integrated Network (PDI‑Net), a low‑latency framework that integrates infrared reconstruction with object detection and further embeds optical priors into the learning process. PDI‑Net uses a supervised U‑Net during training, while a semi‑U‑Net encoder shares features directly with a YOLO‑based detector during inference, avoiding full image reconstruction. To bridge the gap between fidelity‑oriented reconstruction features and detection‑oriented semantics, we introduce a physics‑aware large‑small bridge (PALS‑Bridge), which uses field‑dependent point spread function priors to adaptively modulate multiscale convolutional branches. A physics‑informed optical degradation simulation pipeline is also developed for training and validation. The method is deployed on a single‑lens infrared camera, reducing system weight by about 50% compared with traditional multi‑lens designs. On the M3FD benchmark under low‑SNR conditions, PDI‑Net reduces inference time by 84.06% compared with the Rec+Det with pruning strategy while improving mAP@0.5:0.95 by 5.07%. These results demonstrate compact, low‑latency computational infrared imaging for real‑time object detection on resource‑constrained platforms.

Abstract:
The integration of machine learning with domain‑specific physics is transforming the design, monitoring, and control of electricity systems, where data scarcity, limited interpretability, and the need to enforce physical laws constrain purely data‑driven models. Physics‑informed machine learning (PIML) addresses these limitations by embedding governing equations directly into the learning process, yielding accurate, efficient, and scalable solutions for Industry 4.0 applications. This article reviews hybrid PIML architectures for electricity systems, including physics‑informed neural networks (PINNs), Deep Operator Networks (DeepONets), Fourier Neural Operators, Extreme Learning Machine‑enhanced PINNs, graph‑based PINNs (PIGNNs), and domain‑decomposition PINNs. Each approach is examined through case studies spanning field analysis, fault detection, digital twins, surrogate modeling, and control optimization. The review shows that embedding Maxwell's equations and other first‑principles constraints substantially improves predictive accuracy under sparse and noisy data, reduces simulation time by orders of magnitude relative to finite element methods, and enhances generalization across operating regimes. Hybrid frameworks consistently outperform purely data‑driven baselines on parameter sensitivity, dynamic behavior, and robustness, while supporting real‑time digital‑twin calibration and uncertainty quantification. Persistent challenges include training instability for stiff multi‑scale problems, computational cost of high‑fidelity models, and the absence of standardized benchmarks. The findings demonstrate that PIML enables a paradigm shift from black‑box data‑driven methods to transparent, physics‑informed strategies, positioning the field for sustained innovation in resilient and intelligent electricity systems.

Abstract:
Frequency crowding is a fundamental limitation in superconducting quantum architectures, particularly in tunable‑coupler systems. We present a framework that explicitly models both coherent spectator‑induced errors and incoherent lifetime effects through an error budgeting approach. Using this model, we analyze how frequency crowding impacts gate fidelity as module size and connectivity scale, and formulate a constrained optimization problem to assign qubit and coupler frequencies under realistic separation and hardware constraints. We demonstrate scalable frequency allocation strategies that minimize spectator‑induced errors. We further show that increasing qubit count and coupling density within a module leads to a fidelity‑connectivity tradeoff. To explore the benefits at the system scale, we have developed a noise‑aware transpilation approach called FINESSE, which minimizes error by selecting high‑fidelity paths that satisfy connectivity via SWAP insertion while jointly optimizing downstream gate execution. We demonstrate this physics‑informed architecture‑transpilation co‑design approach for a SNAIL‑based third‑order coupler that natively realizes the \sqrtiSWAP basis with frequency aware gate fidelities. On SNAIL architectures, FINESSE achieves an average 8.9% reduction in log‑infidelity cost and 6.8% reduction in circuit depth vs. SABRE. We also compare results on IBM Brisbane's architecture.

Abstract:
Solving partial differential equations with neural operators significantly reduces computational costs but remains bottlenecked by high training data requirements. Active learning offers a natural framework to mitigate this by selectively acquiring the most informative samples in an iterative manner. We introduce physics‑based acquisition ‑ a novel physics‑informed active learning algorithm that leverages the partial differential equation residual to guide data selection. We validate the method by presenting numerical experiments for the 1D Burgers equation and the 2D compressible Navier‑Stokes equations. We show that, in our experiments, physics‑based acquisition consistently outperforms random acquisition and matches the state of the art in data efficiency. At the same time, it has the unique advantage of injecting a physics inductive bias into the training process, ensuring that simulation cost is spent where the model's physical understanding is weakest.

Abstract:
Artificial intelligence (AI) empowers innovative diagnostic tools for common diseases, yet its clinical application in skeletal health evaluation is constrained by unsatisfactory accuracy, owing to the inherent porous and poroelastic biophysical features of bone. To address such bottlenecks amid global population aging, this study targets skeletal health and develops a reliable AI framework for precise bone microstructural characterization. We proposed Biot‑PINN, a physics‑informed neural network embedded with Biot's poroelasticity theory to characterize mechanical responses and wave propagation in poroelastic bone tissues. By decoding photoacoustic signals encoding bone mineral and microstructural features, the framework enables automatic bone microstructural grading. Experimental results reveal that Biot‑PINN reaches an accuracy of 97%, markedly surpassing traditional data‑driven approaches and providing a robust solution for early skeletal health diagnosis.

Abstract:
Accurate classification of breast ultrasound images into benign, malignant, and normal categories is a critical clinical task complicated by speckle noise, acoustic shadowing, and inter‑class visual ambiguity. Existing deep learning methods rely on single‑stream architectures with generic augmentation that ignores ultrasound acquisition physics, and no prior method dedicates a stream to the lesion boundary features identified as the most diagnostically significant visual cue. We propose HADS‑Net, a Hybrid Attention‑Augmented Dual‑Stream Network exploiting global texture and local boundary cues through two parallel pathways. Stream 1 applies physics‑informed augmentation simulating speckle noise, acoustic shadowing, and gain variation before extracting features via pretrained EfficientNet‑B3 projected to 512 dimensions. Stream 2 extracts Sobel edge maps processed by a lightweight CNN projected to the same 512‑dimensional space. A cross‑attention fusion module allows the texture stream to selectively query boundary features, producing a jointly optimised representation classified by an MLP trained with adaptive class‑weighted focal loss. Five‑fold stratified cross‑validation with cosine annealing over 50 epochs is used, with the globally best checkpoint selected by lowest validation loss evaluated on a held‑out test set. On the BUSI dataset, HADS‑Net achieves 96.58% accuracy, macro ROC‑AUC of 0.9978, macro F1 of 0.9654, and per‑class F1‑scores of 0.970, 0.951, and 0.976 for benign, malignant, and normal. No malignant lesion is misclassified as normal. These results confirm that modality‑specific augmentation with cross‑modal attention fusion is an effective strategy for ultrasound‑based breast cancer diagnosis.

Abstract:
Interpolation and smoothing using cubic and generalized splines are fundamental tools in data analysis and statistical modeling. Recently, fast computational algorithms were developed for natural L‑splines of order four, which arise as piecewise solutions to the differential operator L_ξ^2 = (\fracd^2dt^2 ‑ ξ^2)^2. In this paper, we extend this mathematical framework to the important case of clamped (or complete) boundary conditions, where the first derivatives at the interval endpoints are prescribed. We explicitly construct the governing linear system for the interpolation problem and mathematically prove that the resulting tridiagonal matrix is strictly row diagonally dominant, thereby guaranteeing its invertibility and the numerical stability of the fast algorithm. The proposed method is implemented in MATLAB. Furthermore, the developed clamped L‑splines provide a foundation for constructing multivariate clamped polysplines, which serve as a promising alternative to Physics‑Informed Neural Networks (PINNs) for solving partial differential equations in Mathematical Physics.

Abstract:
Accurate simulation of fluid flow in porous media is challenging due to complex pore‑space geometries and the computational cost of solving the Navier‑Stokes equations. This difficulty is particularly important when repeated simulations are required, as standard numerical solvers may converge slowly in intricate porous domains. We present a neural‑network‑based framework for predicting pore‑scale velocity fields directly from sample geometry. The method uses a convolutional encoder‑decoder architecture with skip connections to preserve spatial detail while extracting multi‑scale features. Physical consistency is encouraged through a custom loss function combining velocity reconstruction with incompressibility, no‑flow conditions inside solids, periodicity constraints, and agreement with the global tortuosity index. We analyze the influence of the corresponding loss weights and quantify the contribution of individual loss components to prediction accuracy. Several CNN backbones are evaluated to identify architectures providing accurate and robust predictions. The generalization ability of the trained model is tested on samples outside the training distribution, including changes in obstacle geometry, boundary conditions, porosity, and realistic porous structures. Finally, we demonstrate a practical use of the predicted velocity fields as initial conditions for Lattice‑Boltzmann simulations. This warm‑start strategy accelerates solver convergence, reducing the number of iterations in over 90% of tested cases.

Abstract:
Training very deep neural networks requires controlling the propagation of magnitudes across depth. Without such control, activations and gradients may vanish, explode, or enter unstable regimes that make optimization fail. Modern architectures often mitigate this problem through Batch Normalization, residual connections, or other normalization layers, which repeatedly re‑scale or bypass intermediate representations. However, these mechanisms are not always appropriate. In Physics‑Informed Neural Networks (PINNs), the network represents a continuous physical field and its input derivatives define the training objective, making batch‑dependent normalization problematic because it can introduce non‑local dependencies into the predicted field and its derivatives. We propose StableGrad, an optimizer‑level scale‑control mechanism that corrects layer‑wise weight‑gradient imbalances without modifying the forward model. Because the normalization is applied only after backpropagation and before the optimizer update, the network output, its derivatives, and the physical residual remain unchanged. We analyze the effective training dynamics induced by this rescaling and evaluate StableGrad on deep PINNs as the target application, with BatchNorm‑free convolutional networks serving as a diagnostic stress test. On PINN benchmarks, StableGrad improves matched‑depth solution accuracy and makes deeper models more reliable under standard optimization. On ResNet and EfficientNet architectures, where removing Batch Normalization normally leads to training collapse, StableGrad stabilizes optimization without introducing any other architectural change. These results show that optimizer‑level control of weight‑gradient scale can provide a practical alternative when forward normalization is unavailable or undesirable.

Abstract:
Synthetic sonar datasets offer a scalable alternative to costly real‑world acquisition, yet their utility remains limited by the absence of rigorous quantitative validation. We present ACOUSIM (ACOustic SIMulation and Validation Platform), a physics‑informed framework that evaluates the statistical alignment between synthetic and real sonar imagery without relying on generative models. A Gazebo‑based environment generates sonar‑like images by explicitly controlling seabed texture, illumination‑driven shadowing, platform altitude, and noise. Realism is quantified against two public sonar datasets, SeabedObjects‑KLSG‑II and Sonar Common Target Detection (SCTD), using global intensity and local texture (LBP) distributions assessed via Kullback‑Leibler divergence, Jensen‑Shannon divergence, and Earth Mover's Distance. Results show strong texture alignment (KL < 0.07) across all classes, with plane‑class intensity alignment outperforming ship‑class due to shadow geometry complexity. ACOUSIM establishes a reproducible, distribution‑level baseline for sim‑to‑real sonar evaluation and directly supports reliable dataset validation for underwater image analysis.

Abstract:
Dental aerosol procedures produce sub‑50 micrometre nuclei that can remain airborne for long periods in enclosed clinics, creating pathways for airborne pathogen transmission. Reynolds‑Averaged Navier‑Stokes (RANS) simulations with Euler‑Lagrange particle tracking capture this transport accurately but require very long run times per scenario, which precludes real‑time clinical decision support in 3D. We present the Eulerian‑Lagrangian Graph Interaction Network (ELGIN), a physics‑informed graph surrogate that jointly predicts carrier‑flow dynamics on the OpenFOAM polyhedral mesh and the per‑parcel motion of the polydisperse spray cloud. ELGIN couples a multi‑head Graph Transformer with Jacobi‑preconditioned learnable pressure projection and a turbulence‑closure head to a sigmoid‑gated Lagrangian Interaction Network through differentiable inverse‑distance mesh‑parcel coupling, and advances parcels with a symplectic Stormer‑Verlet integrator. A four‑stage physics‑informed curriculum stabilises 260‑step autoregressive rollouts without gradient explosion. A parameter sweep with foam‑extend 4.1 OpenFOAM reactingParcelFoam across clinically relevant ventilation rates and handpiece spray speeds provides CFD ground truth. This article reports a single‑case demonstration in which both ELGIN and a Lagrangian‑only baseline (M0) are trained and evaluated on Sweep_Case_03 of a twenty‑case sweep; full 16/2/2 retraining is in progress and will replace all reported metrics. On this case, ELGIN tracks the foam‑extend particle cloud much more closely than M0: mean parcel displacement error falls from 19.56% to 16.20% of room width and cloud radius‑of‑gyration error from 9.85% to 6.58%. A 26‑second rollout completes in ~64 s on a 4 GB GPU, approximately 37x faster than the foam‑extend reference pipeline, toward per‑appointment infection‑risk screening once the multi‑case checkpoint is in place.

Abstract:
The rise of Physics‑Informed Neural Networks (PINNs) has popularized the concept of solving differential equations via residual minimization. However, neural networks are often viewed as ``black boxes" requiring significant computational overhead and stochastic optimization. Moreover, PINNs typically treat boundary conditions (BCs) as ``soft constraints" within the loss function and this makes the optimization process struggling to enforce the BCs properly. This paper introduces the Spline‑based Physics‑Informed Numerical Scheme (SPINS), a numerical framework designed to solve both initial and boundary value problems of ordinary differential equations (ODEs). By replacing the neural network architecture of traditional PINNs with a structured spline basis, SPINS achieves high accuracy and interpretability with a minimal parameter set. In addition, the BCs are automatically satisfied from the choice of the splines architecture. Therefore, SPINS provides smooth numerical solutions for ODEs allowing analytical differentiation. Moreover, SPINS benefits from the automatic differentiation where computing the gradient of the physics‑informed loss function is an easy task making the optimization process very fast using gradient‑based optimizers such as the L‑BFGS‑B algorithm. We demonstrate the efficacy of SPINS on nonlinear second order ODEs with several choices of BCs using cubic and quintic interpolating splines and present its natural extension to high order ODEs.

Abstract:
In continuum topology optimization (TO), two essential procedures are involved: structural analysis through the solution of partial differential equations (PDEs) and the subsequent update of design variables. Both procedures can be addressed by training neural networks using the corresponding physical information. Accordingly, Physics‑Informed Neural Network (PINN)‑based algorithms have been developed for TO. However, PINN‑based methods suffer from several notable limitations, including high computational cost, spectral bias, and limited adaptability in solving PDEs.To overcome these challenges, this study proposes a novel algorithm that incorporates two Higher‑Order ReLU‑based Kolmogorov‑Arnold Networks (HRKANs). Specifically, a displacement‑informed HRKAN (d‑HRKAN) is designed to predict PDE solutions, while a sensitivity‑informed HRKAN (s‑HRKAN) is developed to perform sensitivity analysis for updating design variables. For convenience, the proposed approach is referred to as the Dual Physics‑Informed Kolmogorov‑Arnold Networks‑based Topology Optimization (DPIKAN‑TO) method. By leveraging learnable activation functions, the proposed neural networks can accurately approximate the responses of complex structural systems. Moreover, compared with conventional PINN‑based methods, DPIKAN‑TO demonstrates significantly improved computational efficiency and reduced computational cost. Numerical examples show that DPIKAN‑TO can successfully identify optimal material layouts for linear structures, compliant mechanisms, and fluid‑solid coupled systems. Furthermore, owing to the use of learnable activation functions, the proposed framework can be readily extended to structural optimization problems governed by new types of PDEs.

Abstract:
Physics‑informed neural networks (PINNs) offer a mesh‑free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum‑Guided Gaussian Mixture Physics‑Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision‑based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self‑adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time‑varying loss, uniform equivalence between the curriculum‑weighted and standard PDE losses, and a generalization bound with an explicit weighting‑induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection‑dominated, and nonlinear reaction‑diffusion types show that CGMPINN consistently achieves the lowest relative L_2 and maximum absolute errors among all compared methods, reducing relative L_2 error by up to 97.8% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics‑Yang/CGMPINN.

Abstract:
Magnetohydrodynamics (MHD) couples the Navier‑‑Stokes and Maxwell equations into a nonlinear system of partial differential equations governing stellar interiors, astrophysical jets, fusion plasmas, and space weather. Numerical advances, including finite‑volume Godunov schemes, constrained‑transport algorithms, high‑order spectral‑element and discontinuous‑Galerkin discretisations, and adaptive mesh refinement, have made MHD a predictive tool for solar eruptions, tokamak confinement, and magnetised turbulence. A fundamental barrier nevertheless remains. In three‑dimensional MHD turbulence, the degrees of freedom required to resolve all active scales grow as \mathcalO(\mathrmRe^9/4) or faster, where \mathrmRe is the Reynolds number. Direct numerical simulation is therefore intractable at astrophysical and fusion‑relevant parameters, particularly when the Lundquist number S exceeds 10^10 and both viscous and resistive dissipation ranges must be resolved. Kinetic closures, radiation transport, and uncertainty quantification further increase the cost. This chapter examines how AI may help bridge this gap. We review physics‑informed neural networks, Fourier neural operators and physics‑informed neural operators, which learn solution operators across families of MHD problems; and hybrid operator‑diffusion frameworks that combine deterministic surrogates with score‑based generative models to recover broadband turbulent spectra. These developments are set within the wider landscape of exascale high‑order solvers, GPU acceleration, task‑based parallelism, data‑driven sub‑grid closures, and prospective quantum algorithms for implicit linear systems in resistive MHD. The central claim is that physics‑informed AI, integrated with conventional solvers and trained on leadership‑scale simulations, offers a credible route to regimes beyond the reach of classical discretisation alone.

Abstract:
Training and evaluating false data injection attack (FDIA) detectors for power systems is constrained by data scarcity. Operational grid measurements are commercially sensitive, and hand‑crafted attacks fail to capture complex distributional structures imposed by network physics. We present \textscGenAI‑FDIA, a framework benchmarking a pool of P=20 architectures for physics‑compliant FDIA synthesis, spanning Wasserstein GANs, MMD‑VAEs, normalising flows, diffusion models, and cross‑family hybrids. These are evaluated across three IEEE testbeds (14‑bus DC, 30‑bus DC, and 14‑bus AC) under a 60/20/20 chronological split using data‑driven Bad Data Detection (BDD) threshold calibration. Our empirical results verify that these models generate high‑fidelity attacks, with all architectures achieving evasion rates of ε_\textBDD \ge 86.6% on the 14‑bus network; additionally, limiting an attacker's topological knowledge induces a measurable degradation in stealthiness (p \le 0.0022). Crucially, we identify a previously unreported failure mode: applying affine physics projections directly in normalised feature spaces critically displaces the attack vector, collapsing BDD evasion from ～55% to <\!2% on the 30‑bus testbed. We resolve this via a novel inference‑time harmoniser, restoring full stealthiness (ε_\textBDD=100%) across all physics‑informed variants without retraining. Finally, we isolate a covariance‑collapse phenomenon (κ\approx ‑0.076) within advanced hybrid architectures and rectify it through 50‑epoch warm‑up schedules (κ\to 0.785, Δ\textMMD=‑3.1%). Ultimately, \textscGenAI‑FDIA delivers a robust recovery blueprint applicable to any physics‑constrained generative model deployed for power‑system security.

Abstract:
Robotic assembly in architectural construction faces a persistent bottleneck: existing planners are either highly specialized, requiring prohibitive retraining for every new geometric design, or operationally inefficient, treating structural sequencing and kinematic motion as disjoint processes. We present EUPHORIA, a unified framework that achieves universal few‑shot adaptability and dynamic efficiency through a hybrid optimization strategy. To overcome the retraining bottleneck, we propose a Meta‑Geometric Encoder based on Graph Hypernetworks: unlike standard contrastive learning, which performs only feature‑level recognition, our hypernetwork dynamically generates policy parameters from a minimal support set, enabling parameter‑level adaptation to complex topologies (e.g., domes, arches) without gradient‑based retraining. For structural reasoning, we introduce a Physics‑Informed Graph Transformer trained via Soft Actor‑Critic (SAC), with a Physics‑Bias Attention mechanism that modulates attention scores using contact forces from Discrete Element Model (DEM) simulations, guiding the planner toward structurally critical connections. We further ensure operational efficiency through Kinematics‑Aware Sequencing, where the SAC objective penalizes high‑energy transitions. Finally, we bridge the Sim2Real gap via Residual Stability Correction, a differentiable optimization layer that fine‑tunes coarse assembly actions by minimizing a joint energy‑stability cost prior to execution. Experiments show that EUPHORIA significantly reduces energy consumption over decoupled baselines and achieves state‑of‑the‑art success rates on unseen, non‑standard geometries with minimal few‑shot examples, fusing meta‑learning, physics‑informed attention, and residual optimization into a cohesive, generalized planner.

Abstract:
Constrained machine learning enables fairness‑aware training, physics‑informed neural networks, and integration of symbolic domain knowledge into statistical models. Despite its practical importance, no general method exists for the non‑convex, non‑smooth, stochastic setting that arises naturally in deep learning. We propose the Stochastic Penalty‑Barrier Method (SPBM), which extends classical penalty and barrier methods to this setting via exponential dual averaging, a stabilized penalty schedule, and the Moreau envelope to handle non‑smoothness. Experiments across multiple settings show that SPBM matches or outperforms existing constrained optimization baselines while incurring only linear runtime overhead compared to unconstrained Adam for up to 10,000 constraints.

Abstract:
Distributed Acoustic Sensing (DAS) converts existing fibre‑optic cables into dense seismic arrays at near‑zero deployment cost, but measures strain rate rather than particle velocity ‑‑ the quantity required by virtually all seismological analysis tools. Converting strain rate to particle velocity by numerical integration is ill‑posed: the integration constant is undefined and noise accumulates without bound. We present DANTE (DAS‑to‑velocity via physics‑informed neural operator for Acoustic‑wave recoNstruction in heTErogeneous media), a Fourier Neural Operator (FNO) trained entirely on synthetic data that enforces two physics constraints: (i) the exact kinematic relation between DAS strain rate and the spatial gradient of particle velocity, and (ii) the one‑dimensional elastic wave equation. These constraints resolve the undetermined integration constant and suppress noise without requiring co‑located seismometers. On a test set of 200 heterogeneous synthetic wavefields, DANTE achieves a mean output SNR of 15.3 \pm 8.8 dB, Pearson correlation r = 0.907, and SSIM = 0.976, corresponding to a mean SNR improvement of approximately +15 dB over the best conventional baseline (trace stacking, n = 10, 0.02 \pm 0.06 dB), and up to +28.8 dB on the most challenging samples. Zero‑shot inference on seven real microseismic events from the Utah FORGE 2019 DAS dataset yields a kinematic residual of 0.003‑‑0.005, five times lower than the synthetic test baseline, confirming generalisation to real field data with no fine‑tuning and no seismometers.

Abstract:
Presently, there is a steady state approach in Computational fluid dynamics (CFD) to obtain a steady solution directly from the steady state governing equations. Whereas, for obtaining a time‑periodic flow solution, the present unsteady governing equations‑based CFD approach starts from an initial condition and requires a large computational time during the initial non‑periodic transient phase before reaching the periodic state. For obtaining the periodic flow directly, without transient simulations that may not be of interest, our objective is to propose a Physics Informed Neural Network (PINN)‑based periodic CFD approach. The motivation is a substantial reduction in computational time by a meshless PINN‑based periodic CFD solver as compared to the present mesh‑based transient‑to‑periodic solver. Proof‑of‑concept, for the periodic CFD approach, is demonstrated here for 2D periodic heat diffusion and fluid flow problems. The proposed PINN‑based periodic solver primarily focuses on the time‑periodic state, optimizing the neural network model's trainable parameters to precisely fit a smaller time window (one time‑period) rather than the temporal domain starting from the initial condition. After presenting a verification study, effect of the PINN‑related various hyperparameters such as the number of collocation points, neural network architecture, and point spacing for numerical differentiation, on computational time and accuracy are presented. Our results demonstrate that the PINN‑based periodic solver takes substantially less computational time to achieve almost same accuracy as that obtained by the traditional transient‑to‑periodic solver.

Abstract:
Anomaly detection in batch processes is hindered by transient dynamics, scarce fault labels, and reliance on single‑modality sensor data. This work introduces UTOPYA (Unified Temporal Observation for Physics‑Informed Anomaly Detection and Time‑Series Prediction), a 15.2M‑parameter multimodal framework that jointly addresses anomaly detection, time‑series prediction, and phase classification in batch distillation by fusing eight data modalities through Feature‑wise Linear Modulation (FiLM) conditioned cross‑modal attention and gated fusion. A physics‑informed regularisation scheme introduced in this work enforces temporal smoothness and thermodynamic monotonicity, while curriculum learning introduces training samples in order of physical difficulty. On the 119‑experiment multimodal batch distillation dataset of Arweiler et al. (2026), UTOPYA achieves a window‑level test AUROC of 0.832 and 0.874 under multi‑signal experiment‑level scoring, substantially outperforming four external baselines (PCA, autoencoder, Isolation Forest, and LSTM autoencoder) evaluated under identical conditions (+0.147 window‑level AUROC over the best baseline). A multimodal ablation over 15~architectural configurations shows that static context via FiLM conditioning is the key enabler, lifting experiment‑level multi‑signal AUROC by +0.145 over the unimodal baseline (0.729 to 0.874). Separately, a training ablation across 14 design choices reveals that several widely‑adopted techniques, including instance normalisation, Mixup, ensembling, test‑time augmentation, and stochastic weight averaging, fail to improve or actively degrade generalisation in this data‑scarce setting. These negative results expose a fundamental tension between smoothing‑based regularisation and anomaly detection, providing practical guidance for multimodal process monitoring deployment.

Abstract:
The dominant paradigm in computational materials discovery relies on heavily parameterized deep architectures, including message‑passing graph networks and equivariant models, that require millions of DFT‑labeled training structures and produce non‑convex latent representations that complicate continuous optimization for inverse design. These architectures are impractical in data‑scarce regimes, which is the typical case in molecular screening, and exhibit well‑documented limitations in capturing chemically disordered configurations and chiral geometries. This review presents feature engineering based on spatial statistics as a physically rigorous and immediately deployable alternative. Molecular structures are encoded as voxelized scalar fields, and two‑point auto‑ and cross‑correlations are evaluated deterministically via Fast Fourier Transforms, explicitly transferring the burden of spatial pattern recognition from the learning algorithm to a closed‑form, physics‑informed operation. Principal component analysis of the resulting correlation maps yields low‑dimensional, strictly convex representations that support lean neural networks (<100k trainable parameters) and non‑parametric surrogate models, achieving sub‑2% prediction error with as few as 10 training samples. Demonstrated across periodic crystals, chemically disordered high‑entropy alloys, and non‑periodic organic molecules, this framework enables Bayesian active learning and zero‑shot extrapolation on commodity hardware, which current large‑scale architectures cannot replicate at equivalent data budgets.

Abstract:
Digital twins (DTs) rely on continuous synchronization between physical systems and their virtual counterparts through online parameter estimation under uncertainty. In many practical settings, however, this task is challenged by low observability, weak excitation, nonlinear dynamics, and noisy or biased measurements. In this work, we develop a new mathematical framework that integrates Weighted Flow Matching (WFM) generative modeling with physics‑informed nonlinear filtering to enhance parameter estimation in DTs. WFM relies on dynamic reweighting of training samples, which guides the generative model toward parameter regimes most informative of the evolving system state. This generative component is tightly coupled with a physics‑informed filtering architecture based on the Unscented Kalman Filter (UKF), yielding a unified DT framework that combines data‑driven probability transport with physically consistent state and parameter estimation. The effectiveness of the new integrated framework is demonstrated within a spacecraft DT architecture, where stable moment of inertia estimation is achieved under uncertain and noisy sensing, with significant performance improvements over established approaches such as Extended Kalman Filtering (EKF) and Ensemble Kalman Filtering (EnKF). These results highlight the potential of weighted generative modeling as a core mechanism for real‑time DT synchronization in operational and mission‑critical systems.

Abstract:
Solving inverse partial differential equation (PDE) problems is a fundamental topic in scientific research due to its broad significance across a wide range of real‑world applications. Inverse PDE problems arise across medical imaging, geophysics, materials science, and aerodynamics, where the goal is to infer hidden causes, design structures, or control physical states. In this paper, we provide a comprehensive review of recent advances in solving inverse PDE problems using artificial intelligence (AI). We first introduce the basic formulation, key challenges, and traditional numerical foundations of inverse PDE problems, and then organize it into three major categories: inverse problems, inverse design, and control problems. For each category, we further present a methodological paradigms, and review representative state‑of‑the‑art approaches from recent years. We then summarize representative applications across scientific and industrial domains, including mechanical systems, aerodynamic problems, thermal systems, full‑waveform inversion, system identification, and medical imaging. Finally, we discuss open challenges and future prospects, such as physics‑informed architectures, limited real‑world data, uncertainty quantification, and inverse foundation models. This survey aims to provide the first unified and systematic perspective on AI for inverse PDE problems, demonstrating how modern learning‑based methods are reshaping inverse problems, inverse design, and control problems in PDE‑governed systems.

Abstract:
Between the narrow systems we deploy and the general intelligence we speculate about lies an entire regime of machine behavior that has never received its own name. This monograph argues that this regime is not empty: it is where meta‑learning, neural architecture search, AutoML, continual learning, evolutionary computation, and physics‑informed modeling have quietly converged on a common principle, namely the steady removal of the human from the loop of parameter specification. We name this regime Artificial Adaptive Intelligence (AAI) and define it operationally: a system exhibits AAI to the extent that it requires no human‑specified tunable hyperparameters while maintaining competitive performance across a diverse distribution of tasks. To make the definition quantitative, we introduce an adaptivity index that measures progress along an axis orthogonal to scale, combining the fraction of hyperparameters absorbed by the system with the performance ratio against a task‑specialized baseline. We develop the principle of parametric minimality and ground it in the minimum description length framework, showing that the appropriate hyperparameter count is data‑determined rather than designer‑determined. We then organize the field around three pathways to minimality: data‑ and task‑aware configuration, structural and evolutionary morphing, and in‑training self‑adaptation. We analyze their stability, convergence, and governance implications, and illustrate them through case studies spanning aerospace design, financial regime detection, turbulence modeling, ecological dynamics, and vision‑language systems. The thesis is that the path from ANI to AGI passes through AAI, and that naming this stage changes what we measure, what we build, and what we call a success.

Abstract:
Continental‑scale knowledge of subsurface temperature is limited by the cost and sparsity of borehole measurements, but such information is essential for geothermal resource assessment and for understanding heat transport in the shallow crust. The thermal field reflects the interaction between lithology, crustal structure, radiogenic heat production, and advective fluid flow, sometimes producing sharp anomalies that are smoothed by conventional interpolation or difficult to capture with physical models. Here we introduce In‑Context Earth, a transformer‑based model that uses sparse local borehole observations as geological context to predict continuous temperature‑at‑depth fields with calibrated uncertainty. In the contiguous United States, the model achieves a mean absolute error of 4.7 °C, outperforming the physics‑informed Stanford Thermal Model, a model based on AlphaEarth embeddings, the multimodal Transparent Earth model, and universal kriging, while resolving sharper thermal gradients in geothermal provinces. Its uncertainty estimates are well calibrated, with a Kolmogorov‑Smirnov statistic of 2.5%. Without finetuning, the model adapts to Alberta, Australia, and the United Kingdom (UK) using only 20 local observations at inference time, maintaining high accuracy in geologically distinct test regions with a mean absolute error of 2.2 °C in Alberta, 6.2 °C in Australia, and 5.4 °C in the UK. Interpretability analyses show that the model learns internal representations of subsurface properties it never observes during training, including seismic velocities, geochemistry, and crustal structure, and uses these representations in physically consistent ways. More broadly, this work shows that in‑context learning can use sparse borehole observations for continental‑scale subsurface characterization, without requiring dense measurements or region‑specific retraining.

Abstract:
In this paper, we develop a physics‑informed deep operator learning framework for solving multi‑term time‑fractional mixed diffusion‑wave equations (TFMDWEs). We begin by deriving an L_2 approximation, which achieves first‑order accuracy for the Caputo fractional derivative of order β\in (1,2). Building upon this foundation, we propose the fPINN‑DeepONet framework, a novel approach that integrates operator learning with the L_2 approximation to efficiently solve fractional partial differential equations (FPDEs). Our framework is successfully applied to both fixed and variable fractional‑order PDEs, demonstrating the framework's versatility and broad applicability. To evaluate the performance of the proposed model, we conduct a series of numerical experiments that involve dynamically varying fractional orders in both space and time, as well as scenarios with noisy data. These results highlight the accuracy, robustness, and efficiency of the fPINN‑DeepONet framework.

Abstract:
Scientific foundation models are expected to reuse representations under changes in dataset, acquisition protocol, and deployment domain, yet many sequence backbones treat scientific temporal structure as an unconstrained pattern to be fitted. We argue that this misses a central property of natural dynamical systems: neural and atmospheric time series are organized by interacting processes across multiple physical timescales, and failure to preserve this multiscale structure contributes to brittleness under distribution shift. We formalize this failure mode as temporal kernel mismatch, where a model fits in‑distribution dynamics with an effective memory policy that is not anchored to the signal's physical timescales, leading to representation drift and degraded transfer. We propose Physics‑Informed Multi‑Scale Mamba (PIMSM), a state‑space architecture that maps spectrum‑estimated transition points between frequency regimes (knee frequencies) to scale‑specific discretization parameters and anchors them to acquisition time units. On Human Connectome Project fMRI, PIMSM improves robustness and representation stability under severe temporal‑context truncation, extreme low‑resource transfer, and resting‑state‑to‑task‑state generalization. Without modality‑specific adaptation, the same architecture also attains the lowest variable‑wise MAE across all reported horizons and variables on Weather‑5K held‑out‑station spatial out‑of‑distribution forecasting. These results support temporal‑scale alignment as a practical inductive bias for scientific foundation models that must preserve structure, not only fit correlations, under deployment shift.

Abstract:
Accurate seasonal runoff forecasts are critical for managing California's reservoirs and water supply for millions of its residents. Winter snow accumulation provides a strong source of predictability of snowmelt‑based runoff in the spring and summer months, but progressive hydroclimatic changes in the Sierra Nevada are altering its timing and volume. These changes reduce the skill of statistical forecasts trained on historical data, highlighting the need for improved forecasting systems that can capture the changing dynamics of snowmelt. Here we demonstrate that a collaborative workflow between an agentic AI assistant and an automated code‑mutation system, both powered by large language models, can accelerate the development of competitive seasonal runoff forecasting systems. In our framework, the AI agent discovers relevant datasets, synthesizes domain knowledge from prior forecasting competitions and the scientific literature, and explores the space of model architectures, while the code‑mutation system refines each of the solutions explored by the agent through Monte Carlo Tree Search over the code space. The resulting system forecasts monthly Full Natural Flow (FNF) at 1‑ to 6‑month lead times across 23 Sierra Nevada watersheds using an adaptive ensemble of three XGBoost quantile regression sub‑models with physics‑informed feature engineering. Evaluated against California's operational Bulletin 120 forecasts over 2021‑2025, the agent‑evolved model achieves superior skill for early‑season cumulative April‑July runoff predictions, reducing watershed‑averaged quantile forecast error by up to 29%, and offering a new paradigm for AI‑driven scientific model development in the geosciences.

Abstract:
Neural network surrogate models have emerged as a promising approach to model solution fields for a wide variety of boundary value problems encountered in physical modeling. Stochastic problems represent an area of particularly high interest because of the potential to significantly reduce the repeated evaluation of expensive forward models via traditional numerical solvers when conducting parametric analysis. However, many studies found in the literature primarily focus on the ability of neural network surrogate models to represent deterministic samples or mean field solutions and largely overlook surrogate model performance at the tails of the distribution. The present study examines in detail the ability of neural network surrogate models to capture the full distribution of solution fields over the entire probability space, while emphasis is placed at the tails of the distribution. Serving as a canonical problem is the heat conduction equation with a highly stochastic source term, inducing extremely large variation in the thermal solution field. Comparisons are made between a classic feed‑forward fully connected network and a Deep Operator Network architecture, using both data‑driven and physics‑informed loss functions. Results show that the worst‑case prediction errors are an order of magnitude larger than the mean field error, highlighting the importance of the outlier samples. The large errors associated with extreme samples result from the networks having to extrapolate beyond the bounds of the training data. A method for identifying these samples is presented along with a discussion of potential approaches to account of their errors. Among the models considered, the fully connected neural network trained using a weak form residual loss performs best in handling these extrapolated inputs, achieving the highest prediction accuracy for the numerically produced datasets.

Abstract:
Monte Carlo methods are essential across diverse scientific fields, yet their efficiency is frequently hampered by critical slowing down‑a sharp increase in autocorrelation times near phase transitions. Although deep learning approaches, such as neural‑network‑based samplers, have been proposed to alleviate this issue, they face another serious problem: the difficulty of training the models. This difficulty partially stems from the overly general nature of original machine‑learning architectures, which often ignore underlying physical symmetries and force networks to relearn them from scratch. In this paper, we demonstrate that incorporating physical priors into the model significantly enhances performance. Building upon existing strategies that integrate spin‑spin interactions, we propose a framework that utilizes a prior probability distribution as a starting point for training. Our results for the Ising model, as well as for the Edwards‑Anderson spin glass model, suggest that moving away from `blank slate' models in favor of physics‑informed priors reduces the training burden and facilitates the simulation of larger system sizes in discrete spin models.

Abstract:
Physics‑informed neural networks (PINNs) are powerful surrogates for differential equations but are notoriously difficult to train due to spectral bias, stiffness, and poor accuracy on high‑frequency or multiscale solutions. Adversarial training based on generative adversarial networks (GANs) has recently gained surprisingly strong empirical results in improving training, but the underlying mechanisms remain elusive. To this end, we propose a new analysis framework for adversarially trained PINNs, based on the key observation of how the discriminator in GANs can influence the training dynamics of PINNs. The framework first provides a much needed theoretical grounding to why and when adversarial training is effective in PINNs, then presents a unified analysis of GANs variants in such training, and finally leads to a new, practical, efficient training algorithm for PINNs. Empirical results demonstrate that our method can significantly reduce the pathology of PINNs training, thereby providing better models with superior performances, often several magnitudes more accurate than alternative methods.

Abstract:
Driven by rapid advances in artificial intelligence and modern GPU computing capabilities, deep learning methods based on the optimization paradigm have provided new pathways to solve spatiotemporal physical problems, whose mathematical core lies in solving partial differential equations (PDEs). As an emerging class of function‑space learning methods, neural operators (NOs) have exhibited great potential in efficient PDE solving. However, existing mainstream neural operator frameworks suffer from critical bottlenecks when modeling time‑dependent PDEs over long time horizons, including accuracy degradation, insufficient stability, high training costs, and excessive memory consumption, which severely limit their practical deployment. To address these challenges in long‑time prediction with neural operators, we propose a novel spatiotemporally decoupled physics‑informed neural operator architecture, termed the physics‑informed Stone‑Weierstrass neural operator (PI‑SWNO). The design is theoretically grounded in the decoupling paradigm combining time‑invariant spatial basis functions with time‑varying evolution coefficients, as well as the Stone‑Weierstrass approximation theorem. By encoding spatial and temporal information via two separate subnetworks, the framework structurally mitigates the accumulation of errors over extended time intervals. Furthermore, we introduce a time‑marching batch‑wise sampling strategy to resolve the memory bottleneck of full‑range modeling over extended time spans, ensuring continuity and convergence of full‑time‑domain solutions.

Abstract:
Electrochemical impedance spectroscopy (EIS) is the most informative diagnostic for lithium‑ion batteries: its frequency‑resolved spectra decompose cell behavior into distinct electrochemical processes, revealing mechanism‑specific degradation invisible to voltage and resistance measurements. Yet EIS requires dedicated hardware and minutes‑long acquisitions incompatible with field deployment. Here we show that Lie Generator Networks (LGN), a structure‑preserving identification framework, extract electrochemical time constants from 60 seconds of post‑pulse voltage relaxation, data that battery management systems already collect, that encode the same diagnostic and prognostic information as impedance spectra. LGN learns the generator matrix of the relaxation dynamics with stability guaranteed by architecture, yielding time constants precise enough to resolve electrochemical variation that conventional curve fitting cannot detect from identical data. Across five datasets totaling over 850 cells, four institutions, and multiple chemistries, LGN tracks degradation with near‑perfect rank correlation (|ρ_s| = 0.999), enables cross‑validated reconstruction of full Nyquist spectra at 2% median error across 227 cells, predicts which capacity‑matched cells fail first from three early diagnostics, and recovers Arrhenius activation energies with zero physics priors without retraining or cell‑specific tuning. LGN requires no training data, no impedance hardware, and no chemistry‑specific calibration, converting any existing relaxation pulse into an impedance‑grade diagnostic. This enables real‑time health monitoring, rapid second‑life grading, production‑line quality control, and physics‑informed prognosis from minutes of measurement.

Abstract:
Scaling Scientific Machine Learning (SciML) toward universal foundation models is bottlenecked by negative transfer: the simultaneous co‑training of disparate partial differential equation (PDE) regimes can induce gradient conflict, unstable optimization, and plasticity loss in dense neural operators. In particular, broadband open‑channel fluid dynamics and boundary‑dominated porous media flows impose incompatible spectral and geometric demands on a single dense parameter path. We introduce Shodh‑MoE, a sparse‑activated latent transformer architecture for multi‑physics transport. Shodh‑MoE operates on compressed 16^3 physical latents produced by a physics‑informed autoencoder with an intra‑tokenizer Helmholtz‑style velocity parameterization, restricting decoded states to divergence‑free velocity manifolds. The model guarantees exact mass conservation, achieving a physically verifiable velocity divergence of ~2.8 x 10^‑10 (evaluated post‑hoc in FP64) on 128^3 grids. A Top‑1 soft‑semantic router dynamically assigns localized latent patches to expert subnetworks, enabling specialized parameter paths for distinct physical mechanisms while preserving shared experts for universal symmetries. In a 20,000‑step distributed pretraining run over mixed three‑dimensional physical tensors, routing telemetry shows autonomous domain bifurcation: held‑out validation tokens from the open‑channel domain route exclusively to Expert 0, while porous‑media tokens route exclusively to Expert 1. The model converges simultaneously across both regimes, achieving latent validation MSEs of 2.46 x 10^‑5 and 9.76 x 10^‑6, and decoded physical MSEs of 2.48 x 10^‑6 and 1.76 x 10^‑6. These results support sparse expert routing as a practical architectural mechanism for mitigating multi‑physics interference in universal neural operators.

Abstract:
The application of quantum computing to data management has attracted growing interest, yet remains constrained by a limited understanding of how the physical behaviour of quantum devices relates to the structure and difficulty of database problems. In particular, evaluating quantum annealing approaches for combinatorial optimisation, which is central to many data management tasks, poses significant challenges beyond the scope of conventional empirical and complexity‑theoretic methods. We present a computational toolbox for the systematic numerical analysis of quantum annealing processes derived from data management problem formulations. Adopting a physics‑informed perspective, the toolbox enables the study of spectral and dynamical properties ‑‑ such as energy gaps and eigenstate structure ‑‑ that are inaccessible through direct hardware measurements, yet essential for understanding computational hardness and scaling behaviour. Our approach further provides derived quantities and visualisation techniques that support the interpretation of optimisation dynamics, the identification of structural similarities to canonical physical models, and the construction of reduced effective descriptions. By bridging methodological gaps between quantum computing and database systems research, this work establishes a principled foundation for evaluating quantum approaches and guiding future co‑design efforts.

Abstract:
The high penetration of voltage source converters in modern smart microgrids enhances operational flexibility while introducing complex cyber‑physical vulnerabilities. Existing cyber‑attack studies either require detailed knowledge of system topology and controller dynamics or depend on repeated online interactions, which may compromise practicality by generating operationally infeasible or limit‑violating commands. This article investigates a dispatch command manipulation attack and develops an admittance‑guided framework to identify the vulnerable inverter and the worst‑case dispatch command that most severely degrades system stability. A compromised inverter is utilized to inject controlled harmonic perturbations for sparse admittance measurement, and a physics‑informed neural network is then employed to reconstruct the operating‑point‑dependent admittance of target inverters over the feasible dispatch region. Based on the reconstructed admittance, a stability‑margin‑oriented optimization is formulated to locate the most vulnerable inverter and the corresponding worst‑case dispatch command. Controller hardware‑in‑the‑loop experiments on a five‑inverter microgrid demonstrate that the identified command can drive the system into severe sub‑synchronous oscillations while remaining within nominal dispatch bounds, highlighting the need for stability‑aware command screening beyond static limit checking.

Abstract:
We present a physics‑informed framework for system identification based on randomized stable atomic features. Impulse responses are represented as random superpositions of stable atoms, namely damped complex exponentials associated with poles sampled inside a prescribed disk. Identification is then cast as a convex regularized least‑squares problem with optional linear, second‑order‑cone, and KYP constraints. The approach generalizes random Fourier and random Laplace features to the damped, nonstationary regime relevant to engineering systems while retaining modal interpretability and scalable finite‑dimensional computation. The main analytic point is an operator‑theoretic Disk‑Bochner viewpoint: positive measures over stable poles generate positive‑definite kernels with a radius‑dependent shift defect, while a converse scalar disk moment representation for an arbitrary kernel is characterized by subnormality of the canonical shift. We prove this statement, establish an RKHS‑to‑l1 embedding, show that sampled poles induce a valid finite atomic gauge, discuss random‑feature convergence, and state sparse‑recovery guarantees conditionally on the restricted‑eigenvalue properties of the realized disk‑Vandermonde or input‑output design matrix. We also connect the normalized transfer function problem to Nevanlinna‑Pick interpolation and LFT set‑membership. The framework directly encodes stability margins, modal localization, DC‑gain bounds, monotonicity, passivity, relative degree, settling‑time targets, and time/frequency‑domain error bounds. Numerical comparisons illustrate how physically meaningful priors can compensate for poor excitation and improve constrained impulse‑response recovery in an under‑informative data setting.

Abstract:
The steady incompressible Navier‑‑Stokes equations pose significant computational challenges due to their nonlinear convective terms and pressure‑‑velocity coupling. Physics‑informed neural networks (PINNs) provide a mesh‑free framework for approximating such systems, but classical PINNs can experience optimization difficulties in nonlinear flow regimes. In this work, we propose a quantum physics‑informed neural network (QPINN) framework with a quantum neural network (QNN)‑based trainable embedding for the lid‑driven cavity problem. The proposed approach uses a QNN to learn data‑adaptive quantum feature maps that encode spatial coordinates before they are processed by a variational quantum circuit within a physics‑informed loss formulation. Numerical experiments show that the proposed QNN‑TE‑QPINN exhibits stable training behavior and competitive solution accuracy compared with classical PINNs and hybrid quantum models using classical embeddings, while requiring significantly fewer trainable parameters. Rather than claiming computational speedup, these results highlight the potential of trainable quantum embeddings for parameter‑efficient physics‑informed learning. The findings suggest that embedding design plays an important role in quantum‑assisted PDE solvers and support further investigation of QNN‑based trainable embeddings for nonlinear fluid dynamics benchmarks.

Abstract:
This work studies lung tumor growth prediction from sparse and irregular longitudinal computed tomography (CT) observations with measurement variability. A Bayesian physics‑informed neural network is developed by combining Gompertz growth dynamics with low‑dimensional Bayesian inference in the log‑volume domain. The framework employs a two‑stage inference strategy combining maximum a posteriori (MAP) estimation and Hamiltonian Monte Carlo (HMC) sampling to estimate posterior predictive distributions and uncertainty intervals. The method was evaluated on longitudinal data from the National Lung Screening Trial (30 patients). Results show that the model captures heterogeneous tumor growth patterns while maintaining reasonable prediction accuracy under limited observations. Compared with deterministic modeling approaches, the proposed approach additionally provides calibrated uncertainty estimates. The inferred posterior parameter correlations were consistent with expected biological growth behavior. The proposed framework achieved a cohort‑level log‑space RMSE of approximately 0.20 together with well‑calibrated 95% credible interval coverage across 30 patients. These findings suggest that Bayesian physics‑informed modeling may be useful for uncertainty‑aware tumor growth assessment when only limited longitudinal follow‑up scans are available.

Abstract:
Neural ordinary differential equations (Neural ODEs) often fit training trajectories while generalizing poorly to unseen initial conditions and long horizons. We propose MPINeuralODE, which combines a soft physics‑informed residual with a Multiple‑Initial‑Condition (MIC) multiple‑shooting curriculum whose ingredients are structurally complementary: the physics term anchors the vector‑field magnitude on the support that MIC enlarges. We evaluate along three axes: out‑of‑sample error, long‑horizon stability, and Hamiltonian drift, which together expose whether the learned dynamics recover the underlying vector field. On Lotka‑Volterra, MPINeuralODE achieves the lowest out‑of‑sample and long‑horizon MSE among data‑driven methods, with a 26% reduction over the baseline Neural ODE, while essentially matching the PINN ablation on Hamiltonian drift.

Abstract:
Product formulas for Trotter Suzuki simulation remain a practical route to Hamiltonian evolution on noisy intermediate scale quantum (NISQ) hardware, yet their accuracy hinges on three coupled choices: term grouping, product formula order, and timestep allocation. Toolchains such as Qiskit and Paulihedral lean on hand tuned heuristics, while the discrete nature of grouping and order makes naive gradient based optimization awkward. We describe a generate and evaluate loop: a conditional diffusion model proposes strategies, a physics informed neural network (PINN) supplies differentiable fidelity feedback, and a graph neural network (GNN) encodes commutator structure. Training spans a hybrid space (discrete grouping and order, continuous time steps); the closed loop uses REINFORCE and a Pareto tracker. On the transverse field Ising model (TFIM), under our primary comparison setup, the method reaches 85.6% of the fidelity of a fourth order Qiskit baseline (0.856) at roughly 21.8% of the circuit depth and 19.2% of the baseline CNOT count. Under an equal depth budget, fine tuning in the loop reached a best observed fidelity of 0.9994. Updated ablations show that, for a fixed training budget and default guidance knobs, module contributions depend on the training recipe and guidance hyperparameters CFG in particular needs to be tuned jointly with compute budget. Overall, the results suggest that "generative model and physics supervision" is a viable angle for NISQ oriented compilation, though where it wins still depends on the operating point.

Abstract:
Physics‑Informed Neural Networks (PINNs) and their variational counterparts (VPINNs) are neural networks that incorporate physical laws, making them useful for scientific problems. Existing generalization analyses for PINNs and VPINNs remain limited, often requiring restrictive assumptions such as stability conditions or linear ellipticity. In this paper, we derive generalization bounds for neural networks that involve differentiation with respect to input variables, covering PINNs and VPINNs under a unified framework. We apply Taylor expansion to represent nonlinear differential operators as linear operators on a high‑dimensional space, enabling the use of Koopman‑based analysis and showing that high‑rank networks can generalize well even in settings involving differential operators. We also show that the nonlinearity of the differential operator exponentially enlarges the bound, highlighting its significant impact on generalization.

Abstract:
In production Wide‑Area Networks (WANs), correlated failures dominate availability losses, forcing operators to reserve large safety margins that leave substantial capacity underutilized. Achieving high utilization under strict availability targets therefore requires risk‑aware Traffic Engineering (TE) over dozens to hundreds of probabilistic failure scenarios‑yet solving this problem at operational timescales remains elusive. We demonstrate that existing risk‑aware formulations can be unified under an embedded Sort‑and‑Select structure, exposing a fundamental trade‑off between expressiveness and tractability: classical optimizers either restrict scenario selection for efficiency or incur prohibitive decomposition costs. While deep learning appears promising, prior Deep TE methods mainly target maximum link utilization and rely on scaling‑based feasibility, which fundamentally breaks under explicit capacity constraints and scenario‑dependent risk. We present NeuroRisk, a physics‑informed deep unrolled optimizer that exploits the structure of Sort‑and‑Select. NeuroRisk enforces feasibility via gated edge‑local reservations and represents scenario sets through permutation‑invariant, gradient‑aligned cues. Evaluations on production‑style WANs show that NeuroRisk achieves small optimality gaps relative to the solver with orders of magnitude speedup (10^2‑ 10^5 ×) on risk objectives, while outperforming neural baselines on nominal throughput.

Abstract:
Modeling concentric tube robots (CTRs) involves complex nonlinear continuum mechanics, and despite recent progress, physics‑based models often lack an accurate representation of the experimental setups. To overcome these limitations, deep neural network‑based models have been explored as alternatives with superior accuracy; however, they often overlook known mechanics, require large training datasets, and typically discard shape estimation of the robot. We present a physics‑informed neural network (PINN) for kinematic modeling of a 6‑DoF CTR with three pre‑curved tubes that embeds the Cosserat rod differential equations and learns from few‑shot observational data, balancing physics priors with data‑driven fitting. PINN enables full‑state estimation of shape, twist angle, torsional strain, bending moment, and orientation. Benchmark tests show a mean shape error below 1% of the robot length and accurately recovered other kinematic states, outperforming a purely physics‑based Cosserat rod model baseline while using a minimal training set. The resulting model is also computationally efficient and robust, making it well‑suited for real‑time control applications.

Abstract:
Hybrid machine learning combines physical knowledge with data‑driven models to enhance interpretability and performance. In this context, Port‑Hamiltonian Systems (PHS), which generalize Hamiltonian mechanics to describe open, non‑autonomous dynamical systems, have been successfully integrated with neural networks under the name Port‑Hamiltonian Neural Networks (PHNNs). While the ability of PHNNs to identify Hamiltonian ordinary differential equation (ODE) systems has already been demonstrated, their application to learning Hamiltonian partial differential equation (PDE) systems remains largely unexplored. This limitation restricts their use in musical acoustics, where instruments are typically modeled as distributed parameter systems governed by PDEs. In this work, we demonstrate how to learn the nonlinear string dynamics from data in a physically‑consistent framework through a PHNN extension to PDEs. By constructing structured neural network architectures based on PHS, we can recover both the Hamiltonian governing the string and the dissipation affecting it. This approach outperforms baseline, non‑physics‑informed methods in terms of both accuracy and interpretability. Numerical experiments using synthetic data demonstrate the ability of the proposed PHNN model to identify and emulate the nonlinear dynamics of the system.

Abstract:
This paper introduces a physics‑informed generative framework that resolves the fundamental conflict between the statistical flexibility of deep learning and the rigorous theoretical constraints of fixed‑income modeling. We demonstrate that standard generative models and unconstrained statistical extrapolations suffer from "manifold collapse" and severe arbitrage violations when forecasting term structures across diverse macroeconomic regimes. To overcome this, we propose a two‑stage architecture. First, a Student‑t Conditional Variational Autoencoder with Dynamic Level Injection (CVAEsT+LS) extracts a robust, heavy‑tailed term structure manifold, effectively decoupling macroeconomic shape dynamics from absolute base rates. Second, the latent dynamic evolution is governed by a continuous‑time Neural Stochastic Differential Equation (SDE) strictly penalized by a No‑Arbitrage Partial Differential Equation (PDE). Empirical results across multiple sovereign currencies (USD, GBP, JPY) confirm that our synergistic approach drastically reduces out‑of‑sample forecasting errors ‑‑ achieving an exceptional 6.58 bps Mean Tenor RMSE ‑‑ and successfully overcomes the massive parallel drift and zero‑lower‑bound violations exhibited by the classical HJM model in extreme environments. Furthermore, through phase space vector field analysis, we demonstrate the model's superior capability in unsupervised macroeconomic regime detection and high‑quality continuous‑time scenario generation. Ultimately, this research provides a highly scalable, mathematically sound evolutionary engine for term structure modeling.

Abstract:
Quantifying hemodynamics in the curved segments of the intracranial internal carotid artery is a core challenge in diagnosing vascular stenosis. Conventional full‑field imaging, such as 4D Flow MRI, is costly and difficult to widely promote. Meanwhile, reconstructing full‑field fluid information from easily accessible and non‑invasive sparse measurement data (such as transcranial Doppler ultrasound/computed tomography angiography) is essentially a highly challenging ill‑posed inverse problem. To overcome the severe optimization difficulties and generalization failures of conventional physics‑informed neural networks (PINNs) in highly tortuous geometries, we propose a dual‑correction physics‑informed neural network (DCP‑INN) framework taking into account a causal decoupling strategy. The proposed DCP‑INN model utilizes a diamond‑shaped main network to capture low‑frequency trends in physical evolution, and employs a parallel wide‑deep correction network to compensate for high‑frequency residuals resulting from complex geometric shapes. Furthermore, the framework introduces a high‑order physical loss function based on Taylor expansion to enhance local continuity under extremely sparse data constraints. To validate the proposed method, we performed computational evaluations on realistic vascular geometries with significant tortuosity. The results demonstrate that the method effectively mitigates optimization challenges and significantly reduces flow field reconstruction error. This study not only achieves physically credible and robust flow field reconstruction in complex morphologies but also provides a highly promising algorithmic foundation for building low‑cost, high‑resolution personalized cardiovascular digital twins in future.

Abstract:
Physics‑informed deep learning (PIDL) neural networks have shown their capability as a useful instrument for transportation practitioners in utilizing the underlying relationship between the state variables for traffic state estimation (TSE). Another efficient traffic management approach is implementing varying speed limits (VSLs) on transportation corridors to control traffic and mitigate congestion. However, the existing training architecture of PIDL in the literature cannot accommodate the changing traffic characteristics on a freeway with VSL. To tackle this challenge, we propose a novel framework integrating teacher‑student ensemble training with PIDL neural networks for TSE under VSL scenarios. The physics of flow conservation law is encoded locally in the teacher models by PIDL, and the student model uses a multi‑layer perceptron classifier (MLP) to identify traffic characteristics and selects the ensemble member of PIDL neural networks for TSE. This integrated framework provides a natural solution for capturing the heterogeneity of VSL and accurately addressing the TSE problem. The case study results validate the proposed ensemble approach, demonstrating its superior performance in TSE compared to other popular baseline methods, as indicated by relative L2 error.

Abstract:
The Gauss‑Newton matrix is widely viewed as a positive semidefinite approximation of the Hessian, yet mounting empirical evidence shows that Gauss‑Newton descent outperforms Newton's method. We adopt a function space perspective to analyze this phenomenon. We show that the generalized Gauss‑Newton (GGN) matrix projects the Newton direction in function space onto the model's tangent space, while a Jacobian‑only variant obtained by applying the least squares Gauss‑Newton matrix to non‑least squares losses projects the function space loss gradient onto this same tangent space. Both projections eliminate distortions from the model's parameterization. Specifically, the evolution of the prediction‑target mismatch depends on the model's parameterization through the matrix JJ^\top where J is the Jacobian of the model with respect to its parameters. The projections effectively replace JJ^\top with the identity. We call this effect error whitening. Once the parameterization is removed, the prediction‑target mismatch evolves according to dynamics dictated by the structure of the loss and the projection produced by the optimizer. Error whitening is a special property of Gauss‑Newton descent that rigorously distinguishes it from Newton's method. We empirically demonstrate that Gauss‑Newton optimizers follow the theoretically predicted function space dynamics and outperforms Newton's method, Adam, and Muon across case studies spanning supervised learning, physics‑informed deep learning, and approximate dynamic programming.

Abstract:
Modern gravitational wave astronomy relies on modeling tasks that often require months of graduate‑level effort, including building fast waveform surrogates from expensive numerical relativity simulations, modeling orbital dynamics of black holes, fitting merger remnant properties and constructing template banks. These problems demand extreme precision to support detection and parameter inference, with state‑of‑the‑art models achieving \lesssim 10^‑4 relative error. We study whether state‑of‑the‑art LLM coding agents can perform such end‑to‑end scientific modeling, where success requires constructing models with stringent accuracy criteria and reasoning about physical systems. We introduce gwBenchmarks, a suite of eight tasks grounded in gravitational wave analytic calculations and numerical simulations collectively representing over 10^8 core‑hours of compute. The tasks span interpolation, regression, and high‑dimensional time‑series modeling, requiring a combination of numerical methods, machine learning, and physics‑informed approaches. In preliminary experiments, agents frequently relied on proxy metrics, partial evaluation, or fabricated results to spuriously complete tasks. We therefore implement an external pre‑defined framework to gauge agent progress. Evaluating twelve coding agents, we find no consistent winner. On the easiest task, multiple agents converge to the same cubic spline solution, with one rediscovering a coordinate transformation widely used in the literature. On harder tasks like analytic waveform modeling, all agents fall 1‑2 orders of magnitude short of domain requirements and exhibit systematic failures, including metric misuse, constraint violations, and result fabrication. Our code, data, and website are publicly available.

Abstract:
Scientific discovery can be modeled as a sequence of probabilistic decisions that map physical problems to numerical solutions. Recent agentic AI systems automate individual scientific tasks by orchestrating LLM‑driven planners, solvers, and evaluators. Each method is a combination of methodological actions, with many viable combinations for any given problem and structural dependencies between choices. However, existing frameworks treat each problem in isolation, with no shared substrate to accumulate methodological experience across domains. Here we show that GRAFT‑ATHENA, a self‑improving agentic framework, learns from past problems and autonomously expands its own action space across diverse domains. GRAFT (Graph Reduction to Adaptive Factored Trees) projects combinatorial decision spaces into factored probabilistic trees in which each method is a single path, taking the parameter footprint from exponential to linear. In the lineage of classical Bayesian networks, the factorization is an I‑map of the policy, and the resulting paths embed as unique fingerprints in a metric space whose closeness lets each new problem learn from similar past ones. On canonical physics‑informed machine learning (PIML) benchmarks, GRAFT‑ATHENA improves over human and prior agentic baselines, and on production solvers, it tackles complex engineering problems such as reconstructing Mach‑10 flow over the Apollo Command Module from a 1968 report and recovering shear‑thinning blood‑cell rheology. Notably, the system grows its own knowledge substrate, autonomously proposing regularization constraints for ill‑posed inverse problems and discovering new numerical methods such as a spectral PINN with exponential convergence. These results provide a foundation for autonomous laboratories that grow more capable with every problem they solve.

Abstract:
Constructing indoor radio maps traditionally requires extensive site surveys with precise user‑location labels, making the calibration process costly and time‑consuming. Existing calibration‑reduction methods either depend on partial location annotations or exploit inertial measurement units (IMUs) to provide relative motion cues; however, IMU‑assisted solutions are constrained by hardware availability, device‑level access restrictions, and accumulated sensor drift. In this paper, we study a location‑label‑free indoor radio mapping problem under known access‑point deployment geometry and a known walkable spatial domain. We propose a physics‑informed trajectory inference framework that uses only Channel State Information (CSI), without relying on user‑location labels or IMU measurements. The key idea is to recover the latent spatial coordinates of CSI measurements by exploiting the local spatial continuity of multipath propagation. To this end, we construct a Power‑Angle‑Delay Profile (PADP) feature distance from MIMO‑OFDM CSI and show that, within a local neighborhood and under quasi‑static multipath conditions, this distance provides a physically meaningful proxy for small spatial displacements. We then incorporate the PADP‑based continuity constraint into a spatially regularized Bayesian inference model for joint trajectory recovery and propagation‑parameter estimation. Experiments on a real‑world industrial CSI dataset demonstrate that the proposed framework achieves an average localization error of 0.88 m and a relative beam map construction error of 6.68%, improving upon representative channel‑embedding and IMU‑assisted baselines.

Abstract:
Physics‑informed neural networks (PINNs) are a simple surrogate‑modelling paradigm for partial differential equations, but their standard strong‑form residual formulation is ill suited to the shallow water equations (SWE). It cannot enforce local conservation, handle discontinuities, or leverage the boundary‑conforming unstructured meshes used in real‑world applications. We introduce ``Data‑Guided FVM‑PINN'', a framework that replaces the strong‑form residual with a differentiable, well‑balanced Roe Riemann‑solver finite‑volume (FVM) loss evaluated on unstructured meshes. The major finding is that physics‑only FVM‑PINN training often fails on realistic 2D problems: the network collapses to a trivial low‑momentum state that nearly satisfies the FVM‑PINN residual but bears no resemblance to the true flow. A loss‑landscape diagnostic shows that the FVM‑PINN loss at zero momentum is only about 7× larger than at the trained solution, a shallow basin that an ordinary optimizer falls into; adding even sparse data turns this into a 310× separation, breaking the degeneracy. On a 2D block‑in‑channel benchmark, just 200 random velocity measurements drop the velocity‑field L_2 error by 22× versus physics‑only; 50 measurements still deliver a 7× reduction. A controlled ablation isolates the contribution of the FVM‑PINN loss: it reduces velocity‑field L_2 by ～23% in the sparse‑data regime and is essentially neutral when dense reference data is available. On a real‑world Savannah River reach (1306 cells, 3600~s simulation, five Manning zones), the framework constructs an accurate surrogate from SRH‑2D anchor data, with time‑window decomposition reducing error monotonically via progressive initial‑condition handoff.

Abstract:
Landslide monitoring and simulation play an important role in urban safety assessment and disaster prevention. Existing landslide simulation pipelines typically rely on digital elevation model and mesh‑based representations, which are suitable for geometric analysis, but often lack visual realism. This limitation reduces their effectiveness in interactive applications, hazard communication, and public education. In this paper, we propose a UAV‑based scan‑to‑simulation framework that bridges photorealistic scene capture and physics‑based landslide simulation through 3DGS. Specifically, our pipeline includes four stages: (1) UAV‑based acquisition of slope imagery, (2) reconstruction of a low‑anisotropy 3DGS scene representation, (3) volumetric conversion of the target simulation region by filling the interior of the surface‑based model, and (4) integration with the Material Point Method (MPM) for landslide simulation. We validate the proposed framework on a real landslide site in Hong Kong that experienced a severe landslide event. The results show that our method supports both realistic visual reconstruction and effective simulation.

Abstract:
Model Predictive Control (MPC) is widely used to operate safety‑critical infrastructure by predicting future trajectories and optimizing control actions. However, nonlinear dynamics, hard safety constraints, and numerical optimization often render individual control moves opaque to human operators, undermining trust and hindering deployment. This paper presents Hierarchical Causal Abduction (HCA), which combines (i) physics‑informed reasoning via domain knowledge graphs, (ii) optimization evidence from Karush‑‑Kuhn‑‑Tucker (KKT) multipliers, and (iii) temporal causal discovery via the PCMCI algorithm to generate faithful, human‑interpretable explanations for control actions computed by nonlinear MPC. Across three diverse control applications (greenhouse climate, building HVAC, chemical process engineering) with expert validation, HCA improves explanation accuracy by 53% over LIME (0.478 vs. 0.311) using a single set of cross‑domain parameters without per‑domain tuning; domain‑specific KKT‑threshold calibration over 2‑‑3 days further increases accuracy to 0.88. Ablation studies confirm that each evidence source is essential, with 32‑‑37% accuracy degradation when any component is removed, and HCA's ranking‑and‑validation methodology generalizes beyond MPC to other prediction‑based decision systems, including learning‑based control and trajectory planning.

Abstract:
Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non‑stationary kernel to an empirical kernel obtained from low‑fidelity simulations. In the second step, we derive a high‑fidelity kernel with estimated hyperparameters, and construct a corresponding high‑fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics‑informed method on the Burgers' equation.

Abstract:
Physics‑informed neural networks (PINNs) train a single neural approximation by minimizing multiple physics‑ and data‑derived losses, but the gradients of these losses often interfere and can stall optimization. Existing remedies typically treat this pathology either through scalar loss balancing or full‑parameter‑space gradient surgery, leaving it unclear which intervention is most appropriate. We show that PINN gradient conflict is not a uniform failure mode with one universal remedy. Instead, we identify distinct PINN gradient‑conflict regimes, each associated with a different intervention class. Persistent directional conflict may require separate loss‑indexed parameter subspaces, magnitude imbalance often favors scalar reweighting, and low or transient conflict may require no extra mitigation. To select between scalar reweighting and a lightweight architectural intervention, we propose a diagnostic‑first framework. It profiles a 1000‑step unmodified PINN run and, when intervention is warranted, uses one low‑rank adapter per loss to create explicit loss‑indexed parameter subspaces attached to a shared PINN trunk, providing each loss with a direct gradient pathway. Across more than 60 PDE configurations, including forward, inverse, multi‑physics, parameter‑varying, and high‑dimensional problems up to 50D, persistent directional conflict dominates standard forward K=3 benchmarks and a natural K=4 thermoelastic system, where adapters combined with reweighting yield significant improvements. In contrast, K=3 inverse problems and natural K=5 and K=6 multi‑physics systems are largely magnitude‑dominated and often favor reweighting alone, while full‑parameter‑space gradient surgery can fail on heterogeneous parameter spaces.

Abstract:
Physics‑informed neural networks (PINNs) are a promising approach for solving partial differential equations (PDEs). Their training, however, is often difficult because multiple loss terms induced by PDE residuals and boundary or initial conditions must be optimized simultaneously. To address this difficulty, existing approaches often construct update directions by explicitly enforcing particular desirable properties, such as scale robustness and simultaneous descent. While effective in many cases, such property‑by‑property designs can make it unclear which conditions are essential, what geometric principle determines the selected update direction, and how different methods are structurally related. In this work, we formulate update‑direction selection for PINN training as a Chebyshev‑center problem in the dual cone. The proposed formulation selects a normalized direction that maximizes the minimum distance to the cone facets. The resulting formulation admits an efficient dual problem in a much lower‑dimensional space and yields a convergence guarantee in the nonconvex setting. It also recovers the key desirable properties targeted by existing approaches without imposing them separately; rather, they follow from the single geometric criterion underlying the formulation. This makes the selected direction interpretable through a single geometric rule and provides a unified basis for systematically comparing related direction‑selection methods. Experiments on several PINN benchmarks further demonstrate strong empirical performance of the proposed method.

Abstract:
Travel‑time tomography forces a trade‑off between mesh resolution and stability in which the regularizer choice dominates what can be recovered. We introduce MIMIR, a differentiable framework that represents the 2D velocity field as a Fourier‑feature neural network, replacing the grid‑based slowness vector with a continuous, infinitely differentiable function. Prior neural‑field tomography has staircased smooth fields under total‑variation (TV) priors or oscillated near interfaces under L^2 Laplacian smoothing. We adopt second‑order total generalized variation (TGV^2) and parametrize its auxiliary vector field as a second neural network jointly optimized with the velocity field, eliminating the inner Chambolle‑Pock primal‑dual loop that classically dominates TGV computation. On three synthetic benchmarks (Gaussian, horizontally layered, curved‑fault inspired by OpenFWI) using cross‑well acquisition, 5% travel‑time noise, and five seeds, MIMIR‑TGV^2 ties a classical FMM‑LSMR baseline with auto‑tuned hyperparameters on the Gaussian (p=0.134, paired t‑test) and significantly outperforms it on layered (p<0.0001, 44% RMSE reduction) and curved‑fault (p=0.0002, 33% reduction). Replacing TGV^2 with TV degrades performance on Gaussian (p=0.004) and layered (p=0.003); curriculum‑annealed TV improves Gaussian RMSE by only 5.4%, confirming that TV's staircase bias is intrinsic to the regularizer rather than a scheduling artifact. The results empirically validate the Bredies‑Kunisch‑Pock prediction that piecewise‑affine priors are better suited to subsurface velocity recovery than piecewise‑constant TV priors. We argue that the central design choice in physics‑informed neural‑field inversion is not the network architecture but the regularizer. The full pipeline reproduces in under one hour on consumer hardware.

Abstract:
Detecting orbital anomalies, such as maneuvers, atmospheric decay, and attitude upsets, across the rapidly growing population of low‑Earth‑orbit (LEO) satellites is a prerequisite for collision avoidance, decay forecasting, and conjunction screening. The bottleneck is not modeling capacity but labels: there is no public ground‑truth corpus of orbital anomalies, manual review does not scale to approximately 10^4 active satellites, and pure rule‑based detectors trade recall for precision so aggressively that they are blind to most behavioral anomalies. We present a multi‑tier labeling cascade that composes three weak supervision sources of increasing fidelity: a fast physics rule set (rule_v1), an Interacting Multiple Model Unscented Kalman Filter (IMM‑UKF) bank, and a supplemental‑element calibration step (supGP), to produce labels at a scale unavailable from any single source. Applied to 232M Two‑Line Element (TLE) records spanning 60 years, the cascade yields 8.6M labeled sequences of length 50 (430M timesteps) over 11 features that include explicit time encoding and full mean‑element state. On overlapping satellites, IMM‑UKF surfaces 42.6x more anomalies than rule_v1 alone. We train a 6.5M‑parameter Transformer in two stages, achieving a maneuver recall of 55.4% and decay recall of 62.8% on a held‑out test set. An ablation on the time‑delta feature alone yields a 107% relative improvement in decay recall. We frame the resulting model as a high‑recall triage classifier whose role is to surface candidate events for downstream filtering, not to issue final attributions, and discuss the path toward a Neural‑ODE‑based orbital world model.

Abstract:
Training neural networks to satisfy universal constraints over continuous domains poses unique challenges. Common examples include Lyapunov Neural Networks (Lyapunov NNs) and Physics‑Informed Neural Networks (PINNs), where analytical solutions are generally either unavailable or overly restrictive. Sample‑based methods are therefore commonly used to enforce these constraints, and the choice of samples has a substantial impact on convergence speed, stability, and solution quality. Most existing methods rely on fixed heuristics or handcrafted rules, and are suboptimal in practice. In this paper, we aim to improve upon them by learning, from data and experience, how to dynamically and iteratively adjust the samples in response to the model's evolving learning performance. Trained by reinforcement learning, the learned policy improves empirical constraint satisfaction on test problems while significantly improving efficiency. We validate the approach on both Lyapunov NNs and PINNs, and demonstrate its broader applicability to domains where adaptive input selection is essential for effective training.

Abstract:
Artificial intelligence (AI) is reshaping education, scientific training, and materials discovery. In materials science, AI models increasingly support property prediction, experiment prioritization, and hypothesis generation; however, the limiting factor is no longer only algorithmic capability but also whether students and educators can use AI with domain‑specific scientific judgment. This workshop‑informed white paper and curriculum‑oriented position article argues that AI education for AI‑powered materials discovery must move beyond tool access and surface‑level interaction with generative AI systems toward a workflow‑aligned model of AI literacy. We connect AI literacy to materials‑informatics competencies: data provenance, domain‑specific featurization, model validation, uncertainty quantification, physics informed reasoning, reproducibility, and experimental feedback. We also emphasize outcome‑oriented equity: institutions should evaluate not only access, participation, and engagement, but also whether AI‑enabled instruction produces comparable learning gains, transfer of learning, confidence calibration, defined as the alignment with students confidence and the quality or correctness of their work, persistence, and research readiness across student subgroups. The paper synthesizes relevant evidence, identifies risks for learners such as cognitive off‑loading and cognitive surrender, and provides a dual‑track curriculum model and implementation recommendations such as curriculum guides and an assessment plan for courses, bootcamps, workshops, and program‑level reform. The central goal is to prepare students to become better scientists, not merely more efficient users of AI tools.

Abstract:
Machine learning has become a powerful tool in high‑energy collider experiments, which enables the studies based on data‑driven approaches to complex reconstruction and regression tasks. The study of identified hadron spectra in pseudorapidity regions beyond detector acceptance, which is limited to mid‑rapidity regions, carries important information about particle production, yet remains unmeasured. In this work, we develop a physics‑informed neural network, trained on PYTHIA8 pp collisions at \sqrts=13.6 TeV, to infer p_\rm T spectra of π^\pm, K^\pm, p/\barp, Λ/\barΛ, and K^0_\mathrms in different rapidity regions. Physics‑motivated constraints, including particle yield ratios, spectral shape, and smoothness, are incorporated into the loss function. A staged hyperparameter optimization strategy is used to ensure stability. The model achieves yield uncertainties of ～1.5%, 1.8%, and 5.83% in the training, interpolation, and extrapolation regimes, respectively, outperforming XGBoost and LightGBM. It further reproduces key observables such as particle yield ratios, the multiplicity dependence of \langle p_\rm T \rangle, and kinetic freeze‑out parameters, indicating that the model captures the underlying physics and provides reliable predictions beyond the measured phase space.

Abstract:
Deep learning paradigms, such as PINNs and neural operators, have significantly advanced the solving of PDEs. However, they often struggle to capture the continuous integral nature of physical systems, relying either on pointwise residuals that ignore the integral perspective or on pre‑discretized temporal grids. Drawing inspiration from MeanFlow, a continuous‑time integrator recently developed to efficiently solve generative ODEs, we introduce Spatio‑Temporal MeanFlow, which functions as a novel PDE solver learning the finite‑interval evolution of physical states. By substituting the generative velocity field with the physical PDE operator, we transform multi‑step numerical integration into an efficient prediction with a freely controllable integration length. Crucially, we extend the original MeanFlow constraint from the temporal to the spatio‑temporal domain, coupling time evolution with spatial consistency. This yields a unified framework naturally accommodating both time‑dependent and stationary PDEs. Comprehensive experiments on benchmarks demonstrate that our approach achieves superior accuracy and inference efficiency over representative baselines. Furthermore, the proposed integral constraint enables excellent generalization to out‑of‑distribution initial conditions and varying spatial resolutions.

Abstract:
Recent work has shown that strong‑form physics‑informed neural networks (PINNs) based on pointwise enforcement of differential operators can be ill‑posed due to the combination of sufficiently expressive neural network trial spaces with finitely many measurements. In this work, we develop an abstract analytical framework that isolates this finite‑information mechanism and extends its applicability beyond strong‑form formulations. We apply the framework to three representative variational neural discretizations: the Deep Ritz method, neural network discretizations of variational regularization functionals, and weak PINNs. Despite their differing formulations, these methods constrain the neural trial function only through finitely many linear measurements, such as quadrature evaluations or finite‑dimensional test spaces. We show that this structural feature leads to ill‑posed discrete optimization problems, manifested by non‑uniqueness or degeneracy of minimizers, independently of the well‑posedness of the underlying continuous variational problem.

Abstract:
We study the posterior contraction rate of Bayesian Physics‑Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus on learning of the elliptic equation with a non‑homogeneous Dirichlet boundary condition from independent and noisy measurements collected both inside the domain and on the boundary. Assuming that the PDE admits a strong solution in a Hölder space and using with a suitably constructed prior on the neural network weights, we prove that the posterior distribution concentrates around the exact solution at a near‑minimax rate. Furthermore, the chosen prior is rate‑adaptive: the posterior contracts at an (almost) optimal rate without prior knowledge of the smoothness level of the exact solution. Our results provide statistical guarantees for uncertainty quantification of PDEs via Bayesian PINNs.

Abstract:
We describe an approach for estimating the statistical risk of deep networks that contain a mix of learned and known operators. Building on the maximal training error bounds previously established for known operator learning, we derive a deep risk estimator that connects the expected error of a layered network to the size of the training sample. The estimator decomposes the total risk into a sum over learned layers; every known operator contributes zero to this sum, while every learned layer adds an approximation term inspired by Barron's classic work and an estimation term that decreases with the number of training samples. We are able to show that the bound shrinks whenever a learned layer is replaced by a known operator and that the corresponding sample requirement scales with the number of trainable parameters of the layer that is replaced. As an application, we use computed tomography as an example and compare an operator‑aware filtered backprojection network with a fully connected substitute that collapses the entire reconstruction pipeline into a single learned dense matrix. The predicted parameter ratio coincides with the structural sparsity that the analytic decomposition into a circulant filter and a sparse backprojection exposes. We confirm the predicted scaling on CPU at small image scale and on GPU at medium image scale, all on the same scaling law. Beyond CT reconstruction, the estimator applies to physics‑informed neural networks that hardcode a known physical operation in its architecture, and we expect the result to be of interest for a broad community working on operator‑aware deep learning. Calibrating the per‑layer constants on each sweep yields a bound that tracks the empirical test MSE within a factor of two at every training‑set size, so the estimator can be inverted to predict how many training samples are required to reach a target error.

Abstract:
Physics‑informed neural networks (PINNs) provide a flexible framework for solving forward and inverse problems governed by partial differential equations (PDEs), but standard PINN training typically relies on soft penalty formulations that combine PDE residuals, data mismatch, and initial/boundary conditions using manually chosen weights. This often leads to ill‑conditioning, sensitivity to loss weights, and poor constraint satisfaction. In this work, we reformulate PINN training as an equality‑constrained optimization problem and propose a novel Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN (AdamFLIP). The key idea is to view the constraint residuals as the output of a controlled dynamical system and to compute the Lagrange multiplier as a feedback input that locally drives these residuals toward stable linear contraction dynamics. AdamFLIP then applies Adam‑style first‑ and second‑moment adaptation to the resulting feedback‑linearized Lagrangian gradient, combining principled constraint handling with the scalability and robustness of adaptive neural‑network optimization. We test AdamFLIP on a range of benchmark forward and inverse PDE problem, and it consistently outperforms both the standard soft‑constrained PINN and state‑of‑the‑art constrained optimizers. Specifically, on the Navier‑‑Stokes equations benchmark, AdamFLIP reduces relative L_2 error by more than two thirds for the predicted solution compared to the next best method. Our AdamFLIP framework provides an effective and computationally scalable hard constraint optimization method for PINN training.

Abstract:
We study the problem of \empharchitecture selection for deep learning models trained to solve partial differential equations (PDEs), asking when transformer‑based architectures with learned attention outperform Fourier‑domain neural operators. We introduce the Multi‑Scale Attention Transformer (\msat), a deep learning architecture that encodes spatiotemporal solution histories as token sequences and trains end‑to‑end via a composite supervised objective with optional physics‑informed regularization terms. We conduct a comprehensive empirical evaluation against nine baselines ‑‑ including physics‑informed neural networks (PINNs), neural operators (FNO, DeepONet, GNOT), and state‑space models (Mamba‑NO) ‑‑ across five benchmark problems from the PINNacle suite, using identical train/test splits and reference data for all methods. \msat achieves state‑of‑the‑art generalization on complex geometry problems (L^2_\mathrmrel = 0.0101 on Heat2D‑CG, a 3.7× improvement over FNO) at 34\,\mathrms total inference vs.\ 120,812\,\mathrms for Mamba‑NO. Ablation studies over the physics regularization component reveal a precise inductive bias tradeoff: physics priors reduce test error on diffusion‑dominated problems but degrade generalization on chaotic and recirculating‑flow regimes, directly characterizing the prior misspecification boundary. Approximation error bounds as a function of domain boundary complexity κ provide a theoretical basis for these empirical findings and a principled rule for architecture selection.

Abstract:
Positron Emission Tomography (PET) image reconstruction is inherently challenged by Poisson noise and physical degradation factors, which are further exacerbated in limited‑angle acquisitions. While deep learning methods demonstrate promising performance, their generalization to unseen clinical data distributions remains limited without extensive retraining. We propose PET‑Adapter, a test‑time domain adaptation framework for generative PET reconstruction models pretrained solely on phantom data. Our method enables adaptation to clinical datasets with varying anatomies, tracers, and scanner configurations without requiring paired ground truth. PET‑Adapter introduces layer‑wise low‑rank anatomical conditioning during adaptation and Ordered Subset Expectation Maximization‑based warm‑starting that initializes the generation from physics‑informed reconstructions, reducing diffusion steps from 50 to 2 without compromising quality. Experiments across multiple clinical datasets demonstrate superior 3D reconstruction performance in both full‑angle and limited‑angle settings, highlighting the clinical feasibility and computational efficiency of the proposed approach.

Abstract:
Traffic state estimation from sparse fixed sensors is challenging because physics‑informed neural networks (PINNs) tend to over‑smooth the shockwaves admitted by the Lighthill‑Whitham‑Richards (LWR) model. This study proposes Adaptive Domain Decomposition Physics‑Informed Neural Networks (ADD‑PINN), a two‑stage residual‑guided framework for LWR‑based offline speed‑field reconstruction. A coarse global PINN is first trained; its spatial residual profile is then used to place subdomain boundaries and initialize child subnetworks in a decomposition‑enabled mode, while a data‑driven shock indicator can retain a single‑domain fallback when localized evidence of transition is weak. The primary offline I‑24 MOTION evaluation spans five days, five sensor configurations, and ten seeds per configuration, yielding 1,500 runs in total. Against neural and physics‑informed baselines, ADD‑PINN attains the lowest relative L2 error in 18 of 25 configurations and in 14 of 15 sparse‑sensing cases, while training 2.4 times faster than the extended PINN (XPINN) baseline. An ablation study supports spatial‑only decomposition as an effective default for fixed‑sensor traffic reconstruction in the evaluated settings. Supplementary Next Generation Simulation (NGSIM) experiments serve as a negative control: the shock indicator suppresses decomposition in all 50 runs, and the default single‑domain fallback ranks first across all sensor configurations. These results support residual‑guided spatial decomposition as an effective PINN‑family design for offline reconstruction when sparse fixed sensing coincides with localized transition regions.

Abstract:
Turbulence is a phenomena that is \it locally and statistically characterized by measurements, but it is caused by \it nonlocal energy cascades associated with the environment. The presence of turbulence coincides with fluctuations in the refractive index, which impact optical sensing, imaging, and signaling applications. Here, we study the machine learning models that predict near‑surface optical turbulence strength C_n^2, derived from anemometer‑based surface flux measurements through Monin‑Obukhov similarity theory, using ERA5 reanalysis data as model inputs. We evaluate the model's ability to perform temporal extrapolation by training on one year of co‑located C_n^2 observations and ERA5 data, and applying the model to ERA5 data from other years at the same site to reconstruct a multi‑year time series. We compare the predictions across Southern California and New York. In spite of varying weather and terrain, the ML models show consistent performance and seasonal behavior across training years. All models show greater correlation, faster convergence, and lower prediction errors in the summer. However, some ERA5 features drive predictions in New York but not California and vice versa, and such feature dependence depends on the season. Seasonal error and feature trends suggest that turbulence is affected by atmospheric composition or other seasonal environmental considerations that are not currently monitored by ERA5. We find, regardless of terrain, the primary feature of importance to turbulence prediction is solar radiation, which underlines the central role of radiative energy transfer in driving atmospheric turbulence. We point toward physics‑informed ML translation and feature selection as tools for improving the generalizability of data‑driven models.

Abstract:
Physics‑informed operator learning is an attractive candidate for surrogate modeling of microstructures, especially in multiscale finite‑element simulations. Its practical use, however, is often limited by the high cost of loss evaluation. We address this bottleneck by combining the Equilibrium Neural Operator (EquiNO) with the QR‑based discrete empirical interpolation method (Q‑DEIM). EquiNO learns only the modal coefficients of reduced displacement‑fluctuation and first Piola‑Kirchhoff stress representations built from periodic and divergence‑free bases, thereby enforcing periodicity and mechanical equilibrium by construction. Q‑DEIM then identifies a small set of spatial points through a column‑pivoted QR factorization of the stress basis and restricts constitutive evaluations during training to these points alone. This makes full‑batch second‑order optimization practical for three‑dimensional representative volume elements (RVEs). Homogenized first Piola‑Kirchhoff stresses are recovered directly from the offline‑averaged reduced stress modes, without the need to reconstruct the full stress field at inference time. We validate the framework on two three‑dimensional finite‑strain hyperelastic RVEs. Q‑DEIM reduces the per‑step training cost by roughly three orders of magnitude relative to full‑field loss evaluation, while reduced homogenization achieves speed‑up factors of order 10^3 to 10^4 over direct full‑field computations. Despite relying on only a small number of offline snapshot loading paths for basis construction, the method accurately interpolates and extrapolates both microscopic stress fields and homogenized stresses, with prediction quality improving systematically as more snapshots are added.

Abstract:
Physics‑based digital twins aim to predict the dynamics of real‑world objects under interaction, enabling real‑to‑sim‑to‑real applications in robotics. Current approaches reconstruct such twins as explicit physical models (such as spring‑mass systems) to predict the dynamics, but the resulting models often inherit the resolution of the visual reconstruction rather than being reduced to the physical complexity required to reproduce task‑relevant dynamics. This mismatch introduces redundant topology, making repeated forward‑dynamics rollouts unnecessarily expensive. To address this challenge, we present PhySPRING, an fully differentiable GNN‑based method to reduce complexity in spring‑‑mass digital twins. PhySPRING jointly learns a hierarchy of coarsened graph topologies and their mechanical parameters from observations. At each reduction level, PhySPRING merges nodes with similar learned dynamic responses to optimize the topology, while maintaining every reduced layer as an explicit spring‑‑mass system. On the PhysTwin benchmark, PhySPRING improves dense reconstruction and prediction accuracy over PhysTwin, while reduced models retain stable physical and visual fidelity with up to a 2.30 times speed‑up. We further demonstrate the effectiveness of PhySPRING in a Real2Sim robot policy‑evaluation pipeline, where the reduced models are substituted zero‑shot into ACT and π_0 evaluations, maintaining comparable manipulation success rates across downsampling levels while improving action‑sampling effectiveness. Together, PhySPRING enables efficient and structure‑preserving spring‑‑mass reduction without sacrificing fidelity or robotic utility.

Abstract:
The simulation of fluid flows is computationally expensive due to the complexity of its governing partial differential equations. Machine learning models offer a potential surrogate, enabling learning from simulations and significantly faster predictions of flow fields. However, these models require large training datasets, which introduces a trade‑off between dataset generation cost and predictive accuracy. In this work, we investigate the relationship between the size of the training‑set and accuracy of the prediction when learning steady flow fields in an industrial‑scale stirred vessel. A data set of steady flows is generated using Reynolds Averaged Navier Stokes (RANS) simulations in a range of realistic operating conditions, including impeller speeds and liquid heights. We train implicit neural representations of flow fields and compare purely data‑driven and constrained variants. Model performance is evaluated using global mean squared error (MSE), qualitative spatial comparisons of predicted and reference flow fields, and tracer transport simulations. We find that the prediction error decreases monotonically with increasing training data, but also that it exhibits clear diminishing returns beyond moderate dataset sizes. Physics‑based constraints significantly improve accuracy and reduce variability across training runs in low‑data regimes, and they lead to more stable tracer‑transport behavior. Furthermore, reasonable interpolation can be achieved over different impeller speeds and liquid heights. However, these benefits come with an increase in the complexity of training, and their relative advantage diminishes as the training set grows.

Abstract:
Phase‑field simulations provide mechanistic descriptions of microstructure evolution, but repeated high‑fidelity integration over long horizons and broad parameter spaces remains computationally expensive. We present PFNet, a physics‑informed neural operator framework that advances microstructural states by learning conditional evolution operators rather than direct correlations. PFNet combines a diffusion‑inspired U‑Net with periodic padding, entropy‑based state conditioning and thermodynamic‑parameter modulation to encode boundary consistency, instantaneous ordering state and changes in the free‑energy landscape. For Cahn‑Hilliard coarsening, PFNet achieves accurate one‑step prediction and stable autoregressive rollouts across composition, gradient‑energy coefficient, coarsening stage and morphology class, with errors concentrated near diffuse interfaces and topology‑changing regions. The same framework extends to a four‑channel martensitic‑transformation benchmark without martensite‑specific redesign. These results indicate that physics‑informed operator learning can provide transferable surrogates for phase‑field dynamics and broader energy‑dissipative dynamical systems.

Abstract:
The fundamental quantity governing the mechanical and thermodynamic properties of a crystalline solid is its electronic charge density. Yet, its direct use for the rapid prediction of materials properties remains challenging due to its high dimensionality. Here, we present a physics‑informed deep learning framework that directly predicts mechanical and thermodynamic properties from the three‑dimensional electronic charge density derived from density functional theory (DFT). The proposed approach first utilizes a three‑dimensional convolutional autoencoder for unsupervised dimensionality reduction, compressing a high‑resolution charge‑density grid (128 x 128 x 128) into a compact latent representation (16 x 16 x 16 x 16) while preserving physically meaningful features, as confirmed by negligible reconstruction errors across diverse crystal systems. The compressed latent‑space representation of charge density is then used by two different regression models for property prediction: Light Gradient Boosting Machine (LightGBM) and Attention‑based 3D Convolutional Neural Networks (Att CNN), and their performance is compared. Combining composition‑based descriptors (Material Agnostic Platform for Informatics and Exploration or MAGPIE) with electronic charge density data further improves the model accuracy. Using a dataset of about 6059 inorganic compounds spanning multiple crystal symmetries, the models achieve strong predictive performance for bulk modulus K (R2 = 0.94), Young's modulus E (R2 = 0.88), shear modulus G (R2 = 0.87), formation energy Eform (R2 = 0.96), and Debye temperature Θ (R2 = 0.89). This work establishes electronic charge density as a transferable, physics‑grounded descriptor for materials property prediction, requiring ~ 1/25 the computational resources of full‑fledged DFT calculations.

Abstract:
Rapid aerodynamic screening of turbomachinery blades across wide operating envelopes remains a major computational bottleneck in preliminary design, particularly for energy‑conversion and storage systems such as emerging Carnot batteries. Physics‑informed neural networks (PINNs) offer a mesh‑free alternative to conventional CFD, yet convergence and accuracy often deteriorate for complex blade geometries and off‑design flows. We propose a progressive Euler‑PINN framework that (i) gradually relaxes boundary conditions from tunnel flow without a blade to full outlet static pressure, and (ii) employs a geometry‑aware dynamic loss‑weighting scheme that intensifies residual penalties near highly curved boundaries. To the best of our knowledge, this is the first study to deploy a single PINN workflow for large‑scale, engineering‑grade screening of turbomachinery blade families across multiple operating conditions, covering ten NACA6 variants and 30 subsonic operating points. The proposed framework achieves CFD‑comparable accuracy for pressure and velocity fields while reducing the computational cost required for family‑wide blade screening. These results establish the method as a practical surrogate for two‑dimensional turbomachinery blade pre‑design and optimisation.

Abstract:
Physics‑informed neural solvers offer a promising route to model‑based reinforcement learning in continuous time, where optimal feedback synthesis is governed by Hamilton‑‑Jacobi‑‑Bellman (HJB) equations. Practical implementations often occupy a regime that is neither a classical grid method nor a continuous‑PDE PINN: the value function is represented by a neural network, finite‑difference HJB policy‑evaluation operators are evaluated by network queries at shifted points, and residuals are minimized by random continuous collocation. This regime preserves the stabilized finite‑difference policy‑evaluation structure while avoiding grid‑based value unknowns. We develop an error theory for this hybrid regime. Interpreting finite differences as shift operators acting on neural networks, we prove a population L^2 stability estimate for one policy‑evaluation step with learned dynamics. The bound separates residual error, initial and exterior‑collar mismatch, policy mismatch, and model‑identification error, with an explicit gradient amplification factor for learned dynamics, while the underlying linear evaluation stability remains free of hidden inverse‑viscosity blow‑up. We further give a finite‑sample collocation certificate and a conditional multi‑step propagation result through greedy policy improvement. Experiments on compact‑control LQR upto 64 dimensions, Allen‑‑Cahn control, pendulum, Hopper, and 3D quadrotor benchmarks compare against representative model‑based and model‑free RL baselines, demonstrating the predicted residual, policy‑mismatch, and learned‑model error trends.

Abstract:
Physics‑informed neural networks (PINNs) provide a mesh‑free framework for solving PDE‑constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically meaningful prior assumptions are more naturally expressed in function space. In this study, we introduce a unified framework, termed functional‑prior‑based approaches to Bayesian PDE‑constrained inversion using physics‑informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN‑based inversion. We consider two complementary approaches. The first is a functional‑prior‑informed Bayesian PINN (FPI‑BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior, and Bayesian inference is subsequently performed in weight space. The second is function‑space particle‑based variational inference for PINNs (fParVI‑PINN), which performs Bayesian estimation using ParVI directly in function space. We also show that random Fourier features (RFF) play an important role in representing Gaussian functional priors with neural networks and in improving posterior approximation. We applied the proposed approaches to one‑dimensional seismic traveltime tomography and two‑dimensional Darcy‑flow permeability inversion. These numerical experiments showed that both approaches accurately estimated posterior distributions, highlighting the significance of introducing physically interpretable functional priors into Bayesian PINN‑based inverse problems. We also identified the contrasting advantages of FPI‑BPINN and fParVI‑PINN, namely flexibility and accuracy, respectively.

Abstract:
This paper introduces the MAESTRO workflow, that enables the coupling of the PORTALS framework [P. Rodriguez‑Fernandez et al, Nucl. Fusion 2024] with external solvers for the plasma equilibrium, pedestal physics, divertor constraints and heating. The surrogate‑based optimization nature of the transport solver is ideally suited for external coupling, allowing efficient steady‑state predictions of plasma profiles with full physics models. Improvements in the surrogate modeling of quasilinear transport models with PORTALS are presented, which enable the efficient handling of discontinuities in the transport fluxes that can arise from numerical issues or physical instabilities with extreme stiffness. The combination of physics‑informed methods and advanced numerical techniques allows the MAESTRO workflow to provide accurate and efficient predictions of steady‑state plasma profiles, which are critical for fusion reactor design and optimization.

Abstract:
We augment the Slotine‑‑Li adaptive controller for Euler‑‑Lagrange systems with three learned components: a structured‑quadratic Lyapunov function \(V_ψ\) whose positive‑definiteness follows from a Cholesky parameterization, a residual Soft Actor‑‑Critic policy that adds bounded torque corrections to the analytic baseline, and a physics‑informed neural network that estimates unmodeled dynamics. A closed‑form safety filter, derived from the single affine constraint \(\dot V_ψ+ αV_ψ\le 0\), projects every policy output onto the safe set without requiring an online QP solver. We prove: global feasibility of the filter under a drift‑decay condition on the control‑degeneracy set; exponential stability under exact shielding, with a robust extension whose margin depends on the PINN approximation error; almost‑sure convergence of the three‑timescale policy‑‑certificate‑‑multiplier updates to a KKT point; and a PAC generalization bound for the certificate over compacts. On a 2‑DOF manipulator with nonlinear friction and variable payload, the learned certificate accounts for most of the empirical gain: tracking error drops by 41% on nominal friction and 24% on aggressive friction at the centroid of the training distribution. A 7‑DOF scalability study on a Franka Emika Panda confirms clean convergence of the full pipeline at industrial scale, identifies the conditions under which gains over exact model‑based baselines should and should not be expected, and documents a warm‑start pathology of the learned certificate that has practical implications for deployment.

Abstract:
We introduce a modular quantum circuit primitive to model entanglement dynamics in QCD parton splitting and use it as a composable building block for data‑driven, physics‑consistent event generation. For the pure‑gluon channel, we derive an analytic expression for the helicity entanglement generated at the splitting vertex, quantified via the concurrence, and construct a two‑qubit circuit whose measurement outcomes encode the momentum shared between outgoing gluons while reproducing the QCD‑predicted entanglement structure. Calibrating the circuit parameters to LHC jet substructure data maps, reconstructed momentum‑sharing fractions are directly related to circuit rotation angles. Composing multiple splitting primitives yields multi‑prong momentum‑fraction distributions; we validate the three‑ and four‑prong cases against experimental data and find good agreement. For the three‑prong configuration, we execute the circuit on superconducting quantum hardware and obtain results consistent with simulation after standard quality cuts, enabled by the low qubit count and shallow circuit depth. This work provides a concrete framework for quantum‑native parton‑shower modules that encode quantum correlations at the level of splitting dynamics, and offers physics‑informed ansätze for future quantum algorithms for QCD.

Abstract:
Real‑time supervisory control of thermal energy distribution systems requires digital twins that are accurate, interpretable, and uncertainty‑aware, yet remain data and computationally efficient. High‑fidelity simulations alone are costly, while purely data‑driven surrogates often lack robustness. To address these challenges, this work proposes an active learning (AL) framework that couples system‑level Modelica simulations with four simpler physics‑informed and data‑driven surrogate modeling approaches: deterministic Sparse Identification of Nonlinear Dynamics with Control (SINDyC), its probabilistic multivariate‑Gaussian extension (MvG‑SINDyC), feedforward neural network (FNN), and gated recurrent unit (GRU) network. Tailored to each surrogate, model‑specific AL query strategies are employed, including Mahalanobis‑distance sampling in coefficient space for MvG‑SINDyC and error‑based sampling in prediction space for SINDyC, FNN, and GRU, allowing the learning process to prioritize dynamically informative trajectories. The proposed approach is demonstrated on the glycol heat exchanger (GHX) subsystem of the Thermal Energy Distribution System (TEDS) at Idaho National Laboratory. Across key GHX outputs‑‑the bypass mass flow rate \dotm_\mathrmGHX and heat transfer rate Q_\mathrmGHX‑the AL framework achieves comparable predictive accuracy using as few as one‑fifth of the simulation trajectories required by random sampling. Among the evaluated surrogates, the GRU achieves the highest predictive fidelity, while SINDyC remains the most computationally efficient and interpretable. The probabilistic MvG‑SINDyC surrogate further enables uncertainty quantification and exhibits the largest computational gains under AL.

Abstract:
We introduce Geometric Kolmogorov‑‑Arnold Networks (GeoKANs), a family of geometry‑aware KAN‑type models in which approximation is carried out in learned, geometry‑adapted coordinates rather than in fixed Euclidean input coordinates. GeoKAN achieves this by learning a diagonal Riemannian metric that warps the input before basis expansion and feature mixing. The learned metric provides a geometric inductive bias through local length scaling and volume distortion, and in physics‑informed settings it also affects the differential structure seen by the model. Within this framework, we develop three main variants, namely GeoKAN‑NNMetric, GeoKAN‑γ, and LM‑KAN. For LM‑KAN, we further consider three basis‑specific versions, LM‑KAN‑RBF, LM‑KAN‑Wav, and LM‑KAN‑Fourier. These variants allow us to study geometry‑aware KAN models both as general function approximators and as surrogates in physics‑informed learning. By stretching regions with rapid variation and compressing smoother regions, GeoKAN reallocates representational resolution in a task‑dependent manner, allowing the model to place capacity where it is most needed. As a result, GeoKAN is well suited to sharp, stiff, localized, and strongly non‑uniform regimes arising in scientific machine learning and differential‑equation problems.

Abstract:
Evaluation protocols for learned intrinsic image decomposition on MPI Sintel have been inconsistent. Several prior works split the dataset by frames, which allows spatially similar frames of the same scene to appear in both train and test partitions. We quantify this leakage effect for the first time, across three architectures: a frame‑level split inflates test R_PSNR by 1.6 to 2.0 dB (p less than 0.01 for all three, paired t‑test across 3 seeds) relative to a scene‑level split, confirming an architecture‑independent protocol effect. A three‑point gradient (random/temporal/scene) shows the gap is continuous, and under extended training the frame‑level inflation exceeds 10 dB. We advocate scene‑level splits as the community standard and provide reference numbers for six representative models under this protocol. As a case study within the corrected protocol, we present a physics‑informed decomposition I = R composed with S + N with a source‑separable three‑way heteroscedastic uncertainty head. We empirically verify channel specialization: the non‑Lambertian uncertainty channel shows r = 0.67 cross‑correlation with non‑Lambertian residual error, more than 4 times the texture channel's correlation. We further demonstrate downstream utility: filtering out the 75% highest‑uncertainty pixels reduces reconstruction MSE by 77% on retained pixels, whereas random filtering produces no improvement. The specialization also holds on out‑of‑distribution real photographs. We report negative results for a more elaborate variant combining frequency decomposition, cross‑task supervision, evidential learning, contrastive loss, and test‑time adaptation. Our method reaches 15.98 plus or minus 0.41 dB R_PSNR, within 0.8 dB of a 5‑member Deep Ensemble at one‑fifth the cost, with the unique capability of source‑separated uncertainty.

Abstract:
In this paper, we introduce INEUS, a meshfree iterative neural solver for partial integro‑differential equations (PIDEs). The method replaces the explicit evaluation of nonlocal jump integrals with single‑jump sampling and reformulates PIDE solving as a sequence of recursive regression problems. Like Physics‑Informed Neural Networks (PINNs), INEUS learns global solutions over the entire space‑time domain, yet it offers a more efficient treatment of nonlocal terms and avoids the computationally expensive differentiation of full PIDE residuals. These features make INEUS particularly well suited for high‑dimensional PDEs and PIDEs. Supported by a contraction‑based convergence proof for linear PIDEs, our numerical experiments show that INEUS delivers accurate and scalable solutions for various high‑dimensional linear and nonlinear examples.

Abstract:
Optimizing Reconfigurable Intelligent Surfaces (RIS) is a high‑dimensional combinatorial challenge. Current quantum algorithms often simplify this problem by ignoring physical constraints like mutual coupling, which significantly degrades real‑world performance. Rather than targeting a fully realistic RIS description, we embed progressively more physics‑informed models of mutual coupling into Quadratic Unconstrained Binary Optimization (QUBO) formulations. We evaluate four Ising interaction models (J_ij) for the Quantum Approximate Optimization Algorithm (QAOA), ranging from idealized phase‑only to fully dense physical models. Analyzing a 5 × 5 grid, our results expose a critical trade‑off between spatial pointing accuracy and quantum hardware feasibility. While complete global coupling maximizes beamforming precision, dense Hamiltonians introduce prohibitive routing overhead and complicate convergence on near‑term processors. Ultimately, we demonstrate that while physics‑informed quantum optimization is mathematically viable, sparse, distance‑penalized models remain a necessary compromise for execution on current noisy intermediate‑scale quantum (NISQ) devices.

Abstract:
We propose a novel, machine‑learning‑based framework for constructing lattice fermions using Physics‑Informed Neural Networks (PINNs). Our approach treats the formulation of the Dirac operator as an optimization problem guided by physical requirements, such as symmetries, locality and doubler‑decoupling conditions. We first demonstrate that, when trained to satisfy the Ginsparg‑Wilson (GW) relation as a soft constraint, a neural network reproduces the overlap fermion operator to high numerical accuracy and learns an effective sign‑function mapping without explicitly using a prescribed polynomial or rational approximation. Secondly, we extend the framework from operator construction to machine‑assisted algebraic discovery. Within a generalized polynomial ansatz, the network autonomously drives higher‑order terms to zero and recovers the standard Ginsparg‑Wilson relation. Remarkably, by changing the initial search bias, the same framework also finds a distinct solution corresponding to a Fujikawa‑type generalized GW relation.

Abstract:
Deep neural networks owe their expressive power to nonlinear activation functions. The effective field theory of signal propagation at initialization reveals a few distinct universality classes of activations that exhibit different depth scaling. Tuning across these, especially with analytical control, is an open problem. We show that a statistical mixture of activations, where each neuron independently and randomly draws its activation from a two‑component distribution with mixing fraction p, provides a new mechanism for a continuous phase transition. Applied to a mixture of Tanh and Swish, the transition is sharp in the depth scaling of the preactivation variance, separating a variance‑collapsing from a variance‑inflating phase; at p_c, the network acquires statistical scale invariance, with depth‑independent variance, without sacrificing smoothness. This resolves a longstanding tension, where scale‑invariant propagation has previously required the non‑smooth ReLU family, rendering such networks ill‑suited to curvature‑based optimizers, physics‑informed architectures, and neural‑network quantum states. We corroborate the transition through variance propagation, parallel and perpendicular susceptibilities, and Lyapunov exponents. Training multilayer perceptrons on real datasets reveals non‑monotonic test performance as a function of p, with an optimum near the theoretically predicted p_c, confirming that the initialization‑level transition has direct consequences for learned representations. The quenched activation disorder acts as a structural regularizer, suppressing memorization of corrupted labels while preserving generalization. Our framework establishes statistical activation mixtures as a controlled tool for navigating the phase diagram of deep network universality classes.

Abstract:
We propose a self‑supervised physics‑informed neural network (PINN) framework that adaptively balances physics‑based and data‑driven supervision for scientific machine learning under data scarcity. Unlike prior PINNs that rely on fixed or heuristic weighting of physics residuals and data loss, our approach introduces a learnable blending neuron that dynamically adjusts the relative contribution of each term based on their uncertainties. This mechanism enables stable training and improved generalization without manual tuning. To further enhance efficiency, we integrate a transfer learning strategy that reuses representations from related domains and adapts them to new physical systems with limited data. We validate the framework for the prediction of heat transfer in liquid‑metal miniature heat sinks using only 87 CFD datapoints, where the adaptive PINN achieves an error <8%, outperforming shallow neural networks, kernel methods, and physics‑only baselines. Our framework provides a general recipe for embedding physics adaptively into neural networks, offering a robust and reproducible approach for data‑scarce problems across various scientific domains, including fluid dynamics and material modeling.

Abstract:
Neural operators generally demonstrate strong predictive performance on in‑distribution (ID) problems. However, a critical limitation of existing methods is their significant performance degradation when encountering out‑of‑distribution (OOD) data. To address this issue, this work introduces continual learning into physics‑informed neural operators, with particular emphasis on neural operators built upon the Transolver architecture, and proposes a simple yet effective replay‑based continual learning strategy. The proposed method is fully physics‑informed and does not require labeled data, relying solely on input fields together with physical constraints for training. When new OOD data become available, a small number of past data are incorporated through a distillation‑based constraint to preserve previously acquired knowledge and alleviate catastrophic forgetting. Meanwhile, a transfer learning LoRA is employed to enable rapid adaptation to the new data. The proposed framework is systematically validated on three representative physical problems, including the Darcy flow problem in fluid mechanics, a two‑dimensional hyperelastic brain tumor problem in biomechanics, and a three‑dimensional linear elastic Triply Periodic Minimal Surfaces problem in solid mechanics. The results demonstrate that the proposed method effectively mitigates catastrophic forgetting on previously learned data while maintaining fast adaptability to new data. Compared with conventional joint training strategies, the proposed method significantly improves training efficiency while reducing additional memory usage and computational cost.

Abstract:
From neural ODEs to continuous‑time machine learning, differentiable solvers allow physics, optimization, and simulation to become trainable components within deep learning systems. This has opened the path to a new generation of deep learning frameworks for scientific computing, with many promising applications still emerging. In this paper, we integrate a differentiable chemistry solver into a modified physics‑informed neural network to solve parameterized reaction systems that are inherently stiff. The proposed framework introduces several key components required to overcome limitations of standard physics‑informed neural networks. These include a differentiable chemistry solver, a network architecture for parameterized solutions, and residual weighting tailored to stiff reactions. We evaluate the framework on a set of differential equations related to hydrogen combustion, which include initial/boundary value problems, inverse parameter identification, and a parameterized partial differential equation. Our results highlight the ability of the proposed approach to extend physics‑informed neural networks to stiff chemical systems that were previously inaccessible.

Abstract:
Inferring continuum models directly from video is hampered by two facts: the recorded field is uncalibrated image intensity rather than a physical state, and direct numerical differentiation of noisy frames is unstable. We develop a video‑to‑PDE pipeline that converts grayscale recordings of an ink plume into a normalised scalar field u(x,y,t), isolates a bulk drift \mathbfv(t) from intrinsic spreading via the intensity‑weighted centroid, and identifies an effective transport law by weak‑form sparse regression. Conditioning, threshold‑sweep and random‑centre diagnostics show that overcomplete libraries are strongly collinear; the search is therefore restricted to compact gradient‑based libraries. Coefficients are refined by an inverse physics‑informed network and recalibrated against forward rollouts, with a chronological block bootstrap quantifying uncertainty. The selected reduced model u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^2+0.666\,Δu outperforms advection‑‑diffusion baselines on held‑out frames, retains a positive Laplacian coefficient, and admits a Cole‑‑Hopf reduction to a linear advection‑‑diffusion equation. The framework demonstrates that uncalibrated visual data can yield compact, predictive and structurally interpretable continuum models when discovery, calibration and uncertainty are treated as distinct stages.

Abstract:
Physics‑Informed Neural Networks (PINNs) often struggle to train reliably on stiff and oscillatory dynamical systems due to poor optimization conditioning. While prior work has emphasized representational remedies such as spectral parameterizations, the optimization implications of initial‑condition (IC) embeddings in adaptive spectral PINNs have not been well characterized. In this work, we show that the choice of IC gating function induces explicit time‑dependent gradient scaling, which interacts with spectral representations during training. Using a nonlinear stiff spring‑pendulum ODE as a controlled benchmark, we compare exponential and linear IC gates in combination with fixed and adaptive Fourier spectral trunks. We observe stiffness‑dependent changes in relative dominance for adaptive PINNs: at moderate stiffness (k=20), exponential gating often yields lower error but exhibits heterogeneous behavior across random seeds, whereas at higher stiffness (k=60), linear gating becomes preferable, with additional reversals observed at larger k. These trends hold for both relative L^2 error and maximum pointwise error and are confirmed by paired Wilcoxon signed‑rank tests with Holm correction. Overall, our results demonstrate that IC embeddings are not a neutral design choice in PINNs: the induced gradient scaling materially shapes optimization conditioning in stiff regimes, with distinct sensitivity patterns in baseline and adaptive spectral models.

Abstract:
We develop a new class of physics‑informed neural network approximations for the stationary Oseen equations based on stability‑consistent loss constructions. In contrast to standard PINN formulations, which are typically heuristic, the proposed consistent PINN (CPINN) framework is systematically derived from the stability structure of the continuous problem. Within this setting, we introduce two fundamentally new approaches. First, we design standard CPINN formulations that exhibit clear improvements over conventional PINNs. Second, we propose pressure‑robust CPINN formulations that provably eliminate the influence of gradient forces on the velocity approximation, yielding velocity errors that depend solely on the divergence‑free component of the forcing and are independent of the pressure. The framework accommodates both exactly divergence‑free architectures and unconstrained velocity approximations, providing a unified treatment of these two paradigms. Using techniques from optimal recovery theory, we establish, for the first time in the PINN setting for Oseen‑type problems, quantitative recovery estimates and optimal error bounds for both velocity and pressure under suitable Besov regularity assumptions. In particular, we obtain optimal rates for the velocity in \boldsymbolH^1(Ω) and for the pressure in L^2(Ω). The proposed methodology introduces a pressure‑robust CPINN paradigm for incompressible flows, combining structural consistency, robustness with respect to irrotational forces, and rigorous accuracy guarantees. Numerical experiments corroborate the theoretical findings and demonstrate the effectiveness of the approach.

Abstract:
The accurate determination of electron properties is fundamental to low‑temperature plasma simulations, necessitating precise solutions to the spatially inhomogeneous electron Boltzmann equation (EBE). This work explores the use of physics‑informed neural networks (PINNs) for obtaining solutions to the spatially one‑dimensional (1D) EBE subject to a uniform electric field in atomic gases. Employing the two‑term approximation, the resulting equation for the isotropic distribution is solved directly in kinetic energy space without the conventional transformation to total energy. This approach demonstrates the flexibility of the PINN framework in handling diverse equation formulations. To address the convergence difficulties associated with this class of kinetic equations, a new neural network architecture is introduced. It features a Fourier‑feature input layer, adaptive activation functions, and a scaled multiplicative gating mechanism. It is demonstrated that this formulation preserves robust gradient flow throughout the network, which is critical for learning physically correct solutions. Benchmarking against reference data reveals that the present architecture achieves excellent agreement across both microscopic and macroscopic properties of the electrons. Furthermore, the architecture exhibits strong generalization across different gas types and a defined range of electric field strengths without requiring case‑specific hyperparameter tuning. Ultimately, the excellent accuracy achieved here validates the applicability of the present PINN method.

Abstract:
Implicit neural representations (INRs) have emerged as a powerful paradigm for medical imaging via physics‑informed unsupervised learning. Classical INRs optimize an entire network from scratch for each subject, leading to inefficient training and suboptimal imaging quality. Recent initialization‑based approaches attempt to inject population priors into pre‑trained networks, yet they rely on high‑quality images and often suffer from catastrophic forgetting during fine‑tuning. We present DisINR, a novel INR framework that explicitly disentangles shared and subject‑specific representations. DisINR introduces a shared encoder‑decoder pair and subject‑specific encoders, whose features are jointly decoded for image reconstruction. By integrating differentiable forward models, it pre‑trains the shared modules directly from limited raw measurements, removing the need for pre‑acquired high‑quality images. During test‑time adaptation, only the subject‑specific encoder is optimized, while the shared pair remains frozen, effectively preserving learned priors. Extensive evaluations on three representative medical imaging tasks show that DisINR significantly outperforms state‑of‑the‑art INRs in both reconstruction accuracy and efficiency.

Abstract:
Quantum metrology holds the potential to enhance magnetic field sensing beyond current limits. However, in the presence of realistic noise, this advantage degrades to the Standard Quantum Limit. While recent algorithmic and variational techniques attempt to recover this scaling, they are hindered by stringent control requirements on the probe state that are infeasible in the near term, or by barren plateaus and interpretability issues inherent to black‑box variational quantum circuits. Here, we introduce Variational Inference and Sensing with Twin Ansätze (VISTA), a closed‑loop protocol that combines passive sensing, or where the probe state is left to evolve without any active control, with physics‑informed variational optimization. In the VISTA framework, a probe state evolves under a Lindbladian master‑equation, and is compared, via the Swap test, to a parameterized ``quantum twin", a shallow quantum circuit designed to mimic the underlying pure‑state or Lindbladian master‑equation dynamics. By restricting the optimization space to the physical parameters of interest, VISTA circumvents barren plateaus. We demonstrate that by coupling the protocol with a classical optimizer and high shot counts, VISTA can temporarily achieve near‑Heisenberg scaling for moderately noisy qubits over a finite range of system sizes. Furthermore, we introduce a Quasi‑Normalization technique that sharpens the loss gradients, enabling simultaneous extraction of both the coherent signal θ and the environmental noise rate γ with low absolute error. Finally, we extend VISTA to the multi‑parameter vector metrology regime, enabling simultaneous parameter extraction from a transverse‑magnetic‑field Hamiltonian. By eliminating the need for complex, open‑loop control and processing, VISTA offers a highly practical, resource‑efficient framework for near‑ to intermediate‑term quantum sensors.

Abstract:
This paper develops convolutional neural network (CNN) methods for simultaneous approximation and elliptic boundary value problems on compact Riemannian manifolds. We establish simultaneous Sobolev approximation results for single‑ and multichannel CNNs, showing that manifold functions and their derivatives can be approximated with rates governed by the intrinsic dimension and the smoothness gap, rather than by the ambient dimension, thereby mitigating the curse of dimensionality. Building on this approximation theory, we propose a physics‑informed CNN (PICNN) framework specially designed for boundary value problems. The main numerical issue is a boundary‑norm mismatch: standard PINNs usually impose boundary data through low‑order, often L2‑type, penalties, whereas elliptic stability requires Sobolev trace control. We address this by introducing a spectral boundary loss based on the boundary Laplace‑Beltrami operator, which represents trace errors as weighted frequency energies and relates truncation error to boundary eigenvalue decay. This avoids smooth auxiliary constructions required by exact boundary enforcement and singular double integrals arising in Sobolev‑Slobodeckij penalties, while enabling implementations based on Fast Fourier Transforms (FFTs) or precomputed spectral bases on structured boundaries. Numerical experiments demonstrate improved accuracy, convergence, and stability over standard PINNs.

Abstract:
AI data centers experience rapid fluctuations in power demand due to the heterogeneity of computational tasks that they have to support. For example, the power profile of inference and training of large language models (LLMs) is quite distinct and big divergences can result in the instability of the underlying electricity grid. In this paper we propose, to the best of our knowledge, the first physics‑informed DLinear time‑series model that can accurately forecast power utilization of an AI data center 5‑80 minutes (short‑term forecasting) into the future. The physics, based on a multi‑node lumped thermal resistance‑capacitance (RC) network consistent with Newton's law of cooling, is captured using newly derived time‑dependent ordinary differential equations (ODE) that separately models and interlinks power consumption with the GPU compute and memory utilization and temperature. The resulting model, that we refer to as PI‑DLinear, trained and evaluated on a real AI data center dataset and is not only more accurate than the state‑of‑the‑art (SOTA) models tested, but the forecast profile respects the underlying physics under power throttling and load transient events. Relative to the SOTA transformer‑based and non‑transformer‑based models, improvements in forecasting accuracy (averaged across all look‑back and prediction windows) range from 0.782%‑39.08% for MSE, 0.993%‑51.82% for MAE, and 0.370%‑22.28% for RMSE.

Abstract:
Stratocumulus cloud decks exhibit bistability between patterns of high (closed cells) and low (open cells) cloud fraction. Localized transitions between these two states (pockets of open cells) have been observed but their underlying mechanism remains unclear. We model stratocumulus and their interaction with atmospheric aerosol as a data‑driven and physics‑informed stochastic dynamical system with time‑dependent parameters. This allows us to show that pockets of open cells result from noise‑induced transitions between the stratocumulus patterns. We find comparable timescales for these transitions, mesoscale self‑organization into patterns and the evolution of large‑scale parameters. This lack of timescale separation corresponds to an aerosol memory in cloud evolution and means that the sampling of stratocumulus states by polar‑orbiting satellites lacks the encoding of process information that would be present for an asymptotic and ergodic sampling.

Abstract:
The dual norm characterisation of weak solutions of second‑order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the H^‑1 norm of a residual requires a supremum over an infinite‑dimensional function space. We prove that the H^‑1 norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain. Crucially, realisations of this random test function have negative Sobolev regularity for d \geq 2, yet this roughness is not an obstacle: averaging over the distribution exactly recovers the correct weak topology, independently of the differential operator. This equivalence introduces the notion of stochastically weak solutions, which coincide with classical weak solutions, and motivates stochastic variational physics‑informed neural networks (SV‑PINNs): neural networks trained by minimising an empirical approximation of the stochastic norm of the PDE residual. Although instantiated here with neural networks as trial spaces, the underlying principle is independent of the approximation architecture and suggests a broader paradigm for numerical methods based on stochastic rather than deterministic test spaces. The framework extends naturally to higher‑order elliptic, parabolic and hyperbolic equations and to abstract operator equations on Hilbert spaces. As a proof of concept, we present numerical experiments on eight challenging second‑order linear elliptic problems spanning high‑frequency and multi‑scale solutions, indefinite operators, variable coefficients, and non‑standard domains, in which SV‑PINNs consistently and significantly outperform standard PINNs, recovering solutions to within one percent relative error in hundreds of L‑BFGS steps.

Abstract:
Solving inverse problems in dynamical systems governed by high‑dimensional coupled ordinary differential equations (ODEs) is a ubiquitous challenge in scientific machine learning. In many real‑world applications, researchers seek to uncover unknown parameters or model unknown dynamics even as the underlying physics is only partially characterized, and observations are sparse and limited to specific measurable channels. While physics‑informed neural networks (PINNs) are ideal for inverse inference under partial observability, existing PINNs typically rely on task‑specific joint optimization, which suffers from optimization difficulties and poor generalization. In this paper, we propose a meta‑inverse physics‑informed neural network (MI‑PINN) that reformulates inverse modeling as a two‑stage meta‑learning problem. MI‑PINN first learns a physics‑aware representation across multiple tasks, and then performs inverse modeling by optimizing task‑specific unknowns while keeping the learned representation fixed. This two‑stage formulation significantly reduces the parameter search dimension, thereby improving sample efficiency and enabling accurate inference. To handle multi‑scale dynamics common in these high‑dimensional ODE systems, we further introduce an adaptive clustering‑based multi‑branch learning scheme. We demonstrate the effectiveness of MI‑PINN on whole‑body physiologically based pharmacokinetic (PBPK) models with up to 33 coupled ODEs, using paracetamol and theophylline under intravenous and oral dosing scenarios. Experimental results show that MI‑PINN enables accurate recovery of masked kinetic parameters and reconstruction of missing mechanistic terms despite limited clinical observations.

Abstract:
Spatiotemporal forecasting in physical systems, such as large‑scale traffic networks, requires modeling a dual dynamic: continuous macroscopic rhythms and discrete, unpredictable microscopic shocks. While Neural Ordinary Differential Equations (ODEs) excel at capturing smooth evolution, their inherent Lipschitz continuity constraints inevitably cause severe over‑smoothing when confronting abrupt anomalies. Recent physics‑informed methods attempt to bypass this by penalizing numerical integration errors to enforce manifold smoothness. However, we mathematically reveal that such rigid regularization inherently triggers gradient conflicts and ``attention collapse,'' stripping the model of its sensitivity to anomalies. To resolve this continuity‑shock dilemma, we propose Local Truncation Error‑Guided Neural ODEs (LTE‑ODE). Rather than treating numerical error as a nuisance to be eliminated, we innovatively repurpose the Local Truncation Error (LTE) as an unsupervised forward inductive bias. By mapping the LTE into a dynamic spatial attention mask, our architecture gracefully preserves high‑precision continuous ODE evolution in stable regions, while adaptively triggering a discrete compensation branch exclusively at shock points. Trained purely end‑to‑end without manifold penalties, LTE‑ODE achieves state‑of‑the‑art performance on multiple large‑scale benchmarks, exhibiting exceptional robustness against highly non‑linear fluctuations. Furthermore, our ablation on integration steps demonstrates high deployment flexibility, allowing the model to seamlessly adapt to varying hardware memory constraints in real‑world applications.

Abstract:
Pairwise Fisher graphs capture local covariance information, but they cannot distinguish an irreducible multi‑observable radiation pattern from a collection of ordinary pairwise correlations. We show that this missing structure is naturally supplied by higher‑order Fisher tensors. In a finite basis of binned EECs, ECFs, or EFPs, and in the natural exponential‑family coordinates generated by that basis, the same local tensor has three equivalent interpretations: a coefficient in the local Kullback‑Leibler expansion, a connected cumulant of the chosen correlator observables, and a signed weight on a hyperedge linking those observables. This gives an exact Fisher‑correlator‑hypergraph triality in the local exponential‑family embedding. The triality provides a direct construction of physics‑informed hypergraphs from correlator data. Extending the quadratic Fisher matrix to the first non‑trivial higher tensor identifies genuinely connected multi‑observable radiation patterns, supplies hyperedge weights for higher‑order Laplacians and message passing, and gives a principled criterion for compressing observable bases beyond pairwise information. We develop these constructions and spell out why the exact cumulant interpretation is special to natural exponential‑family coordinates. We illustrate the framework in four applications. In a minimal local‑KL study, the cubic Fisher tensor reduces the KL truncation error and isolates the dominant triplet structure. In a two‑versus‑three prong jet substructure benchmark, the hypergraph selector improves compressed‑basis classification. In a 33‑observable basis‑design problem, the Fisher hypergraph retains more third‑order local response at twelve observables. A low‑capacity learning benchmark then shows how the same Fisher hyperedges can be used as an interpretable inductive bias for message passing on correlator observables.

Abstract:
Exchange‑only (EO) spin qubits in silicon realise all‑electrical qubit control through pairwise Heisenberg exchange interactions, making them attractive for scalable quantum computation. Their principal vulnerability is charge noise, which couples multiplicatively to the exchange coupling and degrades gate fidelity. We present a \emphtwo‑stage Physics‑Informed Neural Network (PINN) framework for per‑gate pulse optimisation. In Stage~I (iterations~1‑‑100) the PINN maximises the noise‑averaged gate fidelity toward a threshold of \Fth=0.99; the pulse duration is held fixed at its nominal hardware value. Once the threshold is crossed, Stage~II (iterations~101‑‑250) progressively compresses the total pulse time while maintaining F\geq\Fth via continuous fine‑tuning of the pulse‑shape parameters. The cost function is a Monte‑Carlo ensemble mean‑squared error (MSE) averaged over N_\rm real=2000 quasi‑static Gaussian noise realisations drawn fresh at every iteration. We benchmark the framework on the single‑qubit gate set \X,Y,Z,H\ and the two‑qubit set \X,Y,Z,H,\mathrmCX\ at noise levels \sigmaJ/J\in\1%,5%,10%\. All single‑qubit gates cross \Fth within the first 100 iterations across all noise levels; Stage~II then reduces pulse durations by 20‑‑40% from their nominal values. The two‑qubit gates follow the same two‑phase behaviour, with the CX gate compressing from its nominal \SI31\nano\second to \approx\SI22\nano\second at 1% noise.

Abstract:
This study proposes a novel approach to extract topological properties, specifically the Euler characteristic, from input images using neural networks without relying on large pre‑existing datasets but with a single geometric image. Inspired by solid‑state physics, where topological properties of magnetic structures are derived from spin field analysis, our model generates a unit vector field from an image, interpreted as a spin configuration. The Euler characteristic is then predicted by computing the skyrmion number of this generated spin configuration. Remarkably, the network learns to construct chiral magnetic textures without access to ground‑truth chiral spin configurations, relying instead on only a single, simple geometric image and the straightforward skyrmion number computation. Furthermore, spin configurations generated by independently trained networks can be non‑unique due to inherent degrees of freedom. To constrain these degrees of freedom and further refine the spin configuration, we incorporate a magnetic Hamiltonian, comprising exchange interaction, Dzyaloshinskii‑Moriya (DM) interaction, and anisotropy, as an additional, physics‑informed loss function. We validate the model's efficacy on complex geometrical shapes and demonstrate its applicability to practical tasks.

Abstract:
The lack of analytical models describing diffusion time dependence at intermediate time scales in complex tissue microstructure limits the accurate quantification of extracellular diffusivity and tissue microstructure. We introduce TRACED, a biophysical model that incorporates diffusion time dependence in cell distributions to quantify pathologically‑relevant properties in solid tumors. Neural networks were trained on Monte Carlo diffusion simulations using sphere distribution‑based geometries to enable the rapid computation of time‑dependent diffusion MRI signals in cell populations of variable cell size. Model sensitivity and fit performance were assessed via simulation. Diffusion data from eight mixed‑grade glioma patients was fitted using the TRACED model. Data fitting was performed using a novel physics‑informed transfer learning pipeline, Sim2PINN. In two patients, cell size measurements were compared directly with image‑localized histology. Simulation results indicate improved parameter estimation compared to the simple two‑compartment model. TRACED enabled the simultaneous in vivo quantification of intracellular volume fraction, cell size distribution, extracellular intrinsic diffusivity, and tortuosity in glioma patients. Neural network implementations of diffusion time‑dependence and tortuosity showed behavior consistent with coarse‑graining and effective medium theory, respectively. Future work will explore the clinical utility of TRACED parameters in additional patients.

Abstract:
Physics‑informed neural networks (PINNs) have recently emerged as a promising framework for integrating data‑driven learning with physical knowledge. In this work, we propose a coupled PINN approach for the joint reconstruction of indoor temperature and humidity dynamics in greenhouse environments, together with simultaneous identification of key model parameters. The method incorporates a reduced‑order physically motivated model into the learning process, enabling consistent estimation under sparse and noisy observations. The artificial intelligence contribution lies in the development of a coupled physics‑informed neural learning framework that integrates governing dynamical constraints into neural network training, while the engineering application focuses on greenhouse climate state reconstruction and parameter identification. The proposed framework is evaluated on a controlled synthetic benchmark that mimics diurnal forcing conditions. Compared with a purely data‑driven neural network baseline, the coupled PINN achieves improved reconstruction accuracy, reducing temperature and humidity errors while maintaining high coefficients of determination. The improvement is particularly pronounced in the humidity channel, where latent moisture dynamics are more difficult to infer from limited measurements. In addition to accurate state reconstruction, the method successfully recovers the dominant physical parameters governing the system dynamics, demonstrating its ability to learn interpretable representations beyond data interpolation. These results highlight the potential of physics‑informed learning for greenhouse climate modeling and, more broadly, for data‑scarce environmental systems.

Abstract:
Quantum physics‑informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics‑informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics‑informed neural networks (GQPINNs), a symmetry‑aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum‑circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite‑group and compact Lie‑group symmetries as inductive biases through problem‑specific equivariant generator sets . Using a twirling‑based construction, we derive symmetry‑preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry‑adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry‑aware quantum‑circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum‑enhanced scientific machine learning.

Abstract:
Physics‑informed neural networks provide a mesh‑free framework for solving partial differential equation‑governed problems in solid mechanics. However, most existing formulations in linear elasticity still learn the displacement field directly, which does not explicitly exploit the analytic structure of two‑dimensional elasticity and becomes restrictive for fracture problems with crack face discontinuities and crack tip singularities. Moreover, existing Kolosov‑‑Muskhelishvili informed neural network formulations still rely on residual‑based loss functions with multiple boundary and interface terms, whereas a variational concept has not yet been established. To address these issues, a variational Kolosov‑‑Muskhelishvili informed neural network framework for two‑dimensional linear elastic problems with and without cracks is proposed in this work. The solution is represented by two holomorphic Kolosov‑‑Muskhelishvili potentials and trained through an energy‑based loss function derived from the principle of minimum total potential energy. For crack problems, a discontinuous stress potential representation is further introduced to embed the crack face condition and crack tip singularity directly into the solution ansatz. The proposed framework is validated on a series of benchmark problems with or without crack problems. The results show that variational Kolosov‑‑Muskhelishvili informed neural network can accurately predict stress and displacement field as well as stress intensity factors. Compared with traditional neural network models, it achieves higher accuracy, simpler loss construction, and faster convergence in the considered cases. Overall, the proposed variational Kolosov‑‑Muskhelishvili informed neural network provides an effective and physically consistent variational framework for two‑dimensional linear elastic fracture analysis.

Abstract:
Multiphysics simulation is critical for system‑technology co‑optimization (STCO) in chiplet‑based design, but repeated finite‑element solutions of PDE‑governed problems are computationally expensive in parametric design exploration. This paper proposes a variational matrix‑learning Fourier network (VMLFN) for efficient parametric multiphysics surrogate modeling. VMLFN constructs a log‑space sine neural representation with randomly sampled spectral frequencies, frequency‑dependent decay regulation, and embedded Dirichlet boundary conditions. With fixed hidden‑layer parameters, the output‑layer weights are determined by reformulating the governing PDEs into variational weak forms and enforcing the stationarity condition of the resulting energy functional. This converts physics‑informed training into a linear matrix‑solving problem, requiring only first‑order derivatives and avoiding both high‑order automatic differentiation and penalty‑coefficient tuning. A heuristic frequency‑scanning algorithm is further introduced to select a problem‑adaptive maximum frequency that covers the dominant spectral range of the target problem. The proposed method is validated on heat conduction, solid mechanics, and Helmholtz wave propagation problems. Results from five benchmark cases demonstrate that VMLFN delivers accurate full‑field predictions with substantial speedup over conventional physics‑informed neural networks and repeated finite‑element simulations.

Abstract:
We develop a physics‑informed neural networks (PINNs) framework for the inverse scattering problem in nuclear physics and apply it to the P_3/2 partial wave of neutron‑alpha elastic scattering. The radial potential is represented by a feed‑forward network whose output is multiplied by a Gaussian envelope, embedding the finite‑range condition directly into the architecture rather than through a soft penalty term. This distinction proves essential: without the envelope, the optimizer produces potentials with non‑vanishing tails and the resulting phase shifts remain inconsistent with the data regardless of training duration, demonstrating that hard structural constraints are indispensable for physically meaningful solutions to nuclear inverse problems. Phase shifts are generated at each scattering energy by numerically integrating the variable‑phase equation with a fourth‑order Runge‑Kutta scheme, making the entire pipeline end‑to‑end differentiable.Training converges stably to a loss near 3×10^‑4 and recovers a smooth, purely attractive central potential with a well depth of ‑60.47~MeV. Adding the centrifugal barrier to the learned potential reveals a well‑defined barrier‑well structure that naturally accounts for the P_3/2 resonance. The extracted resonance parameters, E_r = 0.95~MeV and Γ_r = 0.78~MeV, together with the P‑wave effective‑range parameters, are in good agreement with expected values. A leave‑one‑out analysis confirms that the reconstruction is stable against the removal of any single data point. These results establish physics‑guided machine learning as a reliable route to potential reconstruction from nuclear scattering data.

Abstract:
AC optimal power flow (ACOPF) is foundational yet computationally expensive in power grid operations, driving learning‑based surrogates for large‑scale grid analysis. These surrogates, however, often fail to generalize across network topologies, a critical gap for deployment on grids not seen during training and for routine operational what‑if studies. We introduce LUMINA‑Bench, a comprehensive benchmark suite for ACOPF surrogate learning covering multi‑topology pretraining, transfer, and adaptation. The benchmark evaluates homogeneous and heterogeneous architectures under single‑ and multi‑topology learning settings using unified metrics that capture both predictive accuracy and physics‑informed constraint violations. We additionally compare constraint‑aware training objectives, including MSE, augmented Lagrangian, and violation‑based Lagrangian losses, to characterize accuracy‑robustness trade‑offs across settings. Data processing, training, and evaluation frameworks are open‑sourced as the LUMINA suite to support reproducibility and accelerate future research on feasibility‑aware OPF surrogates.

Abstract:
Variational quantum algorithms are promising for near‑term quantum computing, but are severely limited by hardware noise and the substantial circuit overhead required for error mitigation methods such as Zero‑Noise Extrapolation (ZNE). We propose a Physics‑Informed Denoising Network (PIDN) that reduces the cost of ZNE by learning a surrogate model of its optimization dynamics. By viewing the variational update as a trajectory in the parameter space, PIDN is trained to reproduce ZNE‑mitigated expectation values and gradient directions while incorporating a physics‑informed loss that preserves the gradient descent dynamics. Once trained, PIDN replaces repeated multi‑noise evaluations with denoised expectation and gradient estimation directly from the current noisy observation and the historical trajectory, significantly reducing circuit executions. We benchmark the approach on the quantum approximate optimization algorithm for 3‑regular graphs, Sherrington‑Kirkpatrick, and transverse‑field Ising models, as well as the variational quantum eigensolver for LiH, BeH_2 and H_2O. Across all tasks, PIDN attains performance comparable to ZNE, while reducing the number of circuit executions by a factor of approximately 4 to 6. Gradient cosine similarity with ZNE remains above 0.95 throughout training. Robustness analysis shows that PIDN fails only when ZNE itself becomes unreliable, and ablation studies confirm the necessity of the physics‑informed loss for maintaining directional consistency. We further find that PIDN tracks optimization dynamics most accurately when the effective loss landscape retains strong low‑frequency structure. These results establish PIDN as a scalable, resource‑efficient strategy for noise‑resilient variational optimization in the noisy intermediate‑scale quantum regime.

Abstract:
AC Optimal Power Flow (ACOPF) and Security‑Constrained Unit Commitment (SCUC) are fundamental optimization problems in power system operations. ACOPF serves as the physical backbone of grid simulation and real‑time operation, enforcing nonlinear power flow feasibility and network limits, while SCUC represents a core market‑level decision process that schedules generation under operational and security constraints. Although these problems share the same underlying transmission network and physical laws, they differ in decision variables and temporal coupling, and prior learning‑based approaches address them in isolation, resulting in disjoint models and representations.We propose a learning framework that jointly models ACOPF and SCUC through a shared graph‑based backbone that captures grid topology and physical interactions, coupled with task‑specific decoders for static and temporal decision‑making. Training includes solver supervision with physics‑informed objectives to enforce AC feasibility and inter‑temporal operational constraints. To evaluate generalization, we assess cross‑case transfer on unseen grid topologies for ACOPF and SCUC without retraining, and systematic generalization on the UC‑ACOPF problem using unsupervised, physics‑based objectives and a power‑dispatch consensus mechanism. Experiments across multiple grid scales demonstrate improved performance and transferability relative to existing learning‑based baselines, indicating that the model can support learning across heterogeneous power system optimization problems.

Abstract:
Electron‑beam irradiation limits atomic‑resolution characterization of beam‑sensitive hybrid materials, yet quantitative models that connect in situ spectroscopy to dose‑dependent degradation remain scarce. Here we use a physics‑informed neural network (PINN) to model beam‑induced spectral evolution in MIL‑101(Fe) from an in situ low‑loss electron energy‑loss spectroscopy (EELS) dose series. Each spectrum is reduced to fixed‑window low‑loss descriptors, \tilde n_\mathrmeff,j(Φ)=\int_\mathcalW_jS(E,Φ)\,dE, evaluated over nominal π‑‑π^, C‑‑C, C‑‑O, and M‑‑O windows. These descriptors are relative window‑integrated low‑loss spectral areas, not absolute f‑sum‑rule effective electron numbers. For each spectral channel, a latent integrity variable C_i(Φ) obeys the same uncoupled power‑law degradation equation in normalized dose space, dC_i/dϕ=‑k_i C_i^p_i, regularized by monotonicity, boundedness, and a single hierarchy prior k_\mathrmC\text‑O\geq k_\mathrmC\text‑C. Applied to nine dose frames spanning 152‑‑1368~e^‑/Å^2, the ensemble PINN identifies C‑‑O and C‑‑C as the most strongly dose‑sensitive linker‑associated channels, with half‑integrity thresholds of approximately 1.0×10^3~e^‑/Å^2. The 1‑‑3~eV π‑‑π^‑labelled window increases with dose and is therefore interpreted as a mixed low‑energy response, likely involving oscillator‑strength redistribution rather than direct monotonic loss of a single bond population. The framework provides a dose‑dependent, spectroscopy constrained description of MOF degradation while also defining the limits of what fixed‑window low‑loss EELS can assign without independent chemical‑state validation.

Abstract:
Physics‑Informed Neural Networks (PINNs) offer a flexible paradigm for solving differential equations by embedding governing laws into the training objective. A persistent limitation is instance specificity: standard PINNs typically require retraining for each new forcing term, boundary/initial condition, or parameter setting. One‑shot transfer learning (OTL) addresses this bottleneck for linear operators by freezing a pretrained latent representation and computing optimal output weights in closed form, but for nonlinear problems closed‑form adaptation is generally unavailable because the loss is nonconvex in the output layer. In this paper we substantially broaden the class of nonlinearities amenable to one‑shot PINN transfer by combining OTL with Chebyshev polynomial surrogates. We approximate general smooth weakly nonlinear terms by truncated Chebyshev expansions over a prescribed solution range, yielding a polynomial nonlinearity that can be handled by a perturbative decomposition into linear subproblems. A multi‑head PINN learns a reusable latent space associated with the dominant linear operator; at test time, solutions to new instances are obtained via a sequence of closed‑form linear solves in the output layer, without retraining the network body. We provide a unified derivation of the framework for ODEs and PDEs and demonstrate accuracy and fast online adaptation on nonlinear benchmarks, including non‑polynomial and singular ODE nonlinearities as well as a reaction‑diffusion PDE with saturating kinetics, demonstrating the method's utility in many‑query regimes.

Abstract:
The building energy community lacks a foundational thermal model, i.e., a single pretrained model capable of generalizing across diverse buildings, climates, and control strategies without building‑specific calibration. Achieving this vision requires architectural principles that capture universal thermal dynamics rather than memorizing building‑specific patterns. We take a step toward this goal by presenting a physics‑informed transformer architecture that embeds domain knowledge, e.g., derivative enrichment and Euler‑based numerical integration, into a decoder‑only framework. We incorporate static building features extracted from simulation models and employ Rotary Position Embedding attention to capture temporal dependencies. Evaluated on the CityLearn dataset spanning 247 residential buildings across three climate zones, our model achieves one‑step prediction accuracy (RMSE of 0.30°C in Texas, 0.29°C in Vermont) while outperforming both traditional baselines and fine‑tuned Time‑Series Foundation Models. We also demonstrate zero‑shot transferability: models trained on as few as two buildings generalize to unseen buildings and climate zones without fine‑tuning. Despite the limitation of simulated residential buildings, our results establish physics‑informed architectural principles as a promising foundation for universal building thermal models.

Abstract:
To address the initial singularity inherent in solutions to fractional partial differential equations (fPDEs), we propose an accelerated Alikhanov discretization formulation implemented on nonuniform time grids. Based on the physics‑informed neural networks (PINNs) framework, we introduce an Alikhanov‑extended fractional PINNs (XfPINNs) architecture that combines high‑order temporal discretization and deep learning. The nonlocal memory term in fPDEs leads to high computational cost, while the weak singularity near t\to 0^+ can deteriorate accuracy on uniform meshes. To separate temporal discretization effects from optimization and sampling errors, we further develop an auxiliary time‑marching configuration that enables auditable temporal‑convergence studies under controlled training tolerances. This architecture can solve general nonlinear fPDEs. The XfPINNs approach is designed for forward and inverse problems, allowing for data‑driven solution reconstruction and parameter estimation. First, the neural network approximates the solution of nonlinear fPDEs; then, an adaptive activation function accelerates convergence and enhances training efficiency. The optimization framework embeds a variational loss function constructed from the Alikhanov scheme, where the initial and boundary conditions are imposed using a combination of hard and soft constraints. Numerical experiments, including cases with known and unknown exact solutions which demonstrate the robustness, computational efficiency, and significant CPU time savings of the Alikhanov‑XfPINNs method.

Abstract:
Learning‑based fluence map prediction offers a fast alternative to iterative inverse planning in intensity‑modulated radiation therapy (IMRT), but its robustness under realistic distribution shifts remains unclear. We study a two‑stage transformer pipeline that maps anatomy (CT and contours) to dose and then to beamlet fluence maps. We compare fluence‑stage transformer backbones with hierarchical, global, and hybrid attention, trained with a physics‑informed loss enforcing energy consistency. Robustness is evaluated under geometric perturbations, radiometric noise, reduced training data, and domain shifts using a prostate IMRT dataset, with additional evaluation of the dose stage on public datasets. Results show smooth degradation under moderate perturbations but sharp failures under severe rotations and noise. Hierarchical transformers (e.g., SwinUNETR) exhibit slower growth in upper‑quartile energy error, indicating improved robustness. We further show that SSIM alone fails to capture clinically relevant errors, highlighting the need for physics‑informed evaluation.

Abstract:
Form‑finding of unilateral membrane structures is commonly addressed by solving equilibrium equations with Finite Element Methods (FEMs). This paper investigates Physics‑Informed Neural Networks (PINNs) as an alternative, where the equilibrium equation is enforced by minimizing its residual at collocation points during neural‑network training rather than by solving a mesh‑based discretized system. This approach is well suited to form‑finding problems based on Membrane Equilibrium Analysis (MEA), in which the unknown membrane surface is governed by a second‑order elliptic Partial Differential Equation (PDE) with Dirichlet boundary conditions. Two PINN formulations are proposed and compared: a soft‑Boundary Condition (soft‑BC) approach, where the boundary conditions are imposed through a penalty term, and a hard‑BC approach, where they are satisfied exactly by construction through distance and lift functions. The methods are assessed on three case studies with different geometrical complexity, including compression‑only and tension‑only stress states, and combined self‑weight, concentrated vertical loads, and horizontal actions. Both formulations produce membrane surfaces in close agreement with solutions obtained using an FEM‑based PDE solver. The hard‑BC formulation gives smaller errors and a smoother residual distribution, especially near the boundary, showing that exact enforcement of the Dirichlet conditions improves overall accuracy. The soft‑BC formulation still provides structurally meaningful solutions and remains attractive when simpler implementation is preferred and limited relaxation of the boundary data is acceptable. Overall, the results show that PINNs are a viable alternative for MEA‑based form‑finding.

Authors: Jay Lee, Hanqi Su, Marco Macchi, Adalberto Polenghi, Wei Wu, Zhiheng Zhao, George Q. Huang, Kiva Allgood, Devendra Jain, Benedikt Gieger, Vibhor Pandhare, Soumyabrata Bhattacharjee, Ram Mohril, Lingbao Kong, Qiyuan Wang, Xinlan Tang, Sungjong Kim, Chan Hee Park, Byeng D. Youn, Guo Dong Goh, Xi Huang, Wai Yee Yeong, Yung C Shin, He Zhang, Zitong Wang, Fei Tao, Jagjit Singh Srai, Satyandra K. Gupta, Byung Gun Joung, Albin John, John W. Sutherland, Sang Won Lee, Olga Fink, Vinay Sharma, Faez Ahmed, Wei Chen, Mark Fuge, Arild Waaler, Martin G. Skjæveland, Dimitris Kyritsis, Wei Chen, VispiNevile Karkaria, Yi-Ping Chen, Ying-Kuan Tsai, Joseph Cohen, Xun Huan, Jing Lin, Liangwei Zhang, Gregory W. Vogl, Aaron W. Cornelius, Xiaodong Jia, Dai-Yan Ji, Takanobu Minami, Ruoxin Wang

Abstract:
The evolution of artificial intelligence (AI) and machine learning (ML) is reshaping smart manufacturing by providing new capabilities for efficiency, adaptability, and autonomy across industrial value chains. However, the deployment of AI and ML in industrial settings still faces critical challenges, including the complexity of industrial big data, effective data management, integration with heterogeneous sensing and control systems, and the demand for trustworthy, explainable, and reliable operation in high‑stakes industrial environments. In this roadmap, we present a comprehensive perspective on the foundations, applications, and emerging directions of AI and ML in smart manufacturing. It is structured in three parts. The first highlights the foundations and trends that frame the evolution of AI in smart manufacturing. The second focuses on key topics where AI is already enabling advances, including industrial big data analytics, advanced sensing and perception, autonomous systems, additive and laser‑based manufacturing, digital twins, robotics, supply chain and logistics optimization, and sustainable manufacturing. The third section explores non‑traditional ML approaches that are opening new frontiers, such as physics‑informed AI, generative AI, semantic AI, advanced digital twins, explainable AI, RAMS, data‑centric metrology, LLMs, and foundation models for highly connected and complex manufacturing systems. By identifying both opportunities and remaining barriers across these areas, this roadmap outlines the advances needed in methods, integration strategies, and industrial adoption. We hope this roadmap will serve as a guide for researchers, engineers, and practitioners to accelerate innovation, align academic and industrial priorities, and ensure that AI‑driven smart manufacturing delivers reliable, sustainable, and scalable impact for the future of manufacturing ecosystems.

Abstract:
This paper deals with solving the 2D Helmholtz equation on non‑parametric domains, leveraging a physics‑informed neural operator network based on the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary geometry at its center. This inclusion acts as a scatterer for an incoming harmonic wave. The aim is to learn the operator linking the geometry of the scatterer to the resulting scattered field. A signed distance function to the boundary of the inner inclusion, evaluated at several points in the domain, is used to encode its geometry. It serves as input for the branch part of the DeepONet architecture, while local information is used as input for the trunk part. This approach enables the encoding of arbitrary geometries, whether they are parameterized or not. The evaluation of the model on unseen geometries is compared with its finite element method (FEM) equivalent to test its generalization capabilities. The trained network weights implicitly embed the local physics and their interaction with the domain geometry. If the training space sufficiently covers the target evaluation space, the model can generalize accordingly. Furthermore, it can be refined to extend to another region of interest without retraining from scratch. This framework also avoids the need to remesh the domain for each geometry. The proposed approach delivers a computationally lighter surrogate model than FEM alternatives and avoids relying on FEM‑generated training data.

Abstract:
The purpose of the current work is the development of an approach to account for quasi‑static mechanical equilibrium in empirical (i.e., data‑based) models for the stress field employing neural approximations (NAs), which include neural networks (NNs) and neural operators (NOs), in particular Fourier NOs (FNOs). Rather than including such constraints from physics in the loss function as done in the (now standard) physics‑informed approach, the current approach incorporates or "encodes" such constraints directly into the architecture of the NA. As a result, both NA training and output are physically constrained in the physics‑encoded approach, in contrast to the physics‑informed approach, in which only training is physically constrained. For the current constraint of divergence‑free stress, a novel encoding approach based on a stress potential is proposed. As a "proof‑of‑concept" example application of the current approach, a physics‑encoded FNO (PeFNO) is developed for a heterogeneous polycrystalline material consisting of isotropic elastic grains and subject to uniaxial extension. Stress field data for this purpose are obtained from the numerical solution of corresponding boundary‑value problems for quasi‑static mechanical equilibrium. For comparison with the PeFNO, this data is also employed to develop an analogous physics‑guided FNO (PgFNO) and physics‑informed FNO (PiFNO). As expected theoretically, and confirmed by this computational comparison, for comparable accuracy of the stress field itself as compared to the data, the stress field output by the trained and tested PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the PgFNO or the PiFNO.

Abstract:
Physics‑Informed Neural Networks have become a powerful mesh‑free method for solving partial differential equations, but their performance is often limited by spectral bias. Specifically, in standard MLPs used in PINNs, the global parameter coupling causes the model to prioritize learning low‑frequency components, resulting in slow convergence for high‑frequency details. To overcome this limitation, we introduce the Physics‑Informed Local Implicit Representation (PILIR). Our approach separates the global physical domain into a discrete latent feature space and a continuous generative decoder. By using a learnable grid to encode explicit spatial locality, PILIR can capture high‑frequency details locally, preventing dilution by global patterns. A generative neural operator then synthesizes these local latent features into continuous physical fields, allowing accurate reconstruction of fine‑scale structures. Experiments on a range of challenging PDEs show that PILIR effectively mitigates spectral bias, thereby boosting the convergence of high‑frequency details and achieving superior accuracy compared to state‑of‑the‑art methods.

Abstract:
Despite rapid recent advances in quantum machine learning, the field is in many ways stuck. Existing approaches can exhibit serious limitations, and we still lack learning frameworks that are simple, interpretable, scalable, and naturally suited to quantum data. To address this, here we introduce quantum Gaussian processes, a Bayesian framework for learning from quantum systems through priors over unknown quantum transformations. We show that, under suitable conditions, unitary quantum stochastic processes define Gaussian processes, thereby enabling regression, classification, and Bayesian optimization directly on quantum data. The key ingredient in this framework is sufficient knowledge of a quantum process's structure and symmetries to define an informative prior through its corresponding quantum kernel, effectively injecting a strong, physics‑informed inductive bias into the learning model. We then prove that matchgate, or free‑fermionic, evolutions give rise to provable and scalable quantum Gaussian processes, providing the first family in our framework where the unknown unitary acts non‑trivially on all qubits. Finally, we demonstrate accurate long‑range extrapolation, phase‑diagram learning in many‑body systems, and sample‑efficient Bayesian optimization in a quantum sensing task. Our results identify quantum Gaussian processes as a promising route toward simpler and more structured forms of quantum learning.

Abstract:
In recent years, physics‑informed neural networks (PINNs) have gained significant attention for solving differential equations, although they suffer from two fundamental limitations, namely, spectral bias inherent in neural networks and loss imbalance arising from multiscale phenomena. This paper proposes an adaptive wavelet‑based PINN (AW‑PINN) to address the extreme loss imbalance characteristic of problems with localized high‑magnitude source terms. Such problems frequently arise in various physical applications, such as thermal processing, electro‑magnetics, impact mechanics, and fluid dynamics involving localized forcing. The proposed framework dynamically adjusts the wavelet basis function based on residual and supervised loss. This adaptive nature makes AW‑PINN handle problems with high‑scale features effectively without being memory‑intensive. Additionally, AW‑PINN does not rely on automatic differentiation to obtain derivatives involved in the loss function, which accelerates the training process. The method operates in two stages, an initial short pre‑training phase with fixed bases to select physically relevant wavelet families, followed by an adaptive refinement that adapts scales and translations without populating high‑resolution bases across entire domains. Theoretically, we show that under certain assumptions, AW‑PINN admits a Gaussian process limit and derive its associated NTK structure. We evaluate AW‑PINN on several challenging PDEs featuring localized high‑magnitude source terms with extreme loss imbalances having ratios up to 10^10:1. Across these PDEs, including transient heat conduction, highly localized Poisson problems, oscillatory flow equations, and Maxwell equations with a point charge source, AW‑PINN consistently outperforms existing methods in its class.

Abstract:
We propose a noise‑mitigation quantum simulation strategy for near‑term quantum devices based on Quantum Circuit Learning (QCL), which is in particular effective for integrable quantum spin chains. The method trains a shallow variational circuit to approximate a deeper time‑evolution circuit by learning the conserved charges and only a small amount of dynamical information in the system. Under realistic noise models, the learned circuit maintains both conserved quantities and dynamical observables significantly closer to their true values than the noisy simulation of the original circuit. This demonstrates QCL as an effective, physics‑informed error mitigation strategy, producing shorter, more robust circuits without exponential sampling overhead.

Abstract:
Machine learning continues to emerge as an important tool to be utilised within structural engineering and structural health monitoring, due to its ability to accurately and quickly perform both regression and classification tasks. However, a purely data driven approach has its limitations, particularly where we lack data from relevant environmental and operational conditions, a situation that has led to the development of physics‑informed machine learners for structural health monitoring. These "grey‑box" models take into account the physical insight that an engineer would have about the structure they are modelling and have shown promising results in the structural engineering field among many others. This work compares black and grey‑box models through a "green" lens, comparing them in terms of their environmental impact, and investigating how the high extrapolative performance of grey‑box models can reduce their runtimes and therefore carbon emissions. The authors aim to develop physics‑informed models with reduced computational costs, while maintaining high performance, illustrated through a structural health monitoring case study.

Abstract:
Cuffless blood pressure (BP) estimation based on Pulse Transit Time (PTT) has emerged as a promising solution for continuous health monitoring. However, conventional models relying on the Moens‑Korteweg equation often fail during rapid hemodynamic fluctuations, as they assume arterial walls are purely elastic and neglect inherent viscoelasticity. To address this limitation, we propose a physics‑informed framework introducing a viscoelastic compensation mechanism. First, raw photoplethysmogram (PPG) signals undergo high‑fidelity reconstruction using Modified Akima (Makima) interpolation. Second, a robust Intersecting Tangent Method is applied for precise pulse foot localization. Crucially, we utilize Ensemble Empirical Mode Decomposition (EEMD) to isolate high‑frequency Intrinsic Mode Functions (IMFs), defining a ``Viscoelastic Velocity Metric'' to quantify the vascular damping effect (η\cdot \dotε) typically ignored by elastic models. The framework was rigorously validated on a challenging subset of the MIMIC‑II database (364 subjects, 28,525 cardiac cycles) characterized by a high prevalence of hypertension (23.4%). Experimental results demonstrate medical‑grade accuracy, yielding a Root Mean Square Error (RMSE) of 5.22 mmHg for Systolic and 3.65 mmHg for Diastolic BP, with Pearson correlation coefficients (R > 0.97). These findings confirm that incorporating viscoelastic features significantly enhances robustness against vascular hysteresis.

Abstract:
Operational weather prediction has long relied on physics‑based numerical weather prediction (NWP), whose accuracy comes at the cost of substantial compute and complex simulation workflows. Recent transformer‑based forecasters offer efficient data‑driven alternatives, however transformers are physics‑agnostic models. Additionally, standard transformer encoders evolve representations through discrete layer updates that may be less suited to modeling smooth latent dynamics. In this work, we propose a continuous‑depth transformer encoder for weather forecasting that integrates Neural Ordinary Differential Equation (Neural ODE) dynamics within each encoder block. Specifically, we replace discrete residual updates with ODE‑based updates solved using adaptive numerical integration. We also introduce a two‑branch attention module that combines conventional patch‑wise self‑attention with an auxiliary branch that applies a derivative operator to attention logits, providing an additional change‑sensitive interaction signal. To further align forecasts with governing principles, we propose a customized physics‑informed training objective that enforces physical consistency as a soft constraint. We evaluate the proposed method against a standard discrete transformer baseline and an existing continuous‑time Neural ODE forecasting variant, demonstrating the importance of PINN‑Cast in short term weather forecasting.

Abstract:
Thermal plasma properties play a critical role in plasma simulations and plasma‑related applications. However, their strong nonlinear dependence on temperature, pressure, and gas composition makes accurate and efficient evaluation challenging. In this work, an operator learning‑based model, termed DeepPropNet, is proposed for fast prediction of thermodynamic and transport properties of thermal plasmas. Two architectures are developed, including a single‑property model (S‑DeepPropNet) and a Mixture of Experts (MoE)‑based multi‑property model (MoE‑DeepPropNet). The proposed models learn the nonlinear mapping from plasma operating conditions to physical properties based on high‑fidelity datasets. The MoE architecture enables efficient multi‑property prediction within a unified framework. Predictions are performed for binary SF6‑N2 and ternary C4F7N‑CO2‑O2 mixtures. The results show that the proposed models achieve high accuracy, with relative L2 errors on the order of 10‑3 to 10‑2, while maintaining strong generalization capability under unseen conditions. The applicability of DeepPropNet is further demonstrated by coupling with finite volume method (FVM) and physics‑informed neural networks (PINNs). The results indicate that DeepPropNet provides an efficient and scalable approach for plasma property prediction and plasma simulations.

Abstract:
Physics‑informed neural networks (PINNs) approximate solutions of partial differential equations (PDEs) by embedding physical laws into the loss function. In parameterized PDE families, variations in coefficients or boundary/initial conditions define distinct tasks. This makes training individual PINNs for each task computationally prohibitive, while cross‑task transfer can be sensitive to task heterogeneity. While meta‑learning can reduce retraining cost, existing methods often rely on a single global initialization and may suffer from negative transfer, particularly under feature‑scarce coordinate inputs and limited training‑task availability. We propose the Learning‑Affinity Adaptive Modular Physics‑Informed Neural Network (LAM‑PINN), a compositional framework that leverages task‑specific learning dynamics. LAM‑PINN combines PDE parameters with learning‑affinity metrics from brief transfer sessions to construct a task representation and cluster tasks even with coordinate‑only inputs. It decomposes the model into cluster‑specialized subnetworks and a shared meta network, and learns routing weights to selectively reuse modules instead of relying on a single global initialization. Across three PDE benchmarks, LAM‑PINN achieves an average 19.7‑fold reduction in mean squared error (MSE) on unseen tasks using only 10% of the training iterations required by conventional PINNs. These results indicate its effectiveness for generalization to unseen configurations within bounded design spaces of parameterized PDE families in resource‑constrained engineering settings.

Abstract:
Climate change necessitates rapid expansion of renewable energy, with wind energy offering a scalable and low‑impact solution. However, accurate prediction of wind loads and power generation remains challenging due to uncertainties in wind shear and turbulence stresses under atmospheric boundary layer (ABL) conditions. High‑fidelity Large Eddy Simulations (LES) are typically used to reduce these uncertainties but are computationally expensive and impractical for large‑scale or real‑time applications. This work addresses this limitation using generative AI, specifically Conditional Denoising Diffusion Probabilistic Models, to reconstruct high‑resolution turbulent flow fields from coarse inputs. A high‑fidelity dataset is generated using a parallel high‑order finite‑difference solver across varying geostrophic wind speeds, surface roughness conditions aligned with IEC wind classes, and multiple grid resolutions. The diffusion model is trained for super‑resolution across different scale factors and evaluated under interpolation and extrapolation scenarios. Results show accurate recovery of fine‑scale turbulent structures, Reynolds stresses, and statistical properties in interpolation cases, indicating strong physical consistency within the training domain. However, extrapolation to higher wind speeds leads to increased noise and overprediction of turbulent stresses, highlighting limitations in generalization. Overall, the study demonstrates that physics‑informed generative models can significantly reduce computational cost while maintaining acceptable accuracy, enabling faster and more reliable turbulent inflow characterization for wind energy applications.

Abstract:
Accurate and efficient prediction of three‑dimensional (3D) wall‑bounded turbulent flows poses a significant challenge for machine learning methods, particularly in scenarios where flow field data are limited. Physics‑informed neural operator (PINO) combines neural operator and physics constraint methods, and shows great potential for solving a wide range of partial differential equations. Nevertheless, the multi‑scale vortex structures in wall‑bounded turbulence make it difficult for most existing PINO methods to make stable and accurate long‑term predictions at high Reynolds numbers. To address this challenge, we develop the large‑eddy simulation nets (LESnets) that integrates large‑eddy simulation (LES) equations into the factorized Fourier neural operator (F‑FNO) for wall‑bounded turbulence. The LESnets framework does not rely on labeled data for training, which enables it to generate temporal solutions over flexible time horizons during the training process. Moreover, the law of the wall is integrated into the LESnets framework through a wall model for the physics‑informed loss, thus enabling reliable simulations of wall‑bounded turbulence at high Reynolds number using coarse grids. The proposed LESnets methods are demonstrated in turbulent channel flows at three friction Reynolds numbers: 180, 590, and 1000. Numerical experiments show that the performance of the LESnets in terms of prediction accuracy and efficiency is comparable to that of two data‑driven models, namely the implicit U‑Net enhanced Fourier neural operator (IUFNO) and F‑FNO. Meanwhile, the LESnets model achieves prediction accuracy comparable to traditional LES methods while offering a higher computational efficiency. Thus, the LESnets model demonstrates strong potential for efficient and long‑term prediction of wall‑bounded turbulent flows.

Abstract:
Digital twins provide a powerful paradigm for diagnostic and prognostic tasks in the monitoring and control of engineered systems; however, their deployment for complex structures remains challenged by model‑form uncertainty, arising from unknown nonlinear dynamics, and by sparse sensing. These limitations hinder reliable online state estimation using either purely physics‑based or purely data‑driven approaches. This work introduces the Physics‑Guided Graph Neural ODE (PiGGO) framework, a physics‑informed, graph‑based Bayesian state estimation approach in which a learned graph neural ordinary differential equation (GNODE) serves as the continuous‑time state‑transition model within an extended Kalman filter. The graph representation explicitly defines the system state‑space, while physics‑guided inductive biases encode known structural relationships and constrain the learning of nonlinear dynamics. By integrating graph‑native learned dynamics with recursive Bayesian filtering, the proposed PiGGO framework enables online virtual sensing and uncertainty‑aware state estimation for nonlinear systems with unknown model form, while maintaining generalisation across topologically similar structures. Numerical case studies demonstrate improved robustness to model uncertainty and measurement noise, outperforming both open‑loop graph neural models and conventional filtering approaches in online prediction tasks.

Abstract:
Municipal solid waste incineration (MSWI) converts urban waste to energy but simultaneously emits carbon dioxide, carbon monoxide and multiple regulated air pollutants whose formation is tightly coupled within a single combustion system. Controlling these emissions across a network of diverse facilities poses a fundamentally different challenge from optimising a single plant: data‑driven models trained at one site capture local statistical patterns that rarely survive transfer to another, because they lack the physical constraints and regime‑level structure needed to generalise. Here we show that shared emission‑control relationships can be identified across heterogeneous MSWI plants when physical conservation laws, operating‑regime heterogeneity and carbon‑pollutant coupling are treated jointly. We develop a carbon‑pollutant mixture‑of‑experts (CPMoE) model that routes process observations through regime‑specific expert networks under conservation‑based regularisation, and combine it with physics‑informed transfer learning to adapt a reference model to new facilities. Across 13 plants, CPMoE predicts six major pollutants and a composite system‑level risk index with source‑domain R2 of 0.668‑0.904 and 0.666‑0.970, respectively; after transfer to 12 target plants these values remain 0.661‑0.842 and 0.610‑0.841. Expert‑utilisation patterns show that adaptation proceeds through structured regime re‑weighting rather than re‑learning from scratch. Embedding the transferred model in an offline digital twin and screening candidate operating adjustments against historical process records yields consistent risk‑index reductions of 3.6‑6.3% with simultaneous pollutant co‑reductions in 94‑100% of evaluated samples. These findings suggest a practical route toward transferable, system‑level decision support for carbon‑pollutant co‑control in heterogeneous waste‑to‑energy networks.

Abstract:
Lattice metamaterials enable lightweight, multifunctional structures, yet homogenization‑based evaluation of their effective properties remains computationally expensive. Neural surrogates offer speed but often lack the accuracy and stability required for engineering‑grade simulations. We introduce GMT, a Geometric Multigrid Transformer ‑‑ a neural solver with high numerical fidelity for fast and reliable lattice homogenization. GMT achieves architectural alignment with Geometric Multigrid (GMG) by restructuring Point Transformer V3 to operate across sparse GMG hierarchies, capturing long‑range dependencies and cross‑level interactions essential for multigrid convergence. To enforce physical consistency, GMT incorporates physics‑aware positional encoding for strict enforcement of periodicity and predicts both the finest‑level solution and multi‑level residual corrections. These predictions deliver a spectrally‑aligned initialization, enabling end‑to‑end training under physics‑informed and solver‑aware losses and requiring only a single GMG V‑cycle refinement to reach convergence. This fusion of neural prediction and numerical rigor achieves relative residual errors of 10^‑5 with a 160× speedup over state‑of‑the‑art GPU‑based solvers at equivalent accuracy ‑‑ particularly at high resolutions (e.g. 512^3), where traditional methods become most costly. We validate GMT across mechanical and thermal domains, demonstrate robust generalization to unseen geometries and non‑periodic settings, and showcase scalability to high resolutions ‑‑ enabling real‑time design iteration, multi‑scale simulations, high‑throughput material discovery, and inverse design.

Abstract:
We develop a physics‑informed learning framework for energy‑shaping control of port‑Hamiltonian (pH) systems from trajectory data. The proposed approach co‑learns a pH system model and an optimal energy‑balancing passivity‑based controller (EB‑PBC) through alternating optimization with policy‑aware data collection. At each iteration, the system model is refined using trajectory data collected under the current control policy, and the controller is re‑optimized on the updated model. Both components are parameterized by neural networks that embed the pH dynamics and EB‑PBC structure, ensuring interpretability in terms of energy interactions. The learned controller renders the closed‑loop system inherently passive and provably stable, and exploits passive plant dynamics without canceling the natural potential. A dissipation regularization enforces strict energy decay during training, thereby enhancing robustness to sim‑to‑real gaps. The proposed framework is validated on state‑regulation and swing‑up tasks for planar and torsional pendulum systems.

Abstract:
General‑relativistic magnetohydrodynamic (GR‑MHD) simulations are essential for studying black hole accretion, relativistic jets, and magnetic reconnection, yet their computational cost severely limits systematic parameter exploration. We investigate neural operator surrogates for two astrophysically relevant simulation scenarios produced by the Black Hole Accretion Code (\textttBHAC). First, a Physics Informed Fourier Neural Operator (PINO) is trained on the special‑relativistic resistive MHD (SRRMHD) evolution of the Orszag‑Tang vortex over a range of resistivities spanning the Sweet‑Parker and fast reconnection regimes. By embedding the governing equations as an additional loss term evaluated at finer temporal resolution than the available data supervision, the model learns dynamics at time steps where no simulation data is provided, enabling recovery of plasmoid formation that a data‑only baseline trained on the same sparse snapshots fails to reproduce. To our knowledge, the present work is the first application of a physics informed neural operator to special relativistic resistive MHD, and the first to investigate the capability of such models to resolve plasmoid formation in SRRMHD. In a second line of investigation, an OFormer‑style Transformer Neural Operator is trained on the evolution of spine‑sheath relativistic jets created with \textttBHAC, in special‑relativistic MHD (SRMHD). The model is directly applied on the adaptive mesh, highlighting the need for linear attention due to long sequences. The neural surrogate model is capable of capturing most of the major details, especially in early predictions. To our knowledge, this constitutes the first application of a neural operator directly on a high resolution adaptive mesh refinement grid in the context of MHD simulations.

Abstract:
Graph neural networks such as ParticleNet and transformer based networks on point clouds such as ParticleTransformer achieve state‑of‑the‑art performance on jet tagging benchmarks at the Large Hadron Collider, yet the physical reasoning behind their predictions remains opaque. We present different methods, i.e. perturbation‑based (GNNExplainer), Shapley‑value‑based (GNNShap), and gradient‑based (GRADCam); adapted to operate on LundNet's Lund‑plane graph representation. Leveraging the fact that each node in the Lund plane corresponds to a physically meaningful parton splitting, we construct Monte Carlo truth explanation masks and introduce a physics‑informed evaluation framework that goes beyond standard fidelity metrics. We perform the analysis in three transverse‑momentum bins (\mathrmp_T \in [500,700], [800,1000], and the inclusive region [500,1000] GeV), revealing how explanation quality and focus shift between non‑perturbative and perturbative regimes. We further quantify the correlation between explainer‑assigned node importance and classical jet substructure observables ‑‑ N‑subjettiness ratios τ_21 and τ_32 and the energy correlation functions ‑‑ establishing the degree to which the model has learned known QCD features. We find that overall the weight assigned by explainability methods has a correlation with analytic observables, with expected shift across different phase space regimes, indicating that a trained neural network indeed learns some aspects of jet‑substructure moments. Our open‑source implementation enables reproducible explainability studies for graph‑based jet taggers.

Abstract:
Nonlinear dynamical systems with regime transitions are typically described by ordinary differential equations with jumping parameters parameters. Traditional methods often treat change‑point detection and parameter estimation as separate tasks, ignoring the inherent coupling between them. To address this, we propose residual‑loss anomaly analysis of physics‑informed neural networks, a unified framework that leverages dynamical consistency within the physics‑informed learning paradigm. This approach jointly infers piecewise parameters and transition points under a single set of constraints. The method follows a two‑stage strategy: First, local physical residuals are analyzed through overlapping subinterval decomposition. When a subinterval spans a true transition point, the residual exhibits a distinct structural elevation in noise‑free conditions, which has a non‑zero lower bound, enabling effective localization of potential transition intervals. Second, within our framework, change‑point locations and piecewise parameters are integrated into a unified physical loss function for joint optimization, enabling simultaneous identification. Experiments on benchmark nonlinear dynamical systems, including Malthusian and logistic growth models, Van der Pol oscillator, Lotka‑Volterra model and Lorenz system, demonstrate that the proposed method outperforms traditional decoupled approaches in both change‑point localization and parameter estimation accuracy. This study provides an efficient, unified solution for structurally coupled inverse problems in nonlinear dynamical systems with regime switching.

Abstract:
In this paper, we propose a novel Physics‑Informed Neural Network (PINN) framework based on the Cordès condition for solving both linear and fully nonlinear partial differential equations (PDEs) in non‑divergence form, together with their applications. By incorporating the operator structure into the loss function, the proposed method improves the conditioning of the associated optimization problem, thereby enhancing training stability and solution accuracy. The framework is further extended to include Hamilton‑Jacobi‑Bellman and Monge‑Ampère equations, with applications to optimal transport. Numerical experiments demonstrate the effectiveness and robustness of the method, as well as its capability to address high‑dimensional problems, highlighting the promise of learning‑based approaches for tackling challenging PDEs. Owing to its generality and simplicity, the proposed method is expected to be of broad interest to the scientific and engineering communities.

Abstract:
Coherent transition radiation (CTR) spectroscopy is a critical diagnostic for characterizing the longitudinal structure of relativistic electron bunches in laser‑plasma and conventional accelerators. In practice, recovering the bunch profile from a measured CTR spectrum is an ill‑posed phase‑retrieval problem. Traditionally, this is addressed using Gerchberg‑Saxton (GS)‑type iterative algorithms. However, these implementations often rely on explicit inverse propagators, making them difficult to adapt to sophisticated experimental forward models. In this work, we introduce a flexible gradient‑based framework for CTR phase retrieval. By leveraging a differentiable forward model, we propose a phase‑only gradient descent (GD‑Phase) approach that enforces the measured spectral amplitude as a hard constraint while optimizing the Fourier phase under physical real‑space priors. Using synthetic CTR spectra spanning multi‑peaked and strongly modulated profiles, we benchmark GD‑Phase against traditional GS and a real‑space amplitude‑parametrized gradient descent (GD‑Amp) algorithm. Unlike traditional methods, this formulation allows for the seamless inclusion of arbitrary differentiable experimental effects into the reconstruction loop. We demonstrate that this physics‑informed approach not only reproduces the fidelity of GS methods but also establishes a robust baseline for incorporating multi‑diagnostic constraints and uncertainty quantification. This enables the systematic extension to higher‑dimensional, multimodal, and uncertainty‑aware diagnostics, facilitating fast and scalable phase retrieval in realistic experimental settings.

Abstract:
Source‑free test‑time adaptation (TTA) is appealing for mobile and wearable sensing because it enables on‑device personalization from unlabeled test streams without centralizing private data. However, sensor‑based human activity recognition (HAR) poses challenges that are less pronounced in standard vision benchmarks: behavioral inertial streams are temporally correlated and often exhibit within‑session shifts caused by sensor rotation, placement change, and sampling‑rate drift. Under this streaming non‑i.i.d. setting, widely used vision‑style TTA objectives can become unstable, leading to overconfident errors, representation collapse, and catastrophic forgetting. We propose PI‑TTA, a lightweight source‑free adaptation framework that stabilizes online updates through three physics‑consistent constraints: gravity consistency, short‑horizon temporal continuity, and spectral stability. PI‑TTA updates the same small parameter subset as strong source‑free baselines and incurs only modest overhead, making it suitable for on‑device deployment. Experiments on USCHAD, PAMAP2, and mHealth under long‑sequence stress tests and factorized shift protocols show that PI‑TTA mitigates the severe degradation observed in confidence‑driven baselines and preserves stable adaptation under sustained streaming conditions. It improves long‑sequence accuracy by up to 9.13% and reduces physical‑violation rates by 27.5%, 24.1%, and 45.4% on USCHAD, PAMAP2, and mHealth, respectively. These results demonstrate that physics‑informed adaptation can improve accuracy, stability, and deployment reliability for real‑world mobile sensing systems.

Abstract:
Dual‑energy computed tomography (DECT) enables material‑specific imaging through acquisitions at two different X‑ray energy spectra. Material decomposition from DECT data is an ill‑posed inverse problem that is highly sensitive to noise amplification. Conventional methods face challenges regarding accuracy and computational efficiency. We present a novel physics‑informed deep learning (DL) framework for DECT material decomposition that eliminates the requirement for ground‑truth material images during training. Our approach incorporates a polychromatic forward model into the training pipeline, enabling the network to learn the decomposition mapping by minimizing discrepancies in the projection domain. We validate our method on the AAPM DL‑Spectral CT Challenge dataset, comparing performance against three state‑of‑the‑art methods. In the projection domain, our method achieves the lowest root mean squared error (RMSE) across test datasets. For virtual monoenergetic images (VMIs) at 30 keV, 50 keV, and 70 keV, the approach consistently outperforms all conventional methods in both RMSE and structural similarity index (SSIM). These results demonstrate the potential of DL for accurate material decomposition in DECT without requiring labeled training data.

Abstract:
This work addresses the critical problem of tracking fast‑moving objects through strongly scattering media in a low‑light environment. Different from existing approaches that use frame‑based cameras with fixed exposure times, which trade off signal‑to‑noise ratio for temporal resolution, we introduce computational neuromorphic tracking (CNT), a physics‑informed framework that combines asynchronous event sensing with task‑driven speckle analysis for robust motion estimation. We formulate the neuromorphic speckle aggregation as a spatiotemporal speckle representation, jointly optimizing the temporal and spatial parameters to maximize tracking stability under extreme conditions. Extensive experiments demonstrate that our method enables robust motion tracking of 10x faster motion and under 10x dimmer illumination compared to conventional systems. These improvements significantly broaden the operational regime for tracking through scattering media, providing an efficient and scalable solution for demanding scenarios involving rapid motion and low‑light conditions.

Abstract:
We present a physics‑informed Bayesian neural‑network framework to infer neutron‑star equations of state from theoretical priors and to propagate the associated uncertainties to stellar observables. Trained on a large and representative ensemble of hadronic EoSs, the model learns P(ε) via stochastic variational inference, incorporating soft constraints at saturation density and from perturbative QCD, together with penalties enforcing monotonicity and causality. The accepted core EoSs are matched to an SLy4 crust and evolved through a unified Tolman‑Oppenheimer‑Volkoff‑plus‑tidal solver to generate posterior predictions in the mass‑radius (M‑R) and mass‑tidal‑deformability (M‑Λ) planes. The inferred posterior is consistent with NICER radius measurements and the observed 2.0\,M_\odot maximum‑mass constraint, yielding R_1.4=12.1^+1.4_‑0.9\,\mathrmkm, Λ_1.4=580^+520_‑240, and M_\mathrmmax～eq 2.11\pm0.05\,M_\odot (90% CI). The resulting canonical tidal deformability can be assessed \empha posteriori against current gravitational‑wave constraints. Overall, this framework provides a flexible, non‑parametric mapping from microphysical EoS uncertainties to neutron‑star observables.

Abstract:
The transmission electron microscope facilitates the highest‑resolution imaging of any instrument ever created, and its limiting factor is no longer spatial resolution but dose efficiency. Low electron doses avoid sample damage but produce noisy images for which, unlike in classical computer vision, there is no ground truth. Autonomous materials experimentation poses a related problem, since closed‑loop instruments need representations grounded in the microscope state at acquisition. Both demand representations grounded in how an image was acquired. We release 7,330 paired high‑angle annular dark‑field scanning‑TEM (HAADF‑STEM) images and their seven‑dimensional acquisition metadata, and propose Contrastive Image‑Metadata Pre‑training (CIMP), a CLIP‑style encoder that aligns the two modalities and reaches 84.4% Top‑1 cross‑modal retrieval on a held‑out split. All seven parameters are individually recoverable from the frozen visual embedding through a linear probe, and we use the embedding to condition a metadata‑conditioned style‑transfer model that re‑renders experimental images under different acquisition parameters. Virtually scaling dwell time and beam current of low‑dose images turns this model into a physics‑informed denoiser; in a blind user study, experimental microscopists prefer it over the current state‑of‑the‑art denoiser for STEM imagery on 70.2% of trials.

Abstract:
Quantum neural networks generalize classical artificial neural networks into the quantum domain. They are formulated as parameterized quantum circuits which are optimized by measuring and minimizing a suitably chosen loss function. The core challenge in understanding, implementing and ultimately using quantum neural networks is that they represent many‑body systems with an exponentially large Hilbert space, in combination with a large parameter search space. Moreover, noise ‑‑ which is inherent to any quantum measurement ‑‑ sets practical limits for the estimation of training loss. Here, we study physics‑informed large‑scale quantum neural networks that are trained through a finite number of noisy loss function measurements. We show that this architecture permits the construction of nontrivial decision boundaries that enable the classification of quantum states through measuring an order parameter. Our approach can directly process quantum data that is output from quantum simulators and computers and is well suited for implementation on current hardware. Moreover, owed to a close link between the neural network dynamics and the evolution of Markovian open many‑body quantum systems, one may expect a certain robustness to noise, which is ubiquitous in the current NISQ era.

Abstract:
In this chapter, we investigate the bending behavior of a perforated nanobeam subjected to sinusoidal loading using an efficient and computationally robust Physics‑Informed Functional Link Constrained Framework with Domain Mapping (DFL‑TFC) method. Our aim is to determine the relationship between static bending response and dynamic deflection of a perforated nanobeam for various perforation cases. The static bending is obtained using the FL‑TFC with Domain mapped method, whereas dynamic deflection is determined using the Galerkin method. The proposed approach employs the theory of functional connections (TFC) to systematically embed governing differential equation constraints into a constrained expression (CE), which exactly satisfies all prescribed initial and boundary conditions (ICs and BCs) and domain of differential equation is mapped to domain of orthogonal polynomials. Within this framework, the free function appearing in the constrained expression is expressed through a functional link neural network (FLNN). The cost is minimized by the mean square residual of DE, allowing training without requiring complex deep network architectures. Relationship between static and dynamic defection of simply‑supported (S‑S) perforated nanobeams has been investigated here. FL‑TFC with Domain mapped method eliminates the need for deep and complex neural network architectures while ensuring accuracy, efficiency, and strict satisfaction of boundary conditions as compared to standard PINN.

Abstract:
We propose Noise‑Based Spectral Embedding (NBSE), a physics‑informed framework for selecting informative features from high‑dimensional data without greedy search. NBSE constructs a sparse similarity graph on the samples and identifies the Nishimori temperature β_N the critical inverse temperature at which the Bethe Hessian becomes singular. The corresponding smallest eigenvector captures the dominant mode of an intrinsically degree‑corrected diffusion process, naturally reweighting nodes to prevent hub dominance. By transposing the data matrix and applying NBSE in feature space, we obtain a one‑dimensional spectral embedding that reveals groups of redundant or semantically related dimensions; balanced binning then selects one representative per group. We prove that coloured Gaussian perturbations shift β_N by at most O(\barσ^2), guaranteeing robustness to measurement noise. Experiments on ImageNet embeddings from MobileNetV2 and EfficientNet‑B4 show that NBSE preserves classification accuracy even under aggressive compression: on EfficientNet‑B4 the accuracy drop is below 1% when retaining only 30% of features, outperforming ANOVA F‑test and random selection by up to 6.8%.

Abstract:
Accurate estimation of thermospheric mass density is a prerequisite for orbit prediction and space situational awareness, where the upper atmosphere responds nonlinearly to solar and geomagnetic forcing across several orders of magnitude. Physics‑based general circulation models resolve this response but are computationally expensive, while empirical models run cheaply but lack a time‑evolving atmospheric state. This work couples a data‑driven reduced‑order thermospheric model with a Kalman filter that assimilates in situ density observations. An autoregressive Sparse Identification of Nonlinear Dynamics with control (SINDy_c‑AR) reduced‑order model derived from the Thermosphere‑Ionosphere‑Electrodynamics General Circulation Model (TIE‑GCM) captures the dominant modes of variability and their dependence on solar and geomagnetic drivers at a fraction of the parent model's cost. Density observations from CHAMP, GRACE, GRACE‑FO, GOCE, and Swarm are assimilated across a range of orbital configurations and geomagnetic conditions, with a linear DMDc model evaluated as a reference. Assimilation reduces density estimation error relative to open‑loop predictions, most visibly during geomagnetic storms and under single‑satellite coverage. SINDy_c‑AR and DMDc perform comparably on assimilated orbits; on withheld orbits, SINDy_c‑AR is more accurate in the in‑training scenarios while DMDc is better in the out‑of‑training 2024 Swarm‑C case. Benchmarks against NRLMSIS~2.1 and HASDM (2000‑‑2019, where available) show that empirical references can outperform the assimilated model far from the assimilated track, so results are framed as improvements over the open‑loop forecast.

Abstract:
Encoding more information into wave fields is a central goal in imaging, communication, and wave control. Optical holography benefits from polarization multiplexing, but acoustic holography remains largely limited to pressure‑only encoding because sound in fluids lacks naturally independent vector channels. Here, we show that particle velocity can serve as a practical multiplexing degree of freedom despite the intrinsic pressure‑velocity coupling governed by the acoustic Euler equation. We develop a physics‑informed inverse‑design approach that incorporates acoustic propagation and pressure‑velocity coupling to create a binary metasurface for vector‑field acoustic holographic multiplexing. Experiments demonstrate dual‑channel multiplexing on the in‑plane velocity components v_x and v_y, and further extend to three‑channel multiplexing by incorporating pressure p, with high‑fidelity reconstruction and low cross‑talk. This approach adds a new information dimension without reducing spatial or spectral bandwidth and enables broader forms of wave‑based information encoding and multiplexed wave control.

Abstract:
The nonlinear synergy between global warming and urbanization is amplifying extreme climate risks in cities worldwide. While observations and simulations confirm these compounding effects, two fundamental bottlenecks impede predictive understanding: (1) fragmented, case‑specific perspectives that hinder the discovery of universal mechanisms, and (2) a methodological divide between computationally prohibitive high‑resolution models and AI‑based tools that lack physical interpretability at urban scales. This article advocates for a paradigm shift toward the deep integration of physical principles with data intelligence. To this end, we propose a transformative "Classification‑Mechanism‑Inference" (CMI) framework. Classification involves establishing a global urban "climate‑morphology‑development" typology to enable systematic comparison beyond isolated case studies. Mechanism advocates for physics‑informed machine learning (PIML) as the core engine to develop efficient, physics‑constrained surrogate models for uncovering nonlinear interactions. Inference leverages these models for high‑throughput, tailored risk projection to directly inform context‑specific adaptation planning. The CMI framework aims to bridge the cognitive and methodological gaps, thereby advancing urban climate science from phenomenological description towards mechanistic, predictive, and decision‑relevant science, which is crucial for building climate‑resilient cities globally.

Abstract:
The escalating climate crisis and ecosystem degradation demand intelligent, low‑cost sensors capable of robust, long‑term monitoring in real‑world environments. Absolute dissolved oxygen (DO) concentration is a key parameter for predicting climate tipping points. Inexpensive optoelectronic sensors based on microstructured polymer films doped with phosphorescent dyes could be readily deployable; however, signal drift and marine biofouling remain major challenges. Here, we introduce a sensing paradigm that combines camera‑based DO sensors with a visual transformer (ViT)‑based physics‑informed neural network (PINN) for high‑fidelity sensing under biofouling conditions. Training and testing data were obtained from an algae‑laden water tank over 14 days to capture accelerated biofouling. The ViT‑PINN, which embeds the Stern‑Volmer (SV) equation into the loss function, reduces mean average error (MAE) by 92% and 89% compared to classical statistical and ML approaches, achieving ~2 umol/L absolute error. A deep ensemble further quantifies predictive uncertainty, enabling self‑diagnostic sensing.

Abstract:
The rapid growth of large, power‑electronics‑rich data center (DC) loads is creating new operational challenges for bulk power systems. A key risk arises when a DC uninterruptible power supply (UPS) disconnects the facility during voltage/frequency disturbances and then reconnects it while the bulk grid is still dynamically settling to a new equilibrium point. Poorly timed reconnection can amplify electromechanical oscillations, deepen frequency deviations, and lead to repeated connect‑disconnect \emphflapping. In this paper, we develop an analytical framework to characterize the \emphsafe reconnection time for large DC loads after a disturbance‑induced disconnection that avoids flapping. Using a model in the spirit of the classical single‑machine infinite‑bus system, we capture (i) swing dynamics during the disconnection interval and (ii) voltage‑angle coupling at the load bus, which determines the electrical power step at reconnection under constant‑power load assumptions. Using energy function method, we characterize the critical safe reconnection time such that for any reconnection time after the critical safe reconnection time, the post‑reconnection trajectory is guaranteed to remain within operational limits (frequency/angle/voltage) and converge to the post‑reconnection equilibrium, thereby preventing flapping. Time‑domain simulations validate the effectiveness of the proposed analytical approach. The results provide a simple, physics‑informed criterion that can be used to bound reconnection windows for large DC facilities and inform UPS reconnection logic.

Abstract:
The reliability of machine learning in multiscale physical systems depends on how physical structure is embedded into the learning process. We investigate this in the context of turbulent multiphase flows, focusing on the prediction of subgrid interfacial area density, a key quantity governing interphase transport that remains unresolved in large‑eddy simulations. In this work, we develop and evaluate two machine learning subgrid closure models to predict the three‑dimensional subgrid interfacial area density: a purely data‑driven 3D encoder‑decoder network, and a physics‑constrained variant regularized by a fractal geometric prior. Across a range of Weber numbers, the physics‑based model improves predictive accuracy, reduces error variance, and suppresses nonphysical artifacts relative to purely data‑driven approaches. We also show that these gains are regime‑dependent: the embedded inductive bias enhances generalization in corrugation‑dominated regimes where its underlying assumptions hold, but becomes ineffective in fragmentation‑dominated regimes characterized by topology change and droplet breakup. These results reveal a broader principle for scientific machine learning: the utility of physics‑informed models depends not only on the presence of inductive bias, but on its alignment with the governing physical regime. This suggests a path toward regime‑aware learning frameworks for modeling of complex multiscale systems.

Abstract:
Multi‑scale dynamic wake modeling and prediction are essential for the real‑time control and optimization of floating offshore wind turbines (FOWTs). In this study, wakes of FOWTs under coupled surge and pitch motions across a range of Strouhal numbers (St), which can induce wake meandering, are modeled via two novel deep‑learning frameworks: physics‑informed neural networks (PINNs) and Fourier neural operators (FNOs). The high‑fidelity dataset is obtained from large‑eddy simulations with the actuator line model (LES‑AL). The results demonstrate that the dominant large‑scale dynamic structures, such as meandering, can be well modeled by both frameworks; however, FNOs exhibit significant advantages over the PINN model in terms of efficiency (8‑fold computational speedup and 40‑fold faster convergence), long‑term predictive capability, and multi‑scale coherent structural fidelity. Furthermore, the wakes predicted by the PINN model exhibit a smoothing effect that limits the resolution of high‑frequency coherent structures and underestimates turbulent fluctuations in both the wake center and half‑width. Spectral analysis reveals that FNOs resolve the primary meandering frequency (where Stp denotes the frequency induced by the coupled surge and pitch motions), its corresponding higher‑order harmonics (2Stp, 3Stp), and the energy cascade. In contrast, the energy cascade in the PINN predictions dissipates more rapidly in the high‑frequency regime (St > 1.0). Additionally, the pre‑multiplied power spectral density indicates that the energy contained in meandering and the corresponding harmonic frequencies modeled by PINNs is relatively low compared to that in CFD and FNOs. These findings suggest that FNOs are promising for the high‑fidelity, real‑time modeling of FOWT wakes.

Abstract:
Autonomous neutron spectroscopy must solve three distinct tasks: detection (where is the signal?), inference (which Hamiltonian governs it?), and refinement (what are the parameters?). No single controller solves all three equally well. We present TAS‑AI, a hybrid agnostic‑to‑physics‑informed framework for autonomous triple‑axis spin‑wave spectroscopy that separates these tasks explicitly. In blind reconstruction benchmarks, model‑agnostic methods such as random sampling, coarse grids, and Gaussian‑process mappers reach a global error threshold more reliably and with fewer measurements than physics‑informed planning, supporting the claim that discovery and inference are distinct tasks requiring distinct controllers. Once signal structure is localized, the physics‑informed stage performs in‑loop Hamiltonian discrimination and parameter refinement: in a controlled square‑lattice test between nearest‑neighbor‑only and J1‑J2 Hamiltonians, TAS‑AI reaches a decisive AIC‑derived evidence ratio (>100) in fewer than 10 measurements, while motion‑aware scheduling cuts wall‑clock time by 32% at a fixed measurement budget. We also identify a failure mode of posterior‑weighted design, algorithmic myopia, in which the planner over‑refines the current leading model while under‑sampling low‑intensity falsification probes. A constrained falsification channel sharply reduces time spent committed to the wrong model and accelerates correct model selection without modifying the Bayesian inference engine. In controlled two‑model ablations, both a deterministic top‑two max‑disagreement rule and an LLM‑based audit committee achieve this gain under identical constraints. We demonstrate the full workflow in silico using a high‑fidelity digital twin and provide an open‑source Python implementation.

Abstract:
Determining crystal symmetry from powder X‑ray diffraction is a central problem in materials characterization, yet multiple space groups can produce indistinguishable patterns, making automated classification difficult. We show that attention‑based architectures, while superior to convolutional networks for this task, are insufficient on their own: reliable symmetry extraction requires encoding crystallographic knowledge into both the network architecture and the training curriculum. We introduce a physics‑informed transformer that classifies powder patterns into 99 extinction groups, the most specific symmetry classification accessible from diffraction data alone, using an explicit sin^2(theta) coordinate channel, physics‑aware positional encoding, and a structured multi‑task decoder that separates geometric rule learning from holistic pattern recognition. A three‑stage curriculum of balanced synthetic pretraining, realistic fine‑tuning with explicit preferred‑orientation modeling, and Bayesian prior injection proves essential for bridging the synthetic‑to‑real domain gap, while post‑hoc temperature scaling rather than additional training is the key remaining ingredient for robust real‑data transfer. By mapping predictions onto the directed acyclic graph of maximal translationengleiche subgroups, we show that the calibrated model's errors are not random but physically structured: they remain local on the subgroup hierarchy and flow predominantly toward lower‑symmetry descendants, consistent with the physical erasure of systematic‑absence cues by real‑world noise. These results establish that physics‑informed target design, curriculum, and calibrated inference matter as much as model capacity for scientific machine learning on diffraction data.

Abstract:
Deploying machine learning models across diverse well portfolios requires generalisation to wells with design parameters outside the training distribution. Current data‑driven approaches to virtual flow metering (VFM) and bottomhole estimation typically treat each well independently or ignore the influence of well design on operational behaviour. We present WISE (Well Intelligence and Systems Engineering Foundation Model), a design‑aware, physics‑informed multi‑task model that integrates three complementary mechanisms: Feature‑wise Linear Modulation (FiLM) and cross‑modal attention to condition operational embeddings on well design parameters; multi‑task learning for simultaneous prediction of flow rates, bottomhole conditions, and flow regime classification; and structural mass conservation with soft physics constraints derived from well engineering principles. Evaluation on the ManyWells benchmark (2000 simulated wells, 10^6 data points) demonstrates that design‑aware models reduce VFM prediction error by up to 13× compared to design‑unaware baselines, and that physics constraints reduce negative flow predictions by 65%. Flow regime classification achieves 97.7% bottomhole accuracy, providing continuous well integrity monitoring without additional sensors. The methodology transfers to real operational data from five Equinor Volve producers (oil rate R^2 = 0.89, bottomhole pressure R^2 = 0.98, water rate R^2 = 0.97). The trained model additionally serves as a fast surrogate for integrity‑aware well design optimisation over a 24‑dimensional design space, with more than 1000× speedup over drift‑flux simulations. These results demonstrate that design awareness, physics enforcement, and multi‑task learning are essential and complementary ingredients for foundation models intended to operate across large well portfolios.

Abstract:
Deploying quantum machine learning on NISQ devices requires architectures where training overhead does not negate computational advantages. We systematically compare two quantum approaches for chaotic time‑series prediction on the Lorenz system: a variational Quantum Physics‑Informed Neural Network (QPINN) and a Quantum Reservoir Computing (QRC) framework utilizing a fixed transverse‑field Ising Hamiltonian. Under matched resources (4‑‑5 qubits, 2‑‑3 layers), QRC achieves an 81% lower mean‑squared error (test MSE 3.2 \pm 0.6 vs. 47.9 \pm 36.6 for QPINN) while training ～ 52,000× faster (0.2\,s vs. ～ 2.4\,h per seed). Drawing on the classical delay‑embedding principle, we formalize a temporal windowing technique within the QRC pipeline that improves attractor reconstruction by providing bounded, structured input history. Analysis reveals that QPINN instability stems from capacity limitations and competing loss terms rather than barren plateaus; gradient norms remained large (10^3‑‑10^4), ruling out exponential suppression at this scale. These failure modes are absent by construction in the non‑variational QRC approach. We validate robustness across three canonical systems (Lorenz, Rössler, and Lorenz‑96), where QRC consistently achieves low test MSE (3.1 \pm 0.6, 1.8 \pm 0.1, and 12.4 \pm 0.6, respectively) with sub‑second training. Our findings suggest the fixed‑reservoir architecture is a primary driver of QRC's advantage at these scales, warranting further investigation at larger qubit counts and on hardware where quantum‑specific advantages are expected to emerge.

Abstract:
Humanoid robots operating in human‑centered environments (e.g., homes, hospitals, and offices) must mitigate foot‑‑ground impact transients, as impact‑induced vibration and noise degrade user experience and repeated impacts accelerate hardware wear. However, existing low‑noise locomotion training often relies on kinematic proxy objectives or fragile force sensors, and footwear‑induced changes in contact dynamics introduce distribution shifts that hinder policy generalization.We present QuietWalk, a physics‑informed reinforcement learning framework for ground‑reaction‑force‑aware humanoid locomotion under diverse footwear conditions. QuietWalk employs an inverse‑dynamics‑constrained physics‑informed neural network (PINN) to estimate per‑foot vertical ground reaction forces (GRFs) from proprioceptive signals, and integrates the frozen predictor into the RL training loop to penalize predicted impact forces without requiring force sensors at deployment.On a held‑out real‑robot dataset, enforcing inverse‑dynamics consistency reduces vertical GRF prediction errors by 82%‑86% compared with a purely supervised predictor and improves the coefficient of determination from 0.39/0.67 to 0.99/0.99 for the left/right feet. On hardware at 1.2 m/s (barefoot; averaged over four floor materials), QuietWalk reduces mean A‑weighted noise level by 7.17 dB and peak noise level by 4.98 dB under a consistent recording setup. Cross‑footwear experiments (barefoot, skate shoes, athletic sneakers, and high heels) across multiple surfaces further demonstrate robust adaptation to footwear‑induced contact variations.

Abstract:
Accurate modeling of human mobility is critical for tackling urban planning and public health challenges. In undeveloped regions, the absence of comprehensive travel surveys necessitates reconstructing mobility networks from publicly available data. Here we develop neuroGravity, a physics‑informed deep learning model that reliably reconstructs mobility flows from limited observations and transfers to unobserved cities. Using only urban facility and population distributions, we find that neuroGravity's regional representations strongly correlate with socioeconomic and livability status, offering scalable proxies for costly surveys. Furthermore, we uncover that spatial income segregation plays a key role in model transferability: mobility networks are most reliably reconstructed when target cities share similar segregation levels with the source. We design an index to quantify this segregation and accurately predict transferability. Finally, we generate mobility flow proxies for over 1,200 cities worldwide, highlighting neuroGravity's potential to mitigate critical data shortages in resource‑limited, underdeveloped areas.

Abstract:
Spectral problems governed by differential operators underpin a wide range of physical systems, yet remain computationally challenging because their spectra depend sensitively on continuous parameters and often demand repeated evaluations across parameter space. Here we present \textttDeepOPiraKAN, an open source physics informed neural network architecture for spectral analysis. By combining operator learning with enhanced optimization stability, it captures the underlying parameter‑to‑spectrum mapping in a single model, avoiding repeated spectral solutions at isolated points in parameter space. As a representative and stringent benchmark, we apply this framework to the computation of quasinormal modes of Kerr black holes. A single trained network accurately resolves modes with (\ell,m)\in \(2,0),(2,1)\ and overtones up to n=7 across the full spin range, achieving relative errors of \mathcalO(10^‑6) for the fundamental mode and gradually increasing to \mathcalO(10^‑4) for higher overtones, benchmarked against the Leaver's method. This level of accuracy is already significant for black hole spectroscopy and practical ringdown modelling for current and future observatories. More broadly, these results highlight the potential of \textttDeepOPiraKAN as a general and scalable framework for parameter dependent spectral problems across complex physical systems.

Abstract:
Gaussian basis functions provide an efficient and flexible alternative to spline activations in KANs. In this work, we introduce the partition‑of‑unity Gaussian KAN (PU‑GKAN), a Shepard‑type normalized Gaussian KAN in which the Gaussian basis values on each edge are divided by their local sum over fixed centers. This produces a partition‑of‑unity feature map with trainable coefficients, while preserving the standard edge‑based KAN structure. The normalized construction gives exact constant reproduction at the edge level and admits an explicit finite‑feature kernel interpretation. We formulate both the standard Gaussian KAN (GKAN) and PU‑GKAN from a finite‑feature and additive‑kernel viewpoint, making the induced layer kernels and empirical feature matrices explicit. Using the first‑layer feature matrix as the reference object, we adopt a practical scale‑selection interval for \(ε\), with the lower endpoint determined by adjacent‑center overlap and the upper endpoint determined by a conservative conditioning threshold. Numerical experiments show that PU‑GKAN reduces sensitivity to \(ε\), improves validation accuracy for most smooth and moderately non‑smooth targets, and gives more stable training behavior. The benefit persists across sample‑size and center‑number sweeps, higher‑dimensional architectures, Matérn RBF bases, and physics‑informed examples involving Helmholtz and wave equations. These results indicate that Shepard‑type partition‑of‑unity normalization is a simple and effective stabilization mechanism for RBF‑based KANs.

Abstract:
Taiji is a gravitational wave detection mission in space initiated by the Chinese Academy of Sciences, which will open the millihertz window through a heliocentric triangular constellation of three drag‑free spacecraft. Its ultimate sensitivity is determined partly by the residual acceleration noise of the gravitational reference sensors (GRS), within which the coupling between the test‑mass and the fluctuating environmental magnetic field constitutes one of the key stray‑force contributions. Following the path established by the LISA and TianQin teams, high‑precision ground characterization of remanent magnetic moment \vecm_r and volume susceptibility χ of the test masses is a central step in the Taiji pre‑launch test program. A persistent challenge for this characterization is the non‑stationary, colored background noise inherent to torsion‑pendulum facilities, which systematically biases classical Ordinary Least Squares (OLS) and Kalman filter (KF) estimators. We propose an AI‑enhanced Differentiable Weighted Least Squares (AI‑WLS) framework that fuses a dilated one‑dimensional residual network, acting as a dynamic noise evaluator, with a fully differentiable analytical physical solver. This architecture preserves the exact linear mapping from the magnetic parameters to the torque response while autonomously identifying and suppressing contaminated data segments. Validated on real measured noise from the Changchun Institute of Optics, Fine Mechanics and Physics torsion‑pendulum facility developed for Taiji, which achieves a torque sensitivity of order 10^‑13\,\mathrmN\cdot m\,Hz^‑1/2, the AI‑WLS framework bounds the maximum absolute estimation errors at 4.46× 10^‑10\,\mathrmA\cdot m^2 for \vecm_r and 7.8× 10^‑8 for χ, satisfying Taiji's ground‑test requirements on all these parameters simultaneously.

Abstract:
Accurate short‑term electricity load forecasting is a cornerstone of U.S. grid reliability; however, prevailing deep learning models remain opaque, limiting operator trust during extreme weather. A unified, interpretable, physics‑informed ensemble framework is proposed, integrating a Convolutional Neural Network (CNN) branch for local feature extraction and a Transformer branch for long‑range dependency modeling; the branches are fused through a validation‑optimized weighted ensemble and regularized by a physics‑informed loss derived from the piecewise parabolic temperature‑demand relationship of the Electric Reliability Council of Texas (ERCOT) system. Post‑hoc interpretability is provided through SHapley Additive exPlanations (SHAP) with the DeepExplainer backend, yielding global and event‑level attributions. Using eight years of ERCOT hourly load data (2018‑2025) fused with Automated Surface Observing System (ASOS) records from three Texas stations, the framework achieves 713 MW MAE, 812 MW RMSE, and 1.18% MAPE on the test window. For Hampel‑flagged extreme events, MAPE falls by 20.7% relative to its Transformer branch and by 40.5% relative to its CNN branch; an ablation confirms that the parabolic and ramp constraints drive a 14.7% RMSE reduction. SHAP analysis reveals a regime shift: temperature dominates under normal operation, whereas wind speed and precipitation become more influential during cold fronts and heatwaves.

Abstract:
Reconstructing magnetization in nanoscale magnetic thin films is essential for developing next‑generation memory, sensors, and various spintronic technologies. However, this remains challenging due to the ill‑posed nature of the stray field inverse problem, i.e., there are infinitely many magnetization solutions to a given stray field distribution. Here, we demonstrate that a physics‑informed deep image prior (DIP) framework, using a simple convolutional autoencoder conditionally achieves a reasonable qualitative and quantitative reconstruction of complex in‑plane magnetization patterns from scanning NV magnetometry. We find that the orientation of user‑defined masks implemented to restrict the reconstruction solution space dramatically affects convergence. The optimal alignment of the mask improves the reconstruction signal‑to‑noise ratio by up to \SI3\decibel, thereby also serving as a diagnostic tool. The DIP approach requires no pre‑trained datasets and is considered computationally less intensive as compared to supervised learning approaches. We analyze both Landau and dipole domain structures in lithographically patterned Permalloy nanostructures by incorporating experimentally‑guided spatial constraints. Complementary magnetic force microscopy measurements were carried out to support the Scanning NV measurements.

Abstract:
Reconstructing high‑fidelity fluid dynamics from sparse temporal observations is quite challenging, mainly due to the chaotic and non‑linear nature of fluid transport. Standard deep learning‑based interpolation methods often tend to regress to the mean, which results in spatial blurring and temporal strobing, especially noticeable around the observed anchor frames where transitions become discontinuous. In this work, we propose a novel Temporal U‑Net architecture that integrates a VGG‑based perceptual loss along with a Physics‑Informed Bridge to overcome these issues. By introducing time‑weighted feature blending and enforcing a parabolic boundary condition defined by t(1 ‑ t), the model ensures smooth transitions while also maintaining perfect consistency at the endpoints. Experimental results on multi‑channel RGB fluid data show that our method clearly outperforms standard models, both in terms of structural fidelity and texture preservation. In particular, the model achieves a Mean Absolute Error of 0.015, compared to 0.085 for a standard L1 baseline. Further Spatial Power Spectral Density (PSD) analysis reveals that the model is able to retain high‑frequency turbulent details that are usually lost in deterministic reconstructions.

Abstract:
Precisely modeling radio propagation in dynamic wireless environments is fundamental to the realization of wireless digital twins. Traditional ray tracing methods rely on accurate 3D models with detailed environment parameters, while recent neural radiance field approaches learn representations tied to specific static scenes, requiring retraining when environments change. In this paper, we propose RadTwin, a generalizable wireless digital twin framework that explicitly conditions on scene geometry, enabling adaptation to dynamic environments without retraining. RadTwin comprises three key components: 1) a scenario representation network that extracts high‑level latent scene features from point clouds, 2) an electromagnetic ray tracing module that computes physics‑informed sparse attention masks identifying voxels that physically contribute signals toward each query direction, and 3) a neural propagation decoder that aggregates relevant scene features through masked cross‑attention to learn how radio propagation behaves within the given scene geometry. We evaluate RadTwin on a customized dataset of indoor scenes with varying furniture arrangements. Experimental results show that RadTwin achieves 31.6% higher SSIM (0.846 vs. 0.643) and 91.96% lower LPIPS (0.023 vs. 0.286) compared to NeRF2. RadTwin further demonstrates superior cross‑scale performance and high generalization and data efficiency, representing a significant advancement toward practical digital network twins for dynamic wireless environments.

Abstract:
Building models that generalize across physical systems without retraining remains a central challenge in computational science. Here we introduce In‑Context Modeling (ICM), a retrain‑free paradigm that infers physical relationships directly from observational fields. Rather than encoding system‑specific behavior in fixed parameters, ICM assimilates measurements as physical context and performs inference through a single forward pass. Trained in a physics‑informed, label‑free manner using governing equations, a single model generalizes across unseen materials, geometries, and loading conditions. Demonstrated on hyperelasticity, ICM integrates with finite‑element simulations and is validated using experimental full‑field measurements. Moreover, performance improves with increasing data diversity and computational budget, exhibiting favorable scaling behavior analogous to foundation models. By recasting physical modeling as in‑context inference, this work establishes a transferable paradigm for retrain‑free scientific learning and a foundation for scalable modeling across computational science.

Abstract:
In this paper, we propose a Physics‑Informed Neural Network framework for time‑dependent simulations of pollution propagation originating from moving emission sources. We formulate a robust variational framework for the time‑dependent advection‑diffusion problem and establish the boundedness and inf‑sup stability of the corresponding discrete weak formulation. Based on this mathematical foundation, we construct a robust loss function that is directly related to the true approximation error, defined as the difference between the neural network approximation and the (unknown) exact solution. Additionally, a collocation‑based strategy is introduced to speed up neural network training. As a case study, we investigate pollution propagation caused by snowmobile traffic in Longyearbyen, Spitsbergen, supported by detailed in‑field measurements collected using dedicated sensors. The proposed framework is applied to analyze the effects of thermal inversion on pollutant accumulation. Our results demonstrate that thermal inversion traps dense and humid air masses near the ground, significantly enhancing particulate matter (PM) concentration and worsening local air quality.

Abstract:
High‑fidelity simulation of particle‑matter interactions provides the essential theoretical reference for diverse physics disciplines, yet generating synthetic datasets at the scale of current and future experiments has become prohibitive. Here, we introduce PHIN‑GAN, a novel physics‑informed generative adversarial network designed to address this challenge. We derive a set of analytical probability density functions, that effectively describe the ``straggling function'' identified with Landau. For the first time, this enables their continuous evaluation across the entire phase‑space. These analytical forms are leveraged to enforce a parametric distribution‑level learning objective. Rooted in first principles, PHIN‑GAN offers a generalizable, interpretable and scalable proof‑of‑concept approach for a lossless generative model that maintains the high fidelity of the standard‑bearer for simulating such interactions, namely GEANT4, at a fraction of the computational cost.

Abstract:
In many cyber‑physical systems, especially in critical applications such as aeroplanes, data to train anomaly detection and diagnosis algorithms is lacking due to data protection issues and partial observability. To combat this inherent lack of data, we introduce a high‑fidelity, physics‑informed co‑simulation of a common aircraft main‑fuel‑pump system modelled in \textscMATLAB/Simulink Simscape Fluids. We also describe its generated time‑series data with health and fault mode annotations. To show feasibility of our benchmark, we apply an unsupervised Recurrent Variational Autoencoder (RNN‑VAE) for anomaly detection and a SOM‑VAE for operating mode discretization, trained to separate healthy and faulty conditions.

Abstract:
Neutrino oscillations provide crucial insights into fundamental particle physics, with two‑flavor approximations effectively describing reactor and atmospheric phenomena. This paper investigates the application of Physics‑Informed Neural Networks (PINNs), which have several advantages over traditional solvers. Traditional methods typically depend on mesh‑based techniques or dimensionality reduction approaches to solve the governing differential equations for neutrino evolution in vacuum and matter environments. We review the theoretical framework, including vacuum mixing and the Mikheyev‑Smirnov‑Wolfenstein (MSW) effect in matter, and demonstrate PINN implementations for vacuum and constant‑density profiles. This Machine learning based approach for reactor (low‑energy) and atmospheric (high‑energy) neutrinos shows high precision similar to analytical solutions, with mean squared errors of the order of 10^‑3~10^‑4. We have also discussed the robustness of PINNs in solving coupled ODE systems, along with future extensions to three‑flavor effects.

Abstract:
State estimation is a cornerstone of power system control‑center operations, and its robust operation is increasingly a cyber‑physical security concern as modern grids become more digitalized and communication‑intensive. Neural network‑based approaches have gained attention as alternatives to conventional model‑based state estimation methods. Physics‑Informed Neural Networks (PINNs), which embed power‑flow consistency into the learning objective, have shown improved accuracy over existing approaches. This work proposes a PINN‑based model for Power System State Estimation (PSSE) that protects the estimation process against the stealth‑constrained AC False Data Injection Attacks (FDIAs) considered in this study. The model is developed without adversarial training. Instead, a dynamic loss‑weighting formulation based on homoscedastic uncertainty learns the relative scaling of supervised data‑fit and physics‑residual terms during training, reducing sensitivity to manual weight tuning. Robustness is evaluated on the IEEE 118‑bus system using representative stealthy‑FDIA families including state distortion, load redistribution, line overloading, and residual‑constrained stealth corruption. Performance is measured using Mean Absolute Error (MAE) on voltage magnitudes and phase angles. Results demonstrate higher accuracy and stability than existing fixed‑weight PINN variants.

Abstract:
Despite their ubiquity, the rich physics present in a plasma sheath has inhibited the development of a generally applicable description of this critical region. The present study utilizes a physics‑informed neural network (PINN) to evaluate a hierarchy of models of the plasma sheath. Unlike traditional deep learning methods, PINNs use the governing PDEs to constrain the predictions of a neural network, and thus do not require any experimental or simulation data to train. In this work, we utilize a PINN to identify the parametric solution to fluid models of different physics fidelity of the plasma sheath. While the offline training time of the PINN is often longer than a traditional solver, once trained, the PINN is able to efficiently predict the sheath profiles across a broad range of parameter regimes, thus yielding an effective surrogate of the plasma sheath.

Abstract:
Wireless localization of permanent magnets enables occlusion‑free guidance for medical interventions, yet its practical accuracy is fundamentally limited by two coupled challenges: the poor observability of conventional planar sensor arrays and the simulation‑to‑reality (Sim‑to‑Real) gap of learning‑based estimators. To address these issues, this article presents a unified framework that combines information‑theoretic sensor geometry optimization with physics‑aware deep learning. First, a rigorous Fisher Information Matrix (FIM)‑based evaluation framework is established to quantify geometry‑induced observability limitations. The results show that a staggered split‑array topology provides a substantially stronger observability foundation for localization while remaining compatible with practical external deployment. Second, building on this optimized sensing configuration, we propose Phy‑GAANet, a calibration‑free estimator trained entirely on hardware‑aware synthetic data. By incorporating Physics‑Informed Features (PIF) for saturation modeling and Geometry‑Aware Attention (GAA) for preserving cross‑layer vector structure, the network effectively bridges the Sim‑to‑Real gap. Extensive real‑world experiments demonstrate state‑of‑the‑art performance, achieving a position error of 1.84 mm and an orientation error of 3.18 degrees at a refresh rate exceeding 270 Hz. The proposed method consistently outperforms classical Levenberg‑‑Marquardt solvers and generic convolutional baselines, particularly in suppressing catastrophic outliers and maintaining robustness in challenging near‑field boundary regions. Beyond the proposed network, the FIM‑guided analysis also provides a framework for sensor geometry design in magnetic localization systems under practical deployment constraints.

Abstract:
The numerical reconstruction of controls for nonlinear partial differential equations (PDEs) remains a challenging and relatively underdeveloped problem, despite the extensive literature on controllability theory. In this work, we introduce an operator‑decomposed physics‑informed neural network framework, called WeightedPINN, for approximating controls in nonlinear PDE settings. The method is designed for both internal and bilinear control problems and incorporates the governing equation, boundary and initial conditions, and terminal control constraints directly into the training objective. The main feature of WeightedPINN is that the different components of the controlled PDE residual are weighted separately. In particular, the time derivative, directional diffusion terms, nonlinear response, and control term are assigned independent adaptive space‑‑time weights, and the same weighted formulation is applied to the boundary, initial, and terminal constraints. This produces a control‑aware residual metric that is more sensitive to operator‑level imbalance and to the mechanism through which the unknown control enters the equation. We provide a convergence analysis for the proposed method and present numerical experiments for semilinear heat and wave equations with internal and bilinear controls. The high‑dimensional experiments demonstrate improved residual‑based testing errors compared with the standard PINN baseline, while lower‑dimensional manufactured‑solution benchmarks show improved direct reconstruction errors for both the state and the control against several adaptive and control‑oriented PINN methods. The results suggest that WeightedPINN is particularly effective in regimes where componentwise residual imbalance, anisotropy, variable coefficients, or control‑identification sensitivity play a significant role.

Abstract:
The equation of state (EoS) of strongly interacting matter at finite temperature and chemical potentials (baryon, charge, and strangeness) is a crucial input for hydrodynamic simulations of relativistic heavy‑ion collisions. We construct a four‑dimensional EoS using a deep‑learning‑assisted quasi‑particle model (DLQPM) within a physics‑informed neural network (PINN) framework, in which the masses of light quarks, strange quarks, and gluons are parameterized as functions of temperature and chemical potentials (T, μ_B, μ_Q, μ_S). The model is constrained by lattice QCD data at vanishing chemical potentials and provides a thermodynamically consistent extrapolation to finite μ_B,Q,S. The DLQPM accurately reproduces the lattice‑calculated cumulants χ^B,Q,S_i,j,k at μ_B,Q,S=0, and its predicted EoS at various chemical potentials agrees well with results from the generalized T'‑expansion method in lattice QCD. Furthermore, the calculated baryon‑strangeness correlation C_BS is consistent, within uncertainties, with preliminary STAR data. This work offers a reliable EoS for exploring the QCD phase structure in the beam energy scan region.

Abstract:
Physics‑informed neural networks (PINNs) have garnered significant interest for their potential in solving partial differential equations (PDEs) that govern a wide range of physical phenomena. By incorporating physical laws into the learning process, PINN models have demonstrated the ability to learn physical outcomes reasonably well. However, current PINN approaches struggle to predict or solve new PDEs effectively when there is a lack of training examples, indicating they do not generalize well to unseen problem instances. In this paper, we present a transferable learning approach for PINNs premised on a fast Pseudoinverse PINN framework (Pi‑PINN). Pi‑PINN learns a transferable physics‑informed representation in a shared embedding space and enables rapid solving of both known and unknown PDE instances via closed‑form head adaptation using a least‑squares‑optimal pseudoinverse under PDE constraints. We further investigate the synergies between data‑driven multi‑task learning loss and physics‑informed loss, providing insights into the design of more performant PINNs. We demonstrate the effectiveness of Pi‑PINN on various PDE problems, including Poisson's equation, Helmholtz equation, and Burgers' equation, achieving fast and accurate physics‑informed solutions without requiring any data for unseen instances. Pi‑PINN can produce predictions 100‑1000 times faster than a typical PINN, while producing predictions with 10‑100 times lower relative error than a typical data‑driven model even with only two training samples. Overall, our findings highlight the potential of transferable representations with closed‑form head adaptation to enhance the efficiency and generalization of PINNs across PDE families and scientific and engineering applications.

Abstract:
Standard physics‑informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second‑order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green‑Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE‑residual‑based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second‑order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT‑based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI‑based training consistently outperforms PDE‑based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.

Abstract:
Kolmogorov‑‑Arnold Networks (KANs) have recently attracted attention as edge‑based neural architectures in which learnable univariate functions replace conventional fixed activation functions. A key source of flexibility in KANs is the choice of basis functions used to parameterize the learnable edge functions. In this context, Gaussian basis functions provide a simple and efficient alternative to splines. However, their performance depends strongly on the scale (shape) parameter \(ε\), whose role has not been studied systematically. In this paper, we investigate how \(ε\) affects Gaussian KANs through first‑layer feature geometry, conditioning, and approximation behavior. Our central observation is that scale selection is governed primarily by the first layer, since it is the only layer constructed directly on the input domain and any loss of distinguishability introduced there cannot be recovered by later layers. From this viewpoint, we analyze the first‑layer feature matrix and identify a practical operating interval, \[ ε\in \left[\frac1G‑1,\frac2G‑1\right], \] where \(G\) denotes the number of Gaussian centers. We interpret this interval not as a universal optimality result, but as a stable and effective design rule, and validate it through brute‑force sweeps over \(ε\) across function‑approximation problems with different collocation densities, grid resolutions, network architectures, and input dimensions, as well as physics‑informed problems. We further show that this range is useful for fixed‑scale selection, variable‑scale constructions, constrained training of \(ε\), and efficient scale search using early training MSE. In this way, the paper positions scale selection as a practical design principle for Gaussian KANs rather than as an ad hoc hyperparameter choice.

Abstract:
The rapid growth of inverter‑based resources (IBRs) and distributed energy resources (DERs) has fundamentally altered the long‑term voltage stability characteristics of modern power systems. This article leverages the advantages of machine learning (ML) for the online estimation of long‑term voltage stability margin (VSM) and enhancement of VSM through coordinated transmission system operator‑distribution system operator (TSO‑DSO) optimization. An explicit analytical VSM expression is derived from offline T&D co‑simulation data using a physics‑informed ML‑trained model under probabilistic loading and generation mix scenarios, while accounting for unbalanced distribution modeling. The resulting closed‑form VSM representation is linearized and embedded into the TSO optimization problem, enabling real‑time enforcement of minimum VSM constraints. We further enhance operational efficiency by incorporating VSM sensitivities into both transmission and distribution optimization, allowing prioritization of the most influential reactive power resources. Simulation studies conducted on the IEEE 30‑bus transmission network integrated with multiple IEEE 37‑node distribution feeders validate that the proposed framework successfully achieves the desired VSM enhancement while maintaining high estimation accuracy.

Abstract:
Spreading of liquid droplets on solid substrates constitutes a classic multiphysics problem with widespread applications ranging from inkjet printing, spray cooling, to biomedical microfluidic systems. Yet, accurate computational fluid dynamic (CFD) simulations are prohibitively expensive, taking more than 18 to 24 hours for each transient computation. In this paper, Physics‑Informed Laplace Operator Neural Network (PI‑LNO) is introduced, representing a novel architecture where the Laplace integral transform function serves as a learned physics‑informed functional basis. Extensive comparative benchmark studies were performed against five other state‑of‑the‑art approaches: UNet, UNet with attention modules (UNet‑AM), DeepONet, Physics‑Informed UNet (PI‑UNet), and Laplace Neural Operator (LNO). Through complex Laplace transforms, PI‑LNO natively models the exponential transient dynamics of the spreading process. A TensorFlow‑based PI‑LNO is trained on multi‑surface CFD data spanning contact angles θ_s ε[20,160], employing a physics‑regularized composite loss combining data fidelity (MSE, MAE, RMSE) with Navier‑Stokes, Cahn‑Hilliard, and causality constraints.

Abstract:
Accurate reconstruction of localized extreme structures remains a critical bottleneck in the physics‑informed modeling of electro‑thermal‑convective flows. Although conventional physics‑informed neural networks effectively capture smooth global dynamics, they frequently suffer from numerical diffusion and distortion when attempting to resolve sharp charge boundary layers or abrupt multiphase interfaces. To address these limitations, we propose a Residual‑Attention Physics‑Informed Neural Network (RA‑PINN) that embeds gated attention modulation within a residual feature framework to adaptively enhance local sensitivity to steep physical gradients. The proposed architecture is rigorously evaluated against standard and recurrent network baselines using canonical electrohydrodynamic scenarios, encompassing near‑electrode exponential boundary layers and sharply concentrated charge fields. Quantitative analyses demonstrate that the RA‑PINN significantly reduces localized errors and faithfully preserves critical interface topologies without compromising the global consistency dictated by the coupled governing equations. Ultimately, this methodology establishes a highly robust predictive framework for resolving complex interfacial and boundary layer phenomena in advanced fluid dynamics applications.

Abstract:
This paper presents the hybrid solver for a CO_2 sequestration problem. The solver uses the IGA‑ADS (IsoGeometric Analysis Alternating Directions solver) to compute the saturation scalar field update using the explicit method, and CRVPINN (Collocation‑based Robust Variational Physics Informed Neural Networks solver) to compute the pressure scalar field. The study focuses on simulating the physical behavior of CO_2 in porous structures, excluding chemical reactions. The mathematical model is based on Darcy's Law. The CRVPINN is pretrained on the initial pressure configuration, and the time step pressure updates require only 100 iterations of the Adam method per time step. We compare our hybrid IGA‑ADS solver, coupled with the CRVPINN method, with a baseline of the IGA‑ADS solver coupled with the MUMPS direct solver. Our hybrid solver is over 3 times faster on a single computational node from the ARES cluster of ACK CYFRONET. Future work includes extensive testing, inverse problem solving, and potential application to H_2 storage problems.

Abstract:
Accurate prediction of thermal runaway in lithium‑ion batteries is essential for ensuring the safety, efficiency, and reliability of modern energy storage systems. Conventional data‑driven approaches, such as Long Short‑Term Memory (LSTM) networks, can capture complex temporal dependencies but often violate thermodynamic principles, resulting in physically inconsistent predictions. Conversely, physics‑based thermal models provide interpretability but are computationally expensive and difficult to parameterize for real‑time applications. To bridge this gap, this study proposes a Physics‑Informed Long Short‑Term Memory (PI‑LSTM) framework that integrates governing heat transfer equations directly into the deep learning architecture through a physics‑based regularization term in the loss function. The model leverages multi‑feature input sequences, including state of charge, voltage, current, mechanical stress, and surface temperature, to forecast battery temperature evolution while enforcing thermal diffusion constraints. Extensive experiments conducted on thirteen lithium‑ion battery datasets demonstrate that the proposed PI‑LSTM achieves an 81.9% reduction in root mean square error (RMSE) and an 81.3% reduction in mean absolute error (MAE) compared to the standard LSTM baseline, while also outperforming CNN‑LSTM and multilayer perceptron (MLP) models by wide margins. The inclusion of physical constraints enhances the model's generalization across diverse operating conditions and eliminates non‑physical temperature oscillations. These results confirm that physics‑informed deep learning offers a viable pathway toward interpretable, accurate, and real‑time thermal management in next‑generation battery systems.

Abstract:
A physics‑informed neural network (PINN) is developed, for the first time, to learn the time‑dependent quasi‑static magnetohydrodynamic (MHD) equations in axisymmetric tokamak geometry, without any experimental or synthetic data. The initial study considered an ITER‑like tokamak and found that a PINN, after careful treatment, was capable of learning the solution to the MHD system and predict a vertically displacing plasma, where general agreement with ground truth simulation was observed. The proof‑of‑principle demonstration highlights the potential of physics‑constrained deep learning to learn complex plasma behavior.

Abstract:
Neural surrogates for stiff differential‑algebraic equations (DAEs) face two barriers: soft‑constraint methods leave algebraic residuals that stiffness amplifies into errors, and hard‑constraint methods require trajectory data from stiff integrators. We introduce an extended Newton implicit layer that enforces algebraic constraints exactly and reduces fast dynamics to their quasi‑steady‑state values in a single differentiable solve. Embedded in a physics‑informed DeepONet, the layer recovers all fast and algebraic states exactly from slow‑state predictions, removes the per‑window stiffness‑amplification pathway, and yields a stiffness‑scaled Implicit Function Theorem gradient absent from penalty methods. Cascaded implicit layers extend this to multi‑component systems with provable convergence. On a grid‑forming inverter (stiffness ratio of about 4712), extended Newton attains 1.42% error versus 39.3% (penalty) and 57.0% (standard Newton); augmented Lagrangian and feedback linearization diverged. Two independently trained models compose without retraining (0.72% to 1.16% error, exact constraint satisfaction). Cross‑domain validation on the Robertson stiff DAE (stiffness ratio up to 10^5) confirms generalization. Conformal prediction provides 90% coverage with automatic out‑of‑distribution detection.

Abstract:
The aim of this paper is to introduce a Mapping‑based Hard‑constrained Physics‑Informed Neural Network (MH‑PINN) for efficiently and accurately solving unbounded wave problems. First, we propose a coordinate mapping technique that compactifies the infinite physical domain into a finite computational space. This effectively resolves the sampling difficulties inherent to standard PINNs in unbounded regions. Additionally, it avoids the artificial truncation errors introduced by traditional methods such as perfectly matched layers. Second, we design a physics‑based hard‑constrained network structure that automatically satisfies both the inner boundary conditions and the far‑field radiation conditions. This structure eliminates boundary loss terms, yielding high computational efficiency and fast convergence, which effectively addresses the challenges of high‑frequency problems. Third, we introduce an inverse factor correction for boundary coefficients to address the influence of asymptotic factors,which makes the method highly geometrically adaptable. Finally, we present numerical examples covering various acoustic radiation and scattering scenarios as well as elastic dynamics scenarios to demonstrate the efficiency and accuracy of our algorithm.It highlights its potential for broader applications in the field of computational wave dynamics.

Abstract:
Fractional PDEs involving the fractional Laplacian on bounded domains are challenging because of hypersingular nonlocal kernels, exterior Dirichlet constraints, reduced boundary regularity, and the high computational cost in high dimensions. To address these issues, we first adopt a spatially varying radius with directional distance‑to‑boundary information, which yields a geometry‑adaptive three‑part decomposition of the fractional Laplacian: singular near‑field, regular interior far‑field, and analytical exterior far‑field contributions. Then we employ Gauss‑Jacobi quadrature for the singular radial integral, Gauss quadrature for the regular interior radial integral, and Monte Carlo sampling for the angular variables. A feature‑enhanced physics‑informed neural network trial space is finally used to tackle the low‑regularity behavior near the boundary. Through the above steps, we obtain a quadrature‑enhanced Monte Carlo fractional physics‑informed neural network (QE‑MC‑fPINN) method. Numerical experiments on fractional Poisson equations and time‑dependent fractional PDEs show that, on the tested benchmarks, the proposed method outperforms two representative MC‑fPINN discretizations in accuracy and convergence, especially for solutions with strong boundary singularities.

Abstract:
Micromechanics‑based granular models are widely used to predict the failure behavior of porous and particulate materials, including concrete, soils, foams, and biological tissues. Although these models offer considerable flexibility through microstructural parametrization and statistical representation, their mapping to macroscopic responses, particularly failure envelopes, is implicit and requires costly nonlinear, non‑smooth simulations, where each failure point is obtained by following a loading trajectory. This limitation is further amplified in inverse settings, where one seeks microstructure configurations that reproduce a target failure response. In this work, we propose a differentiable neural operator that learns the mapping from microstructure configurations to failure envelopes, enabling efficient forward prediction and inverse identification without repeated micromechanical simulations. To ensure mechanical admissibility, we incorporate a physics‑informed training strategy that enforces convexity of the predicted envelopes, consistent with Drucker's postulate, thereby eliminating potential non‑physical artifacts. We also compare finite difference and automatic differentiation for evaluating the proposed regularization, and find that finite difference provides a favorable practical trade‑off in the present DeepONet‑based setting. The operator is trained on failure envelopes represented as irregular point clouds, allowing learning from data sampled at heterogeneous resolutions. To further reduce computational cost, we introduce an active learning strategy that adaptively queries the micromechanical model in regions of high epistemic uncertainty. This leads to efficient exploration of the parameter space with fewer high‑fidelity simulations. The versatility and performance of the method are demonstrated and benchmarked through several numerical examples.

Abstract:
Large‑scale automated morphometric analysis of brain MRI is limited by the thick‑slice, anisotropic acquisitions prevalent in routine clinical practice. Existing generative super‑resolution (SR) methods produce visually compelling isotropic volumes but often introduce anatomical hallucinations, systematic volumetric overestimation, and structural distortions that compromise downstream quantitative analysis and diagnostic safety. To address this, we propose CAHAL (Clinically Applicable resolution enHAncement for Low‑resolution MRI scans), a hallucination‑robust, physics‑informed resolution enhancement framework that operates directly in the patient's native acquisition space. CAHAL employs a deterministic bivariate Mixture of Experts (MoE) architecture routing each input through specialised residual 3D U‑Net experts conditioned on both volumetric resolution and acquisition anisotropy, two independent descriptors of clinical MRI acquisition. Experts are optimised with a composite loss combining edge‑penalised spatial reconstruction, Fourier‑domain spectral coherence matching, and a segmentation‑guided semantic consistency constraint. Training pairs are generated on‑the‑fly via physics‑based degradation sampled from a large‑scale real‑world database, ensuring robust generalisation. Validated on T1‑weighted and FLAIR sequences against generative baselines, CAHAL achieves state‑of‑the‑art results, improving the best related methods in terms of accuracy and efficiency.

Abstract:
Physics‑informed neural networks (PINNs) provide a powerful framework for learning governing equations of dynamical systems from data. Biologically‑informed neural networks (BINNs) are a variant of PINNs that preserve the known differential operator structure (e.g., reaction‑diffusion) while learning constitutive terms via trainable neural subnetworks, enforced through soft residual penalties. Existing BINN studies are limited to 1\mathrmD+t reaction‑diffusion systems and focus on forward prediction, using the governing partial differential equation as a regulariser rather than an explicit identification target. Here, we extend BINNs to 2\mathrmD+t systems within a PINN framework that combines data preprocessing, BINN‑based equation learning, and symbolic regression post‑processing for closed‑form equation discovery. We demonstrate the framework's real‑world applicability by learning the governing equations of lung cancer cell population dynamics from time‑lapse microscopy data, recovering 2\mathrmD+t reaction‑diffusion models from experimental observations. The proposed framework is readily applicable to other spatio‑temporal systems, providing a practical and interpretable tool for fast analytic equation discovery from data.

Abstract:
Quantum Fisher Information (QFI) sets the ultimate precision limit for parameter estimation and is therefore a central quantity in quantum metrology. In time‑dependent many‑body systems, however, maximizing QFI is a highly non‑trivial task due to the combined effects of non‑commutativity, control complexity, and the exponential growth of the Hilbert space. In this work, we present a physics‑informed neural network (PINN) framework to address this problem through the learning of counter‑diabatic quantum dynamics. Our approach combines a variational PINN formulation with a Magnus‑expansion treatment of time‑ordered evolution, enabling the adiabatic gauge potential and the scheduling function to be inferred directly from the underlying physics while enforcing the Euler‑Lagrange structure of the protocol. The method is applied to several families of driven spin Hamiltonians, including nearest‑neighbor, dipolar, and trapped‑ion‑inspired interactions, for systems of up to six qubits. The numerical results show that the proposed framework systematically improves over reference solutions based only on the Euler‑Lagrange condition, yielding high normalized QFI together with favorable fidelity and extremal‑balance metrics while preserving small phsical residuals. The analysis further shows that learning the scheduling function provides a clear performance advantage in most cases, and reveals non‑trivial finite‑size effects, with q=3 emerging as a particularly challenging regime. Although scalability remains limited by the exponential growth of the operator space and by automatic‑differentiation costs, the results demonstrate that PINNs constitute a viable and physically grounded route for learning metrologically optimal control strategies in interacting quantum systems.

Abstract:
We present a scalable, data‑driven simulation framework for large‑scale heating, ventilation, and air conditioning (HVAC) systems that couples physics‑informed neural ordinary differential equations (PINODEs) with differential‑algebraic equation (DAE) solvers. At the component level, we learn heat‑exchanger dynamics using an implicit PINODE formulation that predicts conserved quantities (refrigerant mass M_r and internal energy E_\texthx) as outputs, enabling physics‑informed training via automatic differentiation of mass/energy balances. Stable long‑horizon prediction is achieved through gradient‑stabilized latent evolution with gated architectures and layer normalization. At the system level, we integrate learned components with DAE solvers (IDA and DASSL) that explicitly enforce junction constraints (pressure equilibrium and mass‑flow consistency), and we use Bayesian optimization to tune solver parameters for accuracy‑‑efficiency trade‑offs. To reduce residual system‑level bias, we introduce a lightweight corrector network trained on short trajectory segments. Across dual‑compressor and scaled network studies, the proposed approach attains multi‑fold speedups over high‑fidelity simulation while keeping errors low (MAPE below a few percent) and scales to systems with up to 16 compressor‑condenser pairs.

Abstract:
Accurate dynamical modeling is essential for simulation and control of embodied systems, yet first‑principles models of electromechanical systems often fail to capture complex dissipative effects such as joint friction, stray losses, and structural damping. While residual‑learning physics‑informed neural networks (PINNs) can effectively augment imperfect first‑principles models with data‑driven components, the residual terms are typically implemented as unconstrained multilayer perceptrons (MLPs), which may inadvertently inject artificial energy into the system. To more faithfully model the dissipative dynamics, we propose DiLaR‑PINN, a dissipative latent residual PINN designed to learn unmodeled dissipative effects in a physically consistent manner. Structurally, the residual network operates only on unmeasurable (latent) state components and is parameterized in a skew‑dissipative form that guarantees non‑increasing energy for any choice of network parameters. To enable stable and data‑efficient training under partial measurability of the state, we further develop a recurrent rollout scheme with a curriculum‑based sequence length extension strategy. We validate DiLaR‑PINN on a real‑world helicopter system and compare it against four baselines: a pure physical model (without a residual network), an unstructured residual MLP, a DiLaR variant with a soft dissipativity constraint, and a black‑box LSTM. The results demonstrate that DiLaR‑PINN more accurately captures dissipative effects and achieves superior long‑horizon extrapolation performance.

Abstract:
The multi‑scale and non‑linear nature of phase‑field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial‑intelligence approaches. Surrogate models based on neural operators could have a lower computational cost than conventional numerical discretization methods. We propose a new neural operator approach that bridges classical convex‑concave splitting schemes with physics‑informed learning to accelerate the simulation of phase‑field models. It consists of a Deep Ritz method, where a neural operator is trained to approximate a variational formulation of the phase‑field model. By training the neural operator with an energy‑splitting variational formulation, we enforce the energy dissipation property of the underlying models. We further introduce a custom Reaction‑Diffusion Neural Operator (RDNO) architecture, adapted to the operators of the model equations. We successfully apply the deep learning approach to the isotropic Allen‑Cahn equation and to anisotropic dendritic growth simulation. We demonstrate that our physically‑informed training provides better generalization in out‑of‑distribution evaluations than data‑driven training, while achieving faster inference than traditional Fourier spectral methods.

Abstract:
Approximation of functions satisfying partial differential equations (PDEs) is paramount for simulation of physical fluid flows and other problems in physics. Recently, physics‑informed machine learning approaches have proven useful as a data‑driven complement to numerical models for partial differential equations, bringing faster responses and allowing us to capitalize on past observations. However, their efficiency and convergence depend on the availability of vast training datasets. For sparse observations, Gaussian process regression or Kriging has emerged as a powerful interpolation model, offering principled estimates and uncertainty quantification. Several attempts have been made to condition Gaussian processes on linear PDEs via artificial or collocation observations and kernel design.These methods suffer from scalability issues in higher dimensions and limited generalizability. The aim of this study is to explore the extension of the Kriging predictor in the presence of linear PDE information at a finite number of collocation points. Two approaches are proposed: 1) A collocated co‑Kriging with primary observations of the physical field and auxiliary differential observations; 2) A constrained Kriging optimization problem strongly satisfying linear PDE constraints at the points of prediction through a Lagrangian formulation. Numerical experiments are given for ordinary differential equations, 2D harmonic PDEs and an application to perfect flows around a cylinder. This work highlights a trade‑off between the computational efficiency of the Lagrange multipliers approach and the strict interpolation of observations.

Abstract:
Off‑policy learning in constrained MDPs with large binary state spaces faces a fundamental tension: causal identification of transition dynamics requires structural assumptions, while sample‑efficient policy learning requires state‑space compression. We introduce PI‑CMDP, a framework for CMDPs whose constraint dependencies form a layered DAG under a Lifecycle Ordering Assumption (LOA). We propose an Identify‑Compress‑Estimate pipeline: (i) Identify: LOA enables backdoor identification of causal edge weights for cross‑layer pairs, with formal partial‑identification bounds when LOA is violated; (ii) Compress: a Markov abstraction compresses state cardinality from 2^(WL) to (W+1)^L under layer‑priority regularity and exchangeability; and (iii) Estimate: a physics‑guided doubly‑robust estimator remains unbiased and reduces the variance constant when the physics prior outperforms a learned model. We instantiate PI‑CMDP on constraint repair in engineering simulation pipelines. On the TPS benchmark (4,206 episodes), PI‑CMDP achieves 76.2% repair success rate with only 300 training episodes versus 70.8% for the strongest baseline (+5.4 pp), narrowing to +2.8 pp (83.4% vs. 80.6%) in the full‑data regime, while substantially reducing cascade failure rates. All improvements are consistent across 5 independent seeds (paired t‑test p < 0.02).

Abstract:
World models, generative AI systems that simulate how environments evolve, are transforming autonomous driving, yet all existing approaches adopt an ego‑vehicle perspective, leaving the infrastructure viewpoint unexplored. We argue that infrastructure‑centric world models offer a fundamentally complementary capability: the bird's‑eye, multi‑sensor, persistent viewpoint that roadside systems uniquely possess. Central to our thesis is a spatio‑temporal complementarity: fixed roadside sensors excel at temporal depth, accumulating long‑term behavioral distributions including rare safety‑critical events, while vehicle‑borne sensors excel at spatial breadth, sampling diverse scenes across large road networks. This paper presents a vision for Infrastructure‑centric World Models (I‑WM) in three phases: (I) generative scene understanding with quality‑aware uncertainty propagation, (II) physics‑informed predictive dynamics with multi‑agent counterfactual reasoning, and (III) collaborative world models for V2X communication via latent space alignment. We propose a dual‑layer architecture, annotation‑free perception as a multi‑modal data engine feeding end‑to‑end generative world models, with a phased sensor strategy from LiDAR through 4D radar and signal phase data to event cameras. We establish a taxonomy of driving world model paradigms, position I‑WM relative to LeCun's JEPA, Li Fei‑Fei's spatial intelligence, and VLA architectures, and introduce Infrastructure VLA (I‑VLA) as a novel unification of roadside perception, language commands, and traffic control actions. Our vision builds upon existing multi‑LiDAR pipelines and identifies open‑source foundations for each phase, providing a path toward infrastructure that understands and anticipates traffic.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a promising framework for solving inverse problems governed by partial differential equations (PDEs), including the reconstruction of turbulent flow fields from sparse data. However, most existing PINN formulations are deterministic and do not provide reliable quantification of epistemic uncertainty, which is critical for ill‑posed problems such as data‑driven Reynolds‑averaged Navier‑Stokes (RANS) modeling. In this work, we develop and systematically evaluate a set of probabilistic extensions of PINNs for uncertainty quantification in turbulence modeling. The proposed framework combines (i) Bayesian PINNs with Hamiltonian Monte Carlo sampling and a tempered multi‑component likelihood, (ii) Monte Carlo dropout, and (iii) repulsive deep ensembles that enforce diversity in function space. Particular emphasis is placed on the role of ensemble diversity and likelihood tempering in improving uncertainty calibration for PDE‑constrained inverse problems. The methods are assessed on a hierarchy of test cases, including the Van der Pol oscillator and turbulent flow past a circular cylinder at Reynolds numbers Re=3,900 (direct numerical simulation data) and Re = 10,000 (experimental particle image velocimetry data). The results demonstrate that Bayesian PINNs provide the most consistent uncertainty estimates across all inferred quantities, while function‑space repulsive ensembles offer a computationally efficient approximation with competitive accuracy for primary flow variables. These findings provide quantitative insight into the trade‑offs between accuracy, computational cost, and uncertainty calibration in physics‑informed learning, and offer practical guidance for uncertainty quantification in data‑driven turbulence modeling.

Abstract:
Magnetic Resonance Imaging (MRI) is vital for prostate cancer (PCa) diagnosis. While advanced techniques such as Hybrid Multi‑dimensional MRI (HM‑MRI) have enhanced diagnostic capabilities, the significant need remains for robust, automated Artificial Intelligence (AI)‑based detection methods. In this study, we combine quantitative HM‑MRI of tissue composition with an AI‑based neural network. We propose the Hadamard‑Bias Network plus ResNet18 (HBR‑Net‑18), a two‑stage AI framework for PCa detection. In the first stage, a Hadamard U‑Net‑based algorithm suppresses intensity inhomogeneities (bias fields) across six parametric HM‑MRI maps generated via a Physics‑Informed Autoencoder (PIA). In the second stage, a Residual Network (ResNet‑18) performs patch‑level classification. The framework utilizes overlapping 11‑by‑11 patches, incorporating both 2D intra‑slice and 3D inter‑slice (adjacent‑slice) information to improve spatial consistency. Our experimental results demonstrate that HB‑Net achieves balanced sensitivity and specificity, significantly outperforming conventional radiomics‑based approaches and baseline CNN models, highlighting its potential for clinical deployment.

Abstract:
We propose Physics‑Informed Tracking (PIT), a video‑based framework for tracking a single particle from video, where a neural network autoencoder localizes a particle as a heatmap peak (landmark) and a differentiable physics module embedded in the autoencoder constrains several landmarks over time (a trajectory) to satisfy known dynamics. The novel Physics‑Informed Landmark Loss (PILL) compares this predicted trajectory back against the landmarks, enforcing physical consistency without labels. Its supervised variant (PILLS) instead compares the prediction against ground‑truth position, velocity, and bounce from simulation, enabling end‑to‑end backpropagation. To support supervised and unsupervised learning, we use an autoencoder with a split bottleneck that separates A) tracking‑related structure via landmark heatmaps from B) background noise and subsequent image reconstruction. We evaluate a replicated 26 factorial design (n = 4 replicates, 64 configurations), showing that PILLS consistently achieves sub‑pixel tracking accuracy for the bilinear and physics‑refined decoder outputs under both clean and noisy conditions.

Abstract:
This thesis develops numerical and theoretical approaches for understanding and analyzing singularity formation in Partial Differential Equations (PDEs). The singularity formation in the Navier‑Stokes Equation (NSE) is famously challenging as one of the seven Clay Prize problems. Unlike simpler equations such as the Nonlinear Heat (NLH) or Keller‑Segel (KS) equations, where formal asymptotics near blowup are better understood, the intrinsic complexity of NSE makes quantitative analytical treatment difficult, if not impossible, without numerical guidance. Building on numerical insights, we introduce a robust analytical framework to simplify and systematize pen‑and‑paper proofs for simpler singular PDEs. We present a novel approach based on enforcing vanishing modulation conditions for perturbations around approximate blowup profiles, complemented by singularly weighted energy estimates. We demonstrate the efficacy of our method on PDEs with complicated asymptotics, such as NLH and the Complex Ginzburg‑Landau (CGL) equation, and address the open problem of singularity formation in the 3D KS equation with logistic damping. We develop and refine numerical approaches that facilitate deeper insights into singularity formation. We demonstrate that machine learning methods significantly enhance our capability to identify and characterize potential blowup solutions with high precision. We improve on existing Physics‑Informed Neural Network (PINN) and Neural Operator (NO) frameworks. Moreover, we present a novel machine learning paradigm, the Kolmogorov‑Arnold Network (KAN) architecture, whose interpretability and excellent scaling properties are achieved through learnable nonlinearities.

Abstract:
Road traffic accidents are a leading cause of fatalities worldwide. In the US, human error causes 94% of crashes, resulting in excess of 7,000 pedestrian fatalities and 500 billion in costs annually. Autonomous Vehicles (AVs) with emergency collision avoidance systems that operate at the limits of vehicle dynamics at a high frequency, a dual constraint of nonlinear kinodynamic accuracy and computational efficiency, further enhance safety benefits during adverse weather and cybersecurity breaches, and to evade dangerous human driving when AVs and human drivers share roads. This paper parameterizes a Deep Reinforcement Learning (DRL) collision avoidance policy Out‑Of‑Distribution (OOD) utilizing race car overtaking, without explicit geometric mimicry reference trajectory guidance, in simulation, with a physics‑informed, simulator exploit‑aware reward to encode nonlinear vehicle kinodynamics. Two policies are evaluated, a default uni‑direction and a reversed heading variant that navigates in the opposite direction to other cars, which both consistently outperform a Model Predictive Control and Artificial Potential Function (MPC‑APF) baseline, with zero‑shot transfer to proportionally scaled hardware, across three intersection collision scenarios, at 31x fewer Floating Point Operations (FLOPS) and 64x lower inference latency. The reversed heading policy outperforms the default racing overtaking policy in head‑to‑head collisions by 30% and the baseline by 50%, and matches the former in side collisions, where both DRL policies evade 10% greater than numerical optimal control.

Abstract:
Non‑local thermodynamic equilibrium (NLTE) calculations remain a major computational bottleneck in radiation‑‑hydrodynamics, while most existing machine‑learning surrogates treat NLTE as a static input‑‑output mapping rather than a kinetic evolution problem. Here, we present a physics‑informed Latent Space Dynamics Identification (pLaSDI) framework specifically designed for NLTE atomic kinetics, which captures the time‑dependent atomic kinetics of non‑equilibrium plasmas through an explicit reduced governing equation. To ensure the physical reliability of the reduced model, we impose physics‑informed loss terms that enforce macroscopic consistency, dynamical stability, and convergence to the correct steady state during long‑time integration. Applied to tin NLTE population data generated along hydrodynamically modeled temperature‑‑density trajectories relevant to extreme ultraviolet (EUV) lithography plasmas, the model accurately reproduces charge‑state evolution and mean charge state with errors below 2%, achieves speedups of approximately 5×10^4‑‑10^5, and remains stable outside the training trajectories by converging toward physically admissible states and the correct steady‑state solution under fixed plasma conditions. These results show that careful physics‑informed design of the latent dynamics, rather than data fitting alone, is essential for constructing fast, stable, and physically reliable extrapolative surrogates for time‑dependent NLTE kinetics.

Abstract:
Shell structures are pivotal in the fields of architecture and engineering, due to their aesthetic appeal and structural efficiency. Recently, 3D concrete printing has reignited the interest in these structures. But, as printed concrete cannot be reinforced with steel, structures built in this way must be designed to withstand primarily pure compression: they must be funicular shells. Nevertheless, a fundamental challenge remains unsolved since Robert Hooke's discovered the catenary arch in 1675: it is not known whether the concept of a funicular polygon can be generalised to three‑dimensional structures. Generative Adversarial Networks (GANs), have shown remarkable success in generating realistic data samples matching the distribution of the training data and have been shown to produce highly convincing synthetic images. This work proposes a physics‑informed generative adversarial framework for the design of funicular shell structures. The approach employs a modified Deep Convolutional Generative Adversarial architecture physically guided by an auxiliary discriminator to generate realistic and structurally efficient shell geometries. Specifically, the model is constrained by the membrane factor to penalize geometries dominated by bending. An additional discriminator is also employed allowing the model to deal with more complex structures. Results show that the developed model is stable and capable of generating physically optimal, previously unseen, funicular shells with smooth forms and high membrane factor distributions.

Abstract:
Accurate phase equilibria are foundational to alloy design because they encode the underlying thermodynamics governing stability, transformations, and processing windows. However, while the CALculation of Phase Diagrams (CALPHAD) provides a rigorous thermodynamic framework, exploring multicomponent composition‑temperature space remains computationally expensive and is typically limited to sparse section. To enable rapid phase mapping and alloy screening, we propose a physics‑informed graph attention network (GAT) that learns element‑aware representations and couples them with thermodynamic constraints for multi‑label phase‑set prediction in the Ag‑Bi‑Cu‑Sn alloy system. Using about 25,000 equilibrium states generated with pycalphad, each composition‑temperature point is represented as a four‑node element graph with atomic fractions and elemental descriptors as node features. The model combines graph attention, global pooling, and a multilayer perceptron to predict nine relevant phases. To improve physical consistency, we incorporate thermodynamic constraints, applied as training penalties or as an inference‑time projection. Across six binary and three ternary subsystems, the baseline model achieves a macro‑F1 score of 0.951 and 93.98% exact‑set match, while physics‑informed decoding improves robustness and raises exact‑set accuracy to about 96% on dense in‑domain grids. The surrogate also generalizes to an unseen ternary section with 99.32% exact‑set accuracy and to a quaternary section at 700 °C with 91.78% accuracy. These results demonstrate that attention‑based graph learning coupled with thermodynamic constraint enforcement provides an effective and physically consistent surrogate for high‑resolution phase mapping and extrapolative alloy screening.

Abstract:
Representing turbulent flow fields in a compact yet physically faithful form remains a central challenge in computational fluid dynamics. We propose a continuous parametric representation based on localized Gaussian primitives, in which the velocity field is modeled as a superposition of kernels with learnable positions, amplitudes, and scales. This formulation yields a compact, grid‑independent encoding while enabling evaluation of derived quantities such as vorticity and enstrophy. The approach is assessed on three‑dimensional Taylor‑Green vortex fields spanning stages from smooth flow to fully developed turbulence. We quantify the compression‑accuracy trade‑off using both primary variables and derivative‑sensitive diagnostics. The baseline isotropic formulation achieves high velocity accuracy at compression ratios exceeding 1e3‑1e4, but exhibits substantial enstrophy degradation due to loss of small‑scale structure. To address this limitation, we investigate structure‑aware extensions including adaptive placement, multi‑resolution kernels, and anisotropic Gaussians. The anisotropic formulation provides the most consistent improvement, better aligning with elongated vortical structures and recovering intermediate‑ and high‑wavenumber content, while other strategies yield modest gains. A compact‑support Beta basis improves enstrophy in some cases but introduces localized artifacts. Overall, the results indicate that the main limitation of baseline Gaussian representations lies in geometric expressiveness rather than parameter count. The proposed framework provides a compact, interpretable, and continuous representation of turbulent flows, and establishes a foundation for structure‑aware and physics‑informed flow compression.

Abstract:
Flow and transport in fractured geological media are strongly controlled by aperture heterogeneity and uncertainty in subsurface characterisation, yet most upscaling approaches rely on deterministic representations of fracture permeability. This study presents a scalable probabilistic workflow that bridges image‑based fracture geometry and uncertainty‑aware hydraulic predictions across scales. The approach integrates Bayesian correction of aperture‑permeability model misspecification, a deep learning surrogate for predicting spatially distributed permeability statistics, and Darcy‑scale flow upscaling to propagate uncertainty to effective transmissivity. The workflow is applied to natural shear fractures from core material in the Little Grand Wash Fault damage zone (Utah) and to simplified geometries derived from the same datasets. The Bayesian component quantifies uncertainty due to measurement errors and imperfect constitutive relations, while a Residual U‑Net learns the effects of local heterogeneity and spatial correlation on predicted permeability uncertainty. Together, these components generate ensembles of permeability fields that are subsequently upscaled to probabilistic macroscopic flow responses. Results show that common empirical aperture‑permeability relations are systematically biased for natural fractures, whereas the proposed probabilistic workflow yields uncertainty‑aware permeability estimates consistent with physics‑based behaviour. The method captures the impact of channelisation, connectivity, and complex 3D void geometries on transmissivity while quantifying the resulting uncertainty bounds. Computational efficiency arises from the proposed hybrid strategy for probabilistic upscaling, which combines physics‑informed and data‑driven approaches, preserves Stokes‑flow consistency and supports uncertainty propagation without repeated high‑fidelity simulations.

Abstract:
Large‑scale Unmanned Aerial Vehicle (UAV) failures can split an unmanned aerial vehicle swarm network into disconnected sub‑networks, making decentralized recovery both urgent and difficult. Centralized recovery methods depend on global topology information and become communication‑heavy after severe fragmentation. Decentralized heuristics and multi‑agent reinforcement learning methods are easier to deploy, but their performance often degrades when the swarm scale and damage severity vary. We present Physics‑informed Graph Adversarial Imitation Learning algorithm (PhyGAIL) that adopts centralized training with decentralized execution. PhyGAIL builds bounded local interaction graphs from heterogeneous observations, and uses physics‑informed graph neural network to encode directional local interactions as gated message passing with explicit attraction and repulsion. This gives the policy a physically grounded coordination bias while keeping local observations scale‑invariant. It also uses scenario‑adaptive imitation learning to improve training under fragmented topologies and variable‑length recovery episodes. Our analysis establishes bounded local graph amplification, bounded interaction dynamics, and controlled variance of the terminal success signal. A policy trained on 20‑UAV swarms transfers directly to swarms of up to 500 UAVs without fine‑tuning, and achieves better performance across reconnection reliability, recovery speed, motion safety, and runtime efficiency than representative baselines.

Abstract:
Always‑on converter health monitoring demands sub‑mW edge inference, a regime inaccessible to GPU‑based physics‑informed neural networks. This work separates spiking temporal processing from physics enforcement: a three‑layer leaky integrate‑and‑fire SNN estimates passive component parameters while a differentiable ODE solver provides physics‑consistent training by decoupling the ODE physics loss from the unrolled spiking loop. On an EMI‑corrupted synchronous buck converter benchmark, the SNN reduces lumped resistance error from 25.8% to 10.2% versus a feedforward baseline, within the \pm 10% manufacturing tolerance of passive components, at a projected ～270× energy reduction on neuromorphic hardware. Persistent membrane states further enable degradation tracking and event‑driven fault detection via a +5.5 percentage‑point spike‑rate jump at abrupt faults. With 93% spike sparsity, the architecture is suited for always‑on deployment on Intel Loihi 2 or BrainChip Akida.

Abstract:
We present PINNACLE, an open‑source computational framework for physics‑informed neural networks (PINNs) that integrates modern training strategies, multi‑GPU acceleration, and hybrid quantum‑classical architectures within a unified modular workflow. The framework enables systematic evaluation of PINN performance across benchmark problems including 1D hyperbolic conservation laws, incompressible flows, and electromagnetic wave propagation. It supports a range of architectural and training enhancements, including Fourier feature embeddings, random weight factorization, strict boundary condition enforcement, adaptive loss balancing, curriculum training, and second‑order optimization strategies, with extensibility to additional methods. We provide a comprehensive benchmark study quantifying the impact of these methods on convergence, accuracy, and computational cost, and analyze distributed data parallel scaling in terms of runtime and memory efficiency. In addition, we extend the framework to hybrid quantum‑classical PINNs and derive a formal estimate for circuit‑evaluation complexity under parameter‑shift differentiation. Results highlight the sensitivity of PINNs to architectural and training choices, confirm their high computational cost relative to classical solvers, and identify regimes where hybrid quantum models offer improved parameter efficiency. PINNACLE provides a foundation for benchmarking physics‑informed learning methods and guiding future developments through quantitative assessment of their trade‑offs.

Abstract:
We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks. Natural gradient descent (NGD) for the optimization of a loss function can be seen as a preconditioned gradient descent where updates in the parameter space are driven by a functional perspective. In a spirit similar to Newton's method, a NGD step uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. This corresponds to a locally optimal update in function space, following a projected gradient onto the tangent space to the manifold. Still, both gradient and natural gradient descent methods get stuck in local minima. Furthermore, when the model class is a nonlinear manifold or the loss function is not ideally conditioned (e.g., the KL‑divergence for density estimation, or a norm of the residual of a partial differential equation in physics informed learning), even the natural gradient might yield non‑optimal directions at each step. This work introduces a natural version of classical inertial dynamic methods like Heavy‑Ball or Nesterov and show how it can improve the learning process when working with nonlinear model classes.

Abstract:
We explore the possibility of solving Partial Differential Equations (PDEs) using discrete weak formulations. We propose a programming environment for defining a discrete computational domain, introducing discrete functions defined over a set of points, constructing discrete inner products, and introducing discrete weak formulations employing Kronecker delta test functions. Building on this setup, we propose a discrete neural network representation, training the solution function defined over a discrete set of points and employing discrete finite difference derivatives in the automatic differentiation procedures. As a challenging computational model example, we focus on Stokes equations in two‑dimensions, defined over a discrete set of points. We train the solution using the discrete weak residual and the Adamax algorithm with discrete automatic differentiation of the discrete gradients. Despite introducing the python environment, we also provide a rigorous mathematical formulation based on discrete weak formulations, proving the well‑posedness and robustness of the loss function. The solution of the discrete weak formulations is based on neural network training employing a robust loss function that is related to the true error. In this way, we have a robust control of the numerical error during the training of the neural networks. Besides the Stokes formulation, we also explain the functionality of the proposed library using the Laplace problem formulation.

Abstract:
Physics‑Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss landscapes. We propose a lightweight curvature‑aware optimization framework that augments existing first‑order optimizers with an adaptive predictive correction based on secant information. Consecutive gradient differences are used as a cheap proxy for local geometric change, together with a step‑normalized secant curvature indicator to control the correction strength. The framework is plug‑and‑play, computationally efficient, and broadly compatible with existing optimizers, without explicitly forming second‑order matrices. Experiments on diverse PDE benchmarks show consistent improvements in convergence speed, training stability, and solution accuracy over standard optimizers and strong baselines, including on the high‑dimensional heat equation, Gray‑‑Scott system, Belousov‑‑Zhabotinsky system, and 2D Kuramoto‑‑Sivashinsky system.

Abstract:
Edge intelligence is constrained by the energy and latency costs of shuttling data through electronic memory hierarchies. Optical systems offer a fundamentally different computational regime: once an input wavefront is launched into a structured medium, propagation, diffraction, and interference jointly enact a linear transformation whose cost is determined by wave physics rather than by clocked arithmetic. This paper develops a rigorous systems‑level treatment of that regime and introduces a hybrid diffractive holographic architecture for image classification. The proposed model couples a Diffractive Optical Neural Network (DONN) with a Holographic Interference‑Based Learning (HIBL) operator a formal map from digitally optimized phase distributions to physically realizable, fabrication‑compatible interference patterns embeddable in passive optical elements. We express the full inference pipeline as a composition of encoding, phase modulation, free‑space propagation, and intensity measurement operators, making explicit which quantities are learned, which are fixed by design, and where nonlinearity enters through photodetection. This operator‑theoretic view resolves a persistent gap in the optical‑ML literature between learning a transformation and physically realizing it. In physics‑informed simulation on MNIST, a three‑layer system with approximately 25,000 phase elements achieves 91.2% test accuracy with propagation‑limited nanosecond‑scale latency. The primary contribution is not a performance claim but a precise computational framework: learned representations can be physically embedded into structured optical media so that inference is executed by wavefront transformation through a passive, fabricated object rather than by sequential electronic multiply accumulate operations.

Abstract:
Nonlinear system identification must balance physical interpretability with model flexibility. Classical methods yield structured, control‑relevant models but rely on rigid parametric forms that often miss complex nonlinearities, whereas Neural ODEs are expressive yet largely black‑box. Physics‑Informed Neural Networks (PINNs) sit between these extremes, but inverse PINNs typically assume a known governing equation with fixed coefficients, leading to identifiability failures when the true dynamics are unknown or state‑dependent. We propose SOLIS, which models unknown dynamics via a \emphstate‑conditioned second‑order surrogate model and recasts identification as learning a Quasi‑Linear Parameter‑Varying (Quasi‑LPV) representation, recovering interpretable natural frequency, damping, and gain without presupposing a global equation. SOLIS decouples trajectory reconstruction from parameter estimation and stabilizes training with a cyclic curriculum and Local Physics Hints windowed ridge‑regression anchors that mitigate optimization collapse. Experiments on benchmarks show accurate parameter‑manifold recovery and coherent physical rollouts from sparse data, including regimes where standard inverse methods fail.

Abstract:
Accurate temperature estimation of pouch cells with indirect liquid cooling is essential for optimizing battery thermal management systems for transportation electrification. However, it is challenging due to the computational expense of finite element simulations and the limitations of data‑driven models. This paper presents a physics‑informed machine learning (PIML) framework for the efficient and reliable estimation of steady‑state temperature profiles. The PIML approach integrates the governing heat transfer equations directly into the neural network's loss function, enabling high‑fidelity predictions with significantly faster convergence than purely data‑driven methods. The framework is evaluated on a dataset of varying cooling channel geometries. Results demonstrate that the PIML model converges more rapidly and achieves markedly higher accuracy, with a 49.1% reduction in mean squared error over the data‑driven model. Validation against independent test cases further confirms its superior performance, particularly in regions away from the cooling channels. These findings underscore the potential of PIML for surrogate modeling and design optimization in battery systems.

Abstract:
Accurate thermal modeling in metal additive manufacturing (AM) is essential for understanding the process‑structure‑performance relationship. While prior studies have explored generalization across unseen process conditions, they often require extensive datasets, costly retraining, or pre‑training. Generalization across different materials also remains relatively unexplored due to the challenges posed by distinct material‑dependent thermal behaviors. This paper introduces a parametric physics‑informed neural network (PINN) framework for zero‑shot generalization across arbitrary materials without labeled data, retraining, or pre‑training. The framework adopts a decoupled parametric PINN architecture that separately encodes material properties and spatiotemporal coordinates, fusing them through conditional modulation to better align with the multiplicative role of material parameters in the governing equation and boundary conditions. Physics‑guided output scaling derived from Rosenthal's analytical solution and a hybrid optimization strategy are further incorporated to enhance physical consistency, training stability, and convergence. Experiments on bare plate laser powder bed fusion (LPBF) across diverse metal alloys, including both in‑distribution and out‑of‑distribution cases, demonstrate effective zero‑shot generalizability along with superior training efficiency. Specifically, the proposed framework achieved up to a 64.2% reduction in relative L2 error compared to the non‑parametric baseline while surpassing its performance within only 4.4% of the baseline training epochs. Ablation studies confirm that the proposed framework's components are broadly applicable to other PINN‑based approaches. Overall, the proposed framework provides an efficient and scalable material‑agnostic solution for zero‑shot thermal modeling, contributing to more flexible and practical deployment in metal AM.

Abstract:
Physics‑informed neural networks (PINNs) are often selected by a single scalar loss even when the quantity of interest is more specific. We study a hybrid design in which the governing PDE residual remains automatic‑differentiation (AD) based, while finite differences (FD) appear only in a weak auxiliary term that penalizes gradients of the sampled residual field. The FD term regularizes the residual field without replacing the PDE residual itself. We examine this idea in two stages. Stage 1 is a controlled Poisson benchmark comparing a baseline PINN, the FD residual‑gradient regularizer, and a matched AD residual‑gradient baseline. Stage 2 transfers the same logic to a three‑dimensional annular heat‑conduction benchmark (PINN3D), where baseline errors concentrate near a wavy outer wall and the auxiliary grid is implemented as a body‑fitted shell adjacent to the wall. In Stage 1, the FD regularizer reproduces the main effect of residual‑gradient control while exposing a trade‑off between field accuracy and residual cleanliness. In Stage 2, the shell regularizer improves the application‑facing quantities, namely outer‑wall flux and boundary‑condition behavior. Across seeds 0‑5 and 100k epochs, the most reliable tested configuration is a fixed shell weight of 5e‑4 under the Kourkoutas‑beta optimizer regime: relative to a matched run without the shell term, it reduces the mean outer‑wall BC RMSE from 1.22e‑2 to 9.29e‑4 and the mean wall‑flux RMSE from 9.21e‑3 to 9.63e‑4. Adam with beta2=0.999 becomes usable when the initial learning rate is reduced to 1e‑3, although its shell benefit is less robust than under Kourkoutas‑beta. Overall, the results support a targeted view of hybrid PINNs: an auxiliary‑only FD regularizer is most valuable when it is aligned with the physical quantity of interest, here the outer‑wall flux.

Abstract:
Most practical engineering design problems involve nonlinear spatio‑temporal dynamical systems. Multi‑physics simulations are often performed to capture the fine spatio‑temporal scales which govern the evolution of these systems. However, these simulations are often high‑fidelity in nature, and can be computationally very expensive. Hence, generating data from these expensive simulations becomes a bottleneck in an end‑to‑end engineering design process. Spatio‑temporal surrogate modeling of these dynamical systems has been a popular data‑driven solution to tackle this computational bottleneck. This is because accurate machine learning models emulating the dynamical systems can be orders of magnitude faster than the actual simulations. However, one key limitation of purely data‑driven approaches is their lack of generalizability to inputs outside the training distribution. In this paper, we propose a physics‑informed spatio‑temporal surrogate modeling (PISTM) framework constrained by the physics of the underlying dynamical system. The framework leverages state‑of‑the‑art advancements in the field of Koopman autoencoders to learn the underlying spatio‑temporal dynamics in a non‑intrusive manner, coupled with a spatio‑temporal surrogate model which predicts the behavior of the Koopman operator in a specified time window for unknown operating conditions. We evaluate our framework on a prototypical fluid flow problem of interest: two‑dimensional incompressible flow around a cylinder.

Abstract:
Steady‑state electrothermal systems involve strongly coupled heat transfer, fluid flow, and electric‑potential transport, creating severe numerical challenges for standard physics‑informed neural networks (PINNs) due to stark disparities in gradient scales and residual stiffnesses across the physical fields. To resolve these multiphysics bottlenecks, we introduce a Long Short‑Term Memory PINN (LSTM‑PINN) framework that utilizes a depth‑recursive memory mechanism to preserve long‑range spatial feature dependencies and maintain strict cross‑field consistency. The proposed architecture is rigorously evaluated against conventional and attention‑based networks across a unified five‑field formulation encompassing four complex convective and drag regimes: Boussinesq electrothermal flow, drift‑potential gauge‑constrained transport, strong buoyancy‑coupled convection, and Brinkman‑‑Forchheimer drift. Quantitative and visual analyses demonstrate that LSTM‑PINN successfully suppresses non‑physical artifacts and structural distortions, yielding the highest thermodynamic fidelity and consistently outperforming state‑of‑the‑art baselines in global error metrics. Ultimately, this memory‑enhanced approach provides a highly robust and accurate computational baseline for capturing localized boundary layers and complex energy‑momentum feedback in advanced electrothermal energy systems.

Abstract:
Accurate methane sorption prediction across heterogeneous coal ranks requires models that combine thermodynamic consistency, efficient knowledge transfer across data‑scarce geological systems, and calibrated uncertainty estimates, capabilities that are rarely addressed together in existing frameworks. We present a physics‑informed transfer learning framework that adapts a hydrogen sorption PINN to methane sorption prediction via Elastic Weight Consolidation, coal‑specific feature engineering, and a three‑phase curriculum that progressively balances transfer preservation with thermodynamic fine‑tuning. Trained on 993 equilibrium measurements from 114 independent coal experiments spanning lignite to anthracite, the framework achieves R2 = 0.932 on held‑out coal samples, a 227% improvement over pressure‑only classical isotherms, while hydrogen pre‑training delivers 18.9% lower RMSE and 19.4% faster convergence than random initialization. Five Bayesian uncertainty quantification approaches reveal a systematic divergence in performance across physics‑constrained architectures. Monte Carlo Dropout achieves well‑calibrated uncertainty at minimal overhead, while deep ensembles, regardless of architectural diversity or initialization strategy, exhibit performance degradation because shared physics constraints narrow the admissible solution manifold. SHAP and ALE analyses confirm that learned representations remain physically interpretable and aligned with established coal sorption mechanisms: moisture‑volatile interactions are most influential, pressure‑temperature coupling captures thermodynamic co‑dependence, and features exhibit non‑monotonic effects. These results identify Monte Carlo Dropout as the best‑performing UQ method in this physics‑constrained transfer learning framework, and demonstrate cross‑gas transfer learning as a data‑efficient strategy for geological material modeling.

Abstract:
Deep neural networks (DNNs) deliver state‑of‑the‑art accuracy on regression and classification tasks, yet two structural deficits persistently obstruct their deployment in safety‑critical, resource‑constrained settings: (i) opacity of the learned function, which precludes formal verification, and (ii) reliance on heterogeneous, library‑bound activation functions that inflate latency and silicon area on edge hardware. The recently introduced Exp‑Minus‑Log (EML) Sheffer operator, eml(x, y) = exp(x) ‑ ln(y), was shown by Odrzywolek (2026) to be sufficient ‑ together with the constant 1 ‑ to express every standard elementary function as a binary tree of identical nodes. We propose to embed EML primitives inside conventional DNN architectures, yielding a hybrid DNN‑EML model in which the trunk learns distributed representations and the head is a depth‑bounded, weight‑sparse EML tree whose snapped weights collapse to closed‑form symbolic sub‑expressions. We derive the forward equations, prove computational‑cost bounds, analyse inference and training acceleration relative to multilayer perceptrons (MLPs) and physics‑informed neural networks (PINNs), and quantify the trade‑offs for FPGA/analog deployment. We argue that the DNN‑EML pairing closes a literature gap: prior neuro‑symbolic and equation‑learner approaches (EQL, KAN, AI‑Feynman) work with heterogeneous primitive sets and do not exploit a single hardware‑realisable Sheffer element. A balanced assessment shows that EML is unlikely to accelerate training, and on commodity CPU/GPU it is also unlikely to accelerate inference; however, on a custom EML cell (FPGA logic block or analog circuit) the asymptotic latency advantage can reach an order of magnitude with simultaneous gain in interpretability and formal‑verification tractability.

Abstract:
Integro‑differential equations arise in a wide range of applications, including transport, kinetic theory, radiative transfer, and multiphysics modeling, where nonlocal integral operators couple the solution across phase space. Such nonlocality often introduces dense coupling blocks in deterministic discretizations, leading to increased computational cost and memory usage, while physics‑informed neural networks may suffer from expensive nonconvex training and sensitivity to hyperparameter choices. In this work, we present randomized neural networks (RaNNs) as a mesh‑free collocation framework for linear integro‑differential equations. Because the RaNN approximation is intrinsically dense through globally supported random features, the nonlocal integral operator does not introduce an additional loss of sparsity, while the approximate solution can still be represented with relatively few trainable degrees of freedom. By randomly fixing the hidden‑layer parameters and solving only for the linear output weights, the training procedure reduces to a convex least‑squares problem in the output coefficients, enabling stable and efficient optimization. As a representative application, we apply the proposed framework to the steady neutron transport equation, a high‑dimensional linear integro‑differential model featuring scattering integrals and diverse boundary conditions. Extensive numerical experiments demonstrate that, in the reported test settings, the RaNN approach achieves competitive accuracy while incurring substantially lower training cost than the selected neural and deterministic baselines, highlighting RaNNs as a robust and efficient alternative for the numerical simulation of nonlocal linear operators.

Abstract:
Physics‑Informed Neural Networks (PINNs) recast PDE solving as an optimisation problem in function space by minimising a residual‑based objective, yet many applications require additional derivative‑based relations that are just as fundamental as the governing equations. In this paper, we present Derivative‑Constrained PINNs (DC‑PINNs), a general framework that treats constrained PDE solving as an optimisation guided by a minimum objective function criterion where the physics resides in the minimum principle. DC‑PINNs embed general nonlinear constraints on states and derivatives, e.g., bounds, monotonicity, convexity, incompressibility, computed efficiently via automatic differentiation, and they employ self‑adaptive loss balancing to tune the influence of each objective, reducing reliance on manual hyperparameters and problem‑specific architectures. DC‑PINNs consistently reduce constraint violations and improve physical fidelity versus baseline PINN variants, representative hard‑constraint formulations on benchmarks, including heat diffusion with bounds, financial volatilities with arbitrage‑free, and fluid flow with vortices shed. Explicitly encoding derivative constraints stabilises training and steers optimisation toward physically admissible minima even when the PDE residual alone is small, providing reliable solutions of constrained PDEs grounded in energy minimum principles.

Abstract:
We present Dehaze‑then‑Splat, a two‑stage pipeline for multi‑view smoke removal and novel view synthesis developed for Track~2 of the NTIRE 2026 3D Restoration and Reconstruction Challenge. In the first stage, we produce pseudo‑clean training images via per‑frame generative dehazing using Nano Banana Pro, followed by brightness normalization. In the second stage, we train 3D Gaussian Splatting (3DGS) with physics‑informed auxiliary losses ‑‑ depth supervision via Pearson correlation with pseudo‑depth, dark channel prior regularization, and dual‑source gradient matching ‑‑ that compensate for cross‑view inconsistencies inherent in frame‑wise generative processing. We identify a fundamental tension in dehaze‑then‑reconstruct pipelines: per‑image restoration quality does not guarantee multi‑view consistency, and such inconsistency manifests as blurred renders and structural instability in downstream 3D reconstruction.Our analysis shows that MCMC‑based densification with early stopping, combined with depth and haze‑suppression priors, effectively mitigates these artifacts. On the Akikaze validation scene, our pipeline achieves 20.98\,dB PSNR and 0.683 SSIM for novel view synthesis, a +1.50\,dB improvement over the unregularized baseline.

Abstract:
Toroidal rotation is crucial for maintaining stable and high performance plasmas in tokamak fusion reactors. Among its driving mechanisms, the neoclassical toroidal viscosity (NTV) torque‑‑induced by three‑dimensional magnetic perturbations‑‑is particularly significant due to its strong impact and controllability, especially for reactor‑scale devices like ITER where conventional momentum injection method becomes less effective. However, traditional first‑principle NTV modeling is computationally expensive, as it requires solving the drift kinetic equation (DKE) in high‑dimensional phase space, therefore precluding any real‑time applications such as active control or nonlinear integrated modeling of tokamak plasma. Although surrogate solver shows promising ability for accelerating scientific computations, obtaining the data required to train such model is still very challenging. In this work, we present a novel, data‑free approach for developing fast surrogate solver of DKE, by training neural network solely based on physical constraints. Such physical constraints are implemented in two ways: First, the loss function is defined based on physical governing equations; Second, the boundary condition is hard‑coded into the predicting model. The proposed model is validated against the dataset generated by first‑principle numerical solver, which is found to achieve accurate DKE solution with significantly reduced time consuming. In particular, physics‑driven surrogate shows higher physical consistency than data‑driven surrogate. In general, our study provides a new idea for developing surrogate solvers in data‑scarce scenarios, and demonstrates the potential of purely physics‑driven neural networks to accelerate demanding scientific computations.

Abstract:
Surrogate modeling of body‑driven fluid flows where immersed moving boundaries couple structural dynamics to chaotic, unsteady fluid phenomena remains a fundamental challenge for both computational physics and machine learning. We present AeTHERON, a heterogeneous graph neural operator whose architecture directly mirrors the structure of the sharp‑interface immersed boundary method (IBM): a dual‑graph representation separating fluid and structural domains, coupled through sparse cross‑attention that reflects the compact support of IBM interpolation stencils. This physics‑informed inductive bias enables AeTHERON to learn nonlinear fluid‑structure coupling in a shared high‑dimensional latent space, with continuous sinusoidal time embeddings providing temporal generalization across lead times. We evaluate AeTHERON on direct numerical simulations of a flapping flexible caudal fin, a canonical FSI benchmark featuring leading‑edge vortex formation, large membrane deformation, and chaotic wake shedding across a 4x5 parameter grid of membrane thickness (h = 0.01‑0.04) and Strouhal number (St = 0.30‑0.50). As a proof‑of‑concept, we train on the first 150 timesteps of a representative case using a 70/30 train/validation split and evaluate on the fully unseen extrapolation window t=150‑200. AeTHERON captures large‑scale vortex topology and wake structure with qualitative fidelity, achieving a mean extrapolation MAE of 0.168 without retraining, with error peaking near flapping half‑cycle transitions where flow reorganization is most rapid ‑‑ a physically interpretable pattern consistent with the nonlinear fluid‑membrane coupling. Inference requires milliseconds per timestep on a single GPU versus hours for equivalent DNS computation. This is a continuously developing preprint; results and figures will be updated in subsequent versions.

Abstract:
Accurate characterization of subsurface heterogeneity is challenging but essential for applications such as reservoir pressure management, geothermal energy extraction and CO_2, H_2, and wastewater injection operations. This challenge becomes especially acute in extreme pressure events, which are rarely observed but can strongly affect operational risk. Traditional history matching and inversion techniques rely on expensive full‑physics simulations, making it infeasible to handle uncertainty and extreme events at scale. Purely data‑driven models often struggle to maintain physics consistency when dealing with sparse observations, complex geology, and extreme events. To overcome these limitations, we introduce a physics‑informed machine learning method that embeds a differentiable subsurface flow simulator directly into neural network training. The network infers heterogeneous permeability fields from limited pressure observations, while training minimizes both permeability and pressure losses through the simulator, enforcing physical consistency. Because the simulator is used only during training, inference remains fast once the model is learned. In an initial test, the proposed method reduces the pressure inference error by half compared with a purely data‑driven approach. We then extend the test over eight distinct data scenarios, and in every case, our method produces significantly lower pressure inference errors than the purely data‑driven model. We also evaluate our method on extreme events, which represent high‑consequence data in the tail of the sample distribution. Similar to the bulk distribution, the physics‑informed model maintains higher pressure inference accuracy in the extreme event regimes. Overall, the proposed method enables rapid, physics‑consistent subsurface inversion for real‑time reservoir characterization and risk‑aware decision‑making.

Abstract:
Purpose: Novel MR sequence developments still today allow generation of new diagnostic tools or novel imaging biomarkers. Programming MRI pulse sequences, however, is time‑consuming and requires deep expertise in sequence design, restrictions by hardware constraints and MRI physics; even small modifications often require substantial debugging and validation. LLMs can assist when given structured prompts and error feedback, but many generated sequences still exhibit physical inconsistencies. We present Agent4MR, an agent‑based framework that automatically generates and refines PyPulseq sequences using a structured, physics‑aware validation report. These agents can perform also autonomous research. Methods: We evaluated Agent4MR on a spin‑echo EPI task across three state‑of‑the‑art LLMs and compared it to a context‑only baseline (LLM4MR) and to a human developer with the same tools. We tested an MR autoresearch on a fluid‑suppressed spin‑echo EPI challenge for three different model generations. Results: Across all models, Agent4MR consistently produced artifact‑free, physically valid sequences in a single user interaction, reducing the number of required interactions below the human baseline while maintaining correct timing and k‑space coverage. Autonomous agents could then improve a sequence to match a given target contrast in an autoresearch approach. Conclusion: An appropriate agentic harness with physics‑based validation can turn general‑purpose LLMs into reliable MRI sequence developers and may ultimately enable non‑experts to refine or innovate MR sequences guided by biological or clinical questions, or let swarms of agents realize sequence programming for them. Keywords: MRI; pulse sequence; PyPulseq; large language models; agents; autoresearch, sequence development.

Abstract:
The motion planning problem requires finding a collision‑free path between start and goal configurations in high‑dimensional, cluttered spaces. Recent learning‑based methods offer promising solutions, with self‑supervised physics‑informed approaches such as Neural Time Fields (NTFields) solving the Eikonal equation to learn value functions without expert demonstrations. However, existing physics‑informed methods struggle to scale in complex, multi‑room environments, where simply increasing the number of samples cannot resolve local minima or guarantee global consistency. We propose Hierarchical Neural Time Fields (H‑NTFields), a weakly‑supervised framework that combines weak supervision from sparse roadmaps with physics‑informed PDE regularization. The roadmap provides global topological anchors through upper and lower bounds on travel times, while PDE losses enforce local geometric fidelity and obstacle‑aware propagation. Experiments on 18 Gibson environments and real robotic platforms show that H‑NTFields substantially improves robustness over prior physics‑informed methods, while enabling fast amortized inference through a continuous value representation.

Abstract:
Satellite sea surface temperature (SST) products underpin global coral bleaching monitoring, yet they measure only the ocean skin. Corals inhabit depths from the shallows to beyond 20 metres, where temperatures can be 1‑3°C cooler than the surface; applying satellite SST uniformly to all depths therefore overestimates subsurface thermal stress. We present a physics‑informed neural network (PINN) that fuses NOAA Coral Reef Watch SST with sparse in‑situ temperature loggers within the one‑dimensional vertical heat equation, enforcing SST as a hard surface boundary condition and jointly learning effective thermal diffusivity (\kappa) and light attenuation (Kd). Validated across four Great Barrier Reef sites (30 holdout experiments), the PINN achieves 0.25‑1.38°C RMSE at unseen depths. Under extreme sparsity (three training depths), the PINN maintains 0.27°C RMSE at the 5 metre holdout and 0.32°C at the 9.1 metre holdout, where statistical baselines collapse to >1.8°C; it outperforms a physics‑only finite‑difference baseline in 90% of experiments. Depth‑resolved Degree Heating Day (DHD) profiles show that thermal stress attenuates with depth: at Davies Reef, DHD drops from 0.29 at the surface to zero by 10.7 metres, consistent with logger observations, while satellite DHD remains constant at 0.31 across all depths. However, the PINN underestimates absolute DHD at shallow depths because its smooth predictions attenuate the short‑duration peaks that drive threshold exceedances; PINN DHD values should be interpreted as conservative lower bounds on depth‑resolved stress. These results demonstrate that physics‑constrained fusion of satellite SST with sparse loggers can extend bleaching assessment to the depth dimension using existing observational infrastructure.

Abstract:
Autonomous vehicles (AVs) are now operating on public roads, which makes their testing and validation more critical than ever. Simulation offers a safe and controlled environment for evaluating AV performance in varied conditions. However, existing simulation tools mainly focus on graphical realism and rely on simple rule‑based models and therefore fail to accurately represent the complexity of driving behaviors and interactions. Artificial intelligence (AI) has shown strong potential to address these limitations; however, despite the rapid progress across AI methodologies, a comprehensive survey of their application to mixed autonomy traffic simulation remains lacking. Existing surveys either focus on simulation tools without examining the AI methods behind them, or cover ego‑centric decision‑making without addressing the broader challenge of modeling surrounding traffic. Moreover, they do not offer a unified taxonomy of AI methods covering individual behavior modeling to full scene simulation. To address these gaps, this survey provides a structured review and synthesis of AI methods for modeling AV and human driving behavior in mixed autonomy traffic simulation. We introduce a taxonomy that organizes methods into three families: agent‑level behavior models, environment‑level simulation methods, and cognitive and physics‑informed methods. The survey analyzes how existing simulation platforms fall short of the needs of mixed autonomy research and outlines directions to narrow this gap. It also provides a chronological overview of AI methods and reviews evaluation protocols and metrics, simulation tools, and datasets. By covering both traffic engineering and computer science perspectives, we aim to bridge the gap between these two communities.

Abstract:
We learn parameterized nonlinear elasticity on curved surfaces using a physics‑informed neural network that enforces governing equations and boundary conditions directly through the loss function, enabling a single trained model to represent a continuous family of elastic equilibria across geometric and material parameters. Nonlinear elasticity on curved manifolds underlies the mechanics of crystalline shells, elastic membranes, and viral capsids, where curvature and topological defects determine equilibrium structure and stability. Traditional exact and finite element solvers rely on symmetry reduction and must be reinitialized for each parameter choice, limiting scalability when symmetry is broken or parameters vary. We validate the proposed learning‑based solver on a benchmark problem from curved elasticity, namely the one‑dimensional single disclination on a spheroidal surface with known exact and numerical solutions. The network accurately reproduces these solutions, including parameter combinations excluded from training, demonstrating generalization across geometry and material regimes. This study establishes a scalable framework for learning nonlinear elastic systems on curved manifolds and lays the groundwork for extensions to fully two‑dimensional and multi‑defect configurations relevant to protein shells and other curved elastic networks.

Abstract:
This paper addresses the limitations of Physics‑Informed Neural Networks for time‑dependent problems by introducing a tangent bundle learning framework. Instead of directly approximating the solution, we parameterize its temporal derivative and reconstruct the state through a Volterra integral operator that enforces initial conditions exactly. This approach eliminates competing soft constraints and naturally amplifies high‑frequency errors through differentiation, countering spectral bias. We prove theoretical equivalence between minimizing the differentiated residual and solving the original partial differential equation. Experiments on advection, Burgers, and Klein‑Gordon equations show that the proposed method achieves 100 to 200 times lower errors than standard approaches using compact three‑layer networks, with superior shock‑capturing and long‑time accuracy.

Abstract:
Hydrogen is a promising energy carrier, yet its practical deployment is limited by the lack of storage materials that simultaneously achieve high storage capacity (w) and practical equilibrium pressure at room temperature (P_\rm eq,RT). Interstitial metal hydrides offer fast kinetics and favorable thermodynamics (high P_\rm eq,RT) but suffer from intrinsically low w. Here, we establish a physically interpretable, data‑driven framework to uncover descriptor‑property relationships in interstitial hydrides using a curated database of pressure‑composition‑temperature measurements (Digital Hydrogen Platform, DigHyd) and white‑box symbolic regression. Strikingly, the analysis reveals a clear separation of governing mechanisms, in which w is governed by geometric and lattice conditions, captured by the average atomic radius (\left\langle r_M \right\rangle) and average thermal conductivity (\left\langleκ\right\rangle), with an optimal regime of r_M ～ 1.47 Å and relatively low \left\langleκ\right\rangle. In contrast, P_\rm eq,RT is governed by elastic properties, captured by the average shear modulus (\left\langle G \right\rangle) and average Poisson's ratio (\left\langle ν\right\rangle), reflecting the role of lattice rigidity and mechanical compliance. These relationships are translated into compositional optimization pathways that follow the descriptor trends above, enabling the design of candidate materials with enhanced w under practical equilibrium conditions (P_\rm eq,RT ～ 0.1 MPa). This work establishes a general, interpretable strategy for physics‑informed design of energy materials systems.

Abstract:
Time‑Resolved Spectroscopy (TRS) is a powerful modality for non‑invasive characterization of turbid media. However, extracting optical properties, absorption μ_a and reduced scattering μ_s', from 3D stochastic measurements remains computationally expensive for real‑time applications. In this paper, we propose a data‑efficient, physics‑informed transfer learning strategy using a Bidirectional Long Short‑Term Memory (Bi‑LSTM) network. By leveraging a fast deterministic solver to establish a physical prior before fine‑tuning on a restricted set of 3D Monte Carlo simulations, our model successfully bridges the analytical‑to‑stochastic domain gap. The proposed method eliminates the systematic bias of analytical models while maintaining a competitive error with near‑instantaneous inference time.

Abstract:
Many safety‑critical scientific and engineering systems evolve according to differential‑algebraic equations (DAEs), where dynamical behavior is constrained by physical laws and admissibility conditions. In practice, these systems operate under stochastically varying environmental inputs, so stability is not a static property but must be reassessed as the context distribution shifts. Repeated large‑scale DAE simulation, however, is computationally prohibitive in high‑dimensional or real‑time settings. This paper proposes a test‑oriented learning framework for stability assessment under distribution shift. Rather than re‑estimating physical parameters or repeatedly solving the underlying DAE, we learn a physics‑informed latent representation of contextual variables that captures stability‑relevant structure and is regularized toward a tractable reference distribution. Trained on baseline data from a certified safe regime, the learned representation enables deployment‑time safety monitoring to be formulated as a distributional hypothesis test in latent space, with controlled Type I error. By integrating neural dynamical surrogates, uncertainty‑aware calibration, and uniformity‑based testing, our approach provides a scalable and statistically grounded method for detecting instability risk in stochastic constrained dynamical systems without repeated simulation.

Abstract:
High‑speed quantitative phase imaging enables non‑intrusive visualization of transient compressible gas flows and energetic phenomena. However, phase maps reconstructed via the transport of intensity equation (TIE) suffer from spatially correlated low‑frequency artifacts introduced by the inverse Laplacian solver, which obscure meaningful flow structures such as jet plumes, shockwave fronts, and density gradients. Conventional filtering approaches fail because signal and noise occupy overlapping spatial frequency bands, and no paired ground truth exists since every frame represents a physically unique, non‑repeatable flow state. We address this by developing a physics‑informed synthetic training dataset where clean targets are procedurally generated using physically plausible gas flow morphologies, including compressible jet plumes, turbulent eddy fields, density fronts, periodic air pockets, and expansion fans, and passed through a forward TIE simulation followed by inverse Laplacian reconstruction to produce realistic noisy phase maps. A U‑Net‑based convolutional denoising network trained solely on this synthetic data is evaluated on real phase maps acquired at 25,000 fps, demonstrating zero‑shot generalization to real parallel TIE recordings, with a 13,260% improvement in signal‑to‑background ratio and 100.8% improvement in jet‑region structural sharpness across 20 evaluated frames.

Abstract:
H‑EFT‑VA established a physics‑informed solution to the Barren Plateau (BP) problem via a hierarchical EFT UV‑cutoff, guaranteeing gradient variance in Omega(1/poly(N)). However, localization restricts the ansatz to a polynomial subspace, creating a reference‑state gap for states distant from |0>^N. We introduce Adaptive H‑EFT‑VA (A‑H‑EFT) to navigate the trainability‑expressibility tradeoff by expanding the reachable Hilbert space along a safe trajectory. Gradient variance is maintained in Omega(1/poly(N)) if sigma(t) <= 0.5/sqrt(LN) (Theorem 1). A Safe Expansion Corollary and Monotone Growth Lemma confirm expansion without discontinuous jumps. Benchmarking across 16 experiments (up to N=14) shows A‑H‑EFT achieves fidelity F=0.54, doubling static H‑EFT‑VA (F=0.27) and outperforming HEA (F~0.01), with gradient variance >= 0.5 throughout. For Heisenberg XXZ (Delta_ref=1), A‑H‑EFT identifies the negative ground state while static methods fail. Results are statistically significant (p < 10^‑37). Robustness over three decades of hyperparameters enables deployment without search. This is the first rigorously bounded trajectory through the VQA landscape.

Abstract:
Accurate battery health prognosis using State of Health (SOH) estimation is essential for the reliability of multi‑scale battery energy storage, yet existing methods are limited in generalizability across diverse battery chemistries and operating conditions. The inability of standard neural networks to capture the complex, high‑dimensional physics of battery degradation is a major contributor to these limitations. To address this, a physics‑informed neural network with the Quantum Feature Mapping(QFM) technique (QPINN) is proposed. QPINN projects raw battery sensor data into a high‑dimensional Hilbert space, creating a highly expressive feature set that effectively captures subtle, non‑linear degradation patterns using Nyström method. These quantum‑enhanced features are then processed by a physics‑informed network that enforces physical constraints. The proposed method achieves an average SOH estimation accuracy of 99.46% across different datasets, substantially outperforming state‑of‑the‑art baselines, with reductions in MAPE and RMSE of up to 65% and 62%, respectively. This method was validated on a large‑scale, multi‑chemistry dataset of 310,705 samples from 387 cells, and further showed notable adaptability in cross‑validation settings, successfully transferring from one chemistry to another without relying on target‑domain SOH labels.

Abstract:
Reliable LiDAR perception requires robustness across sensors, environments, and adverse weather. However, existing datasets rarely provide physically consistent observations of the same scene under varying sensor configurations and weather conditions, limiting systematic analysis of domain shifts. This work presents ReaLiTy, a unified physics‑informed framework that transforms LiDAR data to match target sensor specifications and weather conditions. The framework integrates physically grounded cues with a learning‑based module to generate realistic intensity patterns, while a physics‑based weather model introduces consistent geometric and radiometric degradations. Building on this framework, we introduce the LiDAR Adaptation Dataset Suite (LADS), a collection of physically consistent, transformation‑ready point clouds with one‑to‑one correspondence to original datasets. Experiments demonstrate improved cross‑domain consistency and realistic weather effects. ReaLiTy and LADS provide a reproducible foundation for studying LiDAR adaptation and simulation‑driven perception in intelligent transportation systems.

Abstract:
Hybrid approaches that combine data‑driven learning with physics‑based insight have shown promise for improving the reliability of industrial condition monitoring. This work develops a hybrid condition monitoring framework that integrates primary sensor measurements, lagged temporal features, and physics‑informed residuals derived from nominal surrogate models. Two hybrid integration strategies are examined. The first is a feature‑level fusion approach that augments the input space with residual and temporal information. The second is a model‑level ensemble approach in which machine learning classifiers trained on different feature types are combined at the decision level. Both hybrid approaches of the condition monitoring framework are evaluated on a continuous stirred‑tank reactor (CSTR) benchmark using several machine learning models and ensemble configurations. Both feature‑level and model‑level hybridization improve diagnostic accuracy relative to single‑source baselines, with the best model‑level ensemble achieving a 2.9% improvement over the best baseline ensemble. To assess predictive reliability, conformal prediction is applied to quantify coverage, prediction‑set size, and abstention behavior. The results show that hybrid integration enhances uncertainty management, producing smaller and well‑calibrated prediction sets at matched coverage levels. These findings demonstrate that lightweight physics‑informed residuals, temporal augmentation, and ensemble learning can be combined effectively to improve both accuracy and decision reliability in nonlinear industrial systems.

Abstract:
Autonomous racing without prebuilt maps is a grand challenge for embedded robotics that requires kinodynamic planning from instantaneous sensor data at the acceleration and tire friction limits. Out‑Of‑Distribution (OOD) generalization to various racetrack configurations utilizes Machine Learning (ML) to encode the mathematical relation between sensor data and vehicle actuation for end‑to‑end control, with implicit localization. These comprise Behavioral Cloning (BC) that is capped to human reaction times and Deep Reinforcement Learning (DRL) which requires large‑scale collisions for comprehensive training that can be infeasible without simulation but is arduous to transfer to reality, thus exhibiting greater performance than BC in simulation, but actuation instability on hardware. This paper presents a DRL method that parameterizes nonlinear vehicle dynamics from the spectral distribution of depth measurements with a non‑geometric, physics‑informed reward, to infer vehicle time‑optimal and overtaking racing controls with an Artificial Neural Network (ANN) that utilizes less than 1% of the computation of BC and model‑based DRL. Slaloming from simulation to reality transfer and variance‑induced conservatism are eliminated with the combination of a physics engine exploit‑aware reward and the replacement of an explicit collision penalty with an implicit truncation of the value horizon. The policy outperforms human demonstrations by 12% in OOD tracks on proportionally scaled hardware, by maximizing the friction circle with tire dynamics that resemble an empirical Pacejka tire model. System identification illuminates a functional bifurcation where the first layer compresses spatial observations to extract digitized track features with higher resolution in corner apexes, and the second encodes nonlinear dynamics.

Abstract:
Accurate prediction of intersection turning movements is essential for adaptive signal control but remains difficult due to the high volatility of directional flows. This study proposes HFD‑TM (Hierarchical Flow‑Decomposition for Turning Movement Prediction), a hierarchical deep learning framework that predicts turning movements by first forecasting corridor through‑movements and then expanding these predictions to individual turning streams. This design is motivated by empirical traffic structure, where corridor flows account for 65.1% of total volume, exhibit lower volatility than turning movements, and explain 35.5% of turning‑movement variance. A physics‑informed loss function enforces flow conservation to maintain structural consistency. Evaluated on six months of 15‑minute interval LiDAR (Light Detection and Ranging) data from a six‑intersection corridor in Nashville, Tennessee, HFD‑TM achieves a mean absolute error of 2.49 vehicles per interval, reducing MAE by 5.7% compared to a Transformer and by 27.0% compared to a GRU (Gated Recurrent Unit). Ablation results show that hierarchical decomposition provides the largest performance gain, while training time is 12.8 times lower than DCRNN (Diffusion Convolutional Recurrent Neural Network), demonstrating suitability for real‑time traffic applications.

Abstract:
We propose a hybrid physics‑informed framework for solving families of parametric linear partial differential equations (PDEs) by combining a meta‑learned predictor with a least‑squares corrector. The predictor, termed KAPI (Kernel‑Adaptive Physics‑Informed meta‑learner), is a shallow task‑conditioned model that maps query coordinates and PDE parameters to solution values while internally generating an interpretable, task‑adaptive Gaussian basis geometry. A lightweight meta‑network maps PDE parameters to basis centers, widths, and activity patterns, thereby learning how the approximation space should adapt across the parametric family. This predictor‑generated geometry is transferred to a second‑stage corrector, which augments it with a background basis and computes the final solution through a one‑shot physics‑informed Extreme Learning Machine (PIELM)‑style least‑squares solve. We evaluate the method on four linear PDE families spanning diffusion, transport, mixed advection‑‑diffusion, and variable‑speed transport. Across these cases, the predictor captures meaningful physics through localized and transport‑aligned basis placement, while the corrector further improves accuracy, often by one or more orders of magnitude. Comparisons with parametric PINNs, physics‑informed DeepONet, and uniform‑grid PIELM correctors highlight the value of predictor‑guided basis adaptation as an interpretable and efficient strategy for parametric PDE solving.

Abstract:
This paper establishes an approximation theorem for randomized neural networks (RaNNs) whose hidden‑layer parameters are uniformly sampled from a prescribed bounded domain. Our analysis shows that, for RaNNs of the form \mathop\sum_i W_i σ(A_i, b_i), the size of the sampling domain required to achieve optimal approximation is intrinsically linked to the smoothness of the target function and the number of neurons. Motivated by this theoretical insight, we integrate a partition of unity (PoU) with RaNNs to develop an adaptive physics‑informed randomized neural network (PIRaNN) method for solving partial differential equations with limited local regularity. The proposed adaptive strategy refines the PoU based on a posteriori error indicators, enabling the network to efficiently capture localized solution features. Numerical experiments validate the theoretical results and demonstrate the strong approximation capabilities of RaNNs, confirming the effectiveness of the adaptive PIRaNN method on a range of benchmark problems.

Abstract:
The disintegration of liquid sheets into ligaments and droplets involves highly transient, multi‑scale dynamics that are difficult to quantify from high‑speed shadowgraphy images. Identifying droplets, ligaments, and blobs formed during breakup, along with tracking across frames, is essential for spray analysis. However, conventional multi‑object tracking frameworks impose strict one‑to‑one temporal associations and cannot represent one‑to‑many fragmentation events. In this study, we present a two‑stage deep learning framework for object detection and temporal relationship modeling across frames. The framework captures ligament deformation, fragmentation, and parent‑child lineage during liquid sheet disintegration. In the first stage, a Faster R‑CNN with a ResNet‑50 backbone and Feature Pyramid Network detects and classifies ligaments and droplets in high‑speed shadowgraphy recordings of an impinging Carbopol gel jet. A morphology‑preserving synthetic data generation strategy augments the training set without introducing physically implausible configurations, achieving a held‑out F1 score of up to 0.872 across fourteen original‑to‑synthetic configurations. In the second stage, a Transformer‑augmented multilayer perceptron classifies inter‑frame associations into continuation, fragmentation (one‑to‑many), and non‑association using physics‑informed geometric features. Despite severe class imbalance, the model achieves 86.1% accuracy, 93.2% precision, and perfect recall (1.00) for fragmentation events. Together, the framework enables automated reconstruction of fragmentation trees, preservation of parent‑child lineage, and extraction of breakup statistics such as fragment multiplicity and droplet size distributions. By explicitly identifying children droplets formed from ligament fragmentation, the framework provides automated analysis of the primary atomization mode.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a flexible framework for solving partial differential equations, but their performance on interface problems remains challenging because continuity and flux conditions are typically imposed through soft penalty terms. The standard soft‑constraint formulation leads to imperfect interface enforcement and degraded accuracy near interfaces. We introduce two ansatz‑based hard‑constrained PINN formulations for interface problems that embed the interface physics into the solution representation and thereby decouple interface enforcement from PDE residual minimization. The first, termed the windowing approach, constructs the trial space from compactly supported windowed subnetworks so that interface continuity and flux balance are satisfied by design. The second, called the buffer approach, augments unrestricted subnetworks with auxiliary buffer functions that enforce boundary and interface constraints at discrete points through a lightweight correction. We study these formulations on one‑ and two‑dimensional elliptic interface benchmarks and compare them with soft‑constrained baselines. In one‑dimensional problems, hard constraints consistently improve interface fidelity and remove the need for loss‑weight tuning; the windowing approach attains very high accuracy (as low as O(10^‑9)) on simple structured cases, whereas the buffer approach remains accurate (～ O(10^‑5)) across a wider range of source terms and interface configurations. In two dimensions, the buffer formulation is shown to be more robust because it enforces constraints through a discrete buffer correction, as the windowing construction becomes more sensitive to overlap and corner effects and over‑constrains the problem. This positions the buffer method as a straightforward and geometrically flexible approach to complex interface problems.

Abstract:
Quasars exhibit stochastic variability across wavelengths, typically well described by a Damped Random Walk (DRW). Occasionally, however, they undergo extreme luminosity changes‑‑known as flares‑‑that represent significant departures from this baseline behavior and provide valuable probes of accretion disc dynamics and the physics of supermassive black hole fueling. Although modern transient surveys have spurred growing interest in flare detection, no systematic search has yet been conducted within the legacy SDSS Stripe 82 dataset, which contains 9,258 spectroscopically confirmed quasars observed over a ~10‑year baseline. The principal statistical challenge is distinguishing these rare events from the ever‑present stochastic variability. To address this, we present FLARE (Flare detection via physics‑informed Learning, Anomaly scoring, and Recognition Engine), a modular three‑stage framework for detecting transient flares in quasar light curves. FLARE models baseline DRW behavior, applies statistical anomaly scoring to flag candidate events, and employs a recognition engine to verify detections. For the Stripe 82 implementation, we deploy two complementary baselines‑‑a physics‑informed probabilistic Gated Recurrent Unit (GRU) trained on simulated DRW light curves, and an iterative Ornstein‑Uhlenbeck (OU) process fitted directly to observed data with outlier masking‑‑followed by Extreme Value Theory (EVT) for anomaly scoring. We benchmark twelve open‑weight and proprietary Vision Language Models (VLMs) as recognition engines for final candidate verification. Detection is performed on r‑band light curves, with candidates cross‑checked against g‑band data to rule out instrumental artifacts. Applying this framework, we identify 51 quasars exhibiting distinct flaring activity.

Abstract:
We present ϕ‑DeepONet, a physics‑informed neural operator designed to learn mappings between function spaces that may contain discontinuities or exhibit non‑smooth behavior. Classical neural operators are based on the universal approximation theorem which assumes that both the operator and the functions it acts on are continuous. However, many scientific and engineering problems involve naturally discontinuous input fields as well as strong and weak discontinuities in the output fields caused by material interfaces. In ϕ‑DeepONet, discontinuities in the input are handled using multiple branch networks, while discontinuities in the output are learned through a nonlinear latent embedding of the interface. This embedding is constructed from a \it one‑hot representation of the domain decomposition that is combined with the spatial coordinates in a modified trunk network. The outputs of the branch and trunk networks are then combined through a dot product to produce the final solution, which is trained using a physics‑ and interface‑informed loss function. We evaluate ϕ‑DeepONet on several one‑ and two‑dimensional benchmark problems and demonstrate that it delivers accurate and stable predictions even in the presence of strong interface‑driven discontinuities.

Abstract:
This paper develops a helicity‑aware physics‑informed neural network framework for incompressible non‑Newtonian flow in rotational form. In addition to the energy law and the incompressibility constraint, helicity is a fundamental geometric quantity that characterizes the topology of vortex lines and plays an important role in the physical fidelity of long‑time flow simulations. While helicity‑preserving discretizations have been studied extensively in finite difference, finite element, and other structure‑preserving settings, their realization within neural network solvers remains largely unexplored. Motivated by this gap, we propose a neural formulation in which vorticity is computed directly from the neural velocity field by automatic differentiation rather than learned as an independent output, thereby avoiding compatibility errors that pollute the helicity balance. To improve robustness and scalability, we combine two algorithmic ingredients: an overlapping spatial domain decomposition inspired by finite‑basis physics‑informed neural networks (FBPINNs), and a causal slab‑wise temporal continuation strategy for long‑time transient simulations. The local subnetworks are blended by explicitly normalized super‑Gaussian window functions, which yield a smooth partition of unity, while the temporal evolution is advanced sequentially across time slabs by transferring the converged solution on one slab to the next. The resulting spatiotemporal framework provides a stable and physically meaningful approach for helicity‑aware simulation of incompressible non‑Newtonian flows.

Abstract:
We propose a Physics Informed Learning framework for reconstructing traffic density from sparse trajectory data. The approach combines a second‑order Aw‑Rascle and Zhang model with a first‑order training stage to estimate the equilibrium velocity. The method is evaluated in both equilibrium and transient traffic regimes using SUMO simulations. Results show that while learning the equilibrium velocity improves reconstruction under steady state conditions, it becomes unstable in transient regimes due to the breakdown of the equilibrium assumption. In contrast, the second‑order model consistently provides more accurate and robust reconstructions than first‑order approaches, particularly in nonequilibrium conditions.

Abstract:
Automotive engineering development increasingly relies on heterogeneous 3D data, including finite element (FE) models, body‑in‑white (BiW) representations, CAD geometry, and CFD meshes. At the same time, engineering teams face growing pressure to shorten development cycles, improve performance and accelerate innovation. Although artificial intelligence (AI) is increasingly explored in this domain, many current methods remain task‑specific, difficult to interpret, and hard to reuse across development stages. This paper presents a practical graph learning framework for 3D engineering AI, in which heterogeneous engineering assets are converted into physics‑aware graph representations and processed by Graph Neural Networks (GNNs). The framework is designed to support both classification and prediction tasks. The framework is validated on two automotive applications: CAE vibration mode shape classification and CFD aerodynamic field prediction. For CAE vibration mode classification, a region‑aware BiW graph supports explainable mode classification across vehicle and FE variants under label scarcity. For CFD aerodynamic field prediction, a physics‑informed surrogate predicts pressure and wall shear stress (WSS) across aerodynamic body shape variants, while symmetry preserving down sampling retains accuracy with lower computational cost. The framework also outlines data generation guidance that can help engineers identify which additional simulations or labels are valuable to collect next. These results demonstrate a practical and reusable engineering AI workflow for more trustworthy CAE and CFD decision support.

Abstract:
We present Rhizome OS‑1, a semi‑autonomous operating system for small molecule drug discovery in which multi‑modal AI agents operate as a full multidisciplinary discovery team. These agents function as computational chemists, medicinal chemists, and patent agents: they write and execute analysis code (fingerprint clustering, R‑group decomposition, substructure search), visually triage molecular grids using vision capabilities, formulate explicit medicinal chemistry hypotheses across three strategy tiers, assess patent freedom‑to‑operate, and dynamically adapt generation strategies based on empirical screening feedback. Powered by r1 ‑ a 246M‑parameter graph diffusion model trained on 800 million molecular graphs ‑ the system generates novel chemical matter directly on molecular graphs using fragment masking, scaffold decoration, linker design, and graph editing primitives. In two oncology campaigns (BCL6 BTB domain and EZH2 SET domain), the agent team executed 26 seeds and produced 5,231 novel molecules. Across both targets, 91.9% of generated Murcko scaffolds are absent from ChEMBL, with median Tanimoto similarity of 0.56‑0.69 to the nearest known active. Boltz‑2 binding affinity predictions, calibrated against ChEMBL data, achieved Spearman correlations of ‑0.53 to ‑0.64 and ROC AUC values of 0.88‑0.93. These results demonstrate that semi‑autonomous agent systems, equipped with graph‑native generative tools and physics‑informed scoring, enable a new paradigm for early‑stage drug discovery: scaled, rapid, and adaptive inverse design with embedded medicinal chemistry reasoning.

Abstract:
Accurate knowledge of acoustic surface admittance or impedance is essential for reliable wave‑based simulations, yet its in situ estimation remains challenging due to noise, model inaccuracies, and restrictive assumptions of conventional methods. This work presents a physics‑informed neural operator approach for estimating frequency‑dependent surface admittance directly from near‑field measurements of sound pressure and particle velocity. A deep operator network is employed to learn the mapping from measurement data, spatial coordinates, and frequency to acoustic field quantities, while simultaneously inferring a globally consistent surface admittance spectrum without requiring an explicit forward model. The governing acoustic relations, including the Helmholtz equation, the linearized momentum equation, and Robin boundary conditions, are embedded into the training process as physics‑based regularization, enabling physically consistent and noise‑robust predictions while avoiding frequency‑wise inversion. The method is validated using synthetically generated data from a simulation model for two planar porous absorbers under semi free‑field conditions across a broad frequency range. Results demonstrate accurate reconstruction of both real and imaginary admittance components and reliable prediction of acoustic field quantities. Parameter studies confirm improved robustness to noise and sparse sampling compared to purely data‑driven approaches, highlighting the potential of physics‑informed neural operators for in situ acoustic material characterization.

Abstract:
Autonomous agents operating in dynamic and safety‑critical environments require decision‑making frameworks that are both computationally efficient and physically grounded. However, many existing approaches rely on end‑to‑end learning, which often lacks interpretability and explicit mechanisms for ensuring consistency with physical constraints. In this work, we propose an event‑centric world modeling framework with memory‑augmented retrieval for embodied decision‑making. The framework represents the environment as a structured set of semantic events, which are encoded into a permutation‑invariant latent representation. Decision‑making is performed via retrieval over a knowledge bank of prior experiences, where each entry associates an event representation with a corresponding maneuver. The final action is computed as a weighted combination of retrieved solutions, providing a transparent link between decision and stored experiences. The proposed design enables structured abstraction of dynamic environments and supports interpretable decision‑making through case‑based reasoning. In addition, incorporating physics‑informed knowledge into the retrieval process encourages the selection of maneuvers that are consistent with observed system dynamics. Experimental evaluation in UAV flight scenarios demonstrates that the framework operates within real‑time control constraints while maintaining interpretable and consistent behavior.

Abstract:
Partial differential equations (PDEs) govern nearly every physical process in science and engineering, yet solving them at scale remains prohibitively expensive. Generative AI has transformed language, vision, and protein science, but learned PDE solvers have not undergone a comparable shift. Existing paradigms each capture part of the problem. Physics‑informed neural networks embed residual structure, yet they are often difficult to optimize in stiff, multiscale, or large‑domain regimes. Neural operators amortize across instances, yet they commonly inherit a snapshot‑prediction view of solving and can degrade over long rollouts. Diffusion‑based solvers model uncertainty, yet they are often built on a solver template that still centers on state regression. We argue that the core issue is the abstraction used to train learned solvers. Many models are asked to predict states, while many scientific settings require modeling how uncertainty moves through constrained dynamics. The relevant object is transport over physically admissible futures. This motivates \emphflow learners: models that parameterize transport vector fields and generate trajectories through integration, echoing the continuous dynamics that define PDE evolution. This physics‑to‑physics alignment supports continuous‑time prediction, native uncertainty quantification, and new opportunities for physics‑aware solver design. We explain why transport‑based learning offers a stronger organizing principle for learned PDE solving and outline the research agenda that follows from this shift.

Abstract:
Real‑time supervisory control of advanced reactors requires accurate forecasting of plant‑wide thermal‑hydraulic states, including locations where physical sensors are unavailable. Meeting this need calls for surrogate models that combine predictive fidelity, millisecond‑scale inference, and robustness to partial observability. In this work, we present a physics‑informed message‑passing Graph Neural Network coupled with a Neural Ordinary Differential Equation (GNN‑ODE) to addresses all three requirements simultaneously. We represent the whole system as a directed sensor graph whose edges encode hydraulic connectivity through flow/heat transfer‑aware message passing, and we advance the latent dynamics in continuous time via a controlled Neural ODE. A topology‑guided missing‑node initializer reconstructs uninstrumented states at rollout start; prediction then proceeds fully autoregressively. The GNN‑ODE surrogate achieves satisfactory results for the system dynamics prediction. On held‑out simulation transients, the surrogate achieves an average MAE of 0.91 K at 60 s and 2.18 K at 300 s for uninstrumented nodes, with R^2 up to 0.995 for missing‑node state reconstruction. Inference runs at approximately 105 times faster than simulated time on a single GPU, enabling 64‑member ensemble rollouts for uncertainty quantification. To assess sim‑to‑real transfer, we adapt the pretrained surrogate to experimental facility data using layerwise discriminative fine‑tuning with only 30 training sequences. The learned flow‑dependent heat‑transfer scaling recovers a Reynolds‑number exponent consistent with established correlations, indicating constitutive learning beyond trajectory fitting. The model tracks a steep power change transient and produces accurate trajectories at uninstrumented locations.

Abstract:
We extend the SeQUeNCe discrete‑event simulator with physics‑based models for polarization‑encoded photonic quantum networks. Our framework integrates Jones‑calculus optical components, including an SPDC Bell‑state source, wave plates, and polarizing beam splitters, together with a multi‑section fiber model capturing polarization mode dispersion, chromatic dispersion, and Raman noise from coexisting classical traffic. We validate the simulator by reproducing experimentally reported spectra, polarization correlations, quantum state tomography, and dispersion‑ and Raman‑induced noise. The resulting platform enables hardware‑parameterized prediction of entanglement distribution performance under realistic deployment conditions.

Abstract:
Resolving the three‑dimensional (3D) atomic geometry of free‑standing graphene in real time is essential for understanding how intrinsic rippling governs its electronic properties. However, the low electron doses required to mitigate radiation damage impose severe signal‑to‑noise constraints that limit conventional reconstruction methods. Here, we present a physics‑informed computational framework that reconstructs 3D atomic coordinates of single‑layer graphene from individual low‑dose transmission electron microscopy (TEM) frames (8x10^3 e‑/Ang^2, 1 ms temporal resolution). The approach combines simulated annealing optimisation with molecular dynamics regularisation, achieving sub‑angstrom out‑of‑plane accuracy (sigma_z < 0.45 Ang), validated against ground‑truth simulations. A Kullback‑Leibler divergence‑based calibration aligns the forward model with experimental image statistics, reducing systematic bias. Applied to high‑speed time‑series data, the framework enables simultaneous extraction of real‑time ripple dynamics, strain tensors, surface curvature, bond‑length distributions, and density functional theory (DFT)‑derived electron localisation functions (ELF). We establish quantitative relationships linking local geometry, strain, and bond‑length variations to electron localisation, demonstrating that sub‑angstrom structural fluctuations drive spatially localised, millisecond‑scale electronic modulation. A critical dose threshold is identified below which structural information becomes irrecoverable, providing practical guidance for experimental design. The framework is broadly applicable to beam‑sensitive two‑dimensional materials.

Abstract:
We present two methods to estimate bottom topography in a shallow water flow using only surface deformation measurements. One is based on Physics‑Informed Neural Networks (PINNs) and the other on the Adjoint State Method. We test both methods using synthetic data in 1D and 2D cases. Both are able to successfully reconstruct not only the bottom topography but also the surface velocity. Both also show robustness against noise and data sparsity up to reasonable levels.

Abstract:
This article presents a novel and comprehensive approach for analyzing bending behavior of the tapered perforated beam under an exponential load. The governing differential equation includes important factors like filling ratio (α), number of rows of holes (N), tapering parameters (ϕ and ψ), and exponential loading parameter (γ), providing a realistic and flexible representation of perforated beam configuration. Main goal of this work is to see how well the Domain mapped physics‑informed Functional link Theory of Functional Connection (DFL‑TFC) method analyses bending response of perforated beam with square holes under exponential loading. For comparison purposes, a corresponding PINN‑based formulation is developed. Outcomes clearly show that the proposed DFL‑TFC framework gives better results, including faster convergence, reduced computational cost, and improved solution accuracy when compared to the PINN approach. These findings highlight effectiveness and potential of DFL‑TFC method for solving complex engineering problems governed by differential equations. Within this framework, hidden layer is replaced by a functional expansion block that enriches input representation via orthogonal polynomial basis functions, and the domain of DE mapped to corresponding domain of orthogonal polynomials. A Constrained Expression (CE), constructed through the Theory of Functional Connections (TFC) using boundary conditions, ensures that constraints are exactly satisfied. In CE, free function is represented using a Functional Link Neural Network (FLNN), which learns to solve resulting unconstrained optimization problem. The obtained results are further validated through the Galerkin and PINN solutions.

Abstract:
Protein phosphorylation provides a dynamic readout of cellular signaling yet remains difficult to detect at low abundance and stoichiometry. Single‑molecule surface‑enhanced Raman spectroscopy (SM‑SERS) using particle‑in‑pore plasmonic nanopores offers label‑free molecular detection with submolecular sensitivity. However, reliable identification of subtle post‑translational modifications (PTMs) is hindered by the stochastic nature of SM‑SERS signals, partial excitation of peptide residues within the plasmonic hotspot, and background interference. Here, we introduce a physics‑informed deep learning framework to decode complex SM‑SERS dynamics and identify single‑peptide PTMs. The model integrates multiple‑instance learning with a temporal encoder combining temporal convolutional networks and bidirectional gated recurrent units to capture both local spectral variability and long‑range blinking dynamics. To address diffusion‑driven spectral heterogeneity, long spectral trajectories are segmented using Pearson‑correlation, enabling weakly supervised training under label ambiguity. This framework robustly distinguishes single peptide phosphorylation despite strong background interference and stochastic signal fluctuations. By coupling nanoplasmonic confinement with spatiotemporal deep learning, our approach enables high‑fidelity detection of single‑molecule phosphorylation events and advances ultrasensitive phosphoproteomic analysis.

Abstract:
High‑dimensional hyperspectral imaging (HSI) enables the visualization of ultrafast molecular dynamics and complex, heterogeneous spectra. However, applying this capability to resolve spatially varying vibrational couplings in two‑dimensional infrared (2DIR) spectroscopy, a type of coherent multidimensional spectroscopy (CMDS), necessitates prohibitively long data acquisition, driven by dense Nyquist sampling requirements and the need for extensive signal accumulation. To address this challenge, we introduce a physics‑informed neural representation approach that efficiently reconstructs dense spatially‑resolved 2DIR hyperspectral images from sparse experimental measurements. In particular, we used a multilayer perceptron (MLP) to model the relationship between the sub‑sampled 4D coordinates and their corresponding spectral intensities, and recover densely sampled 4D spectra from limited observations. The reconstruction results demonstrate that our method, using a fraction of the samples, faithfully recovers both oscillatory and non‑oscillatory spectral dynamics in experimental measurement. Moreover, we develop a loss‑aware adaptive sampling method to progressively introduce potentially informative samples for iterative data collection while conducting experiments. Experimental results show that the proposed approach achieves high‑fidelity spectral recovery using only 1/32 of the sampling budget, as opposed to exhaustive sampling, effectively reducing total experiment time by up to 32‑fold. This framework offers a scalable solution for accelerating any experiments with hypercube data, including multidimensional spectroscopy and hyperspectral imaging, paving the way for rapid chemical imaging of transient biological and material systems.

Abstract:
Stellar astrophysics relies critically on accurate descriptions of the physical conditions inside stars. Traditional solvers such as \textttMESA (Modules for Experiments in Stellar Astrophysics), which employ adaptive finite‑difference methods, can become computationally expensive and challenging to scale for large stellar population synthesis (>10^9 stars). In this work, we present an self‑supervised physics‑informed neural network (PINN) framework that provides a mesh‑free and fully differentiable approach to solving the stellar structure equations under hydrostatic and thermal equilibrium. The model takes as input the stellar boundary conditions (at the center and surface) together with the chemical composition, and learns continuous radial profiles for mass M_r(r), pressure P(r), density ρ(r), temperature T(r), and luminosity L_r(r) by enforcing the governing structure equations through physics‑based loss terms. To incorporate realistic microphysics, we introduce auxiliary neural networks that approximate the equation of state and opacity tables as smooth, differentiable functions of the local thermodynamic state. These surrogates replace traditional tabulated inputs and enable end‑to‑end training. Once trained for a given star, the model produces continuous solutions across the entire radial domain without requiring discretization or interpolation. Validation against benchmark \textttMESA models across a range of stellar masses yields a Mean Relative Absolute Error of 3.06% and an average R^2 score of 99.98%. To our knowledge, this is the first demonstration that the stellar structure equations can be solved in a fully self‑supervised and data‑free fashion employing PINNs. This work establishes a foundation for scalable, physics‑informed emulation of stellar interiors and opens the door to future extensions toward time‑dependent stellar evolution.

Abstract:
Efficiently solving the Fokker‑Planck equation (FPE) is central to analyzing complex parameterized stochastic systems. However, current numerical methods lack parallel computation capabilities across varying conditions, severely limiting comprehensive parameter exploration and transient analysis. This paper introduces a deep learning‑based pseudo‑analytical probability solution (PAPS) that, via a single training process, simultaneously resolves transient FPE solutions for arbitrary multi‑modal initial distributions, system parameters, and time points. The core idea is to unify initial, transient, and stationary distributions via Gaussian mixture distributions (GMDs) and develop a constraint‑preserving autoencoder that bijectively maps constrained GMD parameters to unconstrained, low‑dimensional latent representations. In this representation space, the panoramic transient dynamics across varying initial conditions and system parameters can be modeled by a single evolution network. Extensive experiments on paradigmatic systems demonstrate that the proposed PAPS maintains high accuracy while achieving inference speeds four orders of magnitude faster than GPU‑accelerated Monte Carlo simulations. This efficiency leap enables previously intractable real‑time parameter sweeps and systematic investigations of stochastic bifurcations. By decoupling representation learning from physics‑informed transient dynamics, our work establishes a scalable paradigm for probabilistic modeling of multi‑dimensional, parameterized stochastic systems.

Abstract:
Precision Immobilization Technique (PIT) is a potentially effective intervention maneuver for emergency out‑of‑control vehicle, but its automation is challenged by highly nonlinear collision dynamics, strict safety constraints, and real‑time computation requirements. This work presents a PIT‑oriented neural optimal‑control framework built around PicoPINN (Planning‑Informed Compact Physics‑Informed Neural Network), a compact physics‑informed surrogate obtained through knowledge distillation, hierarchical parameter clustering, and relation‑matrix‑based parameter reconstruction. A hierarchical neural‑OCP (Optimal Control Problem) architecture is then developed, in which an upper virtual decision layer generates PIT decision packages under scenario constraints and a lower coupled‑MPC (Model Predictive Control) layer executes interaction‑aware control. To evaluate the framework, we construct a PIT Scenario Dataset and conduct surrogate‑model comparison, planning‑structure ablation, and multi‑fidelity assessment from simulation to scaled by‑wire vehicle tests. In simulation, adding the upper planning layer improves PIT success rate from 63.8% to 76.7%, and PicoPINN reduces the original PINN parameter count from 8965 to 812 and achieves the smallest average heading error among the learned surrogates (0.112 rad). Scaled vehicle experiments are further used as evidence of control feasibility, with 3 of 4 low‑speed controllable‑contact PIT trials achieving successful yaw reversal.

Abstract:
Dexterous robotic manipulation requires more than geometrically valid grasps: it demands physically grounded contact strategies that account for the spatially non‑uniform mechanical properties of the object. However, existing grasp planners typically treat the surface as structurally homogeneous, even though contact in a weak region can damage the object despite a geometrically perfect grasp. We present a pipeline for grasp selection and force regulation in a five‑fingered robotic hand, based on a map of locally admissible contact loads. From an operator command, the system identifies the target object, reconstructs its 3D geometry using SAM3D, and imports the model into Isaac Sim. A physics‑informed geometric analysis then computes a force map that encodes the maximum lateral contact force admissible at each surface location without deformation. Grasp candidates are filtered by geometric validity and task‑goal consistency. When multiple candidates are comparable under classical metrics, they are re‑ranked using a force‑map‑aware criterion that favors grasps with contacts in mechanically admissible regions. An impedance controller scales the stiffness of each finger according to the locally admissible force at the contact point, enabling safe and reliable grasp execution. Validation on paper, plastic, and glass cups shows that the proposed approach consistently selects structurally stronger contact regions and keeps grip forces within safe bounds. In this way, the work reframes dexterous manipulation from a purely geometric problem into a physically grounded joint planning problem of grasp selection and grip execution for future humanoid systems.

Abstract:
Fluid flows are governed by the nonlinear Navier‑Stokes equations, which can manifest multiscale dynamics even from predictable initial conditions. Predicting such phenomena remains a formidable challenge in scientific machine learning, particularly regarding convergence speed, data requirements, and solution accuracy. In complex fluid flows, these challenges are exacerbated by long‑range spatial dependencies arising from distant boundary conditions, which typically necessitate extensive supervision data to achieve acceptable results. We propose the Domain‑Decomposed and Shifted Physics‑Informed Neural Network (DDS‑PINN), a framework designed to resolve such multiscale interactions with minimal supervision. By utilizing localized networks with a unified global loss, DDS‑PINN captures global dependencies while maintaining local precision. The robustness of the approach is demonstrated across a suite of benchmarks, including a multiscale linear differential equation, the nonlinear Burgers' equation, and data‑free Navier‑Stokes simulations of flat‑plate boundary layers. Finally, DDS‑PINN is applied to the computationally challenging backward‑facing step (BFS) problem; for laminar regimes (Re = 100), the model yields results comparable to computational fluid dynamics (CFD) without the need for any data, accurately predicting boundary layer thickness, separation, and reattachment lengths. For turbulent BFS flow at Re = 10,000, the framework achieves convergence to O(10^‑4) using only 500 random supervision points (< 0.3 % of the total domain), outperforming established methods like Residual‑based Attention‑PINN in accuracy. This approach demonstrates strong potential for the super‑resolution of complex turbulent flows from sparse experimental measurements.

Abstract:
An important recurring pattern in scientific breakthroughs is a two‑stage process: an initial phase of undirected experimentation that yields an unexpected finding, followed by a retrospective phase that explains why the finding works and situates it within existing theory. We present ResearchEVO, an end‑to‑end framework that computationally instantiates this discover‑then‑explain paradigm. The Evolution Phase employs LLM‑guided bi‑dimensional co‑evolution ‑‑ simultaneously optimizing both algorithmic logic and overall architecture ‑‑ to search the space of code implementations purely by fitness, without requiring any understanding of the solutions it produces. The Writing Phase then takes the best‑performing algorithm and autonomously generates a complete, publication‑ready research paper through sentence‑level retrieval‑augmented generation with explicit anti‑hallucination verification and automated experiment design. To our knowledge, ResearchEVO is the first system to cover this full pipeline end to end: no prior work jointly performs principled algorithm evolution and literature‑grounded scientific documentation. We validate the framework on two cross‑disciplinary scientific problems ‑‑ Quantum Error Correction using real Google quantum hardware data, and Physics‑Informed Neural Networks ‑‑ where the Evolution Phase discovered human‑interpretable algorithmic mechanisms that had not been previously proposed in the respective domain literatures. In both cases, the Writing Phase autonomously produced compilable LaTeX manuscripts that correctly grounded these blind discoveries in existing theory via RAG, with zero fabricated citations.

Abstract:
Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy ‑‑ faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics‑informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self‑Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high‑speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN‑based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high‑order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi‑Newton optimizers for batched training, enabling efficient and scalable solutions for large data‑driven problems.

Abstract:
While Physics‑Informed Neural Networks offer a promising framework for solving partial differential equations, the standard L^2 loss formulation is fundamentally insufficient when applied to the Bhatnagar‑Gross‑Krook (BGK) model. Specifically, simply minimizing the standard loss does not guarantee accurate predictions of the macroscopic moments, causing the approximate solutions to fail in capturing the true physical solution. To overcome this limitation, we introduce a velocity‑weighted L^2 loss function designed to effectively penalize errors in the high‑velocity regions. By establishing a stability estimate for the proposed approach, we shows that minimizing the proposed weighted loss guarantees the convergence of the approximate solution. Also, numerical experiments demonstrate that employing this weighted PINN loss leads to superior accuracy and robustness across various benchmarks compared to the standard approach.

Abstract:
We study physics‑informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first‑order optimality system. For a class of semilinear parabolic equations, we derive the state equation, the adjoint equation, and the stationarity condition in a form consistent with continuous‑time Pontryagin‑type optimality conditions. We then specialize the framework to an Allen‑Cahn control problem and compare three numerical approaches: (i) a discretize‑then‑optimize adjoint method, (ii) a direct PINN, and (iii) an indirect PINN. Numerical results show that the PINN parameterization has an implicit regularizing effect, in the sense that it tends to produce smoother control profiles. They also indicate that the indirect PINN more faithfully preserves the PDE contraint and optimality structure and yields a more accurate neural approximation than the direct PINN.

Abstract:
Low‑energy electron diffraction (LEED) is a cornerstone technique for determining surface atomic structures[heldStructureDeterminationLowenergy2025], yet the quantitative analysis of electron diffraction intensity as a function of incident electron energy ‑‑ that is, LEED‑I(V) analysis ‑‑ remains a complex inverse problem. In this work, we tackle quantitative LEED‑I(V) analysis based on physics‑informed Bayesian optimization (BO). By embedding multiple scattering LEED forward models directly into a trust‑region BO loop, our approach simultaneously optimizes both structural and experimental parameters, adaptively adjusting trust regions for efficient exploration of complex non‑convex parameter spaces without manual intervention. The robustness and scalability of the approach are demonstrated using the Ag(100)‑(1\time1) and Fe\textsubscript2O\textsubscript3(1\overline102)‑(1\time1) surfaces as examples. Our work establishes a general framework for solving inverse problems in various characterization techniques, unlocking a physics‑informed efficient, reproducible, and autonomous paradigm.

Abstract:
Precision livestock farming requires accurate and timely heat stress prediction to ensure animal welfare and optimize farm management. This study presents a physics‑informed digital twin (DT) framework combined with an uncertainty‑aware, expert‑weighted stacked ensemble for multimodal forecasting of Core Body Temperature (CBT) in dairy cattle. Using the high‑frequency, heterogeneous MmCows dataset, the DT integrates an ordinary differential equation (ODE)‑based thermoregulation model that simulates metabolic heat production and dissipation, a Gaussian process for capturing cow‑specific deviations, a Kalman filter for aligning predictions with real‑time sensor data, and a behavioral Markov chain that models activity‑state transitions under varying environmental conditions. The DT outputs key physiological indicators, such as predicted CBT, heat stress probability, and behavioral state distributions are fused with raw sensor data and enriched through multi‑scale temporal analysis and cross‑modal feature engineering to form a comprehensive feature set. The predictive methodology is designed in a three‑stage stacked ensemble, where stage 1 trains modality‑specific LightGBM 'expert' models on distinct feature groups, stage 2 collects their predictions as meta‑features, and at stage 3 Optuna‑tuned LightGBM meta‑model yields the final CBT forecast. Predictive uncertainty is quantified via bootstrapping and validated using Prediction Interval Coverage Probability (PICP). Ablation analysis confirms that incorporating DT‑derived features and multimodal fusion substantially enhances performance. The proposed framework achieves a cross‑validated R2 of 0.783, F1 score of 84.25% and PICP of 92.38% for 2‑hour ahead forecasting, providing a robust, uncertainty‑aware, and physically principled system for early heat stress detection and precision livestock management.

Abstract:
Single‑pixel imaging (SPI) offers a cost‑effective route to hyperspectral acquisition but struggles to recover high‑fidelity spatial and spectral details under extremely low sampling rates, a severely ill‑posed inverse problem. While deep learning has shown potential, existing data‑driven methods demand large‑scale pretraining datasets that are often impractical in hyperspectral imaging. To overcome this limitation, we propose an end‑to‑end physics‑informed framework that leverages untrained neural networks and RGB guidance for joint hyperspectral reconstruction and super‑resolution without any external training data. The framework comprises three physically grounded stages: (1) a Regularized Least‑Squares method with RGB‑derived Grayscale Priors (LS‑RGP) that initializes the solution by exploiting cross‑modal structural correlations; (2) an Untrained Hyperspectral Recovery Network (UHRNet) that refines the reconstruction through measurement consistency and hybrid regularization; and (3) a Transformer‑based Untrained Super‑Resolution Network (USRNet) that upsamples the spatial resolution via cross‑modal attention, transferring high‑frequency details from the RGB guide. Extensive experiments on benchmark datasets demonstrate that our approach significantly surpasses state‑of‑the‑art algorithms in both reconstruction accuracy and spectral fidelity. Moreover, a proof‑of‑concept experiment using a physical single‑pixel imaging system validates the framework's practical applicability, successfully reconstructing a 144‑band hyperspectral data cube at a mere 6.25% sampling rate. The proposed method thus provides a robust, data‑efficient solution for computational hyperspectral imaging.

Abstract:
Topology optimization is used for the design of high‑performance structures but remains fundamentally limited by its iterative nature, requiring repeated finite element analyses that prevent real‑time deployment and large‑scale design exploration. In this work, we introduce a physics‑informed transformer architecture that directly learns a non‑iterative mapping from boundary conditions, loading configurations, and derived physical fields to optimized structural topologies. By leveraging global self‑attention, the proposed model captures long‑range mechanical interactions that govern structural response, overcoming the locality limitations of convolutional architectures. A conditioning‑token mechanism embeds global problem parameters, while spatially distributed stress and strain energy fields are encoded as patch tokens within a Vision Transformer framework. To ensure physical realism and manufacturability, we incorporate auxiliary loss functions that enforce volume constraints, load adherence, and structural connectivity through a differentiable formulation. The framework is further extended to dynamic loading scenarios using frequency‑domain encoding and transfer learning, enabling efficient generalization from static to time‑dependent problems. Comprehensive benchmarking demonstrates that the proposed model achieves fidelity beyond that of diffusion models, while requiring only a single forward pass, thereby eliminating iterative inference entirely. This establishes topology optimization as a real‑time operator‑learning problem, enabling high‑fidelity structural design with significant reductions in computational cost.

Abstract:
One of the biggest challenges in the optimization of micro‑scale fluid transport phenomena is the prediction of unsteady fluid flow in the presence of rough channel walls. Even though the accuracy of available computational fluid dynamics (CFD) solvers such as the lattice Boltzmann method (LBM) is satisfactory, the computational cost of design exploration is very high due to the diverse range of geometries and flow regimes involved in microchannel flows. The present paper introduces a revolutionary concept of a ground‑breaking physics‑informed neural network (PINN) that utilizes sparse lattice Boltzmann data in combination with the Navier‑Stokes equations for the prediction of unsteady fluid flow in fractal‑rough microchannels. The roughness of the channel walls is represented by the Weierstrass‑Mandelbrot function, considering the characteristics of the surface roughness in real‑life problems. The constraints of the Navier‑Stokes equations are incorporated in the loss function of the PINN concept for achieving accuracy at much lower computational costs of 150‑200 times fewer data points. The validation of the accuracy of the reconstruction of the flow fields is carried out for different Reynolds numbers ranging from Re = 1 to 45 and different amplitude values of the rough channel walls ranging from 5 to 20 lattice units.

Abstract:
We introduce a Lattice‑Boltzmann‑driven kinetic physics‑informed neural network (K‑PINN) for predictive modeling of droplet dynamics on structured surfaces, in which the discrete Boltzmann‑BGK equation is incorporated into the learning framework. Different from traditional PINNs that are restricted by macroscopic continuum equations, the K‑PINN framework is built on the mesoscopic kinetic level, in which the essential Lattice‑Boltzmann physics is preserved in the data‑efficient neural network. The K‑PINN has been successfully employed for modeling non‑trivial droplet phenomena such as contact pinning, anisotropic spreading, and capillary hysteresis on substrates of different morphologies, ranging from random roughness to periodic pillar structures. Moreover, strict physical consistency, such as mass conservation within 1.5%, is ensured in the K‑PINN framework. Furthermore, the U‑Net‑based encoder‑decoder structure of the K‑PINN results in a 50‑75% reduction in error compared to traditional neural networks, achieving almost perfect agreement with high‑resolution Lattice‑Boltzmann simulations L_2 ~ 0.021‑0.026, R^2 ~ 0.999. Robust convergence of the K‑PINN to diverse surface morphologies is ensured through curriculum learning and adaptive two‑phase optimization. Upon convergence, the K‑PINN can perform real‑time prediction with over 10^4 evaluations per second. Through the combination of kinetic theory and physics‑informed learning, this work establishes a new paradigm for fast, physically consistent modeling of multiphase flows on complex surfaces.

Abstract:
Generative artificial intelligence offers a new paradigm to design matter in high‑dimensional spaces. However, its underlying mechanisms remain difficult to interpret and limit adoption in computational mechanics. This gap is striking because its core tools‑diffusion, stochastic differential equations, and inverse problems‑are fundamental to the mechanics of materials. Here we show that diffusion‑based generative AI and computational mechanics are rooted in the same principles. We illustrate this connection using a three‑ingredient burger as a minimal benchmark for material design in a low‑dimensional space, where both forward and reverse diffusion admit analytical solutions: Markov chains with Bayesian inversion in the discrete case and the Ornstein‑Uhlenbeck process with score‑based reversal in the continuous case. We extend this framework to a high‑dimensional design space with 146 ingredients and 8.9x10^43 possible configurations, where analytical solutions become intractable. We therefore learn the discrete and continuous reverse processes using neural network models that infer inverse dynamics from data. We train the models on only 2,260 recipes and generate one million samples that capture the statistical structure of the data, including ingredient prevalence and quantitative composition. We further generate five new burgers and validate them in a blinded restaurant‑based sensory study with n = 101 participants, where three of the AI‑designed burgers outperform the classical Big Mac in overall liking, flavor, and texture. These results establish diffusion‑based generative modeling as a physically grounded approach to design in high‑dimensional spaces. They position generative AI as a natural extension of computational mechanics, with applications from burgers to matter, and establish a path toward data‑driven, physics‑informed generative design.

Abstract:
Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here, we investigate the potential role of quantum circuits for solving PDEs. We design a parameterized quantum circuit (PQC) for implementing a polynomial based on tensor rank decomposition, reducing the quantum resource complexity from exponential to polynomial when the corresponding tensor rank is moderate. Building on this circuit, we develop a Quantum Physics‑Informed Neural Network (QPINN) and a Quantum‑inspired PINN, both of which guarantee the existence of an approximation of the PDE solution, and this approximation is represented as a polynomial that incorporates tensor rank decomposition. Despite using 80 times fewer parameters in experiments, our quantum models achieve higher accuracy and faster convergence than a classical fully connected PINN when solving the PDE for the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk‑free asset. Our quantum models further outperform a classical PINN constructed to share the same inductive bias, providing experimental evidence of quantum‑induced improvement and highlighting a resource‑efficient pathway toward classical and near‑term quantum PDE solvers.

Abstract:
Machine learning, especially physics‑informed neural networks (PINNs) and their neural network variants, has been widely used to solve problems involving partial differential equations (PDEs). The successful deployment of such methods beyond academic research remains limited. For example, PINN methods primarily consider discrete point‑to‑point fitting and fail to account for the potential properties of real solutions. The adoption of continuous activation functions in these approaches leads to local characteristics that align with the equation solutions while resulting in poor extensibility and robustness. A general explicit network (GEN) that implements point‑to‑function PDE solving is proposed in this paper. The "function" component can be constructed based on our prior knowledge of the original PDEs through corresponding basis functions for fitting. The experimental results demonstrate that this approach enables solutions with high robustness and strong extensibility to be obtained.

Abstract:
Thermal behavior has become a first‑order constraint in advanced 2.5D/3D integrated circuits (ICs) and heterogeneous packages. As power densities rise and multiple active dies are vertically integrated, heat removal paths become constricted, elevating junction temperatures, magnifying temperature gradients, and exacerbating reliability risks. This review synthesizes the physical mechanisms, modeling assumptions, and analysis methods that govern multiscale thermal transport in 3D ICs, with emphasis on interface‑dominated conduction, material anisotropy, and strong electrothermal coupling. We unify device‑to‑system scales into a coherent framework, analyzing trade‑offs among compact thermal models (CTMs), finite element/finite difference methods (FEM/FDM), Green's function and semi‑analytical techniques, reduced‑order and multi‑fidelity methods, and physics‑informed machine learning (PIML), while highlighting the central role of thermal boundary resistance (TBR) and variability in thermal interface materials (TIMs), the pitfalls of decoupled electrical/thermal analyses, and the need for rigorous validation against measurements. Finally, we outline practical design guidelines and a forward‑looking research agenda that integrates physics‑based modeling, data‑driven surrogates, and in situ sensing to enable thermally aware co‑optimization across the IC‑package‑system hierarchy.

Abstract:
Accurate assessment of central hemodynamics is essential for diagnosis and risk stratification, yet it still relies largely on invasive measurements or on indirect reconstructions built from population‑averaged transfer functions. While conventional methods are valuable in clinical practice, they face limitations, particularly in personalized medicine. Physics‑informed methods address these by integrating physical principles, reducing the need for extensive data. In this work, a fully noninvasive, patient‑specific framework is developed that combines a validated 1‑D model of the systemic arterial tree with physics‑informed neural networks (PINNs). This model performs the inverse solution of the flow and pressure fields within the arterial network, given minimal noninvasive measurements of pressure from a cuff reading and trains in 4000 iterations, at least 10x faster than the current state‑of‑the‑art models due to several model enhancements. We validate the model predictions against our 1‑D solver, yielding a near perfect correlation, and perform additional tests on a clinical dataset for the identification of important central hemodynamic parameters of cardiac output CO and central systolic blood pressure cSBP, with correlations of r=0.847 and r=0.951, respectively. Moreover, the model is able to tune the patient‑specific coefficients of the terminal resistance R_T and compliance C_T while training, treating them as learnable parameters. The inverse PINN model is able to solve the entire tree of 8 arteries with a single network, costing 5‑10 minutes of computational time. This significant performance boost compared to traditional iterative inverse methods holds promise towards applications of personalized cardiac output monitoring and hemodynamic assessment via noninvasive approaches like wearable devices.

Abstract:
Severe accident analysis using system‑level codes such as MELCOR is indispensable for nuclear safety assessment, yet the computational cost of repeated simulations poses a significant bottleneck for parametric studies and uncertainty quantification. Existing surrogate models accelerate these analyses but depend on large volumes of simulation data, while physics‑informed neural networks (PINNs) enable data‑free training but must be retrained for every change in problem parameters. This study addresses both limitations by developing the Parameterized PINNs coupled with FDM (P2F) method, a node‑assigned hybrid framework for MELCOR's Control Volume Hydrodynamics/Flow Path (CVH/FP) module. In the P2F method, a parameterized Node‑Assigned PINN (NA‑PINN) accepts the water‑level difference, initial velocity, and time as inputs, learning a solution manifold so that a single trained network serves as a data‑free surrogate for the momentum conservation equation across all flow paths without retraining. This PINN is coupled with a finite difference method (FDM) solver that advances the mass conservation equation at each time step, ensuring exact discrete mass conservation while replacing the iterative nonlinear momentum solve with a single forward pass. Verification on a six‑tank gravity‑driven draining scenario yields a water level mean absolute error of 7.85 × 10^‑5 m and a velocity mean absolute error of 3.21 × 10^‑3 m/s under the nominal condition with Δt = 1.0 s. The framework maintains consistent accuracy across time steps ranging from 0.2 to 1.0 s and generalizes to five distinct initial conditions, all without retraining or simulation data. This work introduces a numerical coupling methodology for integrating parameterized PINNs with FDM within a nuclear thermal‑hydraulic system code framework.

Abstract:
Two‑dimensional (2D) carbon networks, from pristine graphene to defect‑rich and amorphous monolayers, exhibit a complex structure‑energy landscape governed not only by local bonding but also by medium‑range order and network topology. Capturing these multi‑scale effects in a compact, interpretable, and data‑efficient manner remains a major challenge for machine learning (ML) in low‑dimensional materials. In this work, we introduce the CARBON‑2D Topological Descriptor (C2DTD), a physically informed structural representation specifically designed for 2D carbon systems. The descriptor integrates local geometric statistics, a compact radial structural signature, and explicit primitive ring topology into a fixed‑length, invariant vector that is both computationally efficient and directly interpretable. Benchmarking on diverse datasets of 2D carbon allotropes and defect‑engineered graphene sheets demonstrates that C2DTD achieves robust predictive performance in small‑data regimes, outperforming generic high‑dimensional featurization schemes while preserving physical transparency. Unsupervised manifold analysis reveals a smoother alignment between descriptor space and the DFT energy landscape, and feature‑importance and ablation studies confirm that ring topology emerges as a dominant energetic driver, particularly under vacancy‑induced reconstruction. Furthermore, controlled simulations with 5‑15% random vacancies show that C2DTD naturally captures the progressive transition from hexagon‑dominated graphene to topologically disordered networks, enabling both dataset‑level and structure‑specific interpretation. Owing to its compactness, interpretability, and strong physics‑based inductive bias, C2DTD provides a fast and generalizable framework for data‑driven modeling, defect analysis, and high‑throughput screening of 2D carbon materials.

Abstract:
The deployment of reinforcement learning (RL)‑based controllers on physical systems is often limited by poor generalization to real‑world scenarios, known as the simulation‑to‑reality (sim‑to‑real) gap. This gap is particularly challenging in spaceflight, where real‑world training data are scarce due to high cost and limited planetary exploration data. Traditional approaches, such as system identification and synthetic data generation, depend on sufficient data and often fail due to modeling assumptions or lack of physics‑based constraints. We propose addressing this data scarcity by introducing physics‑based learning bias in a generative model. Specifically, we develop the Mutual Information‑based Split Variational Autoencoder (MI‑VAE), a physics‑informed VAE that learns differences between observed system trajectories and those predicted by physics‑based models. The latent space of the MI‑VAE enables generation of synthetic datasets that respect physical constraints. We evaluate MI‑VAE on a planetary lander problem, focusing on limited real‑world data and offline RL training. Results show that augmenting datasets with MI‑VAE samples significantly improves downstream RL performance, outperforming standard VAEs in statistical fidelity, sample diversity, and policy success rate. This work demonstrates a scalable strategy for enhancing autonomous controller robustness in complex, data‑constrained environments.

Abstract:
The inherent compositional heterogeneity of multi‑principal element alloys (MPEAs) gives rise to complex, spatially varying mechanical fields that cannot be uniquely determined from coarse‑grained composition descriptors. This non‑uniqueness introduces intrinsically probabilistic structure‑property relationships, posing a fundamental challenge to conventional deterministic modeling and machine learning approaches that collapse such mappings into average predictions. Here, we present AlloyVAE, a physics‑informed generative framework that learns the full conditional distribution of mechanical fields from microstructural inputs. Built upon a conditional variational autoencoder architecture, the model incorporates learned smoothing operators to enhance functional regularity and a self‑consistency mechanism to enforce physical plausibility. Trained on atomistic simulation data, AlloyVAE accurately predicts distributions of residual stress fields from composition and short‑range order, and enables the generation of multiple physically consistent realizations under identical input conditions. Beyond forward prediction, the framework supports inverse design by optimizing composition fields to achieve targeted mechanical responses, and is extensible to coupled mappings involving eigenstrain. By capturing one‑to‑many structure‑property relationships in heterogeneous materials, this work establishes a probabilistic paradigm for materials modeling and design, providing a scalable alternative to conventional simulations for navigating high‑dimensional compositional spaces.

Abstract:
The choice between conservative and non‑conservative formulations is a fundamental dilemma in CFD. While non‑conservative forms offer intuitive modeling in primitive variables, they typically produce erroneous shock speeds. This paper critically analyzes these formulations, contrasting classical failures against the capabilities of Physics‑Informed Neural Networks (PINNs). Using the Adaptive Weight and Viscosity (PINNs‑AWV) architecture, we evaluate cases ranging from shallow water equations to unsteady 1D and 2D Euler equations. Results reveal a significant dichotomy: while PINNs‑AWV restores physical fidelity in scalar and steady systems, standard non‑conservative PINNs fail in unsteady systems like the Sod shock tube. We demonstrate this failure stems from non‑vanishing source terms introduced by viscous regularization, which violate the Rankine‑‑Hugoniot jump conditions. To resolve this, we implement a path‑integral framework based on Dal Maso‑‑LeFloch‑‑Murat (DLM) theory. By incorporating path‑consistent losses in PINNs (PI‑PINN) and using path‑conservative numerical schemes, we successfully recover correct shock speeds within non‑conservative frameworks. Our results prove the path‑integral approach provides a rigorous mathematical bridge for physical accuracy in both classical and machine learning solvers, enabling primitive‑variable formulations in transient, high‑speed simulations.

Abstract:
Wideband channel frequency response (CFR) estimation is challenging in multi‑band wireless systems, especially when one or more sub‑bands are temporarily blocked by co‑channel interference. We present a physics‑informed complex Transformer that reconstructs the full wideband CFR from such fragmented, partially observed spectrum snapshots. The interference pattern in each sub‑band is modeled as an independent two‑state discrete‑time Markov chain, capturing realistic bursty occupancy behavior. Our model operates on the joint time‑frequency grid of T snapshots and F frequency bins and uses a factored self‑attention mechanism that separately attends along both axes, reducing the computational complexity to O(TF^2 + FT^2). Complex‑valued inputs and outputs are processed through a holomorphic linear layer that preserves phase relationships. Training uses a composite physics‑informed loss combining spectral fidelity, power delay profile (PDP) reconstruction, channel impulse response (CIR) sparsity, and temporal smoothness. Mobility effects are incorporated through per‑sample velocity randomization, enabling generalization across different mobility regimes. Evaluation against three classical baselines, namely, last‑observation‑carry‑forward, zero‑fill, and cubic‑spline interpolation, shows that our approach achieves the highest PDP similarity with respect to the ground truth, reaching ρ\geq 0.82 compared to ρ\geq 0.62 for the best baseline at interference occupancy levels up to 50%. Furthermore, the model degrades smoothly across the full velocity range, consistently outperforming all other baselines.

Abstract:
Physics‑Informed Neural Networks (PINNs) are mesh‑free approaches for the numerical approximation of partial differential equations, where a neural network is trained by minimizing a loss function derived from the governing equations and boundary conditions. The Deep Ritz method can be interpreted as a particular variational form of a PINN, where the loss corresponds to the minimization of an energy functional associated with a symmetric positive definite problem. In this work, we study the approximation of the Laplace equation using both the classical PINN formulation and its variational counterpart, the Deep Ritz method, with the objective of accurately estimating prescribed goal functionals. When standard sampling strategies, such as uniform or loss‑based sampling, are employed during training, the convergence of the functional error and the attained minimal functional value can be slow. To address this issue, we introduce a functional‑oriented importance sampling strategy that can be applied to both PINNs and the Deep Ritz method. The key ingredient is the construction of a reliable and accurate estimator for the error in a given quantity of interest. This estimator is derived using concepts from the Dual Weighted Residual (DWR) framework and is implemented entirely within the neural network setting. It is then used to adaptively guide the sampling of training points in the computational domain, focusing computational effort on regions that have the strongest influence on the functional value. Numerical experiments demonstrate that the proposed adaptive sampling strategy significantly accelerates the convergence of the functional error and improves the minimization of the target functional during training for both PINN and Deep Ritz formulations.

Abstract:
Topology control for power grid operation is a challenging sequential decision making problem because the action space grows combinatorially with the size of the grid and action evaluation through simulation is computationally expensive. We propose a physics‑informed Reinforcement Learning framework that combines semi‑Markov control with a Gibbs prior, that encodes the system's physics, over the action space. The decision is only taken when the grid enters a hazardous regime, while a graph neural network surrogate predicts the post action overload risk of feasible topology actions. These predictions are used to construct a physics‑informed Gibbs prior that both selects a small state‑dependent candidate set and reweights policy logits before action selection. In this way, our method reduces exploration difficulty and online simulation cost while preserving the flexibility of a learned policy. We evaluate the approach in three realistic benchmark environments of increasing difficulty. Across all settings, the proposed method achieves a strong balance between control quality and computational efficiency: it matches oracle‑level performance while being approximately 6× faster on the first benchmark, reaches 94.6% of oracle reward with roughly 200× lower decision time on the second one, and on the most challenging benchmark improves over a PPO baseline by up to 255% in reward and 284% in survived steps while remaining about 2.5× faster than a strong specialized engineering baseline. These results show that our method provides an effective mechanism for topology control in power grids.

Abstract:
We propose a consistent physics‑informed neural networks (CPINNs) framework for elliptic obstacle problems formulated as variational inequalities. The method is based on a mixed loss functional that is rigorously aligned with the stability structure of the underlying problem and incorporates obstacle constraints through a consistent treatment of the associated Lagrange multiplier. Relying on optimal recovery theory under Besov regularity assumptions, we establish near‑optimal convergence rates for the simultaneous reconstruction of the solution and the multiplier from pointwise interior and boundary data. To enable practical implementation, we construct discrete counterparts of the continuous stability norms and duality pairings, leading to fully computable and theoretically justified training losses. Numerical experiments on benchmark obstacle problems demonstrate the accuracy, stability, and robustness of the proposed approach, and highlight its clear advantages over standard PINNs.

Abstract:
Reservoir simulation workflows face a fundamental data asymmetry: input parameter fields (geostatistical permeability realizations, porosity distributions) are free to generate in arbitrary quantities, yet existing neural operator surrogates require large corpora of expensive labeled simulation trajectories and cannot exploit this unlabeled structure. We introduce PI‑JEPA (Physics‑Informed Joint Embedding Predictive Architecture), a surrogate pretraining framework that trains \emphwithout any completed PDE solves, using masked latent prediction on unlabeled parameter fields under per‑sub‑operator PDE residual regularization. The predictor bank is structurally aligned with the Lie‑‑Trotter operator‑splitting decomposition of the governing equations, dedicating a separate physics‑constrained latent module to each sub‑process (pressure, saturation transport, reaction), enabling fine‑tuning with as few as 100 labeled simulation runs. On single‑phase Darcy flow, PI‑JEPA achieves 1.9× lower error than FNO and 2.4× lower error than DeepONet at N_\ell=100, with 24% improvement over supervised‑only training at N_\ell=500, demonstrating that label‑free surrogate pretraining substantially reduces the simulation budget required for multiphysics surrogate deployment.

Abstract:
Macroscopic unmanned aerial vehicle (UAV) traffic organization in three‑dimensional airspace faces significant challenges from static wind fields and complex obstacles. A critical difficulty lies in simultaneously capturing the strong anisotropy induced by wind while strictly preserving transport consistency and boundary semantics, which are often compromised in standard physics‑informed learning approaches. To resolve this, we propose a constraint‑preserving hybrid solver that integrates a physics‑informed neural network for the anisotropic Eikonal value problem with a conservative finite‑volume method for steady density transport. These components are coupled through an outer Picard iteration with under‑relaxation, where the target condition is hard‑encoded and strictly conservative no‑flux boundaries are enforced during the transport step. We evaluate the framework on reproducible homing and point‑to‑point scenarios, effectively capturing value slices, induced‑motion patterns, and steady density structures such as bands and bottlenecks. Ultimately, our perspective emphasizes the value of a reproducible computational framework supported by transparent empirical diagnostics to enable the traceable assessment of macroscopic traffic phenomena.

Abstract:
High‑fidelity Monte Carlo simulations and complex inverse problems, such as mapping smeared experimental observations to ground‑truth states, are computationally intensive yet essential for robust data analysis. Conditional Flow Matching (CFM) offers a mathematically robust approach to accelerating these tasks, but we demonstrate its standard training loss is fundamentally misleading. In rigorous physics applications, CFM loss plateaus prematurely, serving as an unreliable indicator of true convergence and physical fidelity. To investigate this disconnect, we designed JetPrism, a configurable CFM framework acting as an efficient generative surrogate for evaluating unconditional generation and conditional detector unfolding. Using synthetic stress tests and a Jefferson Lab kinematic dataset (γp \to ρ^0 p \to π^+π^‑ p) relevant to the forthcoming Electron‑Ion Collider (EIC), we establish that physics‑informed metrics continue to improve significantly long after the standard loss converges. Consequently, we propose a multi‑metric evaluation protocol incorporating marginal and pairwise χ^2 statistics, W_1 distances, correlation matrix distances (D_\mathrmcorr), and nearest‑neighbor distance ratios (R_\mathrmNN). By demonstrating that domain‑specific evaluations must supersede generic loss metrics, this work establishes JetPrism as a dependable generative surrogate that ensures precise statistical agreement with ground‑truth data without memorizing the training set. While demonstrated in nuclear physics, this diagnostic framework is readily extensible to parameter generation and complex inverse problems across broad domains. Potential applications span medical imaging, astrophysics, semiconductor discovery, and quantitative finance, where high‑fidelity simulation, rigorous inversion, and generative reliability are critical.

Abstract:
Accurate LiDAR simulation is crucial for autonomous driving, especially under adverse weather conditions. Existing methods struggle to capture the complex interactions between LiDAR signals and atmospheric phenomena, leading to unrealistic representations. This paper presents a physics‑informed learning framework (PICWGAN) for generating realistic LiDAR data under adverse weather conditions. By integrating physicsdriven constraints for modeling signal attenuation and geometryconsistent degradations into a physics‑informed learning pipeline, the proposed method reduces the sim‑to‑real gap. Evaluations on real‑world datasets (CADC for snow, Boreas for rain) and the VoxelScape dataset show that our approach closely mimics realworld intensity patterns. Quantitative metrics, including MSE, SSIM, KL divergence, and Wasserstein distance, demonstrate statistically consistent intensity distributions. Additionally, models trained on data enhanced by our framework outperform baselines in downstream 3D object detection, achieving performance comparable to models trained on real‑world data. These results highlight the effectiveness of the proposed approach in improving the realism of LiDAR data and enabling robust perception under adverse weather conditions.

Abstract:
Diffusion models have recently emerged as powerful stochastic frameworks for high‑dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics‑informed training strategies, which tightly couple learning with specific governing equations and limit generalization across problem settings. In this work, we propose a diffusion model with physics‑guided inference for solving PDEs, in which the diffusion model is trained using standard data‑driven procedures, while physical laws are incorporated exclusively during the reverse inference stage. The reverse diffusion dynamics is guided by a PDE residual energy function, combined with Gaussian smoothing and explicit boundary enforcement, yielding a physically consistent stochastic iteration that is independent of the training process. From a numerical standpoint, the proposed framework can be interpreted as a diffusion‑inspired implicit solver that converges to the PDE solution even when initialized from random noise and perturbed by stochastic fluctuations. The method is validated on classical PDE equation such as Poisson, Diffusion, and Burgers equations with varying coefficients. Numerical results demonstrate robust convergence, high accuracy, and strong generalization without retraining, highlighting the proposed framework as a unified alternative to classical numerical solvers and physics‑informed neural networks.

Abstract:
Accurate air quality forecasting is crucial for protecting public health and guiding environmental policy, yet it remains challenging due to nonlinear spatiotemporal dynamics, wind‑driven transport, and distribution shifts across regions. Physics‑based models are interpretable but computationally expensive and often rely on restrictive assumptions, whereas purely data‑driven models can be accurate but may lack robustness and calibrated uncertainty. To address these limitations, we propose Neural Dynamic Diffusion‑Advection Fields (NeuroDDAF), a physics‑informed forecasting framework that unifies neural representation learning with open‑system transport modeling. NeuroDDAF integrates (i) a GRU‑Graph Attention encoder to capture temporal dynamics and wind‑aware spatial interactions, (ii) a Fourier‑domain diffusion‑advection module with learnable residuals, (iii) a wind‑modulated latent Neural ODE to model continuous‑time evolution under time‑varying connectivity, and (iv) an evidential fusion mechanism that adaptively combines physics‑guided and neural forecasts while quantifying uncertainty. Experiments on four urban datasets (Beijing, Shenzhen, Tianjin, and Ancona) across 1‑3 day horizons show that NeuroDDAF consistently outperforms strong baselines, including AirPhyNet, achieving up to 9.7% reduction in RMSE and 9.4% reduction in MAE on long‑term forecasts. On the Beijing dataset, NeuroDDAF attains an RMSE of 41.63 μg/m^3 for 1‑day prediction and 48.88 μg/m^3 for 3‑day prediction, representing the best performance among all compared methods. In addition, NeuroDDAF improves cross‑city generalization and yields well‑calibrated uncertainty estimates, as confirmed by ensemble variance analysis and case studies under varying wind conditions.

Abstract:
In this paper, a meshfree method using physics‑informed neural networks (PINNs) is developed for solving two‑phase flow problems with moving interfaces, where two immiscible fluids bearing different material properties, are separated by a dynamically evolving interface and interact with each other through interface conditions. Two kinds of distinct scenarios of interface motion are addressed: the prescribed interface motion whose moving velocity is explicitly given, and the solution‑driven interface motion whose evolution is determined by the velocity field of two‑phase flow. Based upon piecewise deep neural networks and spatiotemporal sampling points/training set in each fluid subdomain, the proposed PINNs framework reformulates the two‑phase flow moving interface problem as a least‑squares (LS) minimization problem, which involves all residuals of governing equations, interface conditions, boundary conditions and initial conditions. Furthermore, approximation properties of the proposed PINNs approach are analyzed rigorously for the presented two‑phase flow model by employing the Reynolds transport theorem in evolving domains, moreover, a comprehensive error estimation is provided to account for additional complexities introduced by the moving interface and the coupling between fluid dynamics and interface evolution. Numerical experiments are carried out to illustrate the effectiveness of the proposed PINNs approach for various configurations of two‑phase flow moving interface problems, and to validate the theoretical findings as well. A practical guidance is thus provided for an efficient training set distribution when applying the proposed PINNs approach to two‑phase flow moving interface problems in practice.

Abstract:
Analyzing nonlinear systems with stabilizable controlled invariant sets (CISs) requires accurate estimation of their domains of stabilization (DOS) together with associated stabilizing controllers. Despite extensive research, estimating DOSs for general nonlinear systems remains challenging due to fundamental theoretical and computational limitations. In this paper, we propose a novel framework for estimating DOSs for controlled input‑constrained discrete‑time systems. The DOS is characterized via newly introduced value functions defined on metric spaces of compact sets. We establish the fundamental properties of these value functions and derive the associated Bellman‑type (Zubov‑type) functional equations. Building on this characterization, we develop a physics‑informed neural network (NN) framework that learns the value functions by embedding the derived functional equations directly into the training process. The proposed methodology is demonstrated through two numerical examples, illustrating its ability to accurately estimate DOSs and synthesize stabilizing controllers from the learned value functions.

Abstract:
When three‑dimensional bodies contain thin features, non‑trivial topology, or scan‑derived surfaces, volumetric meshing can become the dominant bottleneck in simulation workflows. We replace this step with a learned geometric representation: overlapping volumetric coordinate charts, each equipped with a neural decoder and Jacobian, trained from point‑cloud or level‑set data to form a differentiable atlas. Governing equations are pulled back to chart‑local reference coordinates via the Piola identity, and local solutions are coupled through multiplicative Schwarz iterations on the overlap graph. Because the atlas is constructed independently of the downstream discretization, one frozen geometric substrate can support fundamentally different solvers (for example, a meshfree physics‑informed neural network and a conventional finite‑element method) without re‑meshing or re‑parametrization. Benchmark and verification studies show that the learned atlas preserves expected finite‑element convergence behavior and enables both forward and inverse analyses on geometries that would otherwise require solver‑specific volumetric meshing.

Abstract:
Informed Machine Learning has emerged as a viable generalization of Machine Learning (ML) by building a unified conceptual and algorithmic setting for constructing models on a unified basis of knowledge and data. Physics‑informed ML involving physics equations is one of the developments within Informed Machine Learning. This study proposes a novel direction of Knowledge‑Data ML, referred to as KD‑ML, where numeric data are integrated with knowledge tidbits expressed in the form of granular knowledge landmarks. We advocate that data and knowledge are complementary in several fundamental ways: data are precise (numeric) and local, usually confined to some region of the input space, while knowledge is global and formulated at a higher level of abstraction. The knowledge can be represented as information granules and organized as a collection of input‑output information granules called knowledge landmarks. In virtue of this evident complementarity, we develop a comprehensive design process of the KD‑ML model and formulate an original augmented loss function L, which additively embraces the component responsible for optimizing the model based on available numeric data, while the second component, playing the role of a granular regularizer, so that it adheres to the granular constraints (knowledge landmarks). We show the role of the hyperparameter positioned in the loss function, which balances the contribution and guiding role of data and knowledge, and point to some essential tendencies associated with the quality of data (noise level) and the level of granularity of the knowledge landmarks. Experiments on two physics‑governed benchmarks demonstrate that the proposed KD model consistently outperforms data‑driven ML models.

Abstract:
Physics‑Informed Neural Networks (PINNs) frequently encounter difficulties in accurately resolving shock waves within high‑speed compressible flows, a failure largely attributed to the "gradient pathology" arising from extreme stiffness at discontinuities. To overcome this limitation, we propose the Spatio‑Temporal Uncertainty‑Modulated PINN (UM‑PINN), a probabilistic framework that reinterprets the training process as a multi‑task learning problem governed by homoscedastic aleatoric uncertainty. By integrating a gradient‑based spatial mask with learnable variance parameters, our method dynamically balances the conflicting contributions of Partial Differential Equation (PDE) residuals and initial conditions across the spatiotemporal domain, further stabilized by Quasi‑Monte Carlo Sobol sampling. We validate the framework against challenging benchmarks, including the one‑dimensional (1D) Sod shock tube, the high‑frequency Shu‑Osher problem, and the complex two‑dimensional (2D) Riemann interaction, where standard gradient‑based weighting schemes typically fail. Experimental results demonstrate that UM‑PINN achieves orders of magnitude improvement in accuracy and shock resolution compared to baseline methods, establishing a robust new paradigm for mesh‑free Computational Fluid Dynamics in hyperbolic systems.

Abstract:
In this study, we investigate the phase structure of magnetized QCD matter by determining the field‑dependent parameters of the Nambu‑Jona‑Lasinio (NJL) model through a physics‑informed machine learning framework. Specifically, we focus on extracting the optimal functional forms for the running coupling constant G(eB) and the quark anomalous magnetic moment (AMM) ratio v_2(eB), utilizing lattice QCD‑computed quark condensate data as the ``ground truth". By embedding the NJL gap equation as a differentiable physics‑constrained module, our neural network pipeline identifies continuous parameter functions that accurately reproduce the inverse magnetic catalysis (IMC) effect. Our results demonstrate that the magnetic field smoothly suppresses both G and v_2. This approach not only bridges the gap between effective models and lattice data but also provides new microscopic insights into the response of the QCD vacuum to strong magnetic fields.

Abstract:
High‑density through‑substrate vias (TSVs) enable 2.5D/3D heterogeneous integration but introduce significant signal‑integrity and thermal‑reliability challenges due to electrical coupling, insertion loss, and self‑heating. Conventional full‑wave finite‑element method (FEM) simulations provide high accuracy but become computationally prohibitive for large design‑space exploration. This work presents a scalable electro‑thermal modeling and optimization framework that combines physics‑informed analytical modeling, graph neural network (GNN) surrogates, and full‑wave sign‑off validation. A multi‑conductor analytical model computes broadband S‑parameters and effective anisotropic thermal conductivities of TSV arrays, achieving 5%‑10% relative Frobenius error (RFE) across array sizes up to 15x15. A physics‑informed GNN surrogate (TSV‑PhGNN), trained on analytical data and fine‑tuned with HFSS simulations, generalizes to larger arrays with RFE below 2% and nearly constant variance. The surrogate is integrated into a multi‑objective Pareto optimization framework targeting reflection coefficient, insertion loss, worst‑case crosstalk (NEXT/FEXT), and effective thermal conductivity. Millions of TSV configurations can be explored within minutes, enabling exhaustive layout and geometric optimization that would be infeasible using FEM alone. Final designs are validated with Ansys HFSS and Mechanical, showing strong agreement. The proposed framework enables rapid electro‑thermal co‑design of TSV arrays while reducing per‑design evaluation time by more than six orders of magnitude.

Abstract:
Linear dynamical systems are fully characterized by their eigenspectra, accessible directly from the generator of the dynamics. For nonlinear systems governed by partial differential equations, no equivalent theory exists. We introduce Lie Generator Network‑Koopman (LGN‑KM), a neural operator that lifts nonlinear dynamics into a linear latent space and learns the continuous‑time Koopman generator (L_k) through a decomposition L_k = S ‑ D_k, where S is skew‑symmetric representing conservative inter‑modal coupling, and D_k is a positive‑definite diagonal encoding modal dissipation. This architectural decomposition enforces stability and enables interpretability through direct spectral access to the learned dynamics. On two‑dimensional Navier‑‑Stokes turbulence, the generator recovers the known dissipation scaling and a complete multi‑branch dispersion relation from trajectory data alone with no physics supervision. Independently trained models at different flow regimes recover matched gauge‑invariant spectral structure, exposing a gauge freedom in the Koopman lifting. Because the generator is provably stable, it enables guaranteed long‑horizon stability, continuous‑time evaluation at arbitrary time, and physics‑informed cross‑viscosity model transfer.

Abstract:
Singularly perturbed partial differential equations arise in many applications, including magnetohydrodynamic duct flows, chemical reaction transport systems, and Poisson Boltzmann electrostatics. These problems are characterized by sharp boundary layers and pronounced multiscale behavior, posing significant challenges for numerical methods. Existing approaches, particularly machine learning based methods, often rely on explicit asymptotic decompositions or specialized architectures, increasing implementation complexity and leading to optimization imbalance in stiff regimes. In this work, we propose a unified learning framework based on a weighted loss formulation within the standard physics informed neural network setting. The proposed method requires only prior knowledge of the boundary layer thickness, while the boundary layer locations are automatically identified during training. The resulting formulation avoids problem specific architectural modifications and remains applicable across different equation types. Numerical experiments on both scalar and coupled reaction diffusion and convection diffusion reaction systems, defined on regular and irregular domains, demonstrate robust performance for boundary layer thickness as small as 10^‑10 while maintaining high solution accuracy.

Abstract:
Enforcing prescribed global integral constraints in mesh‑free neural PDE solvers is challenging in high‑dimensional domains. Existing projection methods for spatial integrals are often tied to fixed grids or uniform quadrature, which can conflict with randomly sampled physics‑informed neural networks (PINNs) and scale poorly with dimension. High‑order differential operators also increase reverse‑mode automatic differentiation memory costs. We propose Stochastic Dimension Implicit Functional Projection (SDIFP), a quadrature‑level framework for enforcing prescribed first and second spatial moments. SDIFP replaces tensor‑product nodal projection by a global affine correction of the neural‑network output, with two scalar coefficients determined from a weighted quadrature rule. Under positive target variance and nonzero empirical raw variance, this correction is the nearest‑point projection, in the weighted quadrature norm, onto the empirical two‑moment constraint set. Thus, the prescribed moments are exact for the selected quadrature rule, while continuum errors are quadrature errors of the corrected field. For decomposable high‑dimensional linear operators, SDIFP combines affine moment correction with stochastic operator‑subset sampling. With independent residual and derivative sampling and conditionally unbiased coefficient‑gradient estimation, the resulting estimator is unbiased for the specified quadrature‑based residual objective; the shared‑subset fast mode is biased in general. SDIFP avoids tensor‑product quadrature for moment enforcement, separates forward quadrature evaluation from the reverse‑mode graph, and retains pointwise inference efficiency once the affine coefficients are fixed or precomputed.

Abstract:
Nonconvex multi‑well energies in cell‑induced phase transitions give rise to fine‑scale microstructures, low‑regularity transition layers and sharp interfaces, all of which pose numerical challenges for physics‑informed learning. Here we introduce biomimetic physics‑informed neural networks (Bio‑PINNs), which implement a near‑to‑far curriculum by progressively revealing the computational domain away from the cell boundary and combining this schedule with a deformation‑uncertainty proxy that concentrates collocation points near evolving transition layers and tether‑forming regions. Across single‑cell and multicellular benchmarks, Bio‑PINNs recover the densified phase more reliably near cell boundaries and in intercellular gaps, while capturing tether morphology more faithfully than representative ungated and residual‑driven adaptive baselines.

Abstract:
Autonomous experimental systems are increasingly used in materials research to accelerate scientific discovery, but their performance is often limited by low‑quality, noisy data. This issue is especially problematic in data‑intensive structure‑property learning tasks such as Image‑to‑Spectrum (Im2Spec) and Spectrum‑to‑Image (Spec2Im) translations, where standard active learning strategies can mistakenly prioritize poor‑quality measurements. We introduce a gated active learning framework that combines curiosity‑driven sampling with a physics‑informed quality control filter based on the Simple Harmonic Oscillator model fits, allowing the system to automatically exclude low‑fidelity data during acquisition. Evaluations on a pre‑acquired dataset of band‑excitation piezoresponse spectroscopy (BEPS) data from PbTiO3 thin films with spatially localized noise show that the proposed method outperforms random sampling, standard active learning, and multitask learning strategies. The gated approach enhances both Im2Spec and Spec2Im by handling noise during training and acquisition, leading to more reliable forward and inverse predictions. In contrast, standard active learners often misinterpret noise as uncertainty and end up acquiring bad samples that hurt performance. Given its promising applicability, we further deployed the framework in real‑time experiments on BiFeO3 thin films, demonstrating its effectiveness in real autonomous microscopy experiments. Overall, this work supports a shift toward hybrid autonomy in self‑driving labs, where physics‑informed quality assessment and active decision‑making work hand‑in‑hand for more reliable discovery.

Abstract:
This work presents a hybrid physics‑informed and data‑driven modeling framework for predictive control of autonomous off‑road vehicles operating on deformable terrain. Traditional high‑fidelity terramechanics models are often too computationally demanding to be directly used in control design. Modern Koopman operator methods can be used to represent the complex terramechanics and vehicle dynamics in a linear form. We develop a framework whereby a Koopman linear system can be constructed using data from simulations of a vehicle moving on deformable terrain. For vehicle simulations, the deformable‑terrain terramechanics are modeled using Bekker‑Wong theory, and the vehicle is represented as a simplified five‑degree‑of‑freedom (5‑DOF) system. The Koopman operators are identified from large simulation datasets for sandy loam and clay using a recursive subspace identification method, where Grassmannian distance is used to prioritize informative data segments during training. The advantage of this approach is that the Koopman operator learned from simulations can be updated with data from the physical system in a seamless manner, making this a hybrid physics‑informed and data‑driven approach. Prediction results demonstrate stable short‑horizon accuracy and robustness under mild terrain‑height variations. When embedded in a constrained MPC, the learned predictor enables stable closed‑loop tracking of aggressive maneuvers while satisfying steering and torque limits.

Abstract:
Nanoscale thermal transport is governed by the phonon Boltzmann transport equation (BTE). However, simulating the sub‑continuum dynamics remains computationally prohibitive due to the high dimensionality of the phase space and the intrinsic nonlinearity of the scattering collision operator. Traditional numerical solvers and standard physics‑informed neural networks (PINNs) inherently struggle with these integro‑differential equations due to deterministic quadrature limitations, artificial thermalization introduced by the relaxation time approximation (RTA), and multiscale spectral bias. This work introduces a multiscale auxiliary physics‑informed neural network (MTNet) to solve the generalized equation of phonon radiative transfer (GEPRT). By leveraging an auxiliary formulation, this mesh‑free framework recasts the GEPRT into a fully differential system, enabling the analytical evaluation of scattering operators via automatic differentiation and facilitating scalable multi‑GPU parallelization. To circumvent optimization stiffness, the architecture employs a decoupled, shallow neural network explicitly constrained by radiative equilibrium. MTNet is validated by simulating steady‑state cross‑plane transport in a silicon thin film, successfully capturing ballistic‑diffusive regimes and characteristic boundary slips across extreme temperature gradients (ΔT = 100 K) beyond the standard linearization approach. Furthermore, we show that our framework successfully solves a geometric inverse problem in a slab geometry, retrieving the unknown slab thickness based only on interface temperature constraints in the mesoscopic regime. Ultimately, MTNet establishes a robust, fully differentiable foundation for predicting high‑fidelity kinetic transport and extracting material properties in next‑generation nanostructures.

Abstract:
This paper introduces a novel physics‑informed impact identification (Phy‑ID) framework. The proposed method integrates observational, inductive, and learning biases to combine physical knowledge with data‑driven inference in a unified modelling strategy, achieving physically consistent and numerically stable impact identification. The physics‑informed approach structures the input space using physics‑based energy indicators, constrains admissible solutions via architectural design, and enforces governing relations via hybrid loss formulations. Together, these mechanisms limit non‑physical solutions and stabilise inference under degraded measurement conditions. A disjoint inference formulation is used as a representative use case to demonstrate the framework capabilities, in which impact velocity and impactor mass are inferred through decoupled surrogate models, and impact energy is computed by enforcing kinetic energy consistency. Experimental evaluations show mean absolute percentage errors below 8% for inferred impact velocity and impactor mass and below 10% for impact energy. Additional analyses confirm stable performance under reduced data availability and increased measurement noise, as well as generalisation for out‑of‑distribution cases across pristine and damaged regimes when damaged responses are included in training. These results indicate that the systematic integration of physics‑informed biases enables reliable, physically consistent, and data‑efficient impact identification, highlighting the potential of the approach for practical monitoring systems.

Abstract:
Accurate prediction of hydrogen sorption in fine‑grained geological materials is essential for evaluating underground hydrogen storage capacity, assessing caprock integrity, and characterizing hydrogen migration in subsurface energy systems. Classical isotherm models perform well at the individual‑sample level but fail when generalized across heterogeneous populations, with the coefficient of determination collapsing from 0.80‑0.90 for single‑sample fits to 0.09‑0.38 for aggregated multi‑sample datasets. We present a multi‑scale physics‑informed neural network framework that addresses this limitation by embedding classical adsorption theory and thermodynamic constraints directly into the learning process. The framework utilizes 1,987 hydrogen sorption isotherm measurements across clays, shales, coals, supplemented by 224 characteristic uptake measurements. A seven‑category physics‑informed feature engineering scheme generates 62 thermodynamically meaningful descriptors from raw material characterization data. The loss function enforces saturation limits, a monotonic pressure response, and Van't Hoff temperature dependence via penalty weighting, while a three‑phase curriculum‑based training strategy ensures stable integration of competing physical constraints. An architecture‑diverse ensemble of ten members provides calibrated uncertainty quantification, with post‑hoc temperature scaling achieving target prediction interval coverage. The optimized PINN achieves R2 = 0.9544, RMSE = 0.0484 mmol/g, and MAE = 0.0231 mmol/g on the held‑out test set, with 98.6% monotonicity satisfaction and zero non‑physical negative predictions. Physics‑informed regularization yields a 10‑15% cross‑lithology generalization advantage over a well‑tuned random forest under leave‑one‑lithology‑out validation, confirming that thermodynamic constraints transfer meaningfully across geological boundaries.

Abstract:
Deep learning (DL) models have achieved strong performance in an intelligence healthcare setting, yet most existing approaches operate as black boxes and ignore the physical processes that govern tumor growth, limiting interpretability, robustness, and clinical trust. To address this limitation, we propose PhysNet, a physics‑embedded DL framework that integrates tumor growth dynamics directly into the feature learning process of a convolutional neural network (CNN). Unlike conventional physics‑informed methods that impose physical constraints only at the output level, PhysNet embeds a reaction diffusion model of tumor growth within intermediate feature representations of a ResNet backbone. The architecture jointly performs multi‑class tumor classification while learning a latent tumor density field, its temporal evolution, and biologically meaningful physical parameters, including tumor diffusion and growth rates, through end‑to‑end training. This design is necessary because purely data‑driven models, even when highly accurate or ensemble‑based, cannot guarantee physically consistent predictions or provide insight into tumor behavior. Experimental results on a large brain MRI dataset demonstrate that PhysNet outperforms multiple state‑of‑the‑art DL baselines, including MobileNetV2, VGG16, VGG19, and ensemble models, achieving superior classification accuracy and F1‑score. In addition to improved performance, PhysNet produces interpretable latent representations and learned bio‑physical parameters that align with established medical knowledge, highlighting physics‑embedded representation learning as a practical pathway toward more trustworthy and clinically meaningful medical AI systems.

Abstract:
Deterministic channel modeling maps a physical environment to its site‑specific electromagnetic response. Ray tracing produces complete multi‑dimensional channel information but remains prohibitively expensive for area‑wide deployment. We identify line‑of‑sight (LoS) region determination as the dominant bottleneck. To address this, we propose D^2LoS, a physics‑informed neural network that reformulates dense pixel‑level LoS prediction into sparse vertex‑level visibility classification and projection point regression, avoiding the spectral bias at sharp boundaries. A geometric post‑processing step enforces hard physical constraints, yielding exact piecewise‑linear boundaries. Because LoS computation depends only on building geometry, cross‑band channel information is obtained by updating material parameters without retraining. We also construct RayVerse‑100, a ray‑level dataset spanning 100 urban scenarios with per‑ray complex gain, angle, delay, and geometric trajectory. Evaluated against rigorous ray tracing ground truth, D^2LoS achieves 3.28~dB mean absolute error in received power, 4.65^\circ angular spread error, and 20.64~ns delay spread error, while accelerating visibility computation by over 25×.

Abstract:
Nonlinear Partial Differential Equations (PDEs) are ubiquitous in mathematical physics and engineering. Although Physics‑Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDE problems, they typically struggle to identify multiple distinct solutions, since they are designed to find one solution at a time. To address this limitation, we introduce Deflation‑PINNs, a novel framework that integrates a deflation loss with an architecture based on PINNs and Deep Operator Networks (DeepONets). By incorporating a deflation term into the loss function, our method systematically forces the Deflation‑PINN to seek and converge upon distinct finitely many solution branches. We provide theoretical evidence on the convergence of our model and demonstrate the efficacy of Deflation‑PINNs through numerical experiments on the Landau‑de Gennes model of liquid crystals, a system renowned for its complex energy landscape and multiple equilibrium states. Our results show that Deflation‑PINNs can successfully identify and characterize multiple distinct crystal structures.

Abstract:
Reconstructing continuous physical fields from sparse, irregular observations is a central challenge in scientific machine learning, particularly for systems governed by partial differential equations (PDEs). Existing physics‑informed methods typically enforce governing equations as soft penalty terms during optimization, often leading to gradient imbalance, instability, and degraded physical consistency under limited data. We introduce the Physics‑Guided Transformer (PGT), a neural architecture that embeds physical structure directly into the self‑attention mechanism. Specifically, PGT incorporates a heat‑kernel‑derived additive bias into attention logits, encoding diffusion dynamics and temporal causality within the representation. Query coordinates attend to these physics‑conditioned context tokens, and the resulting features are decoded using a FiLM‑modulated sinusoidal implicit network that adaptively controls spectral response. We evaluate PGT on the one‑dimensional heat equation and two‑dimensional incompressible Navier‑Stokes systems. In sparse 1D reconstruction with 100 observations, PGT achieves a relative L2 error of 5.9e‑3, significantly outperforming both PINNs and sinusoidal representations. In the 2D cylinder wake problem, PGT uniquely achieves both low PDE residual (8.3e‑4) and competitive relative error (0.034), outperforming methods that optimize only one objective. These results demonstrate that embedding physics within attention improves stability, generalization, and physical fidelity under data‑scarce conditions.

Abstract:
Resolving transient atomic configurations in non‑crystalline or dynamic environments remains a fundamental bottleneck in the physical sciences. While X‑ray absorption spectroscopy (XAS) is a premier probe of local structure, inverting spectra into structural descriptors is a notoriously ill‑posed problem due to inherent many‑to‑one mapping. Here, we present the Spectral Pattern Translator (SPT), a physics‑informed deep learning framework that establishes a robust bridge between large‑scale theoretical datasets and experimental reality. Our strategy exploits the Fourier duality between spectral energy oscillations and spatial scattering paths to overcome the "simulation‑to‑experiment" gap. By decomposing spectra into frequency domains, SPT effectively isolates robust structural coordination signals from the destabilizing noise inherent in experimental data. Trained on a massive library of diverse atomic environments, this approach achieves state‑of‑the‑art accuracy in resolving continuous phase transitions in battery cathodes and deciphering local order in amorphous materials. With millisecond‑scale latency, SPT removes the primary computational barrier to autonomous materials discovery, establishing a robust, noise‑resilient engine for closed‑loop robotic chemistry.

Abstract:
Accurately and stably solving the incompressible Navier‑‑Stokes equations with physics‑informed neural networks (PINNs) remains challenging, particularly for sparse or noisy observations and for flow regimes in which the local balance among convection, diffusion, and pressure is difficult to capture. To address this issue, we propose a framework, denoted as LVM‑PINN, which incorporates a learnable viscosity modulation (LVM) mechanism into the PINN residual. Specifically, the model predicts a spatiotemporal scalar field that is embedded directly into the viscous diffusion term of the momentum equations, thereby enabling adaptive modulation of the local dissipation strength during training. This modification improves optimization stability while enhancing the representation of complex flow structures. The effect of the proposed mechanism is further examined through a controlled ablation setting with an otherwise unchanged network architecture, as well as through comparisons with GRU‑ and residual‑attention‑based backbone baselines. Numerical experiments on two‑dimensional benchmark problems, including the Kovasznay flow and two manufactured forcing flows, show that the proposed framework yields more stable training behavior and more accurate flow reconstruction under sparse and noisy data conditions.

Abstract:
Photoplasticity, the light‑induced change in plastic deformation, plays a pivotal role in the mechanical durability and manufacturing of semiconductor materials. Yet, its governing mechanisms remain incompletely understood, owing to the interplay of coupled multiphysics factors. Here, we conduct high‑throughput nanoindentation measurements to compile a dataset of paired hardness values in dark and light conditions. Then, we engineer physics‑informed descriptors spanning electrical, mechanical, and optical properties, and identify the ten most informative features, including bandgap, breakdown field, and refractive index, to enable an interpretable machine learning framework that yields transferable design rules for light‑tunable semiconductor mechanics. By identifying and predicting photoplasticity in semiconductors, this work provides a practical pathway for extracting mechanism‑linked, transferable guidelines to engineer light‑responsive mechanical behavior in semiconductor materials and devices.

Abstract:
Disturbance observer‑based control has shown promise in robustifying robotic systems against uncertainties. However, tuning such systems remains challenging due to the strong coupling between controller gains and observer parameters. In this work, we propose MetaTune, a unified framework for joint auto‑tuning of feedback controllers and disturbance observers through differentiable closed‑loop meta‑learning. MetaTune integrates a portable neural policy with physics‑informed gradients derived from differentiable system dynamics, enabling adaptive gain across tasks and operating conditions. We develop an adjoint method that efficiently computes the meta‑gradients with respect to adaptive gains backward in time to directly minimize the cost‑to‑go. Compared to existing forward methods, our approach reduces the computational complexity to be linear in the data horizon. Experimental results on quadrotor control show that MetaTune achieves consistent improvements over state‑of‑the‑art differentiable tuning methods while reducing gradient computation time by more than 50 percent. In high‑fidelity PX4‑Gazebo hardware‑in‑the‑loop simulation, the learned adaptive policy yields 15‑20 percent average tracking error reduction at aggressive flight speeds and up to 40 percent improvement under strong disturbances, while demonstrating zero‑shot sim‑to‑sim transfer without fine‑tuning.

Abstract:
Physics‑Informed Neural Networks (PINNs) and Neural Ordinary Differential Equations (NODEs) represent two distinct machine learning frameworks for modeling nonlinear neuronal dynamics. This study systematically evaluates their performance on the two‑dimensional Morris‑Lecar model across three canonical bifurcation regimes: Hopf, Saddle‑Node on Limit Cycle, and homoclinic orbit. Synthetic time‑series data are generated via numerical integration under controlled conditions, and training is performed using collocation points for PINNs and adaptive solvers for NODEs (Dormand‑Prince method). PINNs incorporate the governing differential equations into the loss function using automatic differentiation, which enforces physical consistency during training. In contrast, NODEs learn the system's vector field directly from data, without prior structural assumptions or inductive bias toward physical laws. Model performance is assessed using standard regression metrics, including Mean Squared Error (MSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and the coefficient of determination. Results indicate that PINNs tend to achieve higher accuracy and robustness in scenarios involving stiffness or sensitive bifurcations, owing to their embedded physical structure. NODEs, while more expressive and flexible, operate as black‑box approximators without structural constraints, which can lead to reduced interpretability and stability in these regimes. Although advanced variants of NODEs (e.g., ANODEs, latent NODEs) aim to mitigate such limitations, their performance under stiff dynamics remains an open question. These findings emphasize the trade‑offs between physics‑informed models, which embed structure and interpretability, and purely data‑driven approaches, which prioritize flexibility at the cost of physical consistency.

Abstract:
High‑dimensional low‑sample‑size (HDLSS) datasets constrain reliable environmental model development, where labeled data remain sparse. Reinforcement learning (RL)‑based adaptive sensing methods can learn optimal sampling policies, yet their application is severely limited in HDLSS contexts. In this work, we present PiCSRL (Physics‑Informed Contextual Spectral Reinforcement Learning), where embeddings are designed using domain knowledge and parsed directly into the RL state representation for improved adaptive sensing. We developed an uncertainty‑aware belief model that encodes physics‑informed features to improve prediction. As a representative example, we evaluated our approach for cyanobacterial gene concentration adaptive sampling task using NASA PACE hyperspectral imagery over Lake Erie. PiCSRL achieves optimal station selection (RMSE = 0.153, 98.4% bloom detection rate, outperforming random (0.296) and UCB (0.178) RMSE baselines, respectively. Our ablation experiments demonstrate that physics‑informed features improve test generalization (0.52 R^2, +0.11 over raw bands) in semi‑supervised learning. In addition, our scalability test shows that PiCSRL scales effectively to large networks (50 stations, >2M combinations) with significant improvements over baselines (p = 0.002). We posit PiCSRL as a sample‑efficient adaptive sensing method across Earth observation domains for improved observation‑to‑target mapping.

Abstract:
Physics‑informed neural networks (PINNs) offer a unified framework for solving both forward and inverse problems of differential equations, yet their performance and physical consistency strongly depend on how governing laws are incorporated. In this work, we present a systematic comparison of different thermodynamic structure‑informed neural networks by incorporating various thermodynamics formulations, including Newtonian, Lagrangian, and Hamiltonian mechanics for conservative systems, as well as the Onsager variational principle and extended irreversible thermodynamics for dissipative systems. Through comprehensive numerical experiments on representative ordinary and partial differential equations, we quantitatively evaluate the impact of these formulations on accuracy, physical consistency, noise robustness, and interpretability. The results show that Newtonian‑residual‑based PINNs can reconstruct system states but fail to reliably recover key physical and thermodynamic quantities, whereas structure‑preserving formulation significantly enhances parameter identification, thermodynamic consistency, and robustness. These findings provide practical guidance for principled design of thermodynamics‑consistency model, and lay the groundwork for integrating more general nonequilibrium thermodynamic structures into physics‑informed machine learning.

Abstract:
Motivation: Generative models for protein backbone design have to simultaneously ensure geometric validity, sampling efficiency, and scalability to long sequences. However, most existing approaches rely on iterative refinement, quadratic attention mechanisms, or post‑hoc geometry correction, leading to a persistent trade‑off between computational efficiency and structural fidelity. Results: We present Physics‑Informed Mamba (PI‑Mamba), a generative model that enforces exact local covalent geometry by construction while enabling linear‑time inference. PI‑Mamba integrates a differentiable constraint‑enforcement operator into a flow‑matching framework and couples it with a Mamba‑based state‑space architecture. To improve optimisation stability and backbone realism, we introduce a spectral initialization derived from the Rouse polymer model and an auxiliary cis‑proline awareness head. Across benchmark tasks, PI‑Mamba achieves 0.0% local geometry violations and high designability (scTM = 0.91\pm 0.03, n = 100), while scaling to proteins exceeding 2,000 residues on a single A5000 GPU (24 GB).

Abstract:
Hyperspectral sensing provides rich spectral information for scene understanding in urban driving, but its high dimensionality poses challenges for interpretation and efficient learning. We introduce Learnable Quantum Efficiency (LQE), a physics‑inspired, interpretable dimensionality reduction (DR) method that parameterizes smooth high‑order spectral response functions that emulate plausible sensor quantum efficiency curves. Unlike conventional methods or unconstrained learnable layers, LQE enforces physically motivated constraints, including a single dominant peak, smooth responses, and bounded bandwidth. This formulation yields a compact spectral representation that preserves discriminative information while remaining fully differentiable and end‑to‑end trainable within semantic segmentation models (SSMs). We conduct systematic evaluations across three publicly available multi‑class hyperspectral urban driving datasets, comparing LQE against six conventional and seven learnable baseline DR methods across six SSMs. Averaged across all SSMs and configurations, LQE achieves the highest average mIoU, improving over conventional methods by 2.45%, 0.45%, and 1.04%, and over learnable methods by 1.18%, 1.56%, and 0.81% on HyKo, HSI‑Drive, and Hyperspectral City, respectively. LQE maintains strong parameter efficiency (12‑‑36 parameters compared to 51‑‑22K for competing learnable approaches) and competitive inference latency. Ablation studies show that low‑order configurations are optimal, while the learned spectral filters converge to dataset‑intrinsic wavelength patterns. These results demonstrate that physics‑informed spectral learning can improve both performance and interpretability, providing a principled bridge between hyperspectral perception and data‑driven multispectral sensor design for automotive vision systems.

Abstract:
In a previous work (Elaarabi et al., 2025b), the Sequence Encoder for online dynamical system identification (Elaarabi et al., 2025a) and its combination with PINN (PINN‑SE) were introduced and tested on both synthetic and real data case scenarios. The sequence encoder is able to effectively encode time series into feature vectors, which the PINN then uses to map to dynamical behavior, predicting system response under changes in parameters, ICs and BCs. Previously (Elaarabi et al., 2025b), the tests on real data were limited to simple 1D problems and only 1D time series inputs of the Sequence Encoder. In this work, the possibility of applying PINN‑SE to a more realistic case is investigated: heating and early cooling of the thermo‑stamping process, which is a critical stage in the forming process of continuous fiber reinforced composite materials with thermoplastic polymer. The possibility of extending the PINN‑SE inputs to multimodal data, such as sequences of temporal 2D images and to scenarios involving variable geometries, is also explored. The results show that combining multiple encoders with the previously proposed method (Elaarabi et al., 2025b) is feasible, we also show that training the model on synthetic data generated based on experimental data can help the model to generalize well for real experimental data, unseen during the training phase.

Abstract:
In practical structural design and solid mechanics simulations, material properties inherently exhibit random variations within bounded intervals. However, evaluating mechanical responses under continuous material uncertainty remains a persistent challenge. Traditional numerical approaches, such as the Finite Element Method (FEM), incur prohibitive computational costs as they require repeated mesh discretization and equation solving for every parametric realization. Similarly, data‑driven surrogate models depend heavily on massive, high‑fidelity datasets, while standard physics‑informed frameworks (e.g., the Deep Energy Method) strictly demand complete retraining from scratch whenever material parameters change. To bridge this critical gap, we propose the Constitutive Parameterized Deep Energy Method (CPDEM). In this purely physics‑driven framework, the strain energy density functional is reformulated by encoding a latent representation of stochastic constitutive parameters. By embedding material parameters directly into the neural network alongside spatial coordinates, CPDEM transforms conventional spatial collocation points into parameter‑aware material points. Trained in an unsupervised manner via expected energy minimization over the parameter domain, the pre‑trained model continuously learns the solution manifold. Consequently, it enables zero‑shot, real‑time inference of displacement fields for unknown material parameters without requiring any dataset generation or model retraining. The proposed method is rigorously validated across diverse benchmarks, including linear elasticity, finite‑strain hyperelasticity, and complex highly nonlinear contact mechanics. To the best of our knowledge, CPDEM represents the first purely physics‑driven deep learning paradigm capable of simultaneously and efficiently handling continuous multi‑parameter variations in solid mechanics.

Abstract:
Accurate risk assessment is essential for safety‑critical autonomous and control systems under uncertainty. In many real‑world settings, stochastic dynamics exhibit asymmetric jumps and long‑range memory, making long‑term risk probabilities difficult to estimate across varying system dynamics, initial conditions, and time horizons. Existing sampling‑based methods are computationally expensive due to repeated long‑horizon simulations to capture rare events, while existing partial differential equation (PDE)‑based formulations are largely limited to Gaussian or symmetric jump dynamics and typically treat memory effects in isolation. In this paper, we address these challenges by deriving a space‑ and time‑fractional PDE that characterizes long‑term safety and recovery probabilities for stochastic systems with both asymmetric Levy jumps and memory. This unified formulation captures nonlocal spatial effects and temporal memory within a single framework and enables the joint evaluation of risk across initial states and horizons. We show that the proposed PDE accurately characterizes long‑term risk and reveals behaviors that differ fundamentally from systems without jumps or memory and from standard non‑fractional PDEs. Building on this characterization, we further demonstrate how physics‑informed learning can efficiently solve the fractional PDEs, enabling accurate risk prediction across diverse configurations and strong generalization to out‑of‑distribution dynamics.

Abstract:
In this work, we present a multi‑fidelity, physics‑informed framework for tritium fuel cycle modelling based on the open‑source PathSim/PathView platform. Three complementary modelling approaches are demonstrated within a unified dynamic simulation environment. First, a zero‑dimensional residence time model is used to reproduce the fuel cycle behaviour of an ARC‑class fusion power plant, providing a baseline system‑level description. Second, an intermediate‑fidelity component model based on coupled one‑dimensional ordinary differential equations is developed to describe tritium mass transfer in a liquid metal bubble column reactor and validated against published literature before integration into the full fuel cycle. Finally, high‑fidelity multi‑dimensional tritium transport models implemented using the finite element code FESTIM are coupled directly to the system model, enabling the inclusion of multi‑dimensional effects, material interfaces, and complex transport phenomena. This work demonstrates how fuel cycle components of varying physical fidelity can be combined consistently within a single, open‑source framework. The proposed approach enables more physically grounded fuel cycle analyses while retaining the flexibility required for system‑level studies and provides a foundation for future integration with neutronics, fluid dynamics, and surrogate modelling tools.

Abstract:
This paper investigates propagation of SH‑waves in a layered composite structure consisting of a pre‑stressed functionally graded magnetoelastic orthotropic layer overlying a pre‑stressed functionally graded orthotropic half‑space under the influence of gravity. The study introduces a physics‑informed neural network (PINN) framework for the dispersion analysis of SH‑waves in the considered composite medium. As a benchmark, an analytical solution to the dispersion relation is derived and used to validate accuracy and reliability of the proposed PINN formulation. In the developed PINN model, the phase velocity corresponding to a prescribed wave number is treated as a trainable parameter, enabling the determination of the dispersion relation associated with the nonlinear eigenvalue problem. The Adam optimizer is employed to minimize the loss function during the training process. In addition, the effects of different activation functions and network architectures, including variations in number of hidden layers and neurons, are systematically investigated to study the performance of the proposed framework. Error analysis is carried out using several norms, namely L_1, L_2, RMSE, relative absolute error, and L_\infty, to assess the accuracy of the predictions. Furthermore, the variation of phase velocity with wave number under different material parameters is investigated. The comparison between the analytical and PINN‑based results demonstrates excellent agreement, confirming the effectiveness of the proposed deep learning approach for analysing dispersion relations in complex layered composite structures.

Abstract:
This paper proposes a physics‑informed learning framework for a class of recurrent neural networks tailored to large‑scale and networked systems. The approach aims to learn control‑oriented models that preserve the structural and stability properties of the plant. The learning algorithm is formulated as a convex optimisation problem, allowing the inclusion of linear matrix inequality constraints to enforce desired system features. Furthermore, when the plant exhibits structural modularity, the resulting optimisation problem can be parallelised, requiring communication only among neighbouring subsystems. Simulation results show the effectiveness of the proposed approach.

Abstract:
This paper proposes a physics‑informed neural operator (PINO) framework for solving inverse scattering problems, enabling rapid and accurate reconstructions under diverse measurement conditions. In the proposed approach, the dielectric property is represented as a learnable tensor, while a neural operator is employed to predict the induced current distribution. A hybrid loss function, consisting of the state loss, data loss and total‑variation (TV) regularization, is constructed to establish a fully differentiable formulation for a joint optimization of network parameters and dielectric property. To demonstrate the framework's generality and flexibility, PINO is implemented using three representative neural operators, i.e., the Fourier Neural Operator (FNO), the enhanced Fourier Neural Operator (U‑FNO) and the Factorized Fourier Neural Operator (F‑FNO). Compared with conventional approaches, the proposed framework offers a simpler formulation and universal modeling capability, making it readily applicable to various measurement scenarios, including multi‑frequency and phaseless inversion. Numerical simulations demonstrate that the proposed PINO achieves high accuracy and robust reconstruction across samples with and without phase information, under single‑frequency and multi‑frequency settings in the presence of noise. The results demonstrate that PINO consistently outperforms conventional contrast‑source inversion (CSI) methods and provides an efficient, unified solution to complex electromagnetic inverse‑scattering problems.

Abstract:
Multiphase flows frequently occur naturally and in manufactured devices. Controlling such phenomena is extremely challenging due to the strongly non‑linear dynamics, rapid phase transitions, and the limited spatial and temporal resolution of available sensors, which can lead to significant inaccuracies in predicting and managing these flows. In most cases, numerical models are the only way to access high spatial and temporal resolution data to an extent that allows for fine control. While embedding numerical models in control algorithms could enable fine control of multiphase processes, the significant computational burden currently limits their practical application. This work proposes a surrogate‑assisted model predictive control (MPC) framework for regulating multiphase processes using learned operators. A Fourier Neural Operator (FNO) is trained to forecast the spatiotemporal evolution of a phase‑indicator field (the volume fraction) over a finite horizon from a short history of recent states and a candidate actuation signal. The neural operator surrogate is then iteratively called during the optimisation process to identify the optimal control variable. To illustrate the approach, we solve an optimal control problem (OCP) on a two‑phase Eulerian bubble column. Here, the controller tracks piecewise‑constant liquid level setpoints by adjusting the gas flow rate introduced into the system. The results we obtained indicate that field‑level forecasting with FNOs are well suited for closed‑loop optimization since they have relatively low evaluation cost. The latter provide a practical route toward MPC for fast multiphase unit operations and a foundation for future extensions to partial observability and physics‑informed operator learning.

Abstract:
Physics‑informed neural networks (PINNs) have been proven as a promising way for solving various partial differential equations, especially high‑dimensional ones and those with irregular boundaries. However, their capabilities in real applications are highly restricted by their poor generalization performance. Inspired by the rigorous mathematical statements on the well‑posedness of PDEs, we develop a novel extension of PINNs by incorporating the additional information on the continuous dependence of PDE solutions with respect to parameters and initial/boundary values (abbreviated as cd‑PINN). Extensive numerical experiments demonstrate that, with limited labeled data, cd‑PINN achieves 1‑3 orders of magnitude lower in test MSE than DeepONet and FNO. Therefore, incorporating the continuous dependence of PDE solutions provides a simple way for qualifying PINNs for operator learning.

Abstract:
Vision‑Language‑Action (VLA) models such as π_0 have demonstrated remarkable generalization across diverse fixed‑base manipulators. However, transferring these foundation models to aerial platforms remains an open challenge due to the fundamental mismatch between the quasi‑static dynamics of fixed‑base arms and the underactuated, highly dynamic nature of flight. In this work, we introduce AirVLA, a system that investigates the transferability of manipulation‑pretrained VLAs to aerial pick‑and‑place tasks. We find that while visual representations transfer effectively, the specific control dynamics required for flight do not. To bridge this "dynamics gap" without retraining the foundation model, we introduce a Payload‑Aware Guidance mechanism that injects payload constraints directly into the policy's flow‑matching sampling process. To overcome data scarcity, we further utilize a Gaussian Splatting pipeline to synthesize navigation training data. We evaluate our method through a cumulative 460 real‑world experiments which demonstrate that this synthetic data is a key enabler of performance, unlocking 100% success in navigation tasks where directly fine‑tuning on teleoperation data alone attains 81% success. Our inference‑time intervention, Payload‑Aware Guidance, increases real‑world pick‑and‑place task success from 23% to 50%. Finally, we evaluate the model on a long‑horizon compositional task, achieving a 62% overall success rate. These results suggest that pre‑trained manipulation VLAs, with appropriate data augmentation and physics‑informed guidance, can transfer to aerial manipulation and navigation, as well as the composition of these tasks.

Abstract:
We propose Weak and Entropy PINNs (WE‑PINNs) for the approximation of entropy solutions to nonlinear hyperbolic conservation laws. Standard physics‑informed neural networks enforce governing equations in strong differential form, an approach that becomes structurally inconsistent in the presence of discontinuities due to the divergence of strong‑form residuals near shocks. The proposed method replaces pointwise residual minimization with a space‑‑time weak formulation derived from the divergence theorem. Conservation is enforced through boundary flux integrals over dynamically sampled space‑‑time control volumes, yielding a mesh‑free control‑volume framework that remains well‑defined for discontinuous solutions. Entropy admissibility is incorporated in integral form to ensure uniqueness and physical consistency of the weak solution. The resulting loss functional combines space‑‑time flux balance and entropy inequalities, without resorting to dual‑norm saddle‑point formulations, auxiliary potential networks, or fixed discretization meshes. This makes the proposed method remarkably easy to implement, requiring only a simple standard neural network architecture. We establish a rigorous convergence analysis linking the network's loss function to the L^1 error towards the entropy solution, providing the first explicit L^1 convergence rate for a mesh‑free control‑volume PINN formulation via the Bouchut‑Perthame framework for scalar conservation laws. Numerical experiments on the Burgers equation, the shallow water equations, and the compressible Euler equations demonstrate accurate shock resolution and robust performance in both smooth and shock‑dominated regimes.

Abstract:
Digital twin technology, when combined with physics‑informed machine learning with simulation results of Aspen, offers transformative capabilities for industrial process monitoring, control, and optimization. In this work, the proposed model presents a Physics‑Informed Neural Network (PINN) digital twin framework for the dynamic, tray‑wise modeling of binary distillation columns operating under transient conditions. The architecture of the proposed model embeds fundamental thermodynamic constraints, including vapor‑liquid equilibrium (VLE) described by modified Raoult's law, tray‑level mass and energy balances, and the McCabe‑Thiele graphical methodology directly into the neural network loss function via physics residual terms. The model is trained and evaluated on a high‑fidelity synthetic dataset of 961 timestamped measurements spanning 8 hours of transient operation, generated in Aspen HYSYS for a binary HX/TX distillation system comprising 16 sensor streams. An adaptive loss‑weighting scheme balances the data fidelity and physics consistency objectives during training. Compared to five data‑driven baselines (LSTM, vanilla MLP, GRU, Transformer, DeepONet), the proposed PINN achieves an RMSE of 0.00143 for HX mole fraction prediction (R^2 = 0.9887), representing a 44.6% reduction over the best data‑only baseline, while strictly satisfying thermodynamic constraints. Tray‑wise temperature and composition profiles predicted under transient perturbations demonstrate that the digital twin accurately captures column dynamics including feed tray responses, reflux ratio variations, and pressure transients. These results establish the proposed PINN digital twin as a robust foundation for real‑time soft sensing, model‑predictive control, and anomaly detection in industrial distillation processes.

Abstract:
Hourly rainfall extremes cause some of the most destructive weather disasters, yet numerical weather prediction models still struggle to forecast them, and a physical basis for their predictability remains unclear. Here, we identify a trivariate clustering of hourly rainfall extremes with surface temperature, phases of the Monsoon Intraseasonal Oscillation (MISO), and precipitable water vapor, establishing a physical foundation for the medium range predictability of these events. This clustering arises from multiscale interactions in which extremes organize into storm systems embedded within mesoscale convective clusters and synoptic low‑pressure systems during active MISO phases. We develop an algorithm to identify, track, and monitor these storm systems. Although rapid error growth limits the prediction of isolated hourly extremes, our results provide basis for a physics informed training of deep learning, data driven models to forecast organized clusters of hourly rainfall extremes more than a week in advance, offering substantial potential to reduce losses from extreme rainfall.

Abstract:
Safe navigation in complex environments remains a central challenge for reinforcement learning (RL) in robotics. This paper introduces Continuous Space‑Time Empowerment for Physics‑informed (C‑STEP) safe RL, a novel measure of agent‑centric safety tailored to deterministic, continuous domains. This measure can be used to design physics‑informed intrinsic rewards by augmenting positive navigation reward functions. The reward incorporates the agents internal states (e.g., initial velocity) and forward dynamics to differentiate safe from risky behavior. By integrating C‑STEP with navigation rewards, we obtain an intrinsic reward function that jointly optimizes task completion and collision avoidance. Numerical results demonstrate fewer collisions, reduced proximity to obstacles, and only marginal increases in travel time. Overall, C‑STEP offers an interpretable, physics‑informed approach to reward shaping in RL, contributing to safety for agentic mobile robotic systems.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a major research focus. However, today's PINNs encounter several limitations. Firstly, during the construction of the loss function using automatic differentiation, PINNs often neglect information from neighboring points, which hinders their ability to enforce physical constraints and diminishes their accuracy. Furthermore, issues such as instability and poor convergence persist during PINN training, limiting their applicability to complex fluid dynamics problems. To address these challenges, a fast physics‑informed neural network framework that integrates a simplified finite volume method (FVM) and residual correction loss term has been proposed, referred to as Fast Finite Volume PINN (FFV‑PINN). FFV‑PINN utilizes a simplified FVM discretization for the convection term, with an accompanying improvement in the dispersion and dissipation behavior. Unlike traditional FVM, the FFV‑PINN outputs can be simply and directly harnessed to approximate values on control surfaces, thereby simplifying the discretization process. Moreover, a residual correction loss term is introduced to significantly accelerates convergence and improves training efficiency. To validate the performance, we solve a series of challenging problems ‑‑ including flow in the two‑dimensional steady and unsteady lid‑driven cavity, three‑dimensional steady lid‑driven cavity, backward‑facing step flows, and natural convection at high Reynolds number and Rayleigh number. Notably, the FFV‑PINN can achieve data‑free solutions for the lid‑driven cavity flow at Re = 10000 and natural convection at Ra = 1e8 for the first time in PINN literature, even while requiring only 680s and 231s. It further highlights the effectiveness of FFV‑PINN in improving both speed and accuracy, marking another step forward in the progression of PINNs as competitive neural PDE solvers.

Abstract:
Near‑field beamfocusing with extremely large aperture arrays can effectively enhance physical layer security. Nevertheless, even small estimation errors of the eavesdropper's location may cause a pronounced focal shift, resulting in a severe degradation of the secrecy rate. In this letter, we propose a physics‑informed robust beamforming strategy that leverages the electromagnetic (EM) caustic effect for near‑field physical layer security provisioning, which can be implemented via phase shifts only. Specifically, we partition the transmit array into caustic and focusing subarrays to simultaneously bypass the potential eavesdropping region and illuminate the legitimate user, thereby significantly improving the robustness against the localization error of eavesdroppers. Moreover, by leveraging the connection between the phase gradient and the EM wave departing angle, we derive the corresponding piece‑wise closed‑form array phase profile for the subarrays. Simulation results demonstrate that the proposed scheme achieves up to an 80% reduction of the worst‑case eavesdropping rate for a localization error of 0.25 m, highlighting its superiority for providing robust and secure communication.

Abstract:
We present the Ising noise filter, a highly portable, graph‑based pre‑filtering algorithm for early‑stage background suppression in particle accelerators and astrophysical detectors. Standard noise rejection methods relying on track fitting suffer from severe combinatorial explosion. Our method bypasses this by mapping individual detector hits to a network of binary spins and minimizing an energy functional. The interaction kernels are physics‑informed, tailored to the underlying physics and geometry of the experiment. We demonstrate the efficacy of this approach in two distinct experimental regimes. Applied to the Baikal‑GVD neutrino telescope the filter yields fast, standard‑quality noise rejection with 96.8% recall for astrophysical neutrinos. For the SPD detector at the NICA collider the filter attains recall of 97% on a toy Monte Carlo sample. Furthermore, when combined with a Peterson‑‑Hopfield network for track finding, our physics‑informed coupling improves the TrackML score from 0.5 to 0.95.

Abstract:
Physics‑informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs) by directly embedding them into the loss function. Despite their notable success, existing PINNs often exhibit training instability and slow convergence when applied to strongly nonlinear fluid dynamics problems. To address these challenges, this paper proposes a novel PINN framework, named as SIMPLE‑PINN, which incorporates velocity and pressure correction loss terms inspired by the semi‑implicit pressure link equation. These correction terms, derived from the momentum and continuity residuals, are tailored for the PINN framework, ensuring velocity‑pressure coupling and reinforcing the underlying physical constraints of the Navier‑Stokes equations. Through this, the framework can effectively mitigate training instability and accelerate convergence to achieve accurate solution. Furthermore, a hybrid numerical‑automatic differentiation strategy is employed to improve the model's generalizability in resolving flows involving complex geometries. The performance of SIMPLE‑PINN is evaluated on a range of challenging benchmark cases, including strongly nonlinear flows, long‑term flow prediction, and multiphysics coupling problems. The numerical results demonstrate SIMPLE‑PINN's high accuracy and rapid convergence. Notably, SIMPLE‑PINN achieves, for the first time, a fully data‑free solution of lid‑driven cavity flow at Re=20000 in just 448s, and successfully captures the onset and long‑time evolution of vortex shedding in flow past a cylinder over t=0‑100. These findings demonstrate SIMPLE‑PINN's potential as a reliable and competitive neural solver for complex PDEs in intelligent scientific computing, with promising engineering applications in aerospace, civil engineering, and mechanical engineering.

Abstract:
Physics‑Informed Neural Networks (PINNs) for high‑dimensional and high‑order partial differential equations (PDEs) are primarily constrained by the \mathcalO(d^k) spatial derivative complexity and the \mathcalO(P) memory overhead of backpropagation (BP). While randomized spatial estimators successfully reduce the spatial complexity to \mathcalO(1), their reliance on first‑order optimization still leads to prohibitive memory consumption at scale. Zeroth‑order (ZO) optimization offers a BP‑free alternative; however, naively combining randomized spatial operators with ZO perturbations triggers a variance explosion of \mathcalO(1/\varepsilon^2), leading to numerical divergence. To address these challenges, we propose the Stochastic Dimension‑free Zeroth‑order Estimator (SDZE), a unified framework that achieves dimension‑independent complexity in both space and memory. Specifically, SDZE leverages \emphCommon Random Numbers Synchronization (CRNS) to algebraically cancel the \mathcalO(1/\varepsilon^2) variance by locking spatial random seeds across perturbations. Furthermore, an \emphimplicit matrix‑free subspace projection is introduced to reduce parameter exploration variance from \mathcalO(P) to \mathcalO(r) while maintaining an \mathcalO(1) optimizer memory footprint. Empirical results demonstrate that SDZE enables the training of 10‑million‑dimensional PINNs on a single NVIDIA A100 GPU, delivering significant improvements in speed and memory efficiency over state‑of‑the‑art baselines.

Abstract:
Symbolic discovery of governing equations is a long‑standing goal in scientific machine learning, yet a fundamental trade‑off persists between interpretability and scalable learning. Classical symbolic regression methods yield explicit analytic expressions but rely on combinatorial search, whereas neural networks scale efficiently with data and dimensionality but produce opaque representations. In this work, we introduce Symbolic Kolmogorov‑Arnold Networks (Symbolic‑KANs), a neural architecture that bridges this gap by embedding discrete symbolic structure directly within a trainable deep network. Symbolic‑KANs represent multivariate functions as compositions of learned univariate primitives applied to learned scalar projections, guided by a library of analytic primitives, hierarchical gating, and symbolic regularization that progressively sharpens continuous mixtures into one‑hot selections. After gated training and discretization, each active unit selects a single primitive and projection direction, yielding compact closed‑form expressions without post‑hoc symbolic fitting. Symbolic‑KANs further act as scalable primitive discovery mechanisms, identifying the most relevant analytic components that can subsequently inform candidate libraries for sparse equation‑learning methods. We demonstrate that Symbolic‑KAN reliably recovers correct primitive terms and governing structures in data‑driven regression and inverse dynamical systems. Moreover, the framework extends to forward and inverse physics‑informed learning of partial differential equations, producing accurate solutions directly from governing constraints while constructing compact symbolic representations whose selected primitives reflect the true analytical structure of the underlying equations. These results position Symbolic‑KAN as a step toward scalable, interpretable, and mechanistically grounded learning of governing laws.

Abstract:
Physics‑informed neural networks (PINNs) are increasingly used in mathematical epidemiology to bridge the gap between noisy clinical data and compartmental models, such as the susceptible‑exposed‑infected‑removed (SEIR) model. However, training these hybrid networks is often unstable due to competing optimization objectives. As established in recent literature on ``gradient pathology," the gradient vectors derived from the data loss and the physical residual often point in conflicting directions, leading to slow convergence or optimization deadlock. While existing methods attempt to resolve this by balancing gradient magnitudes or projecting conflicting vectors, we propose a novel method, conflict‑gated gradient scaling (CGGS), to address gradient conflicts in physics‑informed neural networks for epidemiological modelling, ensuring stable and efficient training and a computationally efficient alternative. This method utilizes the cosine similarity between the data and physics gradients to dynamically modulate the penalty weight. Unlike standard annealing schemes that only normalize scales, CGGS acts as a geometric gate: it suppresses the physical constraint when directional conflict is high, allowing the optimizer to prioritize data fidelity, and restores the constraint when gradients align. We prove that this gating mechanism preserves the standard O(1/T) convergence rate for smooth non‑convex objectives, a guarantee that fails under fixed‑weight or magnitude‑balanced training when gradients conflict. We demonstrate that this mechanism autonomously induces a curriculum learning effect, improving parameter estimation in stiff epidemiological systems compared to magnitude‑based baselines. Our empirical results show improved peak recovery and convergence over magnitude‑based methods.

Abstract:
In fluorescence microscopy, spectral unmixing aims to recover individual fluorophore concentrations from spectral images that capture mixed fluorophore emissions. Since classical methods operate pixel‑wise and rely on least‑squares fitting, their performance degrades with increasingly overlapping emission spectra and higher levels of noise, suggesting that a data‑driven approach that can learn and utilize a structural prior might lead to improved results. Learning‑based approaches for spectral imaging do exist, but they are either not optimized for microscopy data or are developed for very specific cases that are not applicable to fluorescence microscopy settings. To address this, we propose λSplit, a physics‑informed deep generative model that learns a conditional distribution over concentration maps using a hierarchical Variational Autoencoder. A fully differentiable Spectral Mixer enforces consistency with the image formation process, while the learned structural priors enable state‑of‑the‑art unmixing and implicit noise removal. We demonstrate λSplit on 3 real‑world datasets that we synthetically cast into a total of 66 challenging spectral unmixing benchmarks. We compare our results against a total of 10 baseline methods, including classical methods and a range of learning‑based methods. Our results consistently show competitive performance and improved robustness in high noise regimes, when spectra overlap considerably, or when the spectral dimensionality is lowered, making λSplit a new state‑of‑the‑art for spectral unmixing of fluorescent microscopy data. Importantly, λSplit is compatible with spectral data produced by standard confocal microscopes, enabling immediate adoption without specialized hardware modifications.

Abstract:
Efficient thermal management and precise field prediction are critical for the design of advanced energy systems, including electrohydrodynamic transport, microfluidic energy harvesters, and electrically driven thermal regulators. However, the steady‑state simulation of these electrothermal coupled multiphysics systems remains challenging for physics‑informed neural computation due to strong nonlinear field coupling, temperature‑dependent coefficient variability, and complex interface dynamics. This study proposes a Residual Attention Physics‑Informed Neural Network (RA‑PINN) framework for the unified solution of coupled velocity, pressure, electric‑potential, and temperature fields. By integrating a unified five‑field operator formulation with residual‑connected feature propagation and attention‑guided channel modulation, the proposed architecture effectively captures localized coupling structures and steep gradients. We evaluate RA‑PINN across four representative energy‑relevant benchmarks: constant‑coefficient coupling, indirect pressure‑gauge constraints, temperature‑dependent transport, and oblique‑interface consistency. Comparative analysis against Pure‑MLP, LSTM‑PINN, and pLSTM‑PINN demonstrates that RA‑PINN achieves superior accuracy, yielding the lowest MSE, RMSE, and relative L_2 errors across all scenarios. Notably, RA‑PINN maintains high structural fidelity in interface‑dominated and variable‑coefficient settings where conventional PINN backbones often fail. These results establish RA‑PINN as a robust and accurate computational framework for the high‑fidelity modeling and optimization of complex electrothermal multiphysics in sustainable energy applications.

Abstract:
This work establishes rigorous first‑of‑its‑kind upper bounds on the generalization error for the method of approximating solutions to the (d+1)‑dimensional incompressible Navier‑Stokes equations by training depth‑2 neural networks trained via the unsupervised Physics‑Informed Neural Network (PINN) framework. This is achieved by bounding the Rademacher complexity of the PINN risk. For appropriately weight bounded net classes our derived generalization bounds do not explicitly depend on the network width and our framework characterizes the generalization gap in terms of the fluid's kinematic viscosity and loss regularization parameters. In particular, the resulting sample complexity bounds are dimension‑independent. Our generalization bounds suggest using novel activation functions for solving fluid dynamics. We provide empirical validation of the suggested activation functions and the corresponding bounds on a PINN setup solving the Taylor‑Green vortex benchmark.

Abstract:
In complex engineering systems such as electro‑thermal‑fluid coupling, rapid and accurate prediction of multi‑physics fields is essential for advanced applications like digital twins and real‑time condition monitoring. Traditional numerical methods often suffer from high computational latency, whereas standard Physics‑Informed Neural Networks (PINNs) frequently fail to capture critical local features, such as irregular interfaces, localized high‑gradient regions, and multi‑peak transport structures. To address these limitations and provide high‑fidelity intelligent predictions for engineering decision‑making, this paper proposes a Residual‑Attention Physics‑Informed Neural Network (RA‑PINN) as a powerful surrogate modeling engine. The proposed method incorporates residual learning and attention enhancement into the network backbone to improve the representation of oblique transition structures, narrow charge layers, and distributed hotspots while strictly preserving global field consistency. To evaluate its effectiveness as an intelligent prediction framework, three representative benchmark cases are constructed, including an oblique asymmetric interface, a bipolar high‑gradient charge layer, and a multi‑peak Gaussian charge migration field. Under unified training settings, the proposed RA‑PINN is systematically compared with a standard pure PINN and an LSTM‑PINN in terms of average error, local maximum error, structural similarity, and convergence behavior. The results show that RA‑PINN consistently achieves the best overall performance across all benchmark cases, demonstrating its tremendous potential as a highly reliable core inference engine for the condition monitoring and digital twin modeling of complex multi‑physics engineering systems.

Abstract:
Simulation and modeling are essential in product development, integrated into the design and manufacturing process to enhance efficiency and quality. They are typically represented as complex nonlinear differential algebraic equations. The growing diversity of product requirements demands multi‑task optimization, a key challenge in simulation modeling research. A dual physics‑informed neural network architecture has been proposed to decouple constraints and objective functions in parametric differential algebraic equation optimization problems. Theoretical analysis shows that introducing a relaxation variable with a global error bound ensures solution equivalence between the network and optimization problem. A genetic algorithm‑enhanced training framework for physics‑informed neural networks improves training precision and efficiency, avoiding redundant solving of differential algebraic equations. This approach enables generalization for multi‑task objectives with a single, training maintaining real‑time responsiveness to product requirements.

Abstract:
In solving partial differential equations (PDEs), machine learning utilizing physical laws has received considerable attention owing to advantages such as mesh‑free solutions, unsupervised learning, and feasibility for solving high‑dimensional problems. An effective approach is based on physics‑informed neural networks (PINNs), which are based on deep neural networks known for their excellent performance in various academic and industrial applications. However, PINNs struggled with model training owing to significantly slow convergence because of a spectral bias problem. In this study, we propose a PINN‑based method equipped with a coordinate‑encoding layer on linear grid cells. The proposed method improves the training convergence speed by separating the local domains using grid cells. Moreover, it reduces the overall computational cost by using axis‑independent linear grid cells. The method also achieves efficient and stable model training by adequately interpolating the encoded coordinates between grid points using natural cubic splines, which guarantees continuous derivative functions of the model computed for the loss functions. The results of numerical experiments demonstrate the effective performance and efficient training convergence speed of the proposed method.

Abstract:
First‑order Ambisonics (FOA) is a standard spatial audio format based on spherical harmonic decomposition. Its zeroth‑ and first‑order components capture the sound pressure and particle velocity, respectively. Recently, physics‑informed neural networks have been applied to the spatial interpolation of FOA signals, regularizing the network outputs based on soft penalty terms derived from physical principles, e.g., the linearized momentum equation. In this paper, we reformulate the task so that the predicted FOA signal automatically satisfies the linearized momentum equation. Our network approximates a scalar function called velocity potential, rather than the FOA signal itself. Then, the FOA signal can be readily recovered through the partial derivatives of the velocity potential with respect to the network inputs (i.e., time and microphone position) according to physics of sound propagation. By deriving the four channels of FOA from the single‑channel velocity potential, the reconstructed signal follows the physical principle at any time and position by construction. Experimental results on room impulse response reconstruction confirm the effectiveness of the proposed framework.

Abstract:
Data assimilation (DA) for systems governed by partial differential equations (PDE) aims to reconstruct full spatiotemporal fields from sparse high‑fidelity (HF) observations while respecting physical constraints. While full‑grid low‑fidelity (LF) simulations provide informative priors in multi‑fidelity settings, recovering an HF field consistent with both sparse observations and the governing PDE typically requires per‑instance test‑time optimization, which becomes a major bottleneck in time‑critical applications. To alleviate this, amortized reconstruction using generative models has recently been proposed; however, such approaches rely on full‑field HF supervision during training, which is often impractical in real‑world settings. From a more realistic perspective, we propose the Physics‑Informed Conditional Schrödinger Bridge (PICSB), which transports an informative LF prior toward an observation‑conditioned HF posterior without any additional inference‑time guidance. To enable learning without HF endpoints, PICSB employs an iterative surrogate‑endpoint refresh scheme, and directly incorporates PDE residuals into the training objective while enforcing observations via hard conditioning throughout sampling. Experiments on fluid PDE benchmarks demonstrate that PICSB enables extremely fast spatiotemporal field reconstruction while maintaining competitive accuracy under sparse HF supervision.

Abstract:
Foundation models are now increasingly being developed for Earth observation (EO), yet they often rely on stochastic masking that do not explicitly enforce physics constraints; a critical trustworthiness limitation, in particular for predictive models that guide public health decisions. In this work, we propose SpecTM (Spectral Targeted Masking), a physics‑informed masking design that encourages the reconstruction of targeted bands from cross‑spectral context during pretraining. To achieve this, we developed an adaptable multi‑task (band reconstruction, bio‑optical index inference, and 8‑day‑ahead temporal prediction) self‑supervised learning (SSL) framework that encodes spectrally intrinsic representations via joint optimization, and evaluated it on a downstream microcystin concentration regression model using NASA PACE hyperspectral imagery over Lake Erie. SpecTM achieves R^2 = 0.695 (current week) and R^2 = 0.620 (8‑day‑ahead) predictions surpassing all baseline models by (+34% (0.51 Ridge) and +99% (SVR 0.31)) respectively. Our ablation experiments show targeted masking improves predictions by +0.037 R^2 over random masking. Furthermore, it outperforms strong baselines with 2.2x superior label efficiency under extreme scarcity. SpecTM enables physics‑informed representation learning across EO domains and improves the interpretability of foundation models.

Abstract:
Accurate power flow analysis is critical for modern distribution systems, yet classical solvers face scalability issues, and current machine learning models often struggle with generalization. We introduce BOOST‑RPF, a novel method that reformulates voltage prediction from a global graph regression task into a sequential path‑based learning problem. By decomposing radial networks into root‑to‑leaf paths, we leverage gradient‑boosted decision trees (XGBoost) to model local voltage‑drop regularities. We evaluate three architectural variants: Absolute Voltage, Parent Residual, and Physics‑Informed Residual. This approach aligns the model architecture with the recursive physics of power flow, ensuring size‑agnostic application and superior out‑of‑distribution robustness. Benchmarked against the Kerber Dorfnetz grid and the ENGAGE suite, BOOST‑RPF achieves state‑of‑the‑art results with its Parent Residual variant which consistently outperforms both analytical and neural baselines in standard accuracy and generalization tasks. While global Multi‑Layer Perceptrons (MLPs) and Graph Neural Networks (GNNs) often suffer from performance degradation under topological shifts, BOOST‑RPF maintains high precision across unseen feeders. Furthermore, the framework displays linear O(N) computational scaling and significantly increased sample efficiency through per‑edge supervision, offering a scalable and generalizable alternative for real‑time distribution system operator (DSO) applications.

Abstract:
We present a systematic method for exactly enforcing Dirichlet, Neumann, and Robin type conditions on general quadrilateral domains with arbitrary curved boundaries. Our method is built upon exact mappings between general quadrilateral domains and the standard domain, and employs a combination of TFC (theory of functional connections) constrained expressions and transfinite interpolations. When Neumann or Robin boundaries are present, especially when two Neumann (or Robin) boundaries meet at a vertex, it is critical to enforce exactly the induced compatibility constraints at the intersection, in order to enforce exactly the imposed conditions on the joining boundaries. We analyze in detail and present constructions for handling the imposed boundary conditions and the induced compatibility constraints for two types of situations: (i) when Neumann (or Robin) boundary only intersects with Dirichlet boundaries, and (ii) when two Neumann (or Robin) boundaries intersect with each other. We describe a four‑step procedure to systematically formulate the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on general quadrilateral domains. The method developed herein has been implemented together with the extreme learning machine (ELM) technique we have developed recently for scientific machine learning. Ample numerical experiments are presented with several linear/nonlinear stationary/dynamic problems on a variety of two‑dimensional domains with complex boundary geometries. Simulation results demonstrate that the proposed method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries exactly, with the numerical boundary‑condition errors at the machine accuracy.

Abstract:
We study the many‑body localization (MBL) transition in interacting fermionic systems on disordered one‑dimensional lattices using a physics‑informed machine‑learning framework. Instead of feeding full many‑body wave functions into the model, we construct a compact feature representation based on four physically motivated observables: the inverse participation ratio, the Shannon entropy, the many‑body hybridization parameter, and the mean level‑spacing ratio. These quantities capture complementary aspects of localization, entanglement, and spectral correlations, and are used to train a Kolmogorov‑‑Arnold Network (KAN) classifier on eigenstates deep in the weak and strong disorder regimes. The resulting KAN achieves a validation accuracy exceeding 99.9%, comparable to that of convolutional neural networks trained directly on high‑dimensional wave‑function data, while requiring substantially reduced input dimensionality and significantly shorter training time. Applying the trained classifier across the full energy spectrum yields energy‑resolved phase diagrams that reveal a clear many‑body mobility edge and provide a consistent estimate of the critical disorder strength. The approach is inherently extensible: additional physically relevant observables can be incorporated into the feature space in a systematic manner without altering the overall architecture. Our results demonstrate that feature‑based learning with KAN provides an efficient, scalable, and interpretable methodology for identifying many‑body localization transitions, offering a practical alternative to raw‑data‑based neural network approaches.

Abstract:
Energy efficiency remains a critical challenge in deploying physics‑informed operator learning models for computational mechanics and scientific computing, particularly in power‑constrained settings such as edge and embedded devices, where repeated operator evaluations in dense networks incur substantial computational and energy costs. To address this challenge, we introduce the Separable Physics‑informed Neuroscience‑inspired Operator Network (SPINONet), a neuroscience‑inspired framework that reduces redundant computation across repeated evaluations while remaining compatible with physics‑informed training. SPINONet incorporates regression‑friendly neuroscience‑inspired spiking neurons through an architecture‑aware design that enables sparse, event‑driven computation, improving energy efficiency while preserving the continuous, coordinate‑differentiable pathways required for computing spatio‑temporal derivatives. We evaluate SPINONet on a range of partial differential equations representative of computational mechanics problems, with spatial, temporal, and parametric dependencies in both time‑dependent and steady‑state settings, and demonstrate predictive performance comparable to conventional physics‑informed operator learning approaches despite the induced sparse communication. In addition, limited data supervision in a hybrid setup is shown to improve performance in challenging regimes where purely physics‑informed training may converge to spurious solutions. Finally, we provide an analytical discussion linking architectural components and design choices of SPINONet to reductions in computational load and energy consumption.

Abstract:
We address a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solution is sought in the function space spanned by physics‑informed random projection neural networks (PI‑RPNNs), and discretized via a collocation approach. These are single‑hidden‑layer networks with randomly sampled and fixed a priori hidden‑layer weights; only the linear output layer weights are optimized, reducing training to a single least‑squares solve. This linear output structure enables the direct and explicit formulation of the eigenvalue problem governing the linear stability of stationary solutions. This takes a generalized eigenvalue form, which naturally separates the physical domain interior dynamics from the algebraic constraints imposed by boundary conditions, at no additional training cost and without requiring additional PDE solves. However, the random projection collocation matrix is inherently numerically rank‑deficient, rendering naive eigenvalue computation unreliable and contaminating the true eigenvalue spectrum with spurious near‑zero modes. To overcome this limitation, we introduce a matrix‑free shift‑invert Krylov‑Arnoldi method that operates directly in weight space, avoiding explicit inversion of the numerically rank‑deficient collocation matrix and enabling the reliable computation of several leading eigenpairs of the physical Jacobian ‑ the discretized Frechet derivative of the PDE operator with respect to the solution field, whose eigenvalue spectrum determines linear stability. We further prove that the PI‑RPNN‑based generalized eigenvalue problem is almost surely regular, guaranteeing solvability with standard eigensolvers, and that the singular values of the random projection collocation matrix decay exponentially for analytic activation functions.

Abstract:
Stochastic human motion prediction aims to generate diverse, plausible futures from observed sequences. Despite advances in generative modeling, existing methods often produce predictions corrupted by high‑frequency jitter and temporal discontinuities. To address these challenges, we introduce KHMP, a novel framework featuring an adaptiveKalman filter applied in the DCT domain to generate high‑fidelity human motion predictions. By treating high‑frequency DCT coefficients as a frequency‑indexed noisy signal, the Kalman filter recursively suppresses noise while preserving motion details. Notably, its noise parameters are dynamically adjusted based on estimated Signal‑to‑Noise Ratio (SNR), enabling aggressive denoising for jittery predictions and conservative filtering for clean motions. This refinement is complemented by training‑time physical constraints (temporal smoothness and joint angle limits) that encode biomechanical principles into the generative model. Together, these innovations establish a new paradigm integrating adaptive signal processing with physics‑informed learning. Experiments on the Human3.6M and HumanEva‑I datasets demonstrate that KHMP achieves state‑of‑the‑art accuracy, effectively mitigating jitter artifacts to produce smooth and physically plausible motions.

Abstract:
Accurately resolving steady electrohydrodynamic (EHD) flows presents a formidable computational challenge due to the strong nonlinear coupling between charged‑particle density, velocity fields, and electric potential. These interactions frequently induce sharp transition layers, crossing fronts, and multiscale spatial structures, which notoriously degrade the predictive accuracy of standard mesh‑free solvers like Physics‑Informed Neural Networks (PINNs). To systematically address this bottleneck, we formulate a unified four‑variable operator framework and develop a comprehensive benchmark suite for two‑dimensional steady EHD shock‑like problems. The benchmark comprises eight rigorously designed cases featuring diverse front geometries, such as oblique, curved, and intersecting layers, alongside complex multiscale patterns. Under strictly identical configurations, including governing equations, source terms, sampling strategies, and loss formulations, we evaluate a Standard MLP‑based PINN, a Residual Attention PINN (ResAtt‑PINN), and an LSTM‑PINN that leverages pseudo‑sequential spatial encoding. Extensive numerical experiments demonstrate that the LSTM‑PINN consistently achieves the highest predictive accuracy across all eight cases. It successfully reconstructs sharp gradients and intricate multiscale structures where other architectures fail or over‑smooth. Furthermore, the LSTM backbone efficiently captures long‑range spatial correlations while maintaining an exceptionally low computational overhead and GPU memory footprint. These findings not only establish the LSTM‑PINN as a robust and efficient solver for strongly coupled PDEs with shock‑like features, but also provide the computational physics community with a standardized, reproducible benchmark for future algorithmic evaluations.

Abstract:
Designing urban spaces that provide pedestrian wind comfort and safety requires time‑resolved Computational Fluid Dynamics (CFD) simulations, but their current computational cost makes extensive design exploration impractical. We introduce WinDiNet (Wind Diffusion Network), a pretrained video diffusion model that is repurposed as a fast, differentiable surrogate for this task. Starting from LTX‑Video, a 2B‑parameter latent video transformer, we fine‑tune on 10,000 2D incompressible CFD simulations over procedurally generated building layouts. A systematic study of training regimes, conditioning mechanisms, and VAE adaptation strategies, including a physics‑informed decoder loss, identifies a configuration that outperforms purpose‑built neural PDE solvers. The resulting model generates full 112‑frame rollouts in under a second. As the surrogate is end‑to‑end differentiable, it doubles as a physics simulator for gradient‑based inverse optimization: given an urban footprint layout, we optimize building positions directly through backpropagation to improve wind safety as well as pedestrian wind comfort. Experiments on single‑ and multi‑inlet layouts show that the optimizer discovers effective layouts even under challenging multi‑objective configurations, with all improvements confirmed by ground‑truth CFD simulations.

Abstract:
DC‑DC boost converters require advanced control to ensure efficiency and stability under varying loads. Traditional model predictive control (MPC) and data‑driven neural network methods face challenges such as high complexity and limited physical constraint enforcement. This paper proposes a hybrid physics‑informed neural network (PINN) combined with finite control set MPC (FCS‑MPC) for boost converters. The PINN embeds physical laws into neural training, providing accurate state predictions, while FCS‑MPC ensures constraint satisfaction and multi‑objective optimization. The method features adaptive transient recovery, explicit duty‑ratio control, and enhanced dynamic stability. Experimental results on a commercial boost module demonstrate improved transient response, reduced voltage ripple, and robust operation across conduction modes. The proposed framework offers a computationally efficient, physically consistent solution for real‑time control in power electronics.

Abstract:
We present an approximate dynamic programming framework for designing degradation‑aware market participation policies for battery energy storage systems. The approach employs a tailored value function approximation that reduces the state space to state of charge and battery health, while performing dynamic programming along a pseudo‑time axis encoded by state of health. This formulation enables an offline/online computation split that separates long‑term degradation dynamics (months to years) from short‑term market dynamics (seconds to minutes) ‑‑ a timescale mismatch that renders conventional predictive control and dynamic programming approaches computationally intractable. The main computational effort occurs offline, where the value function is approximated via coarse‑grained backward induction along the health dimension. Online decisions then reduce to a real‑time tractable one‑step predictive control problem guided by the precomputed value function. This decoupling allows the integration of high‑fidelity physics‑informed degradation models without sacrificing real‑time feasibility. Backtests on historical market data show that the resulting policy outperforms several benchmark strategies with optimized hyperparameters.

Abstract:
Cascading failures in power grids pose severe risks to infrastructure reliability, yet real‑time prediction of their progression remains an open challenge. Physics‑based simulators require minutes to hours per scenario, while existing graph neural network approaches treat cascading failures as static classification tasks, ignoring temporal evolution and physical laws. This paper proposes Physics‑Informed Graph Neural Jump ODEs (PI‑GN‑JODE), combining an edge‑conditioned graph neural network encoder, a Neural ODE for continuous power redistribution, a jump process handler for discrete relay trips, and Kirchhoff‑based physics regularization. The model simultaneously predicts edge and node failure probabilities, severity classification, and demand not served, while an autoregressive extension enables round‑by‑round temporal cascade prediction. Evaluated on the IEEE 24‑bus and 118‑bus systems with 20,000 scenarios each, PI‑GN‑JODE achieves a Precision‑‑Recall Area Under the Curve of 0.991 for edge failure detection, 0.973 for node failure detection, and a coefficient of determination of 0.951 for demand‑not‑served regression on the 118‑bus system, outperforming a standard graph convolutional network baseline (0.948, 0.925, and 0.912, respectively). Ablation studies reveal that the four components function synergistically, with the physics‑informed loss alone contributing +9.2 points to demand‑not‑served regression. Performance improves when scaling to larger grids, and the architecture achieves the highest balanced accuracy (0.996) on the PowerGraph benchmark using data from a different simulation framework.

Abstract:
Neural networks have demonstrated significant potential in solving partial differential equations (PDEs). While global approaches such as Physics‑Informed Neural Networks (PINNs) offer promising capabilities, they often lack inherent temporal causality, which can limit their accuracy and stability for time‑dependent problems. In contrast, local training frameworks that progressively update network parameters over time are naturally suited for evolving PDEs. However, a critical challenge remains: many physical systems possess intrinsic invariants ‑‑ such as energy or mass ‑‑ that must be preserved to ensure physically meaningful solutions. This paper addresses this challenge by enhancing the Time‑Evolving Natural Gradient (TENG) method, a recently proposed local training framework. We introduce two complementary techniques: (i) a relaxation algorithm that ensures the target solution u_\texttarget preserves both quadratic and general nonlinear invariants of the original system, providing a structure‑preserving learning target; and (ii) a projection technique that maps the updated network parameters θ(t) back onto the invariant manifold, ensuring the final neural network solution strictly adheres to the conservation laws. Numerical experiments on the inviscid Burgers equation, Korteweg‑de Vries equation, and acoustic wave equation demonstrate that our proposed approach significantly improves conservation properties while maintaining high accuracy.

Abstract:
This paper proposes a computable state‑estimation error bound for learning‑based Kazantzis‑‑Kravaris/Luenberger (KKL) observers. Recent work learns the KKL transformation map with a physics‑informed neural network (PINN) and a corresponding left‑inverse map with a conventional neural network. However, no computable state‑estimation error bounds are currently available for this approach. We derive a state‑estimation error bound that depends only on quantities that can be certified over a prescribed region using neural network verification. We further extend the result to bounded additive measurement noise and demonstrate the guarantees on nonlinear benchmark systems.

Abstract:
This work presents a variational physics‑informed deep learning framework for phase‑field modelling of brittle crack propagation in anisotropic media. Previous Deep Ritz Method (DRM) approaches have focused on second‑order, isotropic phase‑field fracture formulations. In contrast, the present work introduces, for the first time within a variational deep learning setting, a family of higher‑order anisotropic phase‑field models through a generalised crack density functional. The resulting fracture problem is solved by minimising the total energy using the DRM. The trial space is enriched with higher‑order B‑spline basis functions to represent higher‑order gradients accurately and stably, thereby eliminating the need for conventional automatic differentiation. The methodology is assessed for isotropic, cubic, and orthotropic fracture surface energy densities. Numerical examples demonstrate direction‑dependent crack growth in anisotropic cases, highlighting the capability of the method to accurately capture this behaviour.

Abstract:
Machine‑learning electronic Hamiltonians achieve orders‑of‑magnitude speedups over density‑functional theory, yet current models omit long‑range Coulomb interactions that govern physics in polar crystals and heterostructures. We derive closed‑form long‑range Hamiltonian matrix elements in a nonorthogonal atomic‑orbital basis through variational decomposition of the electrostatic energy, deriving a variationally consistent mapping from the electron density matrix to effective atomic charges. We implement this framework in HamGNN‑LR, a dual‑channel architecture combining E(3)‑equivariant message passing with reciprocal‑space Ewald summation. Benchmarks demonstrate that physics‑based long‑range corrections are essential: purely data‑driven attention mechanisms fail to capture macroscopic electrostatic potentials. Benchmarks on polar ZnO slabs, CdSe/ZnS heterostructures, and GaN/AlN superlattices show two‑ to threefold error reductions and robust transferability to systems far beyond training sizes, eliminating the characteristic staircase artifacts that plague short‑range models in the presence of built‑in electric fields.

Abstract:
Quantitative high‑resolution transmission electron microscopy (HRTEM) provides an indispensable means to understand the structure‑property relationships of a material in atomic dimensions. Successful quantification requires reliable retrieval of essential atomic structural information despite artifacts arising from unwanted but practically unavoidable imaging imperfections. Experimental observation carried out in tandem with model‑based iterative image simulation shows vast applications in quantitative structural and chemical determination of objects spanning zero to three dimensions [Prog. Mater. Sci. 133, 101037, 2023]. However, the large number of parameters involved in the simulations make the current multi‑step, user‑guided iterative approach highly time consuming, thereby restricting its application primarily to small sample areas and to experienced users. In this work, we implement and apply a physics‑informed Bayesian optimization (BO) framework to advance HRTEM quantification towards full automation and large‑field‑of‑view analysis. Unlike conventional optimization approaches, our method adopts a stepwise strategy that fully leverages the strength of BO in handling high‑dimensional parameters, while its probabilistic engine rigorously and efficiently refines the parameter space to enable rapid quantification. Using a BaTiO3 single crystal that contains heavy, medium and light elements as a model system, we demonstrate that the three‑dimensional crystal structure can be determined from a single HRTEM image with a three to four order‑of‑magnitude improvement in time efficiency. This approach thus opens new avenues for fast and automated image quantification over larger sample volumes and, potentially, in the time domain.

Abstract:
With society entering the Internet era, the volume and speed of data and information have been increasing. Predicting the popularity of information cascades can help with high‑value information delivery and public opinion monitoring on the internet platforms. The current state‑of‑the‑art models for predicting information popularity utilize deep learning methods such as graph convolution networks (GCNs) and recurrent neural networks (RNNs) to capture early cascades and temporal features to predict their popularity increments. However, these previous methods mainly focus on the micro features of information cascades, neglecting their general macroscopic patterns. Furthermore, they also lack consideration of the impact of information heterogeneity on spread popularity. To overcome these limitations, we propose a physics‑informed neural network with adaptive clustering learning mechanism, PIACN, for predicting the popularity of information cascades. Our proposed model not only models the macroscopic patterns of information dissemination through physics‑informed approach for the first time but also considers the influence of information heterogeneity through an adaptive clustering learning mechanism. Extensive experimental results on three real‑world datasets demonstrate that our model significantly outperforms other state‑of‑the‑art methods in predicting information popularity.

Abstract:
Porous materials ‑‑ natural or engineered ‑‑ often exhibit dual pore‑network structures that govern processes such as mineral exploration and hydrocarbon recovery from tight shales. Double porosity/permeability (DPP) mathematical models describe incompressible fluid flow through two interacting pore networks with inter‑network mass exchange. Despite significant advances in numerical methods, there remains a need for computational frameworks that enable rapid forecasting, data assimilation, and reliable inverse analysis. To address this, we present a physics‑informed neural network (PINN) framework for forward and inverse modeling of DPP systems. The proposed approach encodes the governing equations in mixed form, along with boundary conditions, directly into the loss function, with adaptive weighting strategies to balance their contributions. Key features of the framework include adaptive weight tuning, dynamic collocation point selection, and the use of shared trunk neural architectures to efficiently capture the coupled behavior of the dual pore networks. It is inherently mesh‑free, making it well‑suited for complex geometries typical of porous media. It accurately captures discontinuities in solution fields across layered domains without introducing spurious oscillations commonly observed in classical finite element formulations. Importantly, the framework is well‑suited for inverse analysis, enabling robust parameter identification in scenarios where key physical quantities ‑‑ such as the mass transfer coefficient in DPP models ‑‑ are difficult to measure directly. In addition, a systematic convergence analysis is provided to rigorously assess the stability, accuracy, and reliability of the method. The effectiveness and computational advantages of the approach are demonstrated through a series of representative numerical experiments.

Abstract:
Many core problems in nonlinear systems analysis and control can be recast as solving partial differential equations (PDEs) such as Lyapunov and Hamilton‑Jacobi‑Bellman (HJB) equations. Physics‑informed neural networks (PINNs) have emerged as a promising mesh‑free approach for approximating their solutions, but in most existing works there is no rigorous guarantee that a small PDE residual implies a small solution error. This paper develops verifiable error bounds for approximate solutions of Lyapunov and HJB equations, with particular emphasis on PINN‑based approximations. For both the Lyapunov and HJB PDEs, we show that a verifiable residual bound yields relative error bounds with respect to the true solutions as well as computable a posteriori estimates in terms of the approximate solutions. For the HJB equation, this also yields certified upper and lower bounds on the optimal value function on compact sublevel sets and quantifies the optimality gap of the induced feedback policy. We further show that one‑sided residual bounds already imply that the approximation itself defines a valid Lyapunov or control Lyapunov function. We illustrate the results with numerical examples.

Abstract:
Bacterial cancer therapy exploits anaerobic bacteria's ability to target hypoxia tumor regions, yet the interactions among tumor growth, bacterial colonization, oxygen levels, immunosuppressive cytokines, and bacterial communication remain poorly quantified. We present a mathematical model of five coupled nonlinear reaction‑diffusion equations in a two‑dimensional tissue domain. We proved the global well‑posedness of the model and identified its steady states to analyze stability. Furthermore, a physics‑informed neural network (PINN) solves the system without a mesh and without requiring extensive data. It provides convergence guarantees by combining residual stability and Sobolev approximation error bounds. This results in an overall error rate of O(n^‑2 ln^4(n) + N^‑1/2), which depends on the network width n and the number of collocation points N. We conducted several numerical experiments, including predicting the tumor's response to therapy. We also performed a sensitivity analysis of certain parameters. The results suggest that long‑term therapeutic efficacy may require the maintenance of hypoxia regions in the tumor, or using bacteria that tolerate oxygen better, may be necessary for long‑lasting tumor control.

Abstract:
The discovery of constitutive laws for complex materials has historically faced a dichotomy between high‑fidelity data‑driven approaches, which demand prohibitive full‑field experimental data, and traditional engineering fitting, which often yields numerically unstable models outside calibration regimes. In this work, we propose an Engineering‑Oriented Symbolic Regression (EO‑SR) framework that bridges this gap by leveraging Large Language Models (LLMs) as "Physics‑Informed Agents." Unlike unconstrained symbolic regression, our framework utilizes an LLM Agent to zero‑shot synthesize executable physical constraints ‑‑ specifically thermodynamic consistency and frame indifference ‑‑ transforming the search process from mathematical curve‑fitting into a physics‑governed discovery engine. We validate this approach on the hyperelastic modeling of rubber‑like materials using standard Treloar datasets. The framework autonomously identifies a novel hybrid constitutive law that combines a Mooney‑Rivlin linear base with a rational locking term. This discovered model not only achieves high predictive accuracy across multi‑axial deformation modes (including zero‑shot prediction of pure shear) but also guarantees unconditional convexity. Finite element validation demonstrates that while industry‑standard models (e.g., Ogden N=3) fail due to numerical singularities under severe transverse compression, the EO‑SR‑discovered model maintains robust convergence. This study establishes a generalized, low‑barrier pathway for discovering simulation‑ready constitutive closures that satisfy both data accuracy and rigorous physical laws.

Abstract:
Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics‑informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution‑space error. We prove that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.

Abstract:
Traditional weak gravitational lensing shear estimators are carefully calibrated but struggle to fully capture realistic galaxy morphologies, point‑spread‑function (PSF) effects, blending, and noise in deep surveys, while blindly trained machine learning (ML) models can introduce significant calibration biases. Here we construct a fully D_4‑equivariant deep neural network for galaxy shape measurement whose architecture enforces symmetry under 90^\circ rotations and mirror transformations, and adopt the Analytical Calibration framework (AnaCal) to calibrate the model using its backpropagated gradients. For isolated galaxies in LSST‑like single‑band simulations, we demonstrate that our approach achieves ～10% lower shape noise than the traditional moment‑based Fourier Power Function Shapelets estimator in the high‑noise regime, equivalent to a ～20% gain in effective galaxy number density, while simultaneously achieving multiplicative biases consistent with zero across a wide range of noise levels, PSF sizes and ellipticities, and magnitude selection cuts, with all measurements satisfying |m| < 10^‑3 (i.e., within the 0.2% LSST requirement) and most at the ～10^‑4 level. We demonstrate this framework on isolated single‑band galaxy images with Gaussian noise and known PSF, establishing a rigorous, physics‑informed foundation for future extensions of ML‑based shear estimation to blended sources and multi‑band observations in Stage‑IV surveys. All codes and data products will be made publicly available upon acceptance.

Abstract:
With the advent of system‑in‑package (SiP) chiplet‑based design and heterogeneous 2.5D/3D integration, thermal‑induced warpage has become a critical reliability concern. While conventional numerical approaches can deliver highly accurate results, they often incur prohib‑ itively high computational costs, limiting their scalability for complex chiplet‑package systems. In this paper, we present WarPGNN, an ef‑ ficient and accurate parametric thermal warpage analysis framework powered by Graph Neural Networks (GNNs). By operating directly on graphs constructed from the floorplans, WarPGNN enables fast warpage‑aware floorplan exploration and exhibits strong transfer‑ ability across diverse package configurations. Our method first en‑ codes multi‑die floorplans into reduced Transitive Closure Graphs (rTCGs), then a Graph Convolution Network (GCN)‑based encoder extracts hierarchical structural features, followed by a U‑Net inspired decoder that reconstructs warpage maps from graph feature embed‑ dings. Furthermore, to address the long‑tailed pattern of warpage data distribution, we developed a physics‑informed loss and revised a message‑passing encoder based on Graph Isomorphic Network (GIN) that further enhance learning performance for extreme cases and expressiveness of graph embeddings. Numerical results show that WarPGNN achieves more than 205.91x speedup compared with the 2‑D efficient FEM‑based method and over 119766.64x acceleration with 3‑D FEM method COMSOL, respectively, while maintaining comparable accuracy at only 1.26% full‑scale normalized RMSE and 2.21% warpage value error. Compared with recent DeepONet‑based model, our method achieved comparable prediction accuracy and in‑ ference speedup with 3.4x lower training time. In addition, WarPGNN demonstrates remarkable transferability on unseen datasets with up to 3.69% normalized RMSE and similar runtime.

Abstract:
The information recoverable from galaxy spectra depends fundamentally on spectral resolution, yet assembling large samples at high resolution remains observationally expensive. We present a deep‑learning framework for spectral super‑resolution that enhances low‑resolution galaxy spectra by a factor of ～10 in resolving power (R～100 to R～1000). The model is trained on 1,187 paired JWST/NIRSpec observations from the JADES program, where low‑resolution prism spectra are matched with medium‑resolution grating spectra (G140M, G235M, G395M) combined into a unified reference covering 1‑5 μm. Our three‑stage architecture performs an initial super‑resolution, infers the redshift from the coarse reconstruction, and then applies a physics‑informed residual refinement that uses attention across emission‑line tokens to learn inter‑line relationships and predict parametric line profiles, alongside a convolutional branch for continuum corrections. Evaluated on a 20% held‑out sample, the model achieves noise‑limited residuals over most of the spectral range and systematically improves the signal‑to‑noise ratio of key diagnostic lines including [OII], Hβ, [OIII], and Hα, often by factors of several. The super‑resolved spectra successfully deblend features that are entirely unresolved at prism resolution, such as the [OIII] λ\lambda4959,5007 doublet and Hβ. As a proof of concept using JWST data, this approach is readily extensible to the low‑resolution grism spectroscopy that will be delivered by Euclid and the Roman Space Telescope, potentially enabling population‑level diagnostics across millions of galaxy spectra that would otherwise be inaccessible at grism resolution.

Abstract:
Physics‑Informed Neural Networks(PINNs) are a powerful and flexible learning framework that has gained significant attention in recent years. It has demonstrated strong performance across a wide range of scientific and engineering problems. In parallel, wavelets have been extensively used as efficient computational tools due to their strong approximation capabilities. Motivated by the common failure modes observed in standard PINNs, this work introduces a novel family of adaptive wavelet‑based activation functions. The proposed activation functions significantly improve training stability and expressive power by combining trainable wavelet functions with either trainable or fixed hyperbolic tangent and softplus functions. Five distinct activation functions are developed within the PINN framework and systematically evaluated across four representative classes of partial differential equations (PDEs). Comprehensive comparisons using bar plots demonstrate improved robustness and accuracy compared to traditional activation functions. Furthermore, the proposed approach is validated through direct comparisons with baseline PINNs, transformer‑based architectures such as PINNsFormer, and other deep learning models, highlighting its effectiveness and generality.

Abstract:
Quantum kernels offer a valid procedure for learning quantum phase transitions on quantum processing devices, yet issues on the scalability of the learning strategy in connection with the symmetry of the critical model have not been clarified. We derive a link between model symmetry and fidelity‑kernel resource scaling. We quantify the measurement resources required to estimate fidelity‑based quantum kernels for many‑body ground states while preserving the structure of the resulting Gram matrix under finite‑shot sampling. Crucially, we show that increasing symmetry in the underlying spin model systematically amplifies these shot requirements. Moving from the \mathbbZ_2‑symmetric Ising/XY regimes to the U(1)‑symmetric XX (and XXZ) regimes leads to stronger kernel concentration and therefore substantially larger shot costs under the same bounds. We consider a tunable one‑dimensional spin‑\tfrac12 Hamiltonian spanning the transverse‑field Ising, XY, XX, and XXZ limits, and define the kernel as the ground‑state fidelity. Kernel entries are estimated using a SWAP‑test estimator with S shots, and we adapt the ensemble spread and concentration‑avoidance shot bounds to obtain practical shot requirements in terms of the interquartile range of kernel values and a representative kernel magnitude. For the free‑fermion XY/XX family, we use the closed‑form Bogoliubov‑angle fidelity, while for the interacting XXZ chain we compute fidelities by exact diagonalization and benchmark shot‑noise effects. Our symmetry‑aware bounds provide a pragmatic procedure for physics‑informed quantum machine learning.

Abstract:
Deep learning models in quantitative finance often operate as black boxes, lacking interpretability and failing to incorporate fundamental economic principles such as no‑arbitrage constraints. This paper introduces ARTEMIS (Arbitrage‑free Representation Through Economic Models and Interpretable Symbolics), a novel neuro‑symbolic framework combining a continuous‑time Laplace Neural Operator encoder, a neural stochastic differential equation regularised by physics‑informed losses, and a differentiable symbolic bottleneck that distils interpretable trading rules. The model enforces economic plausibility via two novel regularisation terms: a Feynman‑Kac PDE residual penalising local no‑arbitrage violations, and a market price of risk penalty bounding the instantaneous Sharpe ratio. We evaluate ARTEMIS against six strong baselines on four datasets: Jane Street, Optiver, Time‑IMM, and DSLOB (a synthetic crash regime). Results demonstrate ARTEMIS achieves state‑of‑the‑art directional accuracy, outperforming all baselines on DSLOB (64.96%) and Time‑IMM (96.0%). A comprehensive ablation study confirms each component's contribution: removing the PDE loss reduces directional accuracy from 64.89% to 50.32%. Underperformance on Optiver is attributed to its long sequence length and volatility‑focused target. By providing interpretable, economically grounded predictions, ARTEMIS bridges the gap between deep learning's power and the transparency demanded in quantitative finance.

Abstract:
Choosing how much noise to add in Langevin dynamics is essential for making these algorithms effective in challenging optimization problems. One promising approach is to determine this noise by solving Hamilton‑Jacobi‑Bellman (HJB) equations and their exploratory variants. Though these ideas have been demonstrated to work well in one dimension, extension to high‑dimensional minimization has been limited by two unresolved numerical challenges: setting reliable control bounds and stably computing the second‑order information (Hessians) required by the equations. These issues and the broader impact of HJB parameters have not been systematically examined. This work provides the first such investigation. We introduce principled control bounds and develop a physics‑informed neural network framework that embeds the structure of exploratory HJB equations directly into training, stabilizing computation, and enabling accurate estimation of state‑dependent noise in high‑dimensional problems. Numerical experiments demonstrate that the resulting method remains robust and effective well beyond low‑dimensional test cases.

Abstract:
Physics‑informed machine learning surrogates are increasingly explored to accelerate dynamic simulation of generators, converters, and other power grid components. The key question, however, is not only whether a surrogate matches a stand‑alone component model on average, but whether it remains accurate after insertion into a differential‑algebraic simulator, where the surrogate outputs enter the algebraic equations coupling the component to the rest of the system. This paper formulates that in‑simulator use as a verification and validation (V\&V) problem. A finite‑horizon bound is derived that links allowable component‑output error to algebraic‑coupling sensitivity, dynamic error amplification, and the simulation horizon. Two complementary settings are then studied: model‑based verification against a reference component solver, and data‑based validation through conformal calibration of the component‑output variables exchanged with the simulator. The framework is general, but the case study focuses on physics‑informed neural‑network surrogates of second‑, fourth‑, and sixth‑order synchronous‑machine models. Results show that good stand‑alone surrogate accuracy does not by itself guarantee accurate in‑simulator behavior, that the largest discrepancies concentrate in stressed operating regions, and that small equation residuals do not necessarily imply small state‑trajectory errors.

Abstract:
Tensegrity structures possess intrinsic geometric symmetries that govern their dynamic behavior. However, most existing physics‑informed neural network (PINN) approaches for tensegrity dynamics do not explicitly exploit these symmetries, leading to high computational complexity and unstable optimization. In this work, we propose a symmetry‑reduced physics‑informed neural network (SymPINN) framework that embeds group‑theory‑based symmetry directly into both the solution expression and the neural network architecture to predict tensegrity dynamics. By decomposing nodes into symmetry orbits and representing free nodal coordinates using a symmetry basis, the proposed method constructs a reduced coordinate representation that preserves geometric symmetry of the structure. The full coordinates are then recovered via symmetry transformations of the reduced solution learned by the network, ensuring that the predicted configurations automatically satisfy the symmetry constraints. In this framework, equivariance is enforced through orbit‑based coordinate generation, symmetry‑consistent message passing, and physics residual constraints. In addition, SymPINN improves training effectiveness by encoding initial conditions as hard constraints, incorporating Fourier feature encoding to enhance the representation of dynamic motions, and employing a two‑stage optimization strategy. Extensive numerical experiments on symmetric T‑bars and lander structures demonstrate significantly improved prediction accuracy and computational efficiency compared to standard physics‑informed models, indicating the great potential of symmetry‑aware learning for structure‑preserving modeling of tensegrity dynamics.

Abstract:
Contactless fingerprint recognition enables hygienic and convenient biometric authentication but poses new challenges for spoof detection due to the absence of physical contact and traditional liveness cues. Most existing methods rely on single‑image acquisition and appearance‑based features, which often generalize poorly across devices, capture conditions, and spoof materials. In this work, we study paired flash‑non‑flash contactless fingerprint acquisition as a lightweight active sensing mechanism for spoof detection. Through a preliminary empirical analysis, we show that flash illumination accentuates material‑ and structure‑dependent properties, including ridge visibility, subsurface scattering, micro‑geometry, and surface oils, while non‑flash images provide a baseline appearance context. We analyze lighting‑induced differences using interpretable metrics such as inter‑channel correlation, specular reflection characteristics, texture realism, and differential imaging. These complementary features help discriminate genuine fingerprints from printed, digital, and molded presentation attacks. We further examine the limitations of paired acquisition, including sensitivity to imaging settings, dataset scale, and emerging high‑fidelity spoofs. Our findings demonstrate the potential of illumination‑aware analysis to improve robustness and interpretability in contactless fingerprint presentation attack detection, motivating future work on paired acquisition and physics‑informed feature design. Code is available in the repository.

Abstract:
Accurate prediction of permeability tensors from pore‑scale microstructure images is essential for subsurface flow modeling, yet direct numerical simulation requires hours per sample, fundamentally limiting large‑scale uncertainty quantification and reservoir optimization workflows. A physics‑informed deep learning framework is presented that resolves this bottleneck by combining a MaxViT hybrid CNN‑Transformer architecture with progressive transfer learning and differentiable physical constraints. MaxViT's multi‑axis attention mechanism simultaneously resolves grain‑scale pore‑throat geometry via block‑local operations and REV‑scale connectivity statistics through grid‑global operations, providing the spatial hierarchy that permeability tensor prediction physically requires. Training on 20000 synthetic porous media samples spanning three orders of magnitude in permeability, a three‑phase progressive curriculum advances from an ImageNet‑pretrained baseline with D4‑equivariant augmentation and tensor transformation, through component‑weighted loss prioritizing off‑diagonal coupling, to frozen‑backbone transfer learning with porosity conditioning via Feature‑wise Linear Modulation (FiLM). Onsager reciprocity and positive definiteness are enforced via differentiable penalty terms. On a held‑out test set of 4000 samples, the framework achieves variance‑weighted R2 = 0.9960 (R2_Kxx = 0.9967, R2_Kxy = 0.9758), a 33% reduction in unexplained variance over the supervised baseline. The results offer three transferable principles for physics‑informed scientific machine learning: large‑scale visual pretraining transfers effectively across domain boundaries; physical constraints are most robustly integrated as differentiable architectural components; and progressive training guided by diagnostic failure‑mode analysis enables unambiguous attribution of performance gains across methodological stages.

Authors: Zhehui Wang, Justin C. Burton, Niklas Dormagen, Cheng-Ran Du, Yan Feng, John E. Foster, Susan S. Glenn, Max Klein, Christina A. Knapek, Lorin Matthews, André Melzer, Edward Thomas, Chuji Wang, Jalaan Avritte, Shan Chang, Neeraj Chaubey, Pubuduni Ekanayaka, John A. Goree, Truell Hyde, Chen Liang, Zhuang Liu, Zhuang Ma, Ilya Nemenman, Elon Price, A. S. Schmitz, Mike Schwarz, Saikat C. Thakur, M. H. Thoma, Hubertus M. Thomas, L. Wimmer, Wei Yang, Zimu Yang, Xiaoman Zhang

Abstract:
Dusty plasmas are ubiquitous throughout the universe, spanning laboratory and industrial plasmas, fusion devices, planetary environments, cometary comae, and interstellar media. Despite decades of research, many aspects of their behavior remain poorly understood within a unified framework. While numerous theoretical and numerical models describe specific phenomena, such as dust charging, transport, waves, and self‑organization, fully predictive models across the wide range of spatial and temporal scales in both laboratory and natural systems remain elusive. Conventional plasma descriptions rely on coupled differential equations for particle densities, momenta, and energies, but their solutions are often limited by computational cost, numerical uncertainties, and incomplete knowledge of boundary conditions and transport processes. Recent advances in machine learning (ML), particularly deep neural networks, offer new opportunities to complement traditional physics‑based modeling. Here we review ML and artificial intelligence (AI) approaches, termed bottom‑up data‑driven methods, for dusty plasma research. Central to this effort is Dust Neural nEtworks Technology (DustNET), a community‑driven dataset initiative inspired by ImageNet, integrating experimental, simulation, and synthetic data to enable predictive modeling, uncertainty quantification, and multi‑scale analysis. DustNET‑trained models may also be deployed in real‑time experimental settings under edge computing constraints. Combined with emerging multi‑modal AI foundation models and autonomous agents, this framework provides a pathway toward a unified, physics‑informed understanding of dusty plasmas across laboratory, industrial, space, and astrophysical environments.

Abstract:
Advanced autonomous driving systems require accurate vehicle dynamics modeling. However, identifying a precise dynamics model remains challenging due to strong nonlinearities and the coupled longitudinal and lateral dynamic characteristics. Previous research has employed physics‑based analytical models or neural networks to construct vehicle dynamics representations. Nevertheless, these approaches often struggle to simultaneously achieve satisfactory performance in terms of system identification efficiency, modeling accuracy, and compatibility with linear control strategies. In this paper, we propose a fully data‑driven dynamics modeling method tailored for complex distributed electric‑drive trucks (DETs), leveraging Koopman operator theory to represent highly nonlinear dynamics in a lifted linear embedding space. To achieve high‑precision modeling, we first propose a novel dual‑branch encoder which encodes dynamic states and provides a powerful basis for the proposed Koopman‑based methods entitled KODE. A physics‑informed supervision mechanism, grounded in the geometric consistency of temporal vehicle motion, is incorporated into the training process to facilitate effective learning of both the encoder and the Koopman operator. Furthermore, to accommodate the diverse driving patterns of DETs, we extend the vanilla Koopman operator to a mixture‑of‑Koopman operator framework, enhancing modeling capability. Simulations conducted in a high‑fidelity TruckSim environment and real‑world experiments demonstrate that the proposed approach achieves state‑of‑the‑art performance in long‑term dynamics state estimation.

Abstract:
Urban flood emergency response increasingly relies on infrastructure impact forecasts rather than hazard variables alone. However, real‑time predictions are unreliable due to biased rainfall, incomplete flood knowledge, and sparse observations. Conventional open‑loop forecasting propagates impacts without adjusting the system state, causing errors during critical decisions. This study presents CRAF (Crowdsourcing‑Enhanced Real‑Time Awareness and Forecasting), a physics‑informed, closed‑loop framework that converts sparse human‑sensed evidence into rolling, decision‑grade impact forecasts. By coupling physics‑based simulation learning with crowdsourced observations, CRAF infers system conditions from incomplete data and propagates them forward to produce multi‑step, real‑time predictions of zone‑level building functionality loss without online retraining. This closed‑loop design supports continuous state correction and forward prediction under weakly structured data with low‑latency operation. Offline evaluation demonstrates stable generalization across diverse storm scenarios. In operational deployment during Typhoon Haikui (2023) in Fuzhou, China, CRAF reduces 1‑3 hour‑ahead forecast errors by 84‑95% relative to fixed rainfall‑driven forecasting and by 73‑80% relative to updated rainfall‑driven forecasting, while limiting computation to 10 minutes per update cycle. These results show that impact‑state alignment‑rather than hazard refinement alone‑is essential for reliable real‑time decision support, providing a pathway toward operational digital twins for resilient urban infrastructure systems.

Abstract:
International shipping produces approximately 3% of global greenhouse gas emissions, yet voyage routing remains dominated by heuristic methods. We present PIER (Physics‑Informed, Energy‑efficient, Risk‑aware routing), an offline reinforcement learning framework that learns fuel‑efficient, safety‑aware routing policies from physics‑calibrated environments grounded in historical vessel tracking data and ocean reanalysis products, requiring no online simulator. Validated on one full year (2023) of AIS data across seven Gulf of Mexico routes (840 episodes per method), PIER reduces mean CO2 emissions by 10% relative to great‑circle routing. However, PIER's primary contribution is eliminating catastrophic fuel waste: great‑circle routing incurs extreme fuel consumption (>1.5x median) in 4.8% of voyages; PIER reduces this to 0.5%, a 9‑fold reduction. Per‑voyage fuel variance is 3.5x lower (p<0.001), with bootstrap 95% CI for mean savings [2.9%, 15.7%]. Partial validation against observed AIS vessel behavior confirms consistency with the fastest real transits while exhibiting 23.1x lower variance. Crucially, PIER is forecast‑independent: unlike A path optimization whose wave protection degrades 4.5x under realistic forecast uncertainty, PIER maintains constant performance using only local observations. The framework combines physics‑informed state construction, demonstration‑augmented offline data, and a decoupled post‑hoc safety shield, an architecture that transfers to wildfire evacuation, aircraft trajectory optimization, and autonomous navigation in unmapped terrain.

Abstract:
We apply two variants of Physics‑Informed Neural Networks (PINNs) to reconstruct the Type~Ia supernova absolute magnitude M_B(z) from joint BAO and supernova data under four cosmological models (ΛCDM, CPL, GEDE, Λ_sCDM) and two DESI~DR2 fiducial sets. A heteroscedastic single‑network method tested across four constraint configurations establishes that the Etherington distance duality relation is a more fundamental constraint than cosmological model priors, reducing internal inconsistencies by up to an order of magnitude. Under full constraints all models recover M_B \approx ‑19.3~mag with biases below 0.05~mag. A Fisher information‑weighted two‑network variant trains independent networks on BAO and SN data, providing clean probe separation; it finds no significant pointwise M_B evolution in z \in [0.3, 1.5], but reveals a systematic separation of redshift‑binned M_B distributions. The heteroscedastic method identifies a persistent 2‑‑3σ residual at z ～ 0.4‑‑0.5 that is consistent across all four models and both fiducials, implying the same underlying tension. While the origin of this feature remains ambiguous, its model‑independence and cross‑method consistency warrant further investigation with forthcoming data.

Abstract:
Traditional significant wave height (SWH) estima‑ tion from HF radar typically relies on spectral analysis of the received radar signals. This process was previously simplified by establishing a linear relationship between SWH and the standard deviation of received HF radar voltages under first‑ order scattering. Building on this approach, this paper presents a physics‑informed regression model that incorporates second‑ order scattering effects through a quadratic formulation derived from a Neumann expansion. The proposed method is evaluated using HF radar data collected in July 2018 at Argentia, New‑ foundland, with collocated buoy measurements as ground truth. The model achieves a minimum root‑mean‑square error (RMSE) of approximately 19 cm.

Abstract:
Data‑driven model predictive control based on Willems' fundamental lemma has proven effective for linear systems, but extending stability guarantees to nonlinear systems remains an open challenge. In this paper, we establish conditions under which data‑driven MPC, applied directly to input‑output data from a nonlinear system, yields practical exponential stability. The key insight is that the existence of an approximate Koopman linear embedding certifies that the nonlinear data can be interpreted as noisy data from a linear time‑invariant system, enabling the application of existing robust stability theories. Crucially, the Koopman embedding serves only as a theoretical certificate; the controller itself operates on raw nonlinear data without knowledge of the lifting functions. We further show that the proportional structure of the embedding residual can be exploited to obtain an ultimate bound that depends only on the irreducible offset, rather than the worst‑case embedding error. The framework is demonstrated on a synchronous generator connected to an infinite bus, for which we construct an explicit physics‑informed embedding with error bounds.

Abstract:
The transition from optical identification of 2D quantum materials to practical device fabrication requires dynamic reasoning beyond the detection accuracy. While recent domain‑specific Multimodal Large Language Models (MLLMs) successfully ground visual features using physics‑informed reasoning, their outputs are optimized for step‑by‑step cognitive transparency. This yields verbose candidate enumerations followed by dense reasoning that, while accurate, may induce cognitive overload and lack immediate utility for real‑world interaction with researchers. To address this challenge, we introduce OpenQlaw, an agentic orchestration system for analyzing 2D materials. The architecture is built upon NanoBot, a lightweight agentic framework inspired by OpenClaw, and QuPAINT, one of the first Physics‑Aware Instruction Multi‑modal platforms for Quantum Material Discovery. This allows accessibility to the lab floor via a variety of messaging channels. OpenQlaw allows the core Large Language Model (LLM) agent to orchestrate a domain‑expert MLLM,with QuPAINT, as a specialized node, successfully decoupling visual identification from reasoning and deterministic image rendering. By parsing spatial data from the expert, the agent can dynamically process user queries, such as performing scale‑aware physical computation or generating isolated visual annotations, and answer in a naturalistic manner. Crucially, the system features a persistent memory that enables the agent to save physical scale ratios (e.g., 1 pixel = 0.25 μm) for area computations and store sample preparation methods for efficacy comparison. The application of an agentic architecture, together with the extension that uses the core agent as an orchestrator for domain‑specific experts, transforms isolated inferences into a context‑aware assistant capable of accelerating high‑throughput device fabrication.

Abstract:
Epoxy polymers are widely used due to their multifunctional properties, but machine learning (ML) applications remain limited owing to their complex 3D molecular structure, multi‑component nature, and lack of curated datasets. Existing ML studies are largely restricted to simulation data, specific properties, or narrow constituent ranges. To address these limitations, we developed an informed Gaussian Process Regression‑based Knowledge Distillation (GPR‑KD) framework for predicting multiple physical (glass transition temperature, density) and mechanical properties (elastic modulus, tensile strength, compressive strength, flexural strength, fracture energy, adhesive strength) of thermoset epoxy polymers. The model was trained on experimental literature data covering diverse monomer classes (9 resins, 40 hardeners). Individual GPR models serve as teacher models capturing nonlinear feature‑property relationships, while a unified neural network student model learns distilled knowledge across all properties simultaneously. By encoding the target property as an input feature, the student model leverages cross‑property correlations. Molecular‑level descriptors extracted from SMILES representations using RDKit create a physics‑informed model. The framework combines GPR interpretability and robustness with deep learning scalability and generalization. Comparative analysis demonstrates superior prediction accuracy over conventional ML models. Simultaneous multi‑property prediction further improves accuracy through information sharing across correlated properties. The proposed framework enables accelerated design of novel epoxy polymers with tailored properties.

Abstract:
Solving the alternating current power flow equations in real time is essential for secure grid operation, yet classical Newton‑Raphson solvers can be slow under stressed conditions. Existing graph neural networks for power flow are typically trained on a single system and often degrade on different systems. We present PowerModelsGAT‑AI, a physics‑informed graph attention network that predicts bus voltages and generator injections. The model uses bus‑type‑aware masking to handle different bus types and balances multiple loss terms, including a power‑mismatch penalty, using learned weights. We evaluate the model on 14 benchmark systems (4 to 6,470 buses) and train a unified model on 13 of these under N‑2 (two‑branch outage) conditions, achieving an average normalized mean absolute error of 0.89% for voltage magnitudes and R^2 > 0.99 for voltage angles. We also show continual learning: when adapting a base model to a new 1,354‑bus system, standard fine‑tuning causes severe forgetting with error increases exceeding 1000% on base systems, while our experience replay and elastic weight consolidation strategy keeps error increases below 2% and in some cases improves base‑system performance. Interpretability analysis shows that learned attention weights correlate with physical branch parameters (susceptance: r = 0.38; thermal limits: r = 0.22), and feature importance analysis supports that the model captures established power flow relationships.

Abstract:
As spin‑based quantum systems scale, their setup and control complexity increase sharply. In semiconductor quantum dot (QD) experiments, device‑to‑device variability, heterogeneous control‑electronics stacks, and differing operational modalities make it difficult to reuse characterization, calibration, and control logic across laboratories. We present FAlCon, an open‑source software ecosystem for portable, automated characterization and tuning measurement workflows. FAlCon provides (i) a lightweight domain‑specific language for expressing state‑based tuning logic in a hardware‑agnostic form; (ii) specialized transmittable libraries of physics‑informed QD data structures (``tuning vernacula''); and (iii) extensible libraries of shared measurement protocols enabling an interoperable lab‑agnostic measurement stack. By separating algorithm intent from instrument realization, while preserving traceability and supporting typed scripting, FAlCon enables researchers and engineers to exchange, adapt, and deploy characterization and autotuning routines across heterogeneous QD setups. The framework supports all users, ranging from end users running prebuilt algorithms with custom initial conditions to developers extending instrumentation support and contributing new tuning strategies. Although the present release targets QD experiments, other qubit modalities and scientific experiments could reuse FAlCon's modular abstractions by providing new tuning data types and instrument control templates.

Abstract:
Disordered metamaterials are promising for programming physical properties across diverse applications, yet their inverse design remains challenging due to the non‑intuitive structure‑property relationships and large design spaces. Recent generative approaches, particularly diffusion models, have shown potential in high‑dimensional inverse design tasks. However, existing methods typically rely on carefully crafted training objectives, such as conditional data‑driven or physics‑informed loss functions. Because these strategies are inherently task‑specific, the model must be retrained from scratch whenever the design problem changes (e.g., different governing equations, boundary conditions, or design objectives), severely limiting their flexibility and generalization ability. In this work, we propose physics‑guided diffusion models that leverage differentiable physics‑based solvers to instantly guide the generative process for inverse design. Drawing inspiration from classifier guidance, we develop a sampling strategy that directly incorporates physics guidance into the reverse stochastic differential equations. Our approach enables task‑adaptive generation using gradients from differentiable solvers, while the diffusion model itself needs to be trained only once on unlabeled data. Focusing on disordered foam metamaterials, we present three representative design tasks: (1) achieving target effective thermal conductivity, (2) matching desired load‑displacement response, and (3) maximizing energy absorption involving fractures. In each scenario, the proposed method successfully generates foam‑like geometries that fulfill the prescribed physical objectives. These results demonstrate the versatility, efficiency, and practicality of physics‑guided diffusion models for tackling complex inverse design problems in disordered metamaterials and beyond.

Abstract:
Sizing a GPU fleet for LLM inference is harder than it looks. The obvious questions ‑‑ how many GPUs, which type, where to split a two‑pool fleet ‑‑ have no closed‑form answers. They depend on the full token‑length distribution, the routing policy, and queueing dynamics that turn ugly under heavy‑tailed workloads. Existing tools optimize per‑engine configuration for a fixed GPU count; none of them address the upstream question of how many GPUs to buy and how to arrange them. inference‑fleet‑sim fills that gap. It combines analytical M/G/c queueing with discrete‑event simulation (DES) to find the minimum‑cost fleet configuration that empirically meets a P99 TTFT SLO. It includes a physics‑informed GPU performance model covering A10G, A100, and H100 across monolithic, two‑pool‑routed, and disaggregated topologies, all without requiring access to real hardware. We run the tool on seven fleet‑planning scenarios drawn from two public workload traces (LMSYS, Azure) and one synthetic agent‑heavy trace. Each one surfaces a result that simple analysis gets wrong ‑‑ the right split threshold, the cheapest GPU type, whether an apparently idle fleet is actually broken ‑‑ and shows why joint simulation of queueing, routing, and hardware is necessary to find it.

Abstract:
We develop an unsupervised physics‑informed neural network to solve saddle‑point equations (SPEs) governing direct above‑threshold ionization (ATI) within the strong‑field approximation. This setting provides a well‑understood testbed in which the saddle‑point structure is known for tailored driving fields, enabling systematic validation of the proposed solver. The network is trained by minimizing the residual of the SPEs and requires only the definition of the driving‑field shape and its parameters, such as intensity, carrier‑envelope phase, ellipticity, and relative phase. We introduce a window parametrization strategy that maps network outputs to prescribed regions of the complex‑time plane, guiding the optimization toward physically relevant solutions and improving convergence stability. We benchmark the PINN against a conventional solver for a range of fields, demonstrating robust recovery of the dominant complex ionization times over wide parameter ranges. The network tracks changes in ionization event dominance as laser parameters are varied, enabling exploration of regimes where conventional solvers require repeated manual initialization. Using the PINN‑derived solutions, we compute coherent ATI photoelectron momentum distributions and show the symmetries of the driving fields are reflected in both the saddle‑point structure and the resulting spectra. These results establish PINNs as a promising framework for semiclassical strong‑field calculations and provide a foundation for extending machine‑learning solvers to Coulomb‑corrected models or to more complex processes, such as rescattered ATI for which the SPEs are highly nonlinear and the presence of multiple closely‑spaced solutions makes conventional Newton‑type root‑finding highly sensitive to initial guesses, which hinders systematic investigations across laser‑parameter spaces, particularly for tailored fields.

Abstract:
Traditional fluid dynamics simulation pipelines combine numerical solvers with rendering, producing highly realistic results but at considerable computational cost. Diffusion‑based generative video models offer a faster alternative, yet often ignore physical laws and thus fail to capture consistent dynamics. We propose a physics‑informed video diffusion framework that jointly generates visual outputs and physical states. Unlike prior two‑stage approaches that first simulate the physical variables and then render, we directly integrate physics constraints into the generative process, enabling simultaneous prediction of physical states and realistic videos without a separate rendering step. Built on the two‑dimensional shallow water equations with terrain topography, our method produces temporally coherent water flow while maintaining physical plausibility. Experiments show that it outperforms purely data‑driven video diffusion baselines in both realism and physical fidelity, while generating videos significantly faster than traditional simulation‑plus‑rendering pipelines.

Abstract:
Physics‑informed neural networks (PINNs) and neural operators (NOs) for solving the problem of diffraction of Extreme Ultraviolet (EUV) electromagnetic waves from contemporary lithography masks are presented. A novel hybrid Waveguide Neural Operator (WGNO) is introduced, based on a waveguide method with its most computationally expensive components replaced by a neural network. To evaluate performance, the accuracy and inference time of PINNs and NOs are compared against modern numerical solvers for a series of problems with known exact solutions. The emphasis is placed on investigation of solution accuracy by considered artificial neural systems for 13.5 nm and 11.2 nm wavelengths. Numerical experiments on realistic 2D and 3D masks demonstrate that PINNs and neural operators achieve competitive accuracy and significantly reduced prediction times, with the proposed WGNO architecture reaching state‑of‑the‑art performance. The presented neural operator has pronounced generalizing properties, meaning that for unseen problem parameters it delivers a solution accuracy close to that for parameters seen in the training dataset. These results provide a highly efficient solution for accelerating the design and optimization workflows of next‑generation lithography masks.

Abstract:
Physics‑informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post‑hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.

Abstract:
Foundation models for partial differential equations (PDEs) have emerged as powerful surrogates pre‑trained on diverse physical systems, but adapting them to new downstream tasks remains challenging due to limited task‑specific data and distribution shifts. While fine‑tuning has proven transformative in natural language processing, best practices for adapting PDE foundation models remain underexplored. Although physics‑informed training has successfully trained accurate solvers across a wide range of PDE problems, its potential for fine‑tuning data‑based foundation models has not been systematically studied. In this work, we introduce a physics‑informed fine‑tuning framework that adapts pre‑trained PDE foundation models by incorporating physical constraints (PDE residuals and boundary conditions) directly into the fine‑tuning objective. This enables effective adaptation in data‑scarce regimes while promoting physical consistency. We evaluate our method on a downstream task composed of an unseen PDE class and compare it with data‑driven finetuning counterparts. Our results demonstrate that physics‑informed fine‑tuning achieves competitive accuracy without requiring PDE solutions for training. Furthermore, a hybrid fine‑tuning strategy yields superior generalization to out‑of‑distribution scenarios when only minimal training data is available. These findings establish physics‑informed fine‑tuning as a scalable and data‑efficient paradigm, providing a physically interpretable pathway for adapting foundation models in scientific machine learning.

Abstract:
A tandem deep neural network approach is presented for the inverse design of reactively loaded metasurfaces with prescribed far‑field radiation characteristics. The proposed approach integrates a deep neural network (DNN) with a physics‑based microwave network forward solver. The DNN maps target far‑field patterns to distributions of reactive loads across the metasurface unit cells. The predicted distribution of reactive loads is evaluated by the forward solver to compute the resulting radiation pattern and guide the learning process through a cosine‑similarity loss function. The forward solver enables a fast evaluation of the metasurface's electromagnetic response, significantly reducing the computational cost required for training. The proposed approach is applied to a metasurface with aperture‑coupled unit cells loaded with reactances. Several design examples are presented to demonstrate the accurate synthesis of shaped and steered radiation patterns. Full‑wave electromagnetic simulations are performed to validate the accuracy of the designed beamforming metasurfaces.

Abstract:
We propose an efficient and generalizable physics‑informed neural network (PINN) framework for computing traveling wave solutions of n‑dimensional reaction‑diffusion equations with various reaction and diffusion coefficients. By applying a scaling transformation with the traveling wave form, the original problem is reduced to a one‑dimensional scaled reaction‑diffusion equation with unit reaction and diffusion coefficients. This reduction leads to the proposed framework, termed scaled TW‑PINN, in which a single PINN solver trained on the scaled equation is reused for different coefficient choices and spatial dimensions. We also prove a universal approximation property of the proposed PINN solver for traveling wave solutions. Numerical experiments in one and two dimensions, together with a comparison to the existing wave‑PINN method, demonstrate the accuracy, flexibility, and superior performance of scaled TW‑PINN. Finally, we explore an extension of the framework to the Fisher's equation with general initial conditions.

Abstract:
Vision‑Language Models (VLMs) demonstrate strong general‑purpose reasoning but remain limited in physics‑grounded anomaly detection, where causal understanding of dynamics is essential. Existing VLMs, trained predominantly on appearance‑centric correlations, fail to capture kinematic constraints, leading to poor performance on anomalies such as irregular rotations or violated mechanical motions. We introduce a physics‑informed instruction tuning framework that explicitly encodes object properties, motion paradigms, and dynamic constraints into structured prompts. By delivering these physical priors through multi‑turn dialogues, our method decomposes causal reasoning into incremental steps, enabling robust internal representations of normal and abnormal dynamics. Evaluated on the Phys‑AD benchmark, our approach achieves 96.7% AUROC in video‑level detection‑‑substantially outperforming prior SOTA (66.9%)‑‑and yields superior causal explanations (0.777 LLM score). This work highlights how structured physics priors can transform VLMs into reliable detectors of dynamic anomalies.

Abstract:
Accurately simulating extended periodic systems is a central challenge in condensed matter physics. Neural quantum states (NQS) offer expressive wavefunctions for this task but face issues with scalability. In this work, we successfully extend the neural network backflow (NNBF) approach to ab‑initio solid‑state materials. Building on our scalable optimization framework for molecules [Liu et al., PRB 112, 155162 (2025)], we introduce a two‑stage pruning strategy to manage the massive configuration space expansions: by utilizing a computationally cheap, physics‑informed importance proxy, we devote exact NNBF amplitude evaluations solely to the most relevant determinants, significantly improving optimization efficiency, energy estimation, and convergence. Our framework achieves state‑of‑the‑art accuracy across diverse solid‑state benchmarks. For 1D hydrogen chains, NNBF matches or surpasses DMRG and AFQMC, remains robust in strongly correlated bond‑breaking regimes where coupled‑cluster methods fail, and smoothly extrapolates to the thermodynamic limit. We further demonstrate its scalability by computing ground‑state potential energy curves for 2D graphene and 3D silicon. Finally, ablation studies validate the computational savings of our pruning strategy and highlight the dependence of the NNBF energies on basis sets.

Abstract:
Predicting the stress‑strain behaviors of additively manufactured materials is crucial for part qualification in additive manufacturing (AM). Conventional physics‑based constitutive models often oversimplify material properties, while data‑driven machine learning (ML) models often lack physical consistency and interpretability. To address these issues, we propose a physics‑informed machine learning (PIML) framework to improve the predictive performance and physical consistency for predicting the stress‑strain curves of additively manufactured polymers and metals. A polynomial regression model is used to predict the yield point from AM process parameters, then stress‑strain curves are segmented into elastic and plastic regions. Two long short‑term memory (LSTM) models are trained to predict two regions separately. For the elastic region, Hooke's law is embedded into the LSTM model for both polymer and metal. For the plastic region, Voce hardening law and Hollomon's law are embedded into the LSTM model for polymer and metal, respectively. The loss‑based and activation‑based PIML architectures are developed by embedding the physical laws into the loss and activation functions, respectively. The performance of the two PIML architectures are compared with two LSTM‑based ML models, three additional ML models, and a physics‑based constitutive model. These models are built on experimental data collected from two additively manufactured polymers (i.e., Nylon and carbon fiber‑acrylonitrile butadiene styrene) and two additively manufactured metals (i.e., AlSi10Mg and Ti6Al4V). Experimental results demonstrate that two PIML architectures consistently outperform the other models. The segmental predictive model with activation‑based PIML architecture achieves the lowest MAPE of 10.46+/‑0.81% and the highest R^2 of 0.82+/‑0.05 arocss four datasets.

Abstract:
Reinforcement learning (RL) has achieved strong performance in robotic control; however, state‑of‑the‑art policy learning methods, such as actor‑critic methods, still suffer from high sample complexity and often produce physically inconsistent actions. This limitation stems from neural policies implicitly rediscovering complex physics from data alone, despite accurate dynamics models being readily available in simulators. In this paper, we introduce a novel physics‑informed RL framework, called PIPER, that seamlessly integrates physical constraints directly into neural policy optimization with analytical soft physics constraints. At the core of our method is the integration of a differentiable Lagrangian residual as a regularization term within the actor's objective. This residual, extracted from a robot's simulator description, subtly biases policy updates towards dynamically consistent solutions. Crucially, this physics integration is realized through an additional loss term during policy optimization, requiring no alterations to existing simulators or core RL algorithms. Extensive experiments demonstrate that our method significantly improves learning efficiency, stability, and control accuracy, establishing a new paradigm for efficient and physically consistent robotic control.

Abstract:
Accurate and reliable histopathological image classification is essential for breast cancer diagnosis. However, many deep learning models remain sensitive to magnification variability and lack interpretability. To address these challenges, we propose Histo‑MExNet, a unified framework designed for scaleinvariant and uncertainty‑aware classification. The model integrates DenseNet, ConvNeXt, and EfficientNet backbones within a gated multi‑expert architecture, incorporates a prototype learning module for example‑driven interpretability, and applies physics‑informed regularization to enforce morphology preservation and spatial coherence during feature learning. Monte Carlo Dropout is used to quantify predictive uncertainty. On the BreaKHis dataset, Histo‑MExNet achieves 96.97% accuracy under multi‑magnification training and demonstrates improved generalization to unseen magnification levels compared to single‑expert models, while uncertainty estimation helps identify out‑of‑distribution samples and reduce overconfident errors, supporting a balanced combination of accuracy, robustness, and interpretability for clinical decision support.

Abstract:
Partial differential equations (PDEs) encode fundamental physical laws, yet closed‑form analytical solutions for many important equations remain unknown and typically require substantial human insight to derive. Existing numerical, physics‑informed, and data‑driven approaches approximate solutions from data rather than systematically deriving symbolic expressions directly from governing equations. Here we introduce LawMind, a law‑driven symbolic discovery framework that autonomously constructs closed‑form solutions from PDEs and their associated conditions without relying on data or supervision. By integrating structured symbolic exploration with physics‑constrained evaluation, LawMind progressively assembles valid solution components guided solely by governing laws. Evaluated on 100 benchmark PDEs drawn from two authoritative handbooks, LawMind successfully recovers closed‑form analytical solutions for all cases. Beyond known solutions, LawMind further discovers previously unreported closed‑form solutions to both linear and nonlinear PDEs. These findings establish a computational paradigm in which governing equations alone drive autonomous symbolic discovery, enabling the systematic derivation of analytical PDE solutions.

Abstract:
Precise characterization of the local spin environment of single diamond nitrogen‑vacancy (NV) centers is crucial for advancing quantum sensing, quantum networking, and the optimization of quantum materials. However, single NV center fluorescence measurements requires long averaging times to obtain clean data that is suitable for conventional model fitting, and that constitutes a key experimental bottleneck for high‑throughput characterization. To address this, we introduce \textscNVRNet, a physics‑informed simulation‑to‑reality machine learning pipeline that maps minimal‑sweep, noisy Ramsey data to a denoised waveform while directly estimating the hyperfine coupling to proximal ^13\mathrmC nuclear spins. The pipeline's denoiser utilizes a two‑stage time‑frequency U‑Net and an attention‑augmented time‑domain U‑Net, pretrained on Hamiltonian‑based spin‑dynamics simulations with experimentally calibrated noise. To effectively bridge the simulation‑to‑reality gap, parameter‑efficient adapters are attached to the backbone and fine‑tuned on targeted experimental data. Across three distinct NV centers, this experimentally fine‑tuned model reduces the median reconstruction error on held‑out, few‑sweep traces to 0.44\text‑0.67× of the raw experimental noise level. Subsequently, a transformer‑based estimator extracts the underlying hyperfine parameters. Forward reconstructions derived from these inferred parameters faithfully reproduce the dominant experimental time‑ and frequency‑domain features, yielding representative normalized fast Fourier transform (FFT) reconstruction errors of 0.10\text‑0.19. By reducing both the required data volume and acquisition time, \textscNVRNet enables up to ～ 40× acceleration of the measurement process, establishing a fast, hardware‑compatible pathway for robust hyperfine inference and autonomous qubit characterization.

Abstract:
In recent years, deep learning methods, exemplified by Physics‑Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high training costs, and lack of robustness, particularly their inability to preserve the intrinsic physical structures of continuous PDE models, such as the energy dissipation property in gradient flow systems. To address these challenges, we propose a feature‑based Deep Neural Network Galerkin (DNN‑G) framework designed for structure‑preserving simulations of gradient flows. Instead of treating neural networks merely as optimization‑driven solvers, we employ them as adaptive feature generators that define nonlinear trial spaces within a Galerkin projection formulation.This formulation guarantees semi‑discrete energy dissipation and can be naturally combined with energy stable time integration schemes. Several strategies for constructing neural basis functions are investigated, including random features, structured initialization, and problem‑informed pre‑training. Numerical experiments demonstrate that the proposed method preserves robust energy stability in high‑dimensional settings and accurately captures complex topological transitions. With equivalent degrees of freedom, the DNN‑G framework achieves higher accuracy than classical spectral methods, highlighting the effectiveness of neural feature representations for the numerical solution of partial differential equations.

Abstract:
Grid‑scale battery degradation unfolds over multi‑year timescales under coupled electrochemical, thermal, and operational feedbacks difficult to capture using laboratory data or proprietary field datasets. This scarcity limits the development of degradation‑aware algorithms and digital twins that require long‑horizon, physically consistent ground truth. Here we present SAGE (Synthetic Aging for a Grid Environment), an open‑source, physics‑informed simulation framework that generates hour‑resolved, multi‑decade operating histories and degradation trajectories for heterogeneous battery energy storage system (BESS) fleets. The framework couples stochastic environmental drivers, market‑based dispatch, electro‑thermal behavior, aging kinetics, and asset‑level heterogeneity within a transparent, externally parameterized architecture. We validate physical consistency through hierarchical tests, including Arrhenius temperature acceleration, thermal stratification, and emergent wear‑out statistics. Simulations demonstrate how intrinsic heterogeneity in thermal environments and manufacturing naturally produces dispersion in state‑of‑health trajectories without imposed statistical failure assumptions. SAGE serves as a benchmarking platform for optimization, state estimation, and machine learning, enabling reproducible research in grid‑scale energy storage modeling.

Abstract:
Vision‑Language‑Action (VLA) models have emerged as a powerful paradigm for robotic manipulation. However, existing post‑training methods face a dilemma between stability and exploration: Supervised Fine‑Tuning (SFT) is constrained by demonstration quality and lacks generalization, whereas Reinforcement Learning (RL) improves exploration but often induces erratic, jittery trajectories that violate physical constraints. To bridge this gap, we propose SmoothVLA, a novel reinforcement learning fine‑tuning framework that synergistically optimizes task performance and motion smoothness. The technical core is a physics‑informed hybrid reward function that integrates binary sparse task rewards with a continuous dense term derived from trajectory jerk. Crucially, this reward is intrinsic, that computing directly from policy rollouts, without requiring extrinsic environment feedback or laborious reward engineering. Leveraging the Group Relative Policy Optimization (GRPO), SmoothVLA establishes trajectory smoothness as an explicit optimization prior, guiding the model toward physically feasible and stable control. Extensive experiments on the LIBERO benchmark demonstrate that SmoothVLA outperforms standard RL by 13.8% in smoothness and significantly surpasses SFT in generalization across diverse tasks. Our work offers a scalable approach to aligning VLA models with physical‑world constraints through intrinsic reward optimization.

Abstract:
Floods are among the costliest natural catastrophes globally, yet the property and casualty insurance industry's post‑event response remains heavily reliant on manual field inspection: slow, expensive, and geographically constrained. Satellite Synthetic Aperture Radar (SAR) offers cloud‑penetrating, all‑weather imaging uniquely suited to rapid post‑flood assessment, but existing research evaluates SAR flood detection against academic benchmarks such as IoU and F1‑score that do not capture insurance‑workflow requirements. We present ALTIS: a five‑stage pipeline transforming raw Sentinel‑1 GRD and SLC imagery into property‑level impact scores within 24‑48 hours of flood peak. Unlike prior approaches producing pixel‑level maps or binary outputs, ALTIS delivers a ranked, confidence‑scored triage list consumable by claims platforms, integrating (i) multi‑temporal SAR change detection using dual‑polarization VV/VH intensity and InSAR coherence, (ii) physics‑informed depth estimation fusing flood extent with high‑resolution DEMs, (iii) property‑level zonal statistics from parcel footprints, (iv) depth‑damage calibration against NFIP claims, and (v) confidence‑scored triage ranking. We formally define Insurance‑Grade Flood Triage (IGFT) and introduce the Inspection Reduction Rate (IRR) and Triage Efficiency Score (TES). Using Hurricane Harvey (2017) across Harris County, Texas, we present preliminary analysis grounded in validated sub‑components suggesting ALTIS is designed to achieve an IRR of approximately 0.52 at 90% recall of high‑severity claims, potentially eliminating over half of unnecessary dispatches. By blending SAR flood intelligence with the realities of claims management, ALTIS establishes a methodological baseline for translating earth observation research into measurable insurance outcomes.

Abstract:
Physics‑informed neural networks (PINNs) have achieved notable success in modeling dynamical systems governed by partial differential equations (PDEs). To avoid computationally expensive retraining under new physical conditions, parameterized PINNs (P^2INNs) commonly adapt pre‑trained operators using singular value decomposition (SVD) for out‑of‑distribution (OOD) regimes. However, SVD‑based fine‑tuning often suffers from rigid subspace locking and truncation of important high‑frequency spectral modes, limiting its ability to capture complex physical transitions. While parameter‑efficient fine‑tuning (PEFT) methods appear to be promising alternatives, applying conventional adapters such as LoRA to P^2INNs introduces a severe Pareto trade‑off, as additive updates increase parameter overhead and disrupt the structured physical manifolds inherent in operator representations. To address these limitations, we propose Manifold‑Orthogonal Dual‑spectrum Extrapolation (MODE), a lightweight micro‑architecture designed for physics operator adaptation. MODE decomposes physical evolution into complementary mechanisms including principal‑spectrum dense mixing that enables cross‑modal energy transfer within frozen orthogonal bases, residual‑spectrum awakening that activates high‑frequency spectral components through a single trainable scalar, and affine Galilean unlocking that explicitly isolates spatial translation dynamics. Experiments on challenging PDE benchmarks including the 1D Convection‑‑Diffusion‑‑Reaction equation and the 2D Helmholtz equation demonstrate that MODE achieves strong out‑of‑distribution generalization while preserving the minimal parameter complexity of native SVD and outperforming existing PEFT‑based baselines.

Abstract:
To serve as a scalable data source for embodied AI, world models should act as true simulators that infer interaction dynamics strictly from user actions, rather than mere conditional video generators relying on privileged future object states. In this context, egocentric Human‑Object Interaction (HOI) world models are critical for predicting physically grounded first‑person rollouts. However, building such models is profoundly challenging due to rapid head motions, severe occlusions, and high‑DoF hand articulations that abruptly alter contact topologies. Consequently, existing approaches often circumvent these physics challenges by resorting to conditional video generation with access to known future object trajectories. We introduce EgoHOI, an egocentric HOI world model that breaks away from this shortcut to simulate photorealistic, contact‑consistent interactions from action signals alone. To ensure physical accuracy without future‑state inputs, EgoHOI distills geometric and kinematic priors from 3D estimates into physics‑informed embeddings. These embeddings regularize the egocentric rollouts toward physically valid dynamics. Experiments on the HOT3D dataset demonstrate consistent gains over strong baselines, and ablations validate the effectiveness of our physics‑informed design.

Abstract:
Estimating motion from spatiotemporal geoscientific data is a fundamental component of many environmental modeling and forecasting tasks. In this work, we propose a physics‑informed deep learning framework for estimating altitude‑wise motion fields directly from volumetric radar reflectivity data. The model utilizes a fully differentiable semi‑Lagrangian extrapolation operator to process three‑dimensional inputs as independent horizontal slice sequences, enabling efficient inference of horizontal motion across multiple altitude levels. Using a multi‑year radar dataset from Central Europe, we evaluate the impact of altitude‑wise motion estimation on extrapolation‑based precipitation forecasting and conduct a systematic dataset‑scale analysis of inter‑altitude motion consistency. The results show that the estimated motion fields exhibit strong vertical coherence, with high correlation across altitude levels, which results in limited improvement over traditional two‑dimensional approach in this setting. The proposed framework provides a general tool for efficiently analyzing motion structure in volumetric geospatial data. The findings indicate that, in regions dominated by vertically coherent precipitation systems, the added complexity of volumetric motion modeling may offer limited benefit, warranting careful consideration in the design of efficient spatiotemporal advection models.

Abstract:
Low‑dose CT (LDCT) reduces radiation exposure but introduces protocol‑dependent noise and artifacts that vary across institutions. While federated learning enables collaborative training without centralizing patient data, existing methods personalize in image space, making it difficult to separate scanner noise from patient anatomy. We propose ProFed (Projection Guided Personalized Federated Learning), a framework that complements the image space approach by performing dual‑level personalization in the projection space, where noise originates during CT measurements before reconstruction combines protocol and anatomy effects. ProFed introduces: (i) anatomy‑aware and protocol‑aware networks that personalize CT reconstruction to patient and scanner‑specific features, (ii) multi‑constraint projection losses that enforce consistency with CT measurements, and (iii) uncertainty‑guided selective aggregation that weights clients by prediction confidence. Extensive experiments on the Mayo Clinic 2016 dataset demonstrate that ProFed achieves 42.56 dB PSNR with CNN backbones and 44.83 dB with Transformers, outperforming 11 federated learning baselines, including the physics‑informed SCAN‑PhysFed by +1.42 dB.

Abstract:
Predictive maintenance for connected vehicles offers the potential to reduce unexpected breakdowns and improve fleet reliability, but most existing systems rely exclusively on internal diagnostic signals and are validated on simulated or industrial benchmark data. This paper presents a contextual data fusion framework integrating vehicle‑internal sensor streams with external environmental signals ‑‑ road quality, weather, traffic density, and driver behaviour ‑‑ acquired via V2X communication and third‑party APIs, with inference at the vehicle edge. The framework is evaluated across four layers. A feature group ablation study on a physics‑informed synthetic dataset shows contextual features contribute a 2.6‑point F1 improvement; removing all context reduces macro F1 from 0.855 to 0.807. On the AI4I 2020 benchmark (10,000 samples), LightGBM achieves AUC‑ROC 0.973 under 5‑fold stratified cross‑validation with SMOTE confined to training folds. A noise sensitivity analysis shows macro F1 remains above 0.88 at low noise and degrades to 0.74 at high noise. Most critically, the pipeline is validated on real‑world telemetry from five vehicles across three countries (India, Germany, Brazil), comprising 992 trips and 11 evaluable service events identified from component wear resets in the trip logs. Across six wear‑driven events spanning four vehicles, the model achieves 100% detection with mean MAE of 12.2 days. A fine‑tuning ablation shows the base synthetic model already achieves 6/6 binary detection; per‑vehicle adaptation reduces wear‑driven MAE from 25.9 to 12.2 days. SHAP analysis confirms contextual and interaction features rank among the top 15 predictors. Edge‑based inference reduces estimated latency from 3.5 seconds to under 1.0 second relative to cloud‑only processing.

Abstract:
Material parameters such as thermal diffusivity govern how microstructural fields evolve during processing, but difficult to measure directly. The Stability‑Aware Frozen Euler Physics‑Informed Tracking for Continuum Mechanics (SAFE‑PIT‑CM), is an autoencoder that embeds a frozen convolutional layer as a differentiable PDE solver in its latent‑space transition to jointly recover diffusion coefficients and the underlying physical field from temporal observations. When temporal snapshots are saved at intervals coarser than the simulation time step, a single forward Euler step violates the von Neumann stability condition, forcing the learned coefficient to collapse to an unphysical value. Sub‑stepping with SAFE restores stability at negligible cost each sub‑step is a single frozen convolution, far cheaper than processing more frames with recovery error converging monotonically with substep count. Validated on thermal diffusion in metals, the method recovers both the diffusion coefficient and the physical field with near‑perfect accuracy, both with and yet without pre‑training. Backpropagation through the frozen operator supervises an attention‑based parameter estimator without labelled data. The architecture generalises to any PDE with a convolutional finite‑difference discretisation.

Abstract:
Solving Partial Differential Equations (PDEs) using neural networks presents different challenges, including integration errors and spectral bias, often leading to poor approximations. In addition, standard neural network‑based methods, such as Physics‑Informed Neural Networks (PINNs), often lack stability when dealing with PDEs characterized by low‑regularity solutions. To address these limitations, we introduce the Ritz‑‑Uzawa Neural Networks (RUNNs) framework, an iterative methodology to solve strong, weak, and ultra‑weak variational formulations. Rewriting the PDE as a sequence of Ritz‑type minimization problems within a Uzawa loop provides an iterative framework that, in specific cases, reduces both bias and variance during training. We demonstrate that the strong formulation offers a passive variance reduction mechanism, whereas variance remains persistent in weak and ultra‑weak regimes. Furthermore, we address the spectral bias of standard architectures through a data‑driven frequency tuning strategy. By initializing a Sinusoidal Fourier Feature Mapping based on the Normalized Cumulative Power Spectral Density (NCPSD) of previous residuals or their proxies, the network dynamically adapts its bandwidth to capture high‑frequency components and severe singularities. Numerical experiments demonstrate the robustness of RUNNs, accurately resolving highly oscillatory solutions and successfully recovering a discontinuous L^2 solution from a distributional H^‑2 source ‑‑ a scenario where standard energy‑based methods fail.

Abstract:
Computational musicology enables systematic analysis of performative and structural traits in recorded music, yet existing approaches remain largely tailored to notated, score‑based repertoires. This study advances a methodology for analyzing voice‑guitar interaction in Carlos Paredes's vocal collaborations ‑ an oral‑tradition context where compositional and performative layers co‑emerge. Using source‑separated stems, physics‑informed harmonic modelling, and beat‑level audio descriptors, we examine melodic, harmonic, and rhythmic relationships across eight recordings with four singers. Our commonality‑diversity framework, combining multi‑scale correlation analysis with residual‑based detection of structural deviations, reveals that expressive coordination is predominantly piece‑specific rather than corpus‑wide. Diversity events systematically align with formal boundaries and textural shifts, demonstrating that the proposed approach can identify musically salient reorganizations with minimal human annotation. The framework further offers a generalizable computational strategy for repertoires without notated blueprints, extending Music Performance Analysis into oral‑tradition and improvisation‑inflected practices.

Abstract:
Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous‑time latent methods often rely on expensive test‑time auto‑decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics‑informed framework that explicitly separates space, time, and parameters. Instead of unstable auto‑decoding, DLDMF maps PDE parameters directly to a continuous latent embedding through a feed‑forward network. This embedding initializes and conditions a latent state whose evolution is governed by a parameter‑conditioned Neural ODE. We further introduce a dynamic manifold fusion mechanism that uses a shared decoder to combine spatial coordinates, parameter embeddings, and time‑evolving latent states to reconstruct the corresponding spatiotemporal solution. By modeling prediction as latent dynamic evolution rather than static coordinate fitting, DLDMF reduces interference between parameter variation and temporal evolution while preserving a smooth and coherent solution manifold. As a result, it performs well on unseen parameter settings and in long‑term temporal extrapolation. Experiments on several benchmark problems show that DLDMF consistently outperforms state‑of‑the‑art baselines in accuracy, parameter generalization, and extrapolation robustness.

Abstract:
We establish empirical scaling laws for Single‑Layer Physics‑Informed Neural Networks on canonical nonlinear PDEs. We identify a dual optimization failure: (i) a baseline pathology, where the solution error fails to decrease with network width, even at fixed nonlinearity, falling short of theoretical approximation bounds, and (ii) a compounding pathology, where this failure is exacerbated by nonlinearity. We provide quantitative evidence that a simple separable power law is insufficient, and that the scaling behavior is governed by a more complex, non‑separable relationship. This failure is consistent with the concept of spectral bias, where networks struggle to learn the high‑frequency solution components that intensify with nonlinearity. We show that optimization, not approximation capacity, is the primary bottleneck, and propose a methodology to empirically measure these complex scaling effects.

Abstract:
By integrating physics‑informed neural network (PINN) techniques with domain decomposition method, a deep domain decomposition method is presented for solving elliptic variational inequality problems. Based on the Ritz variation method, the elliptic variational inequality problem is firstly reformulated as an optimization problem, and then the subproblem in each subdomain is solved by using the Ritz‑PINN method, which the parameters in the network are updated by the Adam optimizer, and the residual‑adaptive training by introducing a residual‑adaptive dataset update strategy to gradually guide the model to learn more complex regions. Additionally, the impact of overlapping regions on the performance of the new algorithm is explored. Numerical results demonstrate the effectiveness of the proposed algorithm, the mean square error can be reached 1.0e‑07, and the number of iterations is independent of grid length h under uniform overlap conditions.

Abstract:
Variational inequalities are widely applied in mechanical engineering, fluid penetration, transportation, and other fields. In this paper, a Deep Ritz method based on Physics‑Informed Neural Networks (PINNs) is proposed to enhance the accuracy and efficiency of solving elliptic variational inequalities. The Ritz variational method is firstly utilized to transform the variational inequality problem into an optimization problem. Then Bayesian optimization is employed to tune the weights of the loss function, and a residual‑based adaptive dataset update strategy is introduced to improve the convergence and accuracy of the model. Numerical experiments show that the proposed method can effectively approximate the analytical solution.

Abstract:
Speedlines in compressor performance maps (CPMs) are critical for understanding and predicting compressor behavior under various operating conditions. We investigate a physics‑based method for reconstructing compressor performance maps from sparse measurements by fitting each speedline with a superellipse and encoding it as a compact, interpretable vector (surge, choke, curvature, and shape parameters). Building on the formulation of Llamas et al., we develop a robust two‑stage fitting pipeline that couples global search with local refinement. The approach is validated on industrial data‑sets for different turbocharger types. We discuss prediction quality for inter‑ and extrapolation, metric sensitivities and outline opportunities for physics‑informed constraints, alternative function families, and hybrid physics‑ML mappings to improve boundary behavior and, ultimately, enable full CPM reconstruction from limited data.

Abstract:
This report summarizes the discussions and recommendations from the NSF Workshop on Algorithm‑Hardware Co‑design for Medical Applications, held on September 26‑27, 2024, in Pittsburgh, PA. The workshop assembled an interdisciplinary cohort of researchers, clinicians, and industry leaders to examine foundational challenges and develop a strategic roadmap for algorithm‑hardware co‑design in medical computing. The workshop focuses on four thematic areas: (1) teleoperations, telehealth, and surgical operations; (2) wearable and implantable medicine, including implantable living pharmacies; (3) home ICU, hospital systems, and elderly care; and (4) medical sensing, imaging, and reconstruction. This report calls for a fundamental shift in how next‑generation medical technologies are conceived, designed, validated, and translated into practice. The report recommends that NSF sustain investment in shared standardized data infrastructures and compute infrastructures, develop clinic workflow‑aware systems and human‑AI collaboration frameworks, promote scalable validation ecosystems grounded in objective, continuous measures, and physics‑informed, and enable safe, accountable, and resilient platforms, including virtual‑physical healthcare ecosystems, to de‑risk translational pathways. The workshop information can be found on the website: https://sites.google.com/view/nsfworkshop.

Abstract:
We propose a physics‑informed neural particle method (PINN‑‑PM) for the spatially homogeneous Landau equation. The method adopts a Lagrangian interacting‑particle formulation and jointly parameterizes the time‑dependent score and the characteristic flow map with neural networks. Instead of advancing particles through explicit time stepping, the Landau dynamics is enforced via a continuous‑time residual defined along particle trajectories. This design removes time‑discretization error and yields a mesh‑free solver that can be queried at arbitrary times without sequential integration. We establish a rigorous stability analysis in an L^2_v framework. The deviation between learned and exact characteristics is controlled by three interpretable sources: (i) score approximation error, (ii) empirical particle approximation error, and (iii) the physics residual of the neural flow. This trajectory estimate propagates to density reconstruction, where we derive an L^2_v error bound for kernel density estimators combining classical bias‑‑variance terms with a trajectory‑induced contribution. Using Hyvarinen's identity, we further relate the oracle score‑matching gap to the L^2_v score error and show that the empirical loss concentrates at the Monte Carlo rate, yielding computable a posteriori accuracy certificates. Numerical experiments on analytical benchmarks, including the two‑ and three‑dimensional BKW solutions, as well as reference‑free configurations, demonstrate stable transport, preservation of macroscopic invariants, and competitive or improved accuracy compared with time‑stepping score‑based particle and blob methods while using significantly fewer particles.

Abstract:
The modeling of realistic magnetic materials requires the inclusion of defects. Based on the pseudospectral Landau‑Lifshitz description of magnetisation dynamics, we propose a statistical model that takes into account defects, specifically vacancies. This statistical model can be integrated with deep learning techniques that correlate defect thresholds with relevant physical observables. We develop a convolutional neural network and a physics‑informed neural network combined with theory of functional connections to predict the dispersion relation given defect parameters and physical constraints. A two‑branch convolutional neural network is developed to predict domain‑wall widths depending on defects threshold, taking into account the spatial profile and domain‑wall width separately to achieve a prediction. The proposed physics‑informed approaches leverage deep‑learning and achieve statistical predictions measured in physical units. This is a stepping stone towards the discovery of new materials and the determination of minimal defect thresholds required for desired dynamics, states, or topological textures.

Abstract:
LWM‑Temporal is a new member of the Large Wireless Models (LWM) family that targets the spatiotemporal nature of wireless channels. Designed as a task‑agnostic foundation model, LWM‑Temporal learns universal channel embeddings that capture mobility‑induced evolution and are reusable across various downstream tasks. To achieve this objective, LWM‑Temporal operates in the angle‑delay‑time domain and introduces Sparse Spatio‑Temporal Attention (SSTA), a propagation‑aligned attention mechanism that restricts interactions to physically plausible neighborhoods, reducing attention complexity by an order of magnitude while preserving geometry‑consistent dependencies. LWM‑Temporal is pretrained in a self‑supervised manner using a physics‑informed masking curriculum that emulates realistic occlusions, pilot sparsity, and measurement impairments. Experimental results on channel prediction across multiple mobility regimes show consistent improvements over strong baselines, particularly under long horizons and limited fine‑tuning data, highlighting the importance of geometry‑aware architectures and geometry‑consistent pretraining for learning transferable spatiotemporal wireless representations.

Abstract:
Predicting the microstructural and morphological evolution of materials through phase‑field modelling is computationally intensive, particularly for high‑throughput parametric studies. While neural operators such as the Fourier neural operator (FNO) show promise in accelerating the solution of parametric partial differential equations (PDEs), the lack of explicit physical constraints, may limit generalisation and long‑term accuracy for complex phase‑field dynamics. Here, we develop a physics‑informed neural operator framework to learn parametric phase‑field PDEs, namely PF‑PINO. By embedding the residuals of phase‑field governing equations into the data‑fidelity loss function, our framework effectively enforces physical constraints during training. We validate PF‑PINO against benchmark phase‑field problems, including electrochemical corrosion, dendritic crystal solidification, and spinodal decomposition. Our results demonstrate that PF‑PINO significantly outperforms conventional FNO in accuracy, generalisation capability, and long‑term stability. This work provides a robust and efficient computational tool for phase‑field modelling and highlights the potential of physics‑informed neural operators to advance scientific machine learning for complex interfacial evolution problems.

Abstract:
Vision Language Models (VLMs) bridge visual perception and linguistic reasoning. In Autonomous Driving (AD), this synergy has enabled Vision Language Action (VLA) models, which translate high‑level multimodal understanding into driving behaviors, typically represented as future trajectories. However, existing VLA models mainly generate generic collision‑free trajectories. Beyond collision avoidance, adapting to diverse driving styles (e.g., sporty, comfortable) is essential for personalized driving. Moreover, many methods treat trajectory generation as naive token prediction, which can produce kinematically infeasible actions. To address these limitations, we present StyleVLA, a physics‑informed VLA framework for generating diverse and physically plausible driving behaviors. We introduce a hybrid loss that combines a kinematic consistency constraint with a continuous regression head to improve trajectory feasibility. To train StyleVLA, built on Qwen3‑VL‑4B, we construct a large‑scale instruction dataset with over 1.2k scenarios, 76k Bird's Eye View (BEV) samples, and 42k First Person View (FPV) samples, with ground‑truth trajectories for five driving styles and natural‑language instructions. Experiments show that our 4B‑parameter StyleVLA significantly outperforms proprietary models (e.g., Gemini‑3‑Pro) and state‑of‑the‑art VLA models. Using a composite driving score measuring success rate, physical feasibility, and style adherence, StyleVLA achieves 0.55 on BEV and 0.51 on FPV, versus 0.32 and 0.35 for Gemini‑3‑Pro. These results show that a specialized, physics‑informed, lightweight model can surpass closed‑source models on domain‑specific tasks.

Abstract:
Engine sounds originate from sequential exhaust pressure pulses rather than sustained harmonic oscillations. While neural synthesis methods typically aim to approximate the resulting spectral characteristics, we propose directly modeling the underlying pulse shapes and temporal structure. We present the Pulse‑Train‑Resonator (PTR) model, a differentiable synthesis architecture that generates engine audio as parameterized pulse trains aligned to engine firing patterns and propagates them through recursive Karplus‑Strong resonators simulating exhaust acoustics. The architecture integrates physics‑informed inductive biases including harmonic decay, thermodynamic pitch modulation, valve‑dynamics envelopes, exhaust system resonances and derived engine operating modes such as throttle operation and deceleration fuel cutoff (DCFO). Validated on three diverse engine types totaling 7.5 hours of audio, PTR achieves a 21% improvement in harmonic reconstruction and a 5.7% reduction in total loss over a harmonic‑plus‑noise baseline model, while providing interpretable parameters corresponding to physical phenomena. Complete code, model weights, and audio examples are openly available.

Abstract:
(short version abstract, full in article)High‑fidelity flow field reconstruction is important in fluid dynamics, but it is challenged by sparse and spatiotemporally incomplete sensor measurements, as well as failures of pre‑deployed measurement points that can invalidate pre‑trained reconstruction models. Physics‑informed neural networks (PINNs) alleviate dependence on large labeled datasets by incorporating governing physics, yet sensor placement optimization, a key factor in reconstruction accuracy and robustness, remains underexplored. In this study, we propose a PINN with Voronoi‑enhanced Sensor Optimization (VSOPINN). VSOPINN enables differentiable soft Voronoi construction for sparse sensor data rasterization, end‑to‑end fusion of centroidal Voronoi tessellation (CVT) with PINNs for adaptive sensor placement, and unified layout optimization for multi‑condition flow reconstruction through a shared encoder‑multi‑decoder architecture. We validate VSOPINN on three representative problems: lid‑driven cavity flow, vascular flow, and annular rotating flow. Results show that VSOPINN significantly improves reconstruction accuracy across different Reynolds numbers, adaptively learns effective sensor layouts, and remains robust under partial sensor failure. The study clarifies the intrinsic relationship between sensor placement and reconstruction precision in PINN‑based flow field reconstruction.

Abstract:
Physics‑informed neural networks (PINNs) have shown promise in addressing the ill‑posed deconvolution problem in computed tomography perfusion (CTP) imaging for acute ischemic stroke assessment. However, existing PINN‑based approaches remain deterministic and do not quantify uncertainty associated with violations of physics constraints, limiting reliability assessment. We propose Evidential Perfusion Physics‑Informed Neural Networks (EPPINN), a framework that integrates evidential deep learning with physics‑informed modeling to enable uncertainty‑aware perfusion parameter estimation. EPPINN models arterial input, tissue concentration, and perfusion parameters using coordinate‑based networks, and places a Normal‑‑Inverse‑‑Gamma distribution over the physics residual to characterize voxel‑wise aleatoric and epistemic uncertainty in physics consistency without requiring Bayesian sampling or ensemble inference. The framework further incorporates physiologically constrained parameterization and stabilization strategies to promote robust per‑case optimization. We evaluate EPPINN on digital phantom data, the ISLES 2018 benchmark, and a clinical cohort. On the evaluated datasets, EPPINN achieves lower normalized mean absolute error than classical deconvolution and PINN baselines, particularly under sparse temporal sampling and low signal‑to‑noise conditions, while providing conservative uncertainty estimates with high empirical coverage. On clinical data, EPPINN attains the highest voxel‑level and case‑level infarct‑core detection sensitivity. These results suggest that evidential physics‑informed learning can improve both accuracy and reliability of CTP analysis for time‑critical stroke assessment.

Abstract:
Macroscopic traffic flow is stochastic, but the physics‑informed deep learning methods currently used in transportation literature embed deterministic PDEs and produce point‑valued outputs; the stochasticity of the governing dynamics plays no role in the learned representation. This work develops a framework in which the physics constraint itself is distributional and directly derived from stochastic traffic‑flow dynamics. Starting from an Ito‑type Lighthill‑Whitham‑Richards model with Brownian forcing, we derive a one‑point forward equation for the marginal traffic density at each spatial location. The spatial coupling induced by the conservation law appears as an explicit conditional drift term, which makes the closure requirement transparent. Based on this formulation, we derive an equivalent deterministic Probability Flow ODE that is pointwise evaluable and differentiable once a closure is specified. Incorporating this as a physics constraint, we then propose a score network with an advection‑closure module, trainable by denoising score matching together with a Fokker‑Planck residual loss. The resulting model targets a data‑conditioned density distribution, from which point estimates, credible intervals, and congestion‑risk measures can be computed. The framework provides a basis for distributional traffic‑state estimation and for stochastic fundamental‑diagram analysis in a physics‑informed generative setting.

Abstract:
Reliable 3D trajectory estimation of unmanned aerial vehicles (UAVs) is a fundamental requirement for anti‑UAV systems, yet the acquisition of large‑scale and accurately annotated trajectory data remains prohibitively expensive. In this work, we present a novel framework that derives UAV 3D trajectories and category information directly from Internet‑scale UAV videos, without relying on manual annotations. First, language‑driven data acquisition is employed to autonomously discover and collect UAV‑related videos, while vision‑language reasoning progressively filters task‑relevant segments. Second, a training‑free cross‑modal label generation module is introduced to infer 3D trajectory hypotheses and UAV type cues. Third, a physics‑informed refinement process is designed to impose temporal smoothness and kinematic consistency on the estimated trajectories. The resulting video clips and trajectory annotations can be readily utilized for downstream anti‑UAV tasks. To assess effectiveness and generalization, we conduct zero‑shot transfer experiments on a public, well‑annotated 3D UAV benchmark. Results reveal a clear data scaling behavior: as the amount of online video data increases, zero‑shot transfer performance on the target dataset improves consistently, without any target‑domain training. The proposed method closely approaches the current state‑of‑the‑art, highlighting its robustness and applicability to real‑world anti‑UAV scenarios. Code and datasets will be released upon acceptance.

Abstract:
Inferring biophysical parameters and hidden state variables from partial and noisy observations is a fundamental challenge in computational neuroscience. This problem is particularly difficult for fast ‑ slow spiking and bursting models, where strong nonlinearities, multiscale dynamics, and limited observational data often lead to severe sensitivity to initial parameter guesses and convergence failure in the methods replying on the traditional numerical forward solvers. In this work, we developed a physics‑informed neural network (PINN) framework for the joint reconstruction of unobserved state variables and the estimation of unknown biophysical parameters in neuronal models. We demonstrate the effectiveness of the method on biophysical neuron models, including the Morris‑Lecar model across multiple spiking and bursting regimes and a respiratory model neuron. The method requires only partial voltage observations over short observation windows and remains robust even when initialized with non‑informative parameter guesses. These results suggest that PINN can deliver robust and accurate parameter inference and state reconstruction, providing a promising alternative for inverse problems in multiscale neuronal dynamics, where traditional techniques often struggle.

Abstract:
Multi‑Layer Perceptrons (MLPs) rely on pre‑defined, fixed activation functions, imposing a static inductive bias that forces the network to approximate complex topologies solely through increased depth and width. Kolmogorov‑Arnold Networks (KANs) address this limitation through edge‑centric learnable functions, yet their formulation suffers from quadratic parameter scaling and architectural rigidity that hinders the effective integration of standard regularization techniques. This paper introduces the DualFlexKAN (DFKAN), a flexible architecture featuring a dual‑stage mechanism that independently controls pre‑linear input transformations and post‑linear output activations. This decoupling enables hybrid networks that optimize the trade‑off between expressiveness and computational cost. Unlike standard formulations, DFKAN supports diverse basis function families, including orthogonal polynomials, B‑splines, and radial basis functions, integrated with configurable regularization strategies that stabilize training dynamics. Comprehensive evaluations across regression benchmarks, physics‑informed tasks, and function approximation demonstrate that DFKAN outperforms both MLPs and conventional KANs in accuracy, convergence speed, and gradient fidelity. The proposed hybrid configurations achieve superior performance with one to two orders of magnitude fewer parameters than standard KANs, effectively mitigating the parameter explosion problem while preserving KAN‑style expressiveness. DFKAN provides a principled, scalable framework for incorporating adaptive non‑linearities, proving particularly advantageous for data‑efficient learning and interpretable function discovery in scientific applications.

Abstract:
Next‑generation particle accelerators demand advanced beam‑diagnostic capabilities to ensure high performance, operational reliability, and sustainable machine operation. Increasing beam intensities and stored energies make the precise characterization of transverse profiles, phase‑space distributions, and halos often five orders of magnitude below the core but with significant environmental impact essential for loss mitigation and machine protection. Traditional analysis methods struggle with the heterogeneous, noisy, and non‑Gaussian data produced under realistic operating conditions. This work presents a novel tool based on an unsupervised deep‑convolutional neural‑network framework that significantly enhances image denoising and restoration for emittance measurements. The method reconstructs beam‑halo structures with unprecedented resolution, detecting signals at radii beyond seven standard deviations and particle densities below 10 ‑4 of the total beam intensity. Despite very low signal‑to‑noise ratios and small, non‑annotated datasets, the approach preserves fine structures and reveals halo features previously unobserved at pilot installations. Built on a U‑Net architecture with tailored early‑stopping strategies and physics‑informed metrics, the framework operates entirely on CPUs and requires minimal computational resources. The results demonstrate the potential of unsupervised deep learning as an enabling technology for high‑dynamic‑range beam diagnostics and motivate further development of systematic benchmarking and physics‑informed learning strategies.

Abstract:
While Physics‑Informed Neural Networks (PINNs) offer a mesh‑free approach to solving PDEs, standard point‑wise residual minimization suffers from convergence pathologies in topologically complex domains like Triply Periodic Minimal Surfaces (TPMS). The locality bias of point‑wise constraints fails to propagate global information through tortuous channels, causing unstable gradients and conservation violations. To address this, we propose the Multi‑scale Weak‑form PINN (MUSA‑PINN), which reformulates PDE constraints as integral conservation laws over hierarchical spherical control volumes. We enforce continuity and momentum conservation via flux‑balance residuals on control surfaces. Our method utilizes a three‑scale subdomain strategy‑comprising large volumes for long‑range coupling, skeleton‑aware meso‑scale volumes aligned with transport pathways, and small volumes for local refinement‑alongside a two‑stage training schedule prioritizing continuity. Experiments on steady incompressible flow in TPMS geometries show MUSA‑PINN outperforms state‑of‑the‑art baselines, reducing relative errors by up to 93% and preserving mass conservation.

Abstract:
This work integrates the physics‑informed neural network (PINN) approach into the neural quantum state framework to simulate open quantum system dynamics, to circumvent the computationally expensive time‑dependent variational principle required in conventional variational methods. The proposed PINN‑DQME method employs time‑encoded neural networks within a time‑domain decomposition strategy to represent the evolution governed by the dissipaton‑embedded quantum master equation (DQME). We implement and validate this approach in the single‑impurity Anderson model, benchmarking the PINN‑DQME results against the numerically exact hierarchical equations of motion. The PINN‑DQME method demonstrates high accuracy in simulating quantum dissipative dynamics at high temperatures, where non‑Markovian effects are weak. However, for strongly non‑Markovian dynamics at low temperatures, it encounters challenges with error accumulation during time propagation, highlighting an area for future refinement in applying PINNs to complex quantum dynamical settings.

Abstract:
We extract the universal small‑x dipole scattering amplitude N(r,x_B) from a global analysis based on a physics‑informed neural network (PINN), without imposing a priori MV‑type parametrization of the initial condition. The network provides a smooth and differentiable surrogate for N(r,x_B), whose rapidity dependence is constrained by the collinearly improved Balitsky‑‑Kovchegov evolution equation, while its functional form is simultaneously constrained by Deep Inelastic Scattering (DIS) data for the reduced total and charm cross sections, exclusive J/ψ photoproduction measurements, and a positivity requirement for the momentum‑space dipole amplitude. The resulting single universal amplitude consistently describes all fitted observables within a unified framework, alleviating the long‑standing tension between total and charm channels encountered in conventional small‑x fits based on rigid parametric ansätze. Within the fitted kinematic domain, the best extracted PINN solution yields a smooth, non‑negative momentum‑space dipole over the full transverse‑momentum range examined. Our results provide a robust and well‑behaved input for Color Glass Condensate phenomenology across a broad class of high‑energy processes.

Abstract:
Embeddings provide low‑dimensional representations that organize complex function spaces and support generalization. They provide a geometric representation that supports efficient retrieval, comparison, and generalization. In this work we generalize the concept to Physics Informed Neural Networks. We present a method to construct solution embedding spaces of nonlinear partial differential equations using a multi‑head setup, and extract non‑degenerate information from them using principal component analysis (PCA). We test this method by applying it to viscous Burgers' equation, which is solved simultaneously for a family of initial conditions and values of the viscosity. A shared network body learns a latent embedding of the solution space, while linear heads map this embedding to individual realizations. By enforcing orthogonality constraints on the heads, we obtain a principal‑component decomposition of the latent space that is robust to training degeneracies and admits a direct physical interpretation. The obtained components for Burgers' equation exhibit rapid saturation, indicating that a small number of latent modes captures the dominant features of the dynamics.

Abstract:
For robots to navigate safely and efficiently on soft, granular terrains, it is crucial to gather information about the terrain's mechanical properties, which directly affect locomotion performance. Recent research has developed robotic legs that can accurately sense ground reaction forces during locomotion. However, existing tests of granular property estimation often rely on specific foot trajectories, such as vertical penetration or horizontal shear, limiting their applicability during natural locomotion. To address this limitation, we introduce a physics‑informed machine learning framework, Inverse Resistive Force Theory (I‑RFT), which integrates the Granular Resistive Force Theory model with Gaussian Processes to infer terrain properties from proprioceptively measured contact forces under arbitrary gait trajectories. By embedding the granular force model within the learning process, I‑RFT preserves physical consistency while enabling generalization across diverse motion primitives. Experimental results demonstrate that I‑RFT accurately estimates terrain properties across multiple gait trajectories and toe shapes. Moreover, we show that the quantified uncertainty over the terrain resistance stress map could enable robots to optimize foot design and gait trajectories for efficient information gathering. This approach establishes a new foundation for data‑efficient characterization of complex granular environments and opens new avenues for locomotion strategies that actively adapt gait for autonomous terrain exploration.

Abstract:
Coupling physics with machine learning models has shown great potential for solving fluid dynamics problems governed by partial differential equations. However, conventional methods, such as physics‑informed neural networks, often suffer from slow convergence, unstable training, and limited generalization across different flow conditions. To overcome these challenges, this study proposes a novel meta‑learning en‑ hanced physics‑informed neural networks (Meta‑PINNs) framework, which integrates a meta‑optimization strategy into the training process. The approach allows the model to automatically adapt its learning process to varying physical regimes, thereby substantially improving both training efficiency and predictive robustness. The proposed Meta‑PINNs model is evaluated on two representative flow problems: (1) unsteady flow around a circular cylinder at multiple inlet Reynolds numbers, and (2) steady turbulent flow within a compressor cascade passage at various angles of attack. In both cases, the extrapolation performance of the developed framework is comprehensively tested by predicting the flow fields at Reynolds numbers and angles of attack that are not included in the training set. The results demonstrate that Meta‑PINNs achieve a 1‑2 order‑of‑magnitude improvement in accuracy over vanilla physics‑informed neural networks and standard neural networks, while reducing computational cost by up to 95.7 % and 92.1 %, respectively. It successfully captures the sequential patterns of key flow features such as pressure and velocity distributions under unseen conditions. Thus, the findings confirm that the Meta‑PINNs framework offers a notable improvement in convergence and generalization over existing machine learning approaches, providing a promising pathway toward smart simulations of complex turbomachinery flows.

Abstract:
Computational Fluid Dynamics (CFD) is an important approach for analyzing flow phenomena and predicting engineering‑relevant quantities. The governing physics is formulated as partial differential equations(PDEs) and solved numerically on computational grids. Physics‑informed neural networks(PINNs) have emerged as a popular optimization‑based approach for solving PDEs, but they often suffer from ill‑conditioned objectives and the high cost of automatic differentiation. Optimization‑based discretizations such as ODIL mitigate several PINN drawbacks by optimizing discrete variables directly, yet accuracy and efficiency remain limited on body‑fitted geometries and for time‑dependent problems. This paper proposes FDTO, a finite‑difference time‑stepping loss‑optimization solver that defines physics losses from discrete residuals. FDTO couples curvilinear coordinate transforms with body‑fitted structured grids and decomposes long‑horizon evolution into sequential, well‑conditioned subproblems consistent with time marching. The method is primarily evaluated on incompressible Navier‑Stokes flows, including lid‑driven cavity benchmarks, external airfoil aerodynamics (lift/drag consistency), and a cylinder case on a multi‑block structured mesh with cross‑block coherent solutions. Additional validations on diffusion and flow‑mixing problems further demonstrate generality. Compared with representative PINN‑based solvers, FDTO reduces GPU memory by about 82.6% on the lid‑driven cavity case and achieves 3‑5 times lower relative error on the flow‑mixing problem. These results indicate that FDTO enables accurate, stable, and memory‑efficient discrete‑loss optimization for incompressible‑flow solutions, while remaining applicable to other PDE models.

Abstract:
Locomotion on granular slopes such as sand dunes remains a fundamental challenge for legged robots due to reduced shear strength and gravity‑induced anisotropic yielding of granular media. Using a hexapedal robot on a tiltable granular bed, we systematically measure locomotion speed together with slope‑dependent normal and shear granular resistive forces. While normal penetration resistance remains nearly unchanged with inclination, shear resistance decreases substantially as slope angle increases. Guided by these measurements, we develop a simple robot‑terrain interaction model that predicts anchoring timing, step length, and resulting robot speed, as functions of terrain strength and slope angle. The model reveals that slope‑induced performance loss is primarily governed by delayed anchoring and increased backward slip rather than excessive sinkage. By extending the model to generalized terrain conditions, we construct failure phase diagrams that identify sinkage‑ and slippage‑induced failure regimes, enabling quantitative risk estimation for locomotion on granular slopes. This physics‑informed framework provides predictive insight into terrain‑dependent failure mechanisms and offers guidance for safer and more robust robot operation on deformable inclines.

Abstract:
Ferroelectric field‑effect transistors (FeFET)‑based vertical NAND (Fe‑VNAND) has emerged as a promising candidate to overcome z‑scaling limitations with lower programming voltages. However, the data retention of 3D Fe‑VNAND is hindered by the complex interaction between charge detrapping and ferroelectric depolarization. Developing optimized device designs requires exploring an extensive parameter space, but the high computational cost of conventional Technology Computer‑Aided Design (TCAD) tools makes such wide‑scale optimization impractical. To overcome these simulation barriers, we present a Physics‑Informed Neural Operator (PINO)‑based AI surrogate model designed for high‑efficiency prediction of threshold voltage (Vth) shifts and retention behavior. By embedding fundamental physical principles into the learning architecture, our PINO framework achieves a speedup exceeding 10000x compared to TCAD while maintaining physical accuracy. The resulting surrogate provides a physics‑consistent data engine for compact model parameter extraction and look‑up‑table (LUT) generation, directly supporting reliability‑aware SPICE simulation of Fe‑VNAND. This study demonstrates the model's effectiveness on a single FeFET configuration, serving as a pathway toward modeling the retention loss mechanisms.

Abstract:
Data scarcity is a primary obstacle in developing robust Machine Learning (ML) models for detecting rapidly intensifying tropical cyclones. Traditional data augmentation techniques (rotation, flipping, brightness adjustment) fail to preserve the physical consistency and high‑intensity gradients characteristic of rare Category 4‑equivalent events, which constitute only 0.14% of our dataset (202 of 140,514 samples). We propose a physics‑informed diffusion model based on the Context‑UNet architecture to generate synthetic, multi‑spectral satellite imagery of extreme weather events. Our model is conditioned on critical atmospheric parameters such as average wind speed, type of Ocean and stage of development (early, mature, late etc) ‑‑ the known drivers of rapid intensification. Using a controlled pre‑generated noise sampling strategy and mixed‑precision training, we generated 16×16 wind‑field samples that are cropped from multi‑spectral satellite imagery which preserve realistic spatial autocorrelation and physical consistency. Results demonstrate that our model successfully learns discriminative features across ten distinct context classes, effectively mitigating the data bottleneck. Specifically, we address the extreme class imbalance in our dataset, where Class 4 (Ocean 2, early stage with average wind speed 50kn hurricane) contains only 202 samples compared to 79,768 samples in Class 0. This generative framework provides a scalable solution for augmenting training datasets for operational weather detection algorithms. The average Results yield an average Log‑Spectral Distance (LSD) of 4.5dB, demonstrating a scalable framework for enhancing operational weather detection algorithms.

Abstract:
Motion mimicking, i.e., encouraging the control policy to mimic human motion, facilitates the learning of complex tasks via reinforcement learning (RL) for humanoid robots. Although standard RL frameworks demonstrate impressive locomotion agility, they often bypass explicit reasoning about robot dynamics during deployment, which is a design choice that can lead to physically infeasible commands when the robot encounters out‑of‑distribution environments. By integrating model‑based principles, hybrid approaches can improve performance; however, existing methods typically rely on predefined contact timing, limiting their versatility. This paper introduces HybridMimic, a framework in which a learned policy dynamically modulates a centroidal‑model‑based controller by predicting continuous contact states and desired centroidal velocities. This architecture exploits the physical grounding of centroidal dynamics to generate feedforward torques that remain feasible even under domain shift. Using physics‑informed rewards, the policy is trained to efficiently utilize the centroidal controller's optimization by outputting precise control targets and reference torques. Through hardware experiments on the Booster T1 humanoid, HybridMimic reduces the average base position tracking error by 13% compared to a state‑of‑the‑art RL baseline, demonstrating the robustness of dynamics‑aware deployment.

Abstract:
Solving flow through porous media is a crucial step in the topology optimisation of cold plates, a key component in modern thermal management. Traditional computational fluid dynamics (CFD) methods, while accurate, are often prohibitively expensive for large and complex geometries. In contrast, data‑driven surrogate models provide a computationally efficient alternative, enabling rapid and reliable predictions. In this study, we develop a machine‑learning framework for predicting steady‑state flow through porous media governed by the Navier‑Stokes‑Brinkman equations. We implement and compare three model architectures‑convolutional autoencoder (AE), U‑Net, and Fourier Neural Operator (FNO)‑evaluating their predictive performance. To enhance physics consistency, we incorporate physics‑informed loss functions. Our results demonstrate that FNO outperforms AE and U‑Net, achieving a mean squared error (MSE) as low as 0.0017 while providing speedups of up to 1000 times compared to CFD. Additionally, the mesh‑invariant property of FNO emphasizes its suitability for topology optimisation tasks, where varying mesh resolutions are required. This study highlights the potential of machine learning to accelerate fluid flow predictions in porous media, offering a scalable alternative to traditional numerical methods.

Abstract:
Physics‑Informed Neural Networks (PINNs) enforce governing equations by penalizing PDE residuals at interior collocation points, but standard collocation strategies ‑ uniform sampling and residual‑based adaptive refinement ‑ can oversample smooth regions, produce highly correlated point sets, and incur unnecessary training cost. We reinterpret collocation selection as a coreset construction problem: from a large candidate pool, select a fixed‑size subset that is simultaneously informative (high expected impact on reducing PDE error) and diverse (low redundancy under a space‑time similarity notion). We formulate this as a QUBO/BQM objective with linear terms encoding residual‑based importance and quadratic terms discouraging redundant selections. To avoid the scalability issues of dense k‑hot QUBOs, we propose a sparse graph‑based BQM built on a kNN similarity graph and an efficient repair procedure that enforces an exact collocation budget. We further introduce hybrid coverage anchors to guarantee global PDE enforcement. We evaluate the method on the 1D time‑dependent viscous Burgers equation with shock formation and report both accuracy and end‑to‑end time‑to‑accuracy, including a timing breakdown of selection overhead. Results demonstrate that sparse and hybrid formulations reduce selection overhead relative to dense QUBOs while matching or improving accuracy at fixed collocation budgets.

Abstract:
We propose a physics‑informed anomaly detection framework for collider data based on a Bayesian latent diffusion model. Our method combines a probabilistic encoder with diffusion dynamics in the latent space, allowing for stable and flexible density estimation while explicitly enforcing physics constraints, such as mass decorrelation and regularization of latent correlations. We train and test the model on simulated LHC jet data and evaluate its performance using seed‑averaged ROC curves together with discovery‑oriented metrics. Through a series of ablation studies, we show that the diffusion process, Bayesian regularization, and physics‑motivated loss terms each contribute in a complementary way: they help stabilize training and improve generalization, even when the gains in peak performance are moderate. Overall, our results emphasize the importance of incorporating both uncertainty estimates and physics consistency when building reliable anomaly detection methods for new Physics searches in high‑energy physics.

Abstract:
Modeling stiff partial differential equations (PDEs) with sharp gradients remains a significant challenge for scientific machine learning. While Physics‑Informed Neural Networks (PINNs) struggle with spectral bias and slow training times, Physics‑Informed Extreme Learning Machines (PIELMs) offer a rapid, closed‑form linear solution but are fundamentally limited by physics‑agnostic, random initialization. We introduce the Gaussian Mixture Model Adaptive PIELM (GMM‑PIELM), a probabilistic framework that learns a probability density function representing the ``location of physics'' for adaptively sampling kernels of PIELMs. By employing a weighted Expectation‑Maximization (EM) algorithm, GMM‑PIELM autonomously concentrates radial basis function centers in regions of high numerical error, such as shock fronts and boundary layers. This approach dynamically improves the conditioning of the hidden layer without the expensive gradient‑based optimization(of PINNs) or Bayesian search. We evaluate our methodology on 1D singularly perturbed convection‑diffusion equations with diffusion coefficients ν=10^‑4. Our method achieves L_2 errors up to 7 orders of magnitude lower than baseline RBF‑PIELMs, successfully resolving exponentially thin boundary layers while retaining the orders‑of‑magnitude speed advantage of the ELM architecture.

Abstract:
Next‑generation microreactors enable remote deployment and semi‑autonomous operation, but compact, sealed, heterogeneous cores limit conventional safeguard approaches that rely on access and bulk accountancy. Limited inspection access and complex internal geometry reduce sensitivity to localized anomalies such as missing fuel. Here we demonstrate missing‑fuel detection in microreactor scale geometries using muon scattering tomography under realistic cosmic‑ray conditions. We introduce μTRec, a physics‑informed framework that reconstructs event‑level curved muon trajectories by combining a Gaussian multiple Coulomb scattering model with Bayesian updating, then maps scattering density through voxel wise M‑values for core integrity verification. We evaluate a representative hexagonal core containing 61 fuel flakes with embedded control drums and shutdown rods, using both idealized 5 GeV muons and zenith‑angle‑dependent 0‑60 GeV cosmic‑ray spectra. A single missing fuel flake is detected with 3× 10^6 muons at 50 mm voxel resolution. Incorporating per‑muon momentum further increases detectability by up to 149.85% for laser‑driven sources and 105.11% for cosmic‑ray sources relative to momentum‑agnostic reconstruction. The approach remains robust under practical detector limits, with only an 8.88% reduction in detectability for 10 mm spatial resolution and 10% energy resolution. Compared with PoCA, μTRec delivers 326.13% to 392.14% higher detectability at equal muon counts, enabling faster defect identification.

Abstract:
We train machine learning algorithms to infer the entanglement structure of disordered long‑range interacting quantum spin chains by learning from the strong disorder renormalisation group (SDRG) method. The system consists of S=1/2‑quantum spins coupled by antiferromagnetic power‑law interactions with decay exponent α at random positions on a one‑dimensional chain. Using SDRG as a physics‑informed teacher, we compare a Random Forest classifier as a classical baseline with a graph neural network (GNN) that operates directly on the interaction graph and learns a bond‑ranking rule mirroring the SDRG decimation policy. The GNN achieves a disorder‑averaged pairing accuracy close to one and reproduces the entanglement entropy S(\ell) in excellent quantitative agreement with SDRG across all subsystem sizes and interaction exponents. RG flow heat maps confirm that the GNN learns the sequential decimation hierarchy rather than merely fitting final‑state observables. Finite‑temperature entanglement properties are incorporated via the SDRGX framework through a two‑stage strategy, using the zero‑temperature GNN to generate the RG flow and sampling thermal occupations from the canonical ensemble, yielding results in agreement with both numerical SDRGX and analytical predictions without retraining.

Abstract:
Ground‑based solar observations enable unprecedented spatial, spectral, and temporal resolution of the lower solar atmosphere, yet Earths turbulent atmosphere imposes significant limitations, requiring advanced post‑facto image reconstruction. State‑of‑the‑art reconstruction methods are based on restoring a burst of short exposure frames to a single observation. Limitations of these techniques arise due to the sparse information about the atmospheric point spread function (PSF) that degrade the observations and consequently the quality of reconstructions. We develop a novel image reconstruction method to achieve unprecedented spatial resolution from short exposure image bursts. This can provide high‑quality reconstructions and therefore advance the study of the smallest spatial scales from the solar photosphere to the chromosphere. In this study, we present a novel approach for high‑resolution solar image reconstruction based on physics‑informed neural networks. In the training process, the neural network maps coordinate points directly to their corresponding intensity values while simultaneously updating the PSF parameters. The method convolves the true object from the neural network with the estimated PSFs and optimizes the network by minimizing the loss between the synthesized and real short‑exposure image burst. This approach enables the simultaneous estimation of both the degrading PSF and the real high‑resolution intensity distribution. We demonstrate the method on synthetic intensity data derived from a radiative MHD simulation and apply it to high‑resolution observations from GREGOR and DKIST. Our results demonstrate the ability to reconstruct small‑scale solar features that exceed the reconstruction performance of state‑of‑the‑art reconstruction methods. With this approach we lay the foundation for future spatially varying PSFs.

Abstract:
While RAG has greatly enhanced LLMs, extending this paradigm to Time‑Series Foundation Models (TSFMs) remains a challenge. This is exemplified in the Predictive Maintenance of the Pressure Regulating and Shut‑Off Valve (PRSOV), a high‑stakes industrial scenario characterized by (1) data scarcity, (2) short transient sequences, and (3) covariate coupled dynamics. Unfortunately, existing time‑series RAG approaches predominantly rely on generated static vector embeddings and learnable context augmenters, which may fail to distinguish similar regimes in such scarce, transient, and covariate coupled scenarios. To address these limitations, we propose RAG4CTS, a regime‑aware, training‑free RAG framework for Covariate Time‑Series. Specifically, we construct a hierarchal time‑series native knowledge base to enable lossless storage and physics‑informed retrieval of raw historical regimes. We design a two‑stage bi‑weighted retrieval mechanism that aligns historical trends through point‑wise and multivariate similarities. For context augmentation, we introduce an agent‑driven strategy to dynamically optimize context in a self‑supervised manner. Extensive experiments on PRSOV demonstrate that our framework significantly outperforms state‑of‑the‑art baselines in prediction accuracy. The proposed system is deployed in Apache IoTDB within China Southern Airlines. Since deployment, our method has successfully identified one PRSOV fault in two months with zero false alarm.

Abstract:
We present a data‑driven investigation of the exhaustive ensemble of no‑scale type IIB flux vacua constructed in \citeChauhan:2025rdj. Using a combination of linear and non‑linear dimensionality‑reduction techniques, we analyse both flux and moduli spaces and demonstrate that the effective dimensionality of the underlying 12‑dimensional flux space is substantially reduced. A central component of our study is a physics‑informed autoencoder, which provides a non‑linear compression of the flux and moduli data into a low‑dimensional latent space. The learned latent representation organises vacua according to desired features and, in particular, isolates distinguished regions associated with small values of the flux superpotential |W_0|, revealing non‑trivial correlations that are not captured by linear methods. In parallel, we apply tools from topological data analysis, specifically persistent homology, to probe the global structure of the vacuum distribution. This allows us to identify robust, long‑lived topological features in both moduli and flux subspaces. This work is a necessary step for developing foundation models in string phenomenology.

Abstract:
Accurate time series forecasting underpins decision‑making across domains, yet conventional ML development suffers from data scarcity in new deployments, poor adaptability under distribution shift, and diminishing returns from manual iteration. We propose Self‑Evolving Agent for Time Series Algorithms (SEA‑TS), a framework that autonomously generates, validates, and optimizes forecasting code via an iterative self‑evolution loop. Our framework introduces three key innovations: (1) Metric‑Advantage Monte Carlo Tree Search (MA‑MCTS), which replaces fixed rewards with a normalized advantage score for discriminative search guidance; (2) Code Review with running prompt refinement, where each executed solution undergoes automated review followed by prompt updates that encode corrective patterns, preventing recurrence of similar errors; and (3) Global Steerable Reasoning, which compares each node against global best and worst solutions, enabling cross‑trajectory knowledge transfer. We adopt a MAP‑Elites archive for architectural diversity. On the public Solar‑Energy benchmark, SEA‑TS generated code achieves a 40% MAE reduction relative to TimeMixer, surpassing state‑of‑the‑art methods. On proprietary datasets, SEA‑TS generated code reduces WAPE by 8.6% on solar PV forecasting and 7.7% on residential load forecasting compared to human‑engineered baselines, and achieves 26.17% MAPE on load forecasting versus 29.34% by TimeMixer. Notably, the evolved models discover novel architectural patterns‑‑including physics‑informed monotonic decay heads encoding solar irradiance constraints, per‑station learned diurnal cycle profiles, and learnable hourly bias correction‑‑demonstrating that autonomous ML engineering can generate genuinely novel algorithmic ideas beyond manual design.

Abstract:
Algorithmic speedup of training common neural architectures is made difficult by the lack of structure guaranteed by the function compositions inherent to such networks. In contrast to multilayer perceptrons (MLPs), Kolmogorov‑Arnold networks (KANs) provide more structure by expanding learned activations in a specified basis. This paper exploits this structure to develop practical algorithms and theoretical insights, yielding training speedup via multilevel training for KANs. To do so, we first establish an equivalence between KANs with spline basis functions and multichannel MLPs with power ReLU activations through a linear change of basis. We then analyze how this change of basis affects the geometry of gradient‑based optimization with respect to spline knots. The KANs change‑of‑basis motivates a multilevel training approach, where we train a sequence of KANs naturally defined through a uniform refinement of spline knots with analytic geometric interpolation operators between models. The interpolation scheme enables a ``properly nested hierarchy'' of architectures, ensuring that interpolation to a fine model preserves the progress made on coarse models, while the compact support of spline basis functions ensures complementary optimization on subsequent levels. Numerical experiments demonstrate that our multilevel training approach can achieve orders of magnitude improvement in accuracy over conventional methods to train comparable KANs or MLPs, particularly for physics informed neural networks. Finally, this work demonstrates how principled design of neural networks can lead to exploitable structure, and in this case, multilevel algorithms that can dramatically improve training performance.

Abstract:
Neural networks offer powerful tools to solve partial differential equations (PDEs). We present a Variational Physics‑Informed Neural Network (VPINN) designed for parabolic problems. Our approach combines a classical time discretization with a composed loss function, which minimizes the residual's dual norm at every time step. We validate the framework by modeling the freezing of coffee extracts in an industrial cylinder. The simulation accounts for temperature‑dependent properties and experimental data. It successfully captures the thermal dynamics of the process.

Abstract:
Physics‑informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not obvious, with the result that they typically yield moderately accurate solutions. In this article, we propose a method for improving the accuracy of PINNs by coupling them with a post‑processing step that seeks the best approximation in a function space associated with the network. We find that our method yields errors four to five orders of magnitude lower than those of the parent PINNs across architectures and dimensions. Moreover, we can reuse the basis functions for the linear space in more complex settings, such as time‑dependent and nonlinear problems, allowing for transfer learning. Our approach also provides a residual‑based metric that allows us to optimally choose the number of basis functions employed.

Abstract:
Foundation models in general promise to accelerate scientific computation by learning reusable representations across problem instances, yet constrained scientific systems, where predictions must satisfy physical laws and safety limits, pose unique challenges that stress conventional training paradigms. We derive design principles for constrained scientific foundation models through systematic investigation of AC optimal power flow (ACOPF), a representative optimization problem in power grid operations where power balance equations and operational constraints are non‑negotiable. Through controlled experiments spanning architectures, training objectives, and system diversity, we extract three empirically grounded principles governing scientific foundation model design. These principles characterize three design trade‑offs: learning physics‑invariant representations while respecting system‑specific constraints, optimizing accuracy while ensuring constraint satisfaction, and ensuring reliability in high‑impact operating regimes. We present the LUMINA framework, including data processing and training pipelines to support reproducible research on physics‑informed, feasibility‑aware foundation models across scientific applications.

Abstract:
We propose Fluid Logic, a paradigm in which modal logical reasoning, temporal, epistemic, doxastic, deontic, is lifted from discrete Kripke structures to continuous manifolds via Neural Stochastic Differential Equations (Neural SDEs). Each type of modal operator is backed by a dedicated Neural SDE, and nested formulas compose these SDEs in a single differentiable graph. A key instantiation is Logic‑Informed Neural Networks (LINNs): analogous to Physics‑Informed Neural Networks (PINNs), LINNs embed modal logical formulas such as (\Box bounded) and (\Diamond visits\_lobe) directly into the training loss, guiding neural networks to produce solutions that are structurally consistent with prescribed logical properties, without requiring knowledge of the governing equations. The resulting framework, Continuous Modal Logical Neural Networks (CMLNNs), yields several key properties: (i) stochastic diffusion prevents quantifier collapse (\Box and \Diamond differ), unlike deterministic ODEs; (ii) modal operators are entropic risk measures, sound with respect to risk‑based semantics with explicit Monte Carlo concentration guarantees; (iii)SDE‑induced accessibility provides structural correspondence with classical modal axioms; (iv) parameterizing accessibility through dynamics reduces memory from quadratic in world count to linear in parameters. Three case studies demonstrate that Fluid Logic and LINNs can guide neural networks to produce consistent solutions across diverse domains: epistemic/doxastic logic (multi‑robot hallucination detection), temporal logic (recovering the Lorenz attractor geometry from logical constraints alone), and deontic logic (learning safe confinement dynamics from a logical specification).

Abstract:
Inverse problems are the task of calibrating models to match data. They play a pivotal role in diverse engineering applications by allowing practitioners to align models with reality. In many applications, engineers and scientists do not have a complete picture of i) the detailed properties of a system (such as material properties, geometry, initial conditions, etc.); ii) the complete laws describing all dynamics at play (such as friction laws, complicated damping phenomena, and general nonlinear interactions). In this paper, we develop a principled methodology for leveraging data from collections of distinct yet related physical systems to jointly estimate the individual model parameters of each system, and learn the shared unknown dynamics in the form of an ML‑based closure model. To robustly infer the unknown parameters for each system, we employ a hierarchical Bayesian framework, which allows for the joint inference of multiple systems and their population‑level statistics. To learn the closures, we use a maximum marginal likelihood estimate of a neural network embeded within the ODE/PDE formulation of the problem. To realize this framework we utilize the ensemble Metropolis‑Adjusted Langevin Algorithm (MALA) for stable and efficient sampling. To mitigate the computational bottleneck of repetitive forward evaluations in solving inverse problems, we introduce a bilevel optimization strategy to simultaneously train a surrogate forward model alongside the inference. Within this framework, we evaluate and compare distinct surrogate architectures, specifically Fourier Neural Operators (FNO) and parametric Physics‑Informed Neural Network (PINNs).

Abstract:
Cardiac arrhythmogenesis is governed by complex electromechanical interactions that are not directly observable in vivo, motivating the development of non‑invasive computational approaches for reconstructing three‑dimensional activation dynamics. We present a physics‑informed neural network framework for recovering cardiac activation patterns, active tension propagation, deformation fields, and hydrostatic pressure from measurable deformation data in simplified left ventricular geometries. Our approach integrates nonlinear anisotropic constitutive modeling, heterogeneous fiber orientation, weak formulations of the governing mechanics, and finite‑element‑based loss functions to embed physical constraints directly into training. We demonstrate that the proposed framework accurately reconstructs spatiotemporal activation dynamics under varying levels of measurement noise and reduced spatial resolution, while preserving global propagation patterns and activation timing. By coupling mechanistic modeling with data‑driven inference, this method establishes a pathway toward patient‑specific, non‑invasive reconstruction of cardiac activation, with potential applications in digital phenotyping and computational support for arrhythmia assessment.

Abstract:
The numerical simulation of convection‑dominated transient transport phenomena poses significant computational challenges due to sharp gradients and propagating fronts across the spatiotemporal domain. Classical discretization methods often generate spurious oscillations, requiring advanced stabilization techniques. However, even stabilized finite element methods may require additional regularization to accurately resolve localized steep layers. On the other hand, standalone physics‑informed neural networks (PINNs) struggle to capture sharp solution structures in convection‑dominated regimes and typically require a large number of training epochs. This work presents a hybrid computational framework that extends the PINN‑Augmented SUPG with Shock‑Capturing (PASSC) methodology from steady to unsteady problems. The approach combines a semi‑discrete stabilized finite element method with a PINN‑based correction strategy for transient convection‑diffusion‑reaction equations. Stabilization is achieved using the Streamline‑Upwind Petrov‑Galerkin (SUPG) formulation augmented with a YZbeta shock‑capturing operator. Rather than training over the entire space‑time domain, the neural network is applied selectively near the terminal time, enhancing the finite element solution using the last K_s temporal snapshots while enforcing residual constraints from the governing equations and boundary conditions. The network incorporates residual blocks with random Fourier features and employs progressive training with adaptive loss weighting. Numerical experiments on five benchmark problems, including boundary and interior layers, traveling waves, and nonlinear Burgers dynamics, demonstrate significant accuracy improvements at the terminal time compared to standalone stabilized finite element solutions.

Abstract:
Physics‑Informed Neural Networks (PINNs) have been recognized as a mesh‑free alternative to solve partial differential equations where physics information is incorporated. However, in dealing with problems characterized by high stiffness or shock‑dominated dynamics, traditional PINNs have been found to have limitations, including unbalanced training and inaccuracy in solution, even with small physics residuals. In this research, we seek to address these limitations using the viscous Burgers' equation with low viscosity and the Allen‑Cahn equation as test problems. In addressing unbalanced training, we have developed a new adaptive loss balancing scheme using smoothed gradient norms to ensure satisfaction of initial and boundary conditions. Further, to address inaccuracy in the solution, we have developed an adaptive residual‑based collocation scheme to improve the accuracy of solutions in the regions with high physics residuals. The proposed new approach significantly improves solution accuracy with consistent satisfaction of physics residuals. For instance, in the case of Burgers' equation, the relative L2 error is reduced by about 44 percent compared to traditional PINNs, while for the Allen‑Cahn equation, the relative L2 error is reduced by approximately 70 percent. Additionally, we show the trustworthy solution comparison of the proposed method using a robust finite difference solver.

Abstract:
In the present work we construct a novel generative architecture for systems with complex probability distributions. In general, these sampling tasks come with two challenges: resolving sign problems and efficient sampling. The architecture is based on physics‑informed kernels (PIKs) introduced in arXiv:2510.26678, and aims at resolving both challenges. Key to the complex PIK‑architecture is its probability‑weight preserving property, which allows us to map the sampling task to one on a sign‑problem free manifold with a simple distribution and efficient sampling. The potential of this novel architecture is demonstrated within applications to zero‑dimensional field theories with complex couplings, as well as the real‑time evolution of the quantum‑mechanical harmonic oscillator.

Abstract:
Analyzing nonlinear systems with attracting robust invariant sets (RISs) requires estimating their domains of attraction (DOAs). Despite extensive research, accurately characterizing DOAs for general nonlinear systems remains challenging due to both theoretical and computational limitations, particularly in the presence of uncertainties and state constraints. In this paper, we propose a novel framework for the accurate estimation of safe (state‑constrained) and robust DOAs for discrete‑time nonlinear uncertain systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally \ell_p‑stable compact RISs. The notion of uniform \ell_p stability is quite general and encompasses, as special cases, uniform exponential and polynomial stability. The DOAs are characterized via newly introduced value functions defined on metric spaces of compact sets. We establish their fundamental mathematical properties and derive the associated Bellman‑type (Zubov‑type) functional equations. Building on this characterization, we develop a physics‑informed neural network (NN) framework to learn the corresponding value functions by embedding the derived Bellman‑type equations directly into the training process. To obtain certifiable estimates of the safe robust DOAs from the learned neural approximations, we further introduce a verification procedure that leverages existing formal verification tools. The effectiveness and applicability of the proposed methodology are demonstrated through four numerical examples involving nonlinear uncertain systems subject to state constraints, and its performance is compared with existing methods from the literature.

Abstract:
Physics‑Informed Neural Networks (PINNs) incorporate physics into neural networks by embedding partial differential equations (PDEs) into their loss function. Despite their success in learning the underlying physics, PINN models remain difficult to train and interpret. In this work, a novel modeling approach is proposed, which relies on the use of Domain‑aware Fourier Features (DaFFs) for the positional encoding of the input space. These features encapsulate all the domain‑specific characteristics, such as the geometry and boundary conditions, and unlike Random Fourier Features (RFFs), eliminate the need for explicit boundary condition loss terms and loss balancing schemes, while simplifying the optimization process and reducing the computational cost associated with training. We further develop an LRP‑based explainability framework tailored to PINNs, enabling the extraction of relevance attribution scores for the input space. It is demonstrated that PINN‑DaFFs achieve orders‑of‑magnitude lower errors and allow faster convergence compared to vanilla PINNs and RFFs‑based PINNs. Furthermore, LRP analysis reveals that the proposed leads to more physically consistent feature attributions, while PINN‑RFFs and vanilla PINNs display more scattered and less physics‑relevant patterns. These results demonstrate that DaFFs not only enhance PINNs' accuracy and efficiency but also improve interpretability, laying the ground for more robust and informative physics‑informed learning.

Abstract:
We introduce a generalizable framework for learning to identify effective Hamiltonians directly from experimental data in solid‑state quantum systems. Our approach is based on a physics‑informed neural network architecture that embeds physical constraints directly into the model structure. Unlike purely data‑driven supervised schemes, the proposed unsupervised autoencoder‑based method incorporates the governing physics (here, the S‑matrix formalism) within the decoder network, ensuring that the learned representations remain physically meaningful. Through numerical learning experiments, we demonstrate automated characterization of programmable solid‑state simulators from transport measurements, exemplified by a triple quantum dot chain. The trained model generalizes beyond the training domain and accurately infers Hamiltonian parameters from transport data. While the model has finite capacity ‑‑ leading to degraded performance when the parameter space becomes excessively large or structurally diverse ‑‑ we identify regimes in which robust generalization is maintained. We further show how to train the model to handle noisy measurements, reflecting realistic experimental conditions.

Abstract:
Understanding teleconnections of large‑scale modes of climate variability is relevant for seasonal predictability and support a dynamical understanding of climatic changes. While numerical model experiments are the most common approach for investigating counterfactual climate responses, their conclusions are subject to model biases. Data‑driven approaches offer a complementary perspective. Deep learning can extract reduced‑dimensional patterns but usually lacks causal interpretability, while causal methods can disentangle signals in the presence of confounding yet are typically based on simple indices. Treating dimensionality reduction and causal inference separately thereby risks losing the teleconnection signal of interest. This paper introduces DAG‑VAE, a causal representation learning approach that embeds a physics‑informed directed acyclic graph in the latent space of a variational autoencoder. Combining deep learning with causal inference, the method jointly learns nonlinear reduced representations of large‑scale modes of variability and their causal interactions. We apply DAG‑VAE to disentangle the influences of the Pacific and Indian Oceans on the short rains over the Greater Horn of Africa. Trained on seasonal hindcasts, the method identifies dynamically meaningful representations and recovers spatial response patterns consistent with SST‑replacement experiments. Trained on reanalysis data, DAG‑VAE identifies a different response pattern to direct influence of the tropical Pacific, highlighting potential model biases and the value of DAG‑VAE as a complementary, data‑driven approach for estimating spatial causal response patterns from observations. Finally, we demonstrate the ability of the method to generate data‑driven counterfactuals of extreme short rain seasons, with potential applications for forecast‑based early action and scenario planning.

Abstract:
Efficient optimization of molecules with targeted properties remains a significant challenge due to the vast size and discrete nature of chemical compound space. Conventional machine‑learning‑based optimization approaches typically require large datasets to construct accurate surrogate models, limiting their applicability in data‑scarce settings. In this study, we present a Bayesian optimization (BO) framework that identifies optimal molecular structures with high precision using fewer than 2,000 training data points within a chemical subspace containing more than 133,000 molecules. The framework employs a low‑dimensional and physics‑informed molecular descriptor vector that facilitates data‑efficient surrogate modelling and optimization. A key innovation of the proposed framework is a reliable inverse mapping scheme that translates optimized points in the descriptor space back into chemically valid molecular structures, thereby bridging continuous optimization and discrete molecular design. We demonstrate the effectiveness of our approach on the QM9 benchmark dataset, where the framework successfully identifies organic molecules with the target entropy and zero‑point vibrational energy (ZPVE) values.For entropy optimization, our approach achieves a 100% success rate while requiring fewer than 1,000 molecular evaluations in more than 80% of test cases. For ZPVE, the success rate exceeds 80% for molecules containing more than two heavy atoms. These results highlight the critical role of low‑dimensional, interpretable descriptors in enabling data‑efficient optimization and robust inverse molecular design, and establish Bayesian optimization as a practical tool for molecular discovery in small‑data regimes.

Abstract:
Internal charge amplification (ICA) in cryogenic high‑purity germanium (HPGe) can lower detection thresholds by providing gain inside the detector crystal, but reliable operation requires a predictive estimate of the avalanche‑onset \emphcritical electric field \(E_\mathrmcrit\). We present a compact framework for \(E_\mathrmcrit\) at 77~K and 4~K (typical HPGe operating temperatures) that bridges (i) a mobility‑based single‑free‑flight (SFF) upper bound with (ii) a physics‑informed impact‑ionization model incorporating energy‑dependent scattering, nonparabolic (Kane) dispersion, intervalley transfer, and the high‑energy ``lucky‑drift'' tail. This unified treatment yields closed‑form, design‑useful relations, including \(E_\mathrmcrit^(\mathrmPI)=B(T)/\ln[A(T)d]\), and a practical calibration workflow that maps measured low‑field mobility \(μ(T)\) and gain curves \(M(V)\) (Chynoweth analysis) to device‑level bias targets with propagated uncertainty bands. Example electron and hole estimates indicate that realistic transport typically lowers \(E_\mathrmcrit\) relative to SFF and increases the predicted change in \(E_\mathrmcrit\) between 77~K and 4~K. The resulting portable formulas connect materials/transport inputs to geometry, excess noise, and field shaping, providing design‑ready guidance for stable, unipolar‑favored ICA with controlled quenching in Ge and other cryogenic semiconductors.

Abstract:
Large‑scale wave field reconstruction requires precise solutions but faces challenges with computational efficiency and accuracy. The physics‑based numerical methods like Finite Element Method (FEM) provide high accuracy but struggle with large‑scale or high‑frequency problems due to prohibitive computational costs. Pure data‑driven approaches excel in speed but often lack sufficient labeled data for complex scenarios. Physics‑informed neural networks (PINNs) integrate physical principles into machine learning models, offering a promising solution by bridging these gaps. However, standard PINNs embed physical principles only in loss functions, leading to slow convergence, optimization instability, and spectral bias, limiting their ability for large‑scale wave field reconstruction. This work introduces architecture physics embedded (PE)‑PINN, which integrates additional physical guidance directly into the neural network architecture beyond Helmholtz equations and boundary conditions in loss functions. Specifically, a new envelope transformation layer is designed to mitigate spectral bias with kernels parameterized by source properties, material interfaces, and wave physics. Experiments demonstrate that PE‑PINN achieves more than 10 times speedup in convergence compared to standard PINNs and several orders of magnitude reduction in memory usage compared to FEM. This breakthrough enables high‑fidelity modeling for large‑scale 2D/3D electromagnetic wave reconstruction involving reflections, refractions, and diffractions in room‑scale domains, readily applicable to wireless communications, sensing, room acoustics, and other fields requiring large‑scale wave field analysis.

Abstract:
A primary challenge in developing synthetic spatial hearing systems, particularly underwater, is accurately modeling sound scattering. Biological organisms achieve 3D spatial hearing by exploiting sound scattering off their bodies to generate location‑dependent interaural level and time differences (ITD/ILD). While Head‑Related Transfer Function (HRTF) models based on rigid scattering suffice for terrestrial humans, they fail in underwater environments due to the near‑impedance match between water and soft tissue. Motivated by the acoustic anatomy of underwater animals, we introduce a novel, analytically derived, closed‑form forward model for scattering from a semi‑transparent sphere containing two rigid spherical scatterers. This model accurately maps source direction, frequency, and material properties to the pressure field, capturing the complex physics of layered, penetrable structures. Critically, our model is implemented in a fully differentiable setting, enabling its integration with a machine learning algorithm to optimize a cost function for active localization. We demonstrate enhanced convergence for localization under noise using a physics‑informed frequency weighting scheme, and present accurate moving‑source tracking via an Extended Kalman Filter (EKF) with analytically computed Jacobians. Our work suggests that differentiable models of scattering from layered rigid and transparent geometries offer a promising new foundation for microphone arrays that leverage scattering‑based spatial cues over conventional beamforming, applicable to both terrestrial and underwater applications. Our model will be made open source.

Abstract:
Reduced‑order models (ROMs) have become an essential tool for reducing the computational cost of fluid flow simulations. While standard ROMs can efficiently approximate laminar flows, their accuracy often suffers in convection‑dominated regimes due to the truncation of dynamically important modes. To account for the influence of unresolved scales, ROM closure models are commonly introduced. Classical closure strategies are typically based on phenomenological arguments or analogies with large eddy simulation (LES), often formulated within a variational multiscale (VMS) framework, in which the resolved and unresolved scales are explicitly separated and their interactions are systematically modeled. More recently, advances in data‑driven modeling and machine learning have opened new opportunities to construct ROM closures that are both more accurate and more consistent with the underlying physics. In this work, we develop a new ROM closure that combines machine learning with physics‑based modeling principles. The closure term is derived within a VMS framework, where the reduced solution space is decomposed into resolved and unresolved components. This VMS‑derived closure term is then modeled using PhysicsInformed Neural Networks (PINNs) and incorporated into a newly constructed C‑PINN‑ROM. The resulting closure leverages high‑fidelity data while enforcing physical constraints imposed by the reduced‑order equations, thereby ensuring consistency with the underlying dynamics and enhanced robustness in convection‑dominated regimes. Through this PINN‑based framework, we demonstrate how physics‑informed machine learning can substantially improve the accuracy and robustness of ROMs, effectively bridging classical multiscale closure modeling with state‑of‑the‑art data‑driven methodologies.

Abstract:
Detecting water‑surface targets for Unmanned Surface Vehicles (USVs) is challenging due to wave clutter, specular reflections, and weak appearance cues in long‑range observations. Although 4D millimeter‑wave radar complements cameras under degraded illumination, maritime radar point clouds are sparse and intermittent, with reflectivity attributes exhibiting heavy‑tailed variations under scattering and multipath, making conventional fusion designs struggle to exploit radar cues effectively. We propose PhysFusion, a physics‑informed radar‑image detection framework for water‑surface perception. The framework integrates: (1) a Physics‑Informed Radar Encoder (PIR Encoder) with an RCS Mapper and Quality Gate, transforming per‑point radar attributes into compact scattering priors and predicting point‑wise reliability for robust feature learning under clutter; (2) a Radar‑guided Interactive Fusion Module (RIFM) performing query‑level radar‑image fusion between semantically enriched radar features and multi‑scale visual features, with the radar branch modeled by a dual‑stream backbone including a point‑based local stream and a transformer‑based global stream using Scattering‑Aware Self‑Attention (SASA); and (3) a Temporal Query Aggregation module (TQA) aggregating frame‑wise fused queries over a short temporal window for temporally consistent representations. Experiments on WaterScenes and FLOW demonstrate that PhysFusion achieves 59.7% mAP50:95 and 90.3% mAP50 on WaterScenes (T=5 radar history) using 5.6M parameters and 12.5G FLOPs, and reaches 94.8% mAP50 and 46.2% mAP50:95 on FLOW under radar+camera setting. Ablation studies quantify the contributions of PIR Encoder, SASA‑based global reasoning, and RIFM.

Abstract:
Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well‑established numerical methods with new AI‑based techniques, specifically Physics‑Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed‑form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed‑form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.

Abstract:
We propose a multi‑patient inverse modeling framework for identifying effective calcium and citrate diffusion coefficients in hollow‑fiber hemodialysis devices. The approach relies on a coupled forward model combining axisymmetric fluid dynamics with multi‑species convection‑reaction‑diffusion, together with a derivative‑free optimization strategy to estimate membrane transport parameters from outlet concentration measurements. To account for inter‑patient variability, physiological input parameters are first generated from clinical data and complemented by a patient‑specific hydraulic calibration step, ensuring physical consistency across the synthetic cohort. The inverse problem is formulated as a global least‑squares minimization aggregating residuals over multiple patients. Numerical experiments on synthetic data demonstrate multi‑patient identifiability of the diffusion coefficients in the exact‑data setting. Robustness with respect to measurement noise is subsequently assessed by perturbing observable outputs at various noise levels, and sensitivity analyses are performed to quantify the influence of membrane transport parameters on model predictions. The methodology is then applied to real clinical data obtained from an AK200 Gambro/Nikkiso DBB07 dialysis system. The results indicate that aggregating information from several patients substantially improves parameter identifiability and stability compared to single‑patient inversions. Overall, this work provides a physically consistent and computationally tractable framework for multi‑patient parameter estimation in dialysis models, and opens perspectives for large‑scale personalization through physics‑informed surrogate modeling.

Abstract:
This paper presents the CosmicWeb‑21cm array, a novel radio interferometer designed to overcome the key challenges in 21 cm cosmology. Its core innovations include: (1) a multi‑scale nested geometry combining a hexagonal core with logarithmic spiral arms for excellent UV coverage and calibration robustness; (2) an intelligent non‑uniform frequency sampling strategy that adapts resolution to foreground and signal characteristics, reducing data volume while preserving information; and (3) a machine‑learning‑enhanced, physics‑informed processing pipeline that achieves 99.7% foreground removal efficiency; (4) a dual‑polarization crossed dipole integrated with a dielectric lens and cryogenically cooled LNA, achieving stable beam patterns and low noise temperature (<35 K) across 50‑250 MHz. These co‑designed advances enable high sensitivity mapping of the Epoch of Reionization, dark energy constraints and cosmic‑web structure.

Abstract:
In scientific and engineering domains, modeling high‑dimensional complex systems governed by partial differential equations (PDEs) remains challenging in terms of physical consistency and numerical stability. However, existing approaches, such as physics‑informed neural networks (PINNs), typically rely on known physical fields or coefficients and enforce physical constraints via external loss functions, which can lead to training instability and make it difficult to handle high‑dimensional or unobservable scenarios. To this end, we propose PhysFormer, a generative modeling framework that is self‑consistent at both the data and physical levels. PhysFormer leverages a low‑dimensional, physically interpretable latent space to learn key physical quantities directly from data without requiring known high‑dimensional physical field parameters, and embeds the physical process of radiative flux generation within the network to ensure the physical consistency of the generated spectra. In high‑dimensional, degenerate inversion tasks, PhysFormer constrains generation within physical limits and enhances spectral fidelity and inversion stability under varying signal‑to‑noise ratios (SNRs). More broadly, this approach shifts the physical processes from external loss functions into the generative mechanism itself, providing a physically consistent generative modeling paradigm for complex systems involving unknown or unobservable physical quantities.

Abstract:
This work presents a finite element‑guided physics‑informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an operator from the input space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization‑independent prediction beyond the training resolution without relying on labeled simulation data. The present framework for multiphysics problems is implemented in Folax, a JAX‑based operator learning platform, and is verified on nonlinear coupled thermo‑mechanical problems. Two‑ and three‑dimensional representative volume elements with varying heterogeneous microstructures, and a close‑to‑reality industrial casting example under varying boundary conditions are investigated as the example problems. We investigate the potential of several neural operators combined with the proposed finite element‑guided approach, including Fourier neural operators (FNOs), deep operator networks (DeepONets), and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields. The results demonstrate that FNOs yield highly accurate solution operators on regular domains, where the global features can be efficiently learned in the spectral domain, and iFOL offers efficient parametric operator learning capabilities for complex and irregular geometries. Furthermore, studies on training strategies, network decomposition, and training sample quality reveal that a monolithic training strategy using a single network is sufficient for accurate predictions, while training sample quality strongly influences performance. Overall, the present approach highlights the potential of physics‑informed operator learning with a finite element‑based loss as a unified and scalable approach for coupled multiphysics simulations.

Abstract:
This paper presents a fully data‑free Physics‑Informed Neural Network (PINN) capable of solving compressible inviscid flows (ranging from supersonic to hypersonic, up to Ma=15, where Ma is the Mach number) around a circular cylinder. To overcome the spatial blindness of standard Multi‑Layer Perceptrons, a structured hybrid architecture combining radial 1D convolutions with anisotropic azimuthal 2D convolutions is proposed to embed directional inductive biases. For stable optimization across disparate flow regimes, a regime‑dependent, Mach‑number‑guided dynamic residual scaling strategy is introduced. Crucially, this approach scales down residuals to mitigate extreme gradient stiffness in high‑Mach regimes, while applying penalty multipliers to overcome the inherent spectral bias and explicitly enforce weak shock discontinuities in low‑supersonic flows. Furthermore, to establish a global thermodynamic anchor essential for stable shock wave capturing, exact analytical solutions at the stagnation point are embedded into the loss formulation. This is coupled with a novel "Upstream Fixing" boundary loss and a Total Variation (TV) loss to explicitly suppress upstream noise and the non‑physical carbuncle phenomenon. The proposed framework successfully captures the detached bow shock without referential data. While the requisite artificial viscosity yields a slightly thicker shock wave compared to computational fluid dynamics, the proposed method demonstrates unprecedented stability and physical fidelity for data‑free PINNs in extreme aerodynamics.

Abstract:
Accurate weight estimation of commercial and industrial waste is important for efficient operations, yet image‑based estimation remains difficult because similar‑looking objects may have different densities, and the visible size changes with camera distance. Addressing this problem, we propose Multimodal Weight Predictor (MWP) framework that estimates waste weight by combining RGB images with physics‑informed metadata, including object dimensions, camera distance, and camera height. We also introduce Waste‑Weight‑10K, a real‑world dataset containing 10,421 synchronized image‑metadata collected from logistics and recycling sites. The dataset covers 11 waste categories and a wide weight range from 3.5 to 3,450 kg. Our model uses a Vision Transformer for visual features and a dedicated metadata encoder for geometric and category information, combining them with Stacked Mutual Attention Fusion that allows visual and physical cues guide each other. This helps the model manage perspective effects and link objects to material properties. To ensure stable performance across the wide weight range, we train the model using Mean Squared Logarithmic Error. On the test set, the proposed method achieves 88.06 kg Mean Absolute Error (MAE), 6.39% Mean Absolute Percentage Error (MAPE), and an R2 coefficient of 0.9548. The model shows strong accuracy for light objects in the 0‑100 kg range with 2.38 kg MAE and 3.1% MAPE, maintaining reliable performance for heavy waste in the 1000‑2000 kg range with 11.1% MAPE. Finally, we incorporate a physically grounded explanation module using Shapley Additive Explanations (SHAP) and a large language model to provide clear, human‑readable explanations for each prediction.

Abstract:
This paper investigates the power control problem in wireless networks by repurposing pre‑trained large language models (LLMs) as relational reasoning backbones. In hyper‑connected interference environments, traditional optimization methods face high computational cost, while standard message passing neural networks suffer from aggregation bottlenecks that can obscure critical high‑interference structures. In response, we propose PC‑LLM, a physics‑informed framework that augments a pre‑trained LLM with an interference‑aware attention bias. The proposed bias tuning mechanism injects the physical channel gain matrix directly into the self‑attention scores, enabling explicit fusion of wireless topology with pre‑trained relational priors without retraining the backbone from scratch. Extensive experiments demonstrate that PC‑LLM consistently outperforms both traditional optimization methods and state‑of‑the‑art graph neural network baselines, while exhibiting exceptional zero‑shot generalization to unseen environments. We further observe that topology‑relevant relational reasoning is concentrated in shallow layers, whereas deeper layers encode task‑irrelevant semantic noise. Motivated by this finding, we develop a lightweight adaptation strategy that reduces model depth by 50%, significantly lowering inference cost while preserving state‑of‑the‑art spectral efficiency.

Abstract:
Accurately solving time‑dependent partial differential equations (PDEs) with neural networks remains challenging due to long‑time error accumulation and the difficulty of enforcing general boundary conditions. We introduce TENG‑BC, a high‑precision neural PDE solver based on the Time‑Evolving Natural Gradient, designed to perform under general boundary constraints. At each time step, TENG‑BC performs a boundary‑aware optimization that jointly enforces interior dynamics and boundary conditions, accommodating Dirichlet, Neumann, Robin, and mixed types within a unified framework. This formulation admits a natural‑gradient interpretation, enabling stable time evolution without delicate penalty tuning. Across benchmarks over diffusion, transport, and nonlinear PDEs with various boundary conditions, TENG‑BC achieves solver‑level accuracy under comparable sampling budgets, outperforming conventional solvers and physics‑informed neural network (PINN) baselines.

Abstract:
Simulating long‑term, fully dynamic sequences of earthquakes and aseismic slip (SEAS) on geometrically complex fault networks remains computationally demanding due to the cost of resolving elastodynamic interactions. Although high‑performance computing improves feasibility, simulations remain expensive, particularly for multicycle evolution, motivating the widespread use of quasi‑dynamic approximations based on radiation damping. Here we present an efficient numerical framework for fully elastodynamic SEAS simulations on complex fault networks. The method adopts a divide‑and‑conquer strategy in which elastodynamic self‑effects and fault‑to‑fault interactions are treated separately using boundary integral formulations tailored to each interaction type. Self‑interactions along planar faults are computed using a non‑replicating spectral boundary integral formulation that eliminates periodic‑image artifacts, while interactions between arbitrarily oriented faults are evaluated through a fully dynamic space‑time boundary integral representation accelerated by hierarchical matrices (H‑matrices). A key advance is a selective H‑matrix compression strategy based on fault‑wise assembly of independent binary trees, enabling low‑rank approximation of long‑range interactions while preserving near‑field accuracy and excluding self‑effects from the hierarchical structure. Additional efficiency arises from physics‑informed truncation of elastodynamic histories using mode‑dependent time windows and causality‑based kernel truncation. Benchmark multi‑fault simulations validate accuracy against reference uncompressed solutions. The method reduces interaction complexity from O(N^3) to O(N^2 log N), yielding up to three orders of magnitude speedup and an order‑of‑magnitude memory reduction for typical problem sizes (~3e10 degrees of freedom), enabling fully dynamic SEAS simulations on workstation hardware.

Abstract:
Accurate wireless channel estimation is critical for next‑generation wireless systems, enabling precise precoding for effective user separation, reduced interference across cells, and high‑resolution sensing, among other benefits. Traditional model‑based channel estimation methods suffer, however, from performance degradation in complex environments with a limited number of pilots, while purely data‑driven approaches lack physical interpretability, require extensive data collection, and are usually site‑specific. This paper presents a novel physics‑informed neural network (PINN) framework that synergistically combines model‑based channel estimation with a deep network to exploit prior information about environmental propagation characteristics and achieve superior performance under pilot‑constrained scenarios. The proposed approach employs an enhanced U‑Net architecture with transformer modules and cross‑attention mechanisms to fuse initial channel estimates with RSS maps to provide refined channel estimates. Comprehensive evaluation using realistic ray‑tracing data from urban environments demonstrates significant performance improvements, achieving over 5 dB gain in NMSE compared to state‑of‑the‑art methods, with particularly strong performance in pilot‑limited scenarios and robustness across different frequencies and environments with only minimal fine‑tuning. We further extend the decoder for multi‑step temporal prediction, enabling accurate forecasting of several future channel snapshots from a single estimate, useful for proactive beamforming and scheduling in mobile scenarios. The proposed framework maintains practical computational complexity, making it viable for massive MIMO systems in upper‑mid band frequencies. Unlike black‑box neural approaches, the physics‑informed design provides a more interpretable channel estimation method.

Abstract:
This paper proposes a joint multi‑objective optimization framework for strategic sensor placement in power systems to enhance attack detection. A novel physics‑informed graph transformer network (PIGTN)‑based detection model is proposed. Non‑dominated sorting genetic algorithm‑II (NSGA‑II) jointly optimizes sensor locations and the PIGTN's detection performance, while considering practical constraints. The combinatorial space of feasible sensor placements is explored using NSGA‑II, while concurrently training the proposed detector in a closed‑loop setting. Compared to baseline sensor placement methods, the proposed framework consistently demonstrates robustness under sensor failures and improvements in detection performance in seven benchmark cases, including the 14, 30, IEEE‑30, 39, 57, 118 and the 200 bus systems. By incorporating AC power flow constraints, the proposed PIGTN‑based detection model generalizes well to unseen attacks and outperforms other graph network‑based variants (topology‑aware models), achieving improvements up to 37% in accuracy and 73% in detection rate, with a mean false alarms rate of 0.3%. In addition, optimized sensor layouts significantly improve the performance of power system state estimation, achieving a 61%‑‑98% reduction in the average state error.

Abstract:
Data assimilation (DA) for compressible flows with shocks is challenging because many classical DA methods generate spurious oscillations and nonphysical features near uncertain shocks. We focus here on the ensemble Kalman filter (EnKF). We show that the poor performance of the standard EnKF may be attributed to the bimodal forecast distribution that can arise in the vicinity of an uncertain shock location; this violates the assumptions underpinning the EnKF, which assume a forecast which is close to Gaussian. To address this issue we introduce the new neural EnKF. The basic idea is to systematically embed neural function approximations within ensemble DA by mapping the forecast ensemble of shocked flows to the parameter space (weights and biases) of a deep neural network (NN) and to subsequently perform DA in that space. The nonlinear mapping encodes sharp and smooth flow features in an ensemble of NN parameters. Neural EnKF updates are therefore well‑behaved only if the NN parameters vary smoothly within the neural representation of the forecast ensemble. We show that such a smooth variation of network parameters can be enforced via physics‑informed transfer learning, and demonstrate that in so‑doing the neural EnKF avoids the spurious oscillations and nonphysical features that plague the standard EnKF. The applicability of the neural EnKF is demonstrated through a series of systematic numerical experiments with an inviscid Burgers' equation, Sod's shock tube, and a two‑dimensional blast wave.

Abstract:
Offline goal‑conditioned reinforcement learning (GCRL) learns goal‑conditioned policies from static pre‑collected datasets. However, accurate value estimation remains a challenge due to the limited coverage of the state‑action space. Recent physics‑informed approaches have sought to address this by imposing physical and geometric constraints on the value function through regularization defined over first‑order partial differential equations (PDEs), such as the Eikonal equation. However, these formulations can often be ill‑posed in complex, high‑dimensional environments. In this work, we propose a physics‑informed regularization derived from the viscosity solution of the Hamilton‑Jacobi‑Bellman (HJB) equation. By providing a physics‑based inductive bias, our approach grounds the learning process in optimal control theory, explicitly regularizing and bounding updates during value iterations. Furthermore, we leverage the Feynman‑Kac theorem to recast the PDE solution as an expectation, enabling a tractable Monte Carlo estimation of the objective that avoids numerical instability in higher‑order gradients. Experiments demonstrate that our method improves geometric consistency, making it broadly applicable to navigation and high‑dimensional, complex manipulation tasks. Open‑source codes are available at https://github.com/HrishikeshVish/phys‑fk‑value‑GCRL.

Abstract:
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics‑informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non‑linear physical operators while approximating linear operators with fixed finite‑difference convolutions. This modular mixture‑of‑experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right‑hand side, enabling continuous‑in‑time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier‑‑Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter‑efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.

Abstract:
The Bayesian update step poses significant computational challenges in high‑dimensional nonlinear estimation. While log‑homotopy particle flow filters offer an alternative to stochastic sampling, existing formulations usually yield stiff differential equations. Conversely, existing deep learning approximations typically treat the update as a black‑box task or rely on asymptotic relaxation, neglecting the exact geometric structure of the finite‑horizon probability transport. In this work, we propose a physics‑informed neural particle flow, which is an amortized inference framework. To construct the flow, we couple the log‑homotopy trajectory of the prior to posterior density function with the continuity equation describing the density evolution. This derivation yields a governing partial differential equation (PDE), referred to as the master PDE. By embedding this PDE as a physical constraint into the loss function, we train a neural network to approximate the transport velocity field. This approach enables purely unsupervised training, eliminating the need for ground‑truth posterior samples. We demonstrate that the neural parameterization acts as an implicit regularizer, mitigating the numerical stiffness inherent to analytic flows and reducing online computational complexity. Experimental validation on multimodal benchmarks and a challenging nonlinear scenario confirms better mode coverage and robustness compared to state‑of‑the‑art baselines.

Abstract:
Cardiac blood flow patterns contain rich information about disease severity and clinical interventions, yet current imaging and computational methods fail to capture underlying relational structures of coherent flow features. We propose a physics‑informed, latent relational framework to model cardiac vortices as interacting nodes in a graph. Our model combines a neural relational inference architecture with physics‑inspired interaction energy and birth‑death dynamics, yielding a latent graph sensitive to disease severity and intervention level. We first apply this to computational fluid dynamics simulations of aortic coarctation. Learned latent graphs reveal that as the aortic radius narrows, vortex interactions become stronger and more frequent. This leads to a higher graph entropy, correlating monotonically with coarctation severity (R^2=0.78, Spearman |ρ|=0.96). We then extend this method to ultrasound datasets of left ventricles under varying levels of left ventricular assist device support. Again the latent graph representation captures the weakening of coherent vortical structures, thereby demonstrating cross‑modal generalisation. Results show latent interaction graphs and entropy serve as robust and interpretable markers of cardiac disease and intervention.

Abstract:
We introduce Manifold Sobolev Informed Neural Optimization (MSINO), a curvature aware training framework for neural networks defined on Riemannian manifolds. The method replaces standard Euclidean derivative supervision with a covariant Sobolev loss that aligns gradients using parallel transport and improves stability via a Laplace Beltrami smoothness regularization term. Building on classical results in Riemannian optimization and Sobolev theory on manifolds, we derive geometry dependent constants that yield (i) a Descent Lemma with a manifold Sobolev smoothness constant, (ii) a Sobolev Polyak Lojasiewicz inequality giving linear convergence guarantees for Riemannian gradient descent and stochastic gradient descent under explicit step size bounds, and (iii) a two step Newton Sobolev method with local quadratic contraction in curvature controlled neighborhoods. Unlike prior Sobolev training in Euclidean space, MSINO provides training time guarantees that explicitly track curvature and transported Jacobians. Applications include surface imaging, physics informed learning settings, and robotics on Lie groups such as SO(3) and SE(3). The framework unifies value and gradient based learning with curvature aware convergence guarantees for neural training on manifolds.

Abstract:
Crowdsourcing platforms face a four‑way tension between allocation quality, workforce sustainability, operational feasibility, and strategic contractor behaviour‑‑a dilemma we formalise as the Cold‑Start, Burnout, Utilisation, and Strategic Agency Dilemma. Existing methods resolve at most two of these tensions simultaneously: greedy heuristics and multi‑criteria decision making (MCDM) methods achieve Day‑1 quality but cause catastrophic burnout, while bandit algorithms eliminate burnout only through operationally infeasible 100% workforce utilisation.To address this, we introduce FORGE, a physics‑grounded K+1 multi‑agent simulator in which each contractor is a rational agent that declares its own load‑acceptance threshold based on its fatigue state, converting the standard passive Restless Multi‑Armed Bandit (RMAB) into a genuine Stackelberg game. Operating within FORGE, we propose a Neural‑Linear UCB allocator that fuses a Two‑Tower embedding network with a Physics‑Informed Covariance Prior derived from offline simulator interactions. The prior simultaneously warm‑starts skill‑cluster geometry and UCB exploration landscape, providing a geometry‑aware belief state from episode 1 that measurably reduces cold‑start regret.Over T = 200 cold‑start episodes, the proposed method achieves the highest reward of all non‑oracle methods (\textLRew = 0.555 \pm 0.041) at only 7.6% workforce utilisation‑‑a combination no conventional baseline achieves‑‑while maintaining robustness to workforce turnover up to 50% and observation noise up to σ= 0.20.

Abstract:
Current AI weather forecasting models predict conventional atmospheric variables but cannot distinguish between cloud microphysical species critical for aviation safety. We introduce AviaSafe, a hierarchical, physics‑informed neural forecaster that produces global, six‑hourly predictions of these four hydrometeor species for lead times up to 7 days. Our approach addresses the unique challenges of cloud prediction: extreme sparsity, discontinuous distributions, and complex microphysical interactions between species. We integrate the Icing Condition (IC) index from aviation meteorology as a physics‑based constraint that identifies regions where supercooled water fuels explosive ice crystal growth. The model employs a hierarchical architecture that first predicts cloud spatial distribution through masked attention, then quantifies species concentrations within identified regions. Training on ERA5 reanalysis data, our model achieves lower RMSE for cloud species compared to baseline and outperforms operational numerical models on certain key variables at 7‑day lead times. The ability to forecast individual cloud species enables new applications in aviation route optimization where distinguishing between ice and liquid water determines engine icing risk.

Abstract:
Accurate prediction of shaft rotational speed, shaft power, and fuel consumption is crucial for enhancing operational efficiency and sustainability in maritime transportation. Conventional physics‑based models provide interpretability but struggle with real‑world variability, while purely data‑driven approaches achieve accuracy at the expense of physical plausibility. This paper introduces a Physics‑Informed Kolmogorov‑Arnold Network (PI‑KAN), a hybrid method that integrates interpretable univariate feature transformations with a physics‑informed loss function and a leakage‑free chained prediction pipeline. Using operational and environmental data from five cargo vessels, PI‑KAN consistently outperforms the traditional polynomial method and neural network baselines. The model achieves the lowest mean absolute error (MAE) and root mean squared error (RMSE), and the highest coefficient of determination (R^2) for shaft power and fuel consumption across all vessels, while maintaining physically consistent behavior. Interpretability analysis reveals rediscovery of domain‑consistent dependencies, such as cubic‑like speed‑power relationships and cosine‑like wave and wind effects. These results demonstrate that PI‑KAN achieves both predictive accuracy and interpretability, offering a robust tool for vessel performance monitoring and decision support in operational settings.

Abstract:
Stiff differential equations pose a major challenge for Physics‑Informed Neural Networks (PINNs), often causing poor convergence. We propose a simple, hyperparameter‑free method to address stiffness by normalizing loss residuals with the Jacobian. We provide theoretical indications that Jacobian‑based normalization can improve gradient descent and validate it on benchmark stiff ordinary differential equations. We then apply it to a realistic system: the stiff Boltzmann equations (BEs) governing weakly interacting massive particle (WIMP) dark matter (DM). Our approach achieves higher accuracy than attention mechanisms previously proposed for handling stiffness, recovering the full solution where prior methods fail. This is further demonstrated in an inverse problem with a single experimental data point ‑ the observed DM relic density ‑ where our inverse PINNs correctly infer the cross section that solves the BEs in both Standard and alternative cosmologies.

Abstract:
In recent engineering applications using deep learning, physics‑informed neural network (PINN) is a new development as it can exploit the underlying physics of engineering systems. The novelty of PINN lies in the use of partial differential equations (PDE) for the loss function. Most PINNs are implemented using automatic differentiation (AD) for training the PDE loss functions. A lesser well‑known study is the use of finite difference method (FDM) as an alternative. Unlike an AD based PINN, an immediate benefit of using a FDM based PINN is low implementation cost. In this paper, we propose the use of finite difference method for estimating the PDE loss functions in PINN. Our work is inspired by computational analysis in electromagnetic systems that traditionally solve Laplace's equation using successive over‑relaxation. In the case of Laplace's equation, our PINN approach can be seen as taking the Laplacian filter response of the neural network output as the loss function. Thus, the implementation of PINN can be very simple. In our experiments, we tested PINN on Laplace's equation and Burger's equation. We showed that using FDM, PINN consistently outperforms non‑PINN based deep learning. When comparing to AD based PINNs, we showed that our method is faster to compute as well as on par in terms of error reduction.

Abstract:
What mathematical functions do neural network components learn? Symbolic distillation addresses this question by expressing neural network components with interpretable, closed‑form mathematical expressions that expose the functional structure learned during training. We develop symbolic distillation as a systematic, architecture‑agnostic methodology, and release our approach as the open‑source SymTorch package ‑ a PySR‑powered library built natively for the PyTorch ecosystem. Applying this methodology across diverse architectures, we find that SymTorch is successful in the automated discovery of physical laws. Specifically, our approach (1) recovers pairwise interaction forces from graph neural networks trained on empirical n‑body observations, (2) distills the exact closed‑form PDE/ODE solutions of multiple physical systems, including the value of constants, from physics‑informed neural networks trained on sparse data, and (3) uncovers the chaotic dynamics of the Lorenz system from high‑dimensional data, ultimately outperforming the base neural network on downstream prediction tasks. We further demonstrate the utility of our framework for model interpretability by providing an optimized implementation of SLIME ‑ a symbolic extension to the LIME explainability method. SLIME consistently outperforms LIME across predictive metrics across eight popular classification and regression benchmarks, while still providing an interpretable local symbolic model. Lastly, we investigate replacing transformer MLP layers with symbolic surrogates: replacing 1‑7 layers with symbolic approximations yields 2‑19% throughput improvements and up to 18.7% VRAM reduction, with the resulting hybrid models lying on the Pareto front of throughput versus perplexity among open‑source LLMs of comparable scale.

Abstract:
As quantum processors scale beyond 100 qubits, distinguishing software bugs from stochastic hardware noise becomes a critical diagnostic challenge. We present a neuro‑fuzzy framework that addresses this attribution problem by combining Adaptive Neuro‑Fuzzy Inference Systems (ANFIS) with physics‑grounded feature engineering. We introduce the Bhattacharyya Veto, a hard physical constraint grounded in the Data Processing Inequality that prevents the classifier from attributing topologically impossible output distributions to noise. Validated on IBM's 156‑qubit Heron r2 processor (ibm_fez) across 105 circuits spanning 17 algorithm families, the framework achieves 89.5% effective accuracy (+/‑ 5.9% CI). The system implements a safe failure mode, flagging 14.3% of ambiguous cases for manual review rather than forcing low‑confidence predictions. We resolve key ambiguities ‑‑ such as distinguishing correct Grover amplification from bug‑induced collapse ‑‑ and identify fundamental limits of single‑basis diagnostics, including a Z‑basis blind spot where phase‑flip errors remain statistically invisible. This work establishes a robust, interpretable diagnostic layer that prevents error mitigation techniques from being applied to logically flawed circuits.

Abstract:
Magnetic resonance spectroscopy (MRS) is used to quantify metabolites in vivo and estimate biomarkers for conditions ranging from neurological disorders to cancers. Quantifying low‑concentration metabolites such as GABA (γ‑aminobutyric acid) is challenging due to low signal‑to‑noise ratio (SNR) and spectral overlap. We investigate and validate deep learning for quantifying complex, low‑SNR, overlapping signals from MEGA‑PRESS spectra, devise a convolutional neural network (CNN) and a Y‑shaped autoencoder (YAE), and select the best models via Bayesian optimisation on 10,000 simulated spectra from slice‑profile‑aware MEGA‑PRESS simulations. The selected models are trained on 100,000 simulated spectra. We validate their performance on 144 spectra from 112 experimental phantoms containing five metabolites of interest (GABA, Glu, Gln, NAA, Cr) with known ground truth concentrations across solution and gel series acquired at 3 T under varied bandwidths and implementations. These models are further assessed against the widely used LCModel quantification tool. On simulations, both models achieve near‑perfect agreement (small MAEs; regression slopes \approx 1.00, R^2 \approx 1.00). On experimental phantom data, errors initially increased substantially. However, modelling variable linewidths in the training data significantly reduced this gap. The best augmented deep learning models achieved a mean MAE for GABA over all phantom spectra of 0.151 (YAE) and 0.160 (FCNN) in max‑normalised relative concentrations, outperforming the conventional baseline LCModel (0.220). A sim‑to‑real gap remains, but physics‑informed data augmentation substantially reduced it. Phantom ground truth is needed to judge whether a method will perform reliably on real data.

Abstract:
In this work, we present a methodology using Physics Informed Neural Networks (PINNs) to determine the required velocity of a coolant, given inlet and outlet temperatures for a given heat flux in a multilayered metal‑oxide‑semiconductor field‑effect transistor (MOSFET). MOSFETs are integral components of Power Electronic Building Blocks (PEBBs) and experiences the majority of the thermal load. Effective cooling of MOSFETs is therefore essential to prevent overheating and potential burnout. Determining the required velocity for the purpose of effective cooling is of importance but is an ill‑posed inverse problem and difficult to solve using traditional methods. MOSFET consists of multiple layers with different thermal conductivities, including aluminum, pyrolytic graphite sheets (PGS), and stainless steel pipes containing flowing water. We propose an algorithm that employs sequential training of the MOSFET layers in PINNs. Mathematically, the sequential training method decouples the optimization of each layer by treating the parameters of other layers as constants during its training phase. This reduces the dimensionality of the optimization landscape, making it easier to find the global minimum for each layer's parameters and avoid poor local minima. Convergence of the PINNs solution to the analytical solution is theoretically analyzed. Finally we show the prediction of our proposed methodology to be in good agreement with experimental results.

Abstract:
Physics‑informed neural networks (PINNs) provide a promising framework for solving inverse problems governed by partial differential equations (PDEs) by integrating observational data and physical constraints in a unified optimization objective. However, the ill‑posed nature of PDE inverse problems makes them highly sensitive to noise. Even a small fraction of corrupted observations can distort internal neural representations, severely impairing accuracy and destabilizing training. Motivated by recent advances in machine unlearning and structured network pruning, we propose P‑PINN, a selective pruning framework designed to unlearn the influence of corrupted data in a pretrained PINN. Specifically, starting from a PINN trained on the full dataset, P‑PINN evaluates a joint residual‑‑data fidelity indicator, a weighted combination of data misfit and PDE residuals, to partition the training set into reliable and corrupted subsets. Next, we introduce a bias‑based neuron importance measure that quantifies directional activation discrepancies between the two subsets, identifying neurons whose representations are predominantly driven by corrupted samples. Building on this, an iterative pruning strategy then removes noise‑sensitive neurons layer by layer. The resulting pruned network is fine‑tuned on the reliable data subject to the original PDE constraints, acting as a lightweight post‑processing stage rather than a complete retraining. Numerical experiments on extensive PDE inverse‑problem benchmarks demonstrate that P‑PINN substantially improves robustness, accuracy, and training stability under noisy conditions, achieving up to a 96.6% reduction in relative error compared with baseline PINNs. These results indicate that activation‑level post hoc pruning is a promising mechanism for enhancing the reliability of physics‑informed learning in noise‑contaminated settings.

Abstract:
Machine learned interatomic potentials (MLIPs) have emerged as powerful tools for molecular dynamics (MD) simulations with their competitive accuracy and computational efficiency. However, MLIPs are often observed to exhibit un‑physical behavior when encountering configurations which deviate significantly from their training data distribution, leading to simulation instabilities and unreliable dynamics, thus limiting the reliability of MLIPs for materials simulations. We present a physics‑informed pretraining strategy that leverages simple physical potentials which can improve the robustness and stability of graph‑based MLIPs for MD simulations. We demonstrate this approach by deploying a pretraining‑finetuning pipeline where MLIPs are initially pretrained on data labelled with embedded atom model potentials and subsequently finetuned on the quantum mechanical ground truth data. By evaluating across three diverse material systems (phosphorus, silica, and a subset of Materials Project) and three representative MLIP architectures (CGCNN, M3GNet, and TorchMD‑NET), we find that this physics‑informed pretraining consistently improves both prediction accuracy as well as stability in MD compared to the baselines.

Abstract:
Assessing whether two datasets are distributionally consistent is central to modern scientific analysis, particularly as generative artificial intelligence produces synthetic data whose fidelity must be validated against real observations in increasingly high‑dimensional settings. Existing approaches are typically relative: they determine whether one dataset is more consistent with a reference than another, but do not provide a physically grounded absolute standard for fidelity. We propose an information‑theoretic approach in which lossless compression via arithmetic coding provides an operational measure of dataset fidelity under a physics‑informed probabilistic representation. Datasets sharing the same underlying physical correlations admit comparable optimal descriptions, while discrepancies‑arising from miscalibration, mismodeling, or bias‑manifest as an irreducible excess in codelength relative to the Shannon‑optimal limit defined by the physics itself. This excess codelength defines an absolute fidelity metric, quantified directly in bits. Unlike conventional measures, which lack an intrinsic scale, zero excess provides a well‑defined and physically meaningful target corresponding to consistency with the underlying distribution. We show that this metric is global, interpretable, additive across components, and asymptotically optimal, with differences in codelength corresponding to differences in expected negative log‑likelihood under a common reference model. As a byproduct, our approach achieves improved compression relative to standard general‑purpose algorithms such as gzip. These results establish arithmetic coding not merely as a compression tool, but as a measurement instrument for absolute, physics‑grounded assessment of distributional fidelity.

Abstract:
3D imaging of specular surfaces remains challenging in real‑world scenarios, such as in‑line inspection or hand‑held scanning, requiring fast and accurate measurement of complex geometries. Optical metrology techniques such as deflectometry achieve high accuracy but typically rely on multi‑shot acquisition, making them unsuitable for dynamic environments. Fourier‑based single‑shot approaches alleviate this constraint, yet their performance deteriorates when measuring surfaces with high spatial frequency structure or large curvature. Alternatively, polarimetric 3D imaging in computer vision operates in a single‑shot fashion and exhibits robustness to geometric complexity. However, its accuracy is fundamentally limited by the orthographic imaging assumption. In this paper, we propose a physics‑informed deep learning framework for single‑shot 3D imaging of complex specular surfaces. Polarization cues provide orientation priors that assist in interpreting geometric information encoded by structured illumination. These complementary cues are processed through a dual‑encoder architecture with mutual feature modulation, allowing the network to resolve their nonlinear coupling and directly infer surface normals. The proposed method achieves accurate and robust normal estimation in single‑shot with fast inference, enabling practical 3D imaging of complex specular surfaces.

Abstract:
Understanding the conformational evolution of β‑amyloid (Aβ), particularly the Aβ_42 isoform, is fundamental to elucidating the pathogenic mechanisms underlying Alzheimer's disease. However, existing end‑to‑end deep learning models often struggle to capture subtle state transitions in protein trajectories due to a lack of explicit physical constraints. In this work, we introduce PIS, a Physics‑Informed System designed for robust metastable state partitioning. By integrating pre‑computed physical priors, such as the radius of gyration and solvent‑accessible surface area, into the extraction of topological features, our model achieves superior performance on the Aβ_42 dataset. Furthermore, PIS provides an interactive platform that features dynamic monitoring of physical characteristics and multi‑dimensional result validation. This system offers biological researchers a powerful set of analytical tools with physically grounded interpretability. A demonstration video of PIS is available on https://youtu.be/AJHGzUtRCg0.

Authors: Victor-Alexandru Darvariu, Charlotte Z. Reed, Jan Stratmann, Bruno Lacerda, Benjamin Allsup, Stephen Woodward, Elizabeth Siddle, Trishna Saeharaseelan, Owain Jones, Dan Jones, Tobias Ferreira, Chloe Baker, Kevin Chaplin, James Kirk, Ashley Iceton-Morris, Ryan D. Patmore, Jeff Polton, Charlotte Williams, Christopher D. J. Auckland, Rob A. Hall, Alexandra Kokkinaki, Alvaro Lorenzo Lopez, Justin J. H. Buck, Nick Hawes

Abstract:
Underwater glider robots have become indispensable for ocean sampling, yet fully autonomous long‑term operation remains rare in practice. Although stakeholders are calling for tools to manage increasingly large fleets of gliders, existing methods have seen limited adoption due to their inability to account for environmental uncertainty and operational constraints. In this work, we demonstrate that uncertainty‑aware online navigation planning can be deployed in real‑world glider missions at scale. We formulate the problem as a stochastic shortest‑path Markov Decision Process and propose a sample‑based online planner based on Monte Carlo Tree Search. Samples are generated by a physics‑informed simulator calibrated on real‑world glider data that captures uncertain execution of controls and ocean current forecasts while remaining computationally tractable. Our methodology is integrated into an autonomous system for Slocum gliders that performs closed‑loop replanning at each surfacing. The system was validated in two North Sea deployments totalling approximately 3 months and 1000 km, representing the longest fully autonomous glider campaigns in the literature to date. Results demonstrate improvements of up to 9.88% in dive duration and 16.51% in path length compared to standard straight‑to‑goal navigation, including a statistically significant path length reduction of 9.55% in a field deployment.

Abstract:
Solving partial differential equations (PDEs) by neural networks as well as Kolmogorov‑Arnold Networks (KANs), including physics‑informed neural networks (PINNs), physics‑informed KANs (PIKANs), and neural operators, are known to exhibit spectral bias, whereby low‑frequency components of the solution are learned significantly faster than high‑frequency modes. While spectral bias is often treated as an intrinsic representational limitation of neural architectures, its interaction with optimization dynamics and physics‑based loss formulations remains poorly understood. In this work, we provide a systematic investigation of spectral bias in physics‑informed and operator learning frameworks, with emphasis on the coupled roles of network architecture, activation functions, loss design, and optimization strategy. We quantify spectral bias through frequency‑resolved error metrics, Barron‑norm diagnostics, and higher‑order statistical moments, enabling a unified analysis across elliptic, hyperbolic, and dispersive PDEs. Through diverse benchmark problems, including the Korteweg‑de Vries, wave and steady‑state diffusion‑reaction equations, turbulent flow reconstruction, and earthquake dynamics, we demonstrate that spectral bias is not simply representational but fundamentally dynamical. In particular, second‑order optimization methods substantially alter the spectral learning order, enabling earlier and more accurate recovery of high‑frequency modes for all PDE types. For neural operators, we further show that spectral bias is dependent on the neural operator architecture and can also be effectively mitigated through spectral‑aware loss formulations without increasing the inference cost.

Abstract:
This paper further develops the Method of Matched Sections (MMS), a robust numerical framework for the solution of boundary value problems governed by partial differential equations. It demonstrates its unique applicability to the challenges of surface modeling, which lie at the intersection of computational mechanics and computer graphics. This work shows how the MMS successfully bridges this gap. By decomposing the domain into an assembly of 1D directional components matched along their entire boundaries, the method inherently enforces the continuity of all variational parameters, including second‑order (curvature) and third‑order (shear) derivatives. We demonstrate the method's advanced capabilities in high‑fidelity surface reconstruction and blending, showing that it consistently generates energetically optimal, fair surfaces even from complex boundary conditions or sparse internal data points. By advancing the application of the MMS, this research establishes it as a powerful, physics‑informed geometric tool that satisfies the dual demands of rigorous numerical analysis and aesthetic computer‑aided design.

Abstract:
Integrated photonic biosensors provide compact, highly sensitive, and label‑free platforms for biochemical detection, making them attractive for on‑chip and real‑time sensing applications. However, their design remains challenging due to complex resonance behaviour, strong coupling effects, and the computational cost associated with repeated full‑wave electromagnetic simulations. In particular, inverse design of microring resonator‑based sensors requires accurate modelling of geometry‑spectrum relationships while satisfying physical constraints such as resonance conditions and spectral sensitivity requirements. In this work, we propose a physics‑informed graph neural network (PI‑GNN) for the inverse design of a microring resonator biosensor operating in the 1550 nm band. By representing the photonic structure as a graph and embedding resonance‑based physical constraints directly into the learning objective, the model captures both structural connectivity and underlying electromagnetic principles. The proposed approach enables efficient prediction of device geometries that achieve target spectral characteristics, reducing reliance on costly simulations while maintaining physical consistency and competitive design accuracy.

Abstract:
Reconstructing and simulating dynamic 3D scenes with both visual realism and physical consistency remains a fundamental challenge. Existing neural representations, such as NeRFs and 3DGS, excel in appearance reconstruction but struggle to capture complex material deformation and dynamics. We propose PhysConvex, a Physics‑informed 3D Dynamic Convex Radiance Field that unifies visual rendering and physical simulation. PhysConvex represents deformable radiance fields using physically grounded convex primitives governed by continuum mechanics. We introduce a boundary‑driven dynamic convex representation that models deformation through vertex and surface dynamics, capturing spatially adaptive, non‑uniform deformation, and evolving boundaries. To efficiently simulate complex geometries and heterogeneous materials, we further develop a reduced‑order convex simulation that advects dynamic convex fields using neural skinning eigenmodes as shape‑ and material‑aware deformation bases with time‑varying reduced DOFs under Newtonian dynamics. Convex dynamics also offers compact, gap‑free volumetric coverage, enhancing both geometric efficiency and simulation fidelity. Experiments demonstrate that PhysConvex achieves high‑fidelity reconstruction of geometry, appearance, and physical properties from videos, outperforming existing methods.

Abstract:
Physics‑informed neural networks (PINNs) have shown remarkable prospects in solving partial differential equations (PDEs) involving fluid mechanics. However, the method has so far succeeded only in inviscid flows and incompressible viscous flows, while the solution of compressible viscous flows still faces significant challenges. In previous work, we proposed a feature‑enhanced neural network (FENN), which enhances the ability of PINNs to approximate flows by introducing beneficial features into the network inputs, thereby improving the performance in solving PDEs. In this study, we extend FENN to compressible viscous flows, which are governed by the compressible Navier‑Stokes equations including the continuity, momentum, and energy equations. By solving four forward problems under different flow conditions and geometries together with a parametric problem involving angle of attack, we validate the effectiveness of FENN. In contrast, existing advanced methods that are well established for inviscid flows and incompressible viscous flows fail in this scenario. To the best of our knowledge, this is the first time that a PINN‑like method has successfully solved forward and parametric problems involving compressible viscous flows.

Abstract:
Atomic Layer Deposition (ALD) process development is often hindered by time‑consuming and precursor‑intensive tuning cycles required to identify saturation conditions. We introduce a physics‑informed Bayesian active learning (BAL) framework that autonomously tunes precursor pulse times by integrating a Langmuir adsorption model directly into the Gaussian Process (GP) kernel. A key innovation is a two‑stage parameter estimation strategy that decouples noise filtering from physical parameter extraction: the GP first smooths noisy data through standard prediction, then Langmuir parameters are fitted to the noise‑filtered GP predictions. This approach effectively separates signal from experimental noise. We evaluate the framework against a standard data‑driven GP across four simulated regimes, demonstrating convergence within five iterations, up to fourfold improvement in prediction accuracy, and two to fourfold reduction in precursor usage. Experimental validation using TiO2 deposition via Tetrakisdimethylamido Titanium (TDMAT) and ozone confirms that the physics‑informed model accurately identifies saturation times for high‑coverage targets (\geq95%), with observed deviations at lower saturation levels providing valuable insight into non‑ideal desorption behaviors.

Abstract:
The Solid Electrolyte Interphase (SEI) is critical to the performance of lithium‑ion batteries, yet its analysis via Operando Infrared (IR) spectroscopy remains experimentally complex and expensive, which limits its accessibility for standard research facilities. To overcome this bottleneck, we formulate a novel task, Operando IR Prediction, which aims to forecast the time‑resolved evolution of spectral ``fingerprints'' from a single static spectrum. To facilitate this, we introduce OpIRSpec‑7K, the first large‑scale operando dataset comprising 7,118 high‑quality samples across 10 distinct battery systems, alongside OpIRBench, a comprehensive evaluation benchmark with carefully designed protocols. Addressing the limitations of standard spectrum, video, and sequence models in capturing voltage‑driven chemical dynamics and complex composition, we propose Aligned Bi‑stream Chemical Constraint (ABCC), an end‑to‑end physics‑aware framework. It reformulates MeanFlow and introduces a novel Chemical Flow to explicitly model reaction trajectories, employs a two‑stream disentanglement mechanism for solvent‑SEI separation, and enforces physics and spectrum constraints such as mass conservation and peak shifts. ABCC significantly outperforms state‑of‑the‑art static, sequential, and generative baselines. ABCC even generalizes to unseen systems and enables interpretable downstream recovery of SEI formation pathways, supporting AI‑driven electrochemical discovery.

Abstract:
The energy management problem in the context of smart grids is inherently complex due to the interdependencies among diverse system components. Although Reinforcement Learning (RL) has been proposed for solving Optimal Power Flow (OPF) problems, the requirement for iterative interaction with an environment often necessitates computationally expensive simulators, leading to significant sample inefficiency. In this study, these challenges are addressed through the use of Physics‑Informed Neural Networks (PINNs), which can replace conventional and costly smart grid simulators. The RL policy learning process is enhanced so that convergence can be achieved in a fraction of the time required by the original environment. The PINN‑based surrogate is compared with other benchmark data‑driven surrogate models. By incorporating knowledge of the underlying physical laws, the results show that the PINN surrogate is the only approach considered in this context that can obtain a strong RL policy even without access to samples from the true simulator. The results demonstrate that using PINN surrogates can accelerate training by 50% compared to RL training without a surrogate. This approach enables the rapid generation of performance scores similar to those produced by the original simulator.

Abstract:
Physiologically Based Pharmacokinetic (PBPK) modeling is a cornerstone of model‑informed drug development (MIDD), providing a mechanistic framework to predict drug absorption, distribution, metabolism, and excretion (ADME). Despite its utility, adoption is hindered by high computational costs for large‑scale simulations, difficulty in parameter identification for complex biological systems, and uncertainty in interspecies extrapolation. In this work, we propose a unified Scientific Machine Learning (SciML) framework that bridges mechanistic rigor and data‑driven flexibility. We introduce three contributions: (1) Foundation PBPK Transformers, which treat pharmacokinetic forecasting as a sequence modeling task; (2) Physiologically Constrained Diffusion Models (PCDM), a generative approach that uses a physics‑informed loss to synthesize biologically compliant virtual patient populations; and (3) Neural Allometry, a hybrid architecture combining Graph Neural Networks (GNNs) with Neural ODEs to learn continuous cross‑species scaling laws. Experiments on synthetic datasets show that the framework reduces physiological violation rates from 2.00% to 0.50% under constraints while offering a path to faster simulation.

Abstract:
Accurate prediction of main engine power is essential for vessel performance optimization, fuel efficiency, and compliance with emission regulations. Conventional machine learning approaches, such as Support Vector Machines, variants of Artificial Neural Networks (ANNs), and tree‑based methods like Random Forests, Extra Tree Regressors, and XGBoost, can capture nonlinearities but often struggle to respect the fundamental propeller law relationship between power and speed, resulting in poor extrapolation outside the training envelope. This study introduces a hybrid modeling framework that integrates physics‑based knowledge from sea trials with data‑driven residual learning. The baseline component, derived from calm‑water power curves of the form P = cV^n, captures the dominant power‑speed dependence, while another, nonlinear, regressor is then trained to predict the residual power, representing deviations caused by environmental and operational conditions. By constraining the machine learning task to residual corrections, the hybrid model simplifies learning, improves generalization, and ensures consistency with the underlying physics. In this study, an XGBoost, a simple Neural Network, and a Physics‑Informed Neural Network (PINN) coupled with the baseline component were compared to identical models without the baseline component. Validation on in‑service data demonstrates that the hybrid model consistently outperformed a pure data‑driven baseline in sparse data regions while maintaining similar performance in populated ones. The proposed framework provides a practical and computationally efficient tool for vessel performance monitoring, with applications in weather routing, trim optimization, and energy efficiency planning.

Abstract:
Accurate AC power flow (AC‑PF) prediction under domain shift is critical when models trained on medium‑voltage (MV) grids are deployed on high‑voltage (HV) networks. Existing physics‑informed graph neural network (GNN) solvers typically rely on full fine‑tuning for cross‑regime transfer, incurring high retraining cost and offering limited control over the stability‑plasticity trade‑off between target‑domain adaptation and source‑domain retention. We study parameter‑efficient domain adaptation for physics‑informed self‑attention‑based GNNs, encouraging Kirchhoff‑consistent behavior via a physics‑based loss while restricting adaptation to low‑rank updates. Specifically, we apply low‑rank adaptation (LoRA) to attention projections with selective unfreezing of the prediction head to regulate adaptation capacity. This design yields a controllable efficiency‑accuracy trade‑off for physics‑constrained inverse estimation under voltage‑regime shift. Across multiple grid topologies, the proposed LoRA+PHead adaptation recovers near‑full fine‑tuning accuracy with a target‑domain RMSE gap of 2.6 × 10^‑4 while reducing the number of trainable parameters by 85.46%. The physics‑based residual remains comparable to full fine‑tuning; however, relative to Full FT, LoRA+PHead reduces MV source retention by 4.7 percentage points (17.9% vs. 22.6%) under domain shift, while still enabling parameter‑efficient and physically consistent AC‑PF estimation.

Abstract:
The existence of a distinct mass boundary between the heaviest neutron stars and the lightest black holes remains in question. It is an artefact of our ignorance of the properties of matter at supra‑nuclear densities, which exist in the cores of neutron stars. The study addresses these problems with a physics‑informed machine learning approach, guided by astrophysical observations. The Transformer model is trained on an agnostically generated ensemble of equations of state. Two geometric parameters are defined on the mass‑radius sequence of a neutron star‑‑the front bending and the back bending. The transformer provides a two‑step solution: first, the model predicts the maximum mass and radius using the bending parameters. Second, it predicts the square of the sound speed profile, completing the inverse mapping. The prediction is that massive neutron stars form when the sound speed peaks at low density, leading to strong back‑bending and an early phase transition to quark matter. Massive stars favour a stiff equation of state at low density, and the density of matter at the star's core is sufficiently small. The maximum mass for a neutron star predicted by the astrophysical constrained transformer model is 2.477 solar masses, and a minimum radius of about 11.498 km for a neutron star of 1.4 solar masses.

Abstract:
Physics‑informed deep learning models have emerged as powerful tools for learning dynamical systems. These models directly encode physical principles into network architectures. However, systematic benchmarking of these approaches across diverse physical phenomena remains limited, particularly in conservative and dissipative systems. In addition, benchmarking that has been done thus far does not integrate out full trajectories to check stability. In this work, we benchmark three prominent physics‑informed architectures such as Hamiltonian Neural Networks (HNN), Lagrangian Neural Networks (LNN), and Symplectic Recurrent Neural Networks (SRNN) using the DeepChem framework, an open‑source scientific machine learning library. We evaluate these models on six dynamical systems spanning classical conservative mechanics (mass‑spring system, simple pendulum, double pendulum, and three‑body problem, spring‑pendulum) and non‑conservative systems with contact (bouncing ball). We evaluate models by computing error on predicted trajectories and evaluate error both quantitatively and qualitatively. We find that all benchmarked models struggle to maintain stability for chaotic or nonconservative systems. Our results suggest that more research is needed for physics‑informed deep learning models to learn robust models of classical mechanical systems.

Abstract:
Accurate, real‑time, yet non‑destructive estimation of internal states in lithium‑ion batteries is critical for predicting degradation, optimizing usage strategies, and extending operational lifespan. Here, we introduce PINEAPPLE (Physics‑Informed Neuro‑Evolution Algorithm for Prognostic Parameter inference in Lithium‑ion battery Electrodes), a novel framework that integrates physics‑informed neural networks (PINNs) with an evolutionary search algorithm to enable rapid, scalable, and interpretable parameter inference with potential for application to next‑generation batteries. The meta‑learned PINN utilizes fundamental physics principles to achieve accurate zero‑shot prediction of electrode behavior with test errors below 0.1% while maintaining an order‑of‑magnitude speed‑up over conventional solvers. PINEAPPLE demonstrates robust parameter inference solely from voltage‑time discharge curves across multiple batteries from the open‑source CALCE repository, recovering the evolution of key internal state parameters such as Li‑ion diffusion coefficients across usage cycles. Notably, the inferred cycle‑dependent evolution of these parameters exhibit consistent trends across different batteries without any customized degradation physics‑embedded heuristic, highlighting the effective regularizing effect and robustness that can be conferred through incorporation of fundamental physics in PINEAPPLE. By enabling computationally efficient, real‑time parameter estimation, PINEAPPLE offers a promising route towards the non‑destructive, physics‑based characterization of inter‑cell and intra‑cell variability of battery modules and battery packs, thereby unlocking new opportunities for downstream on‑the‑fly needs in next‑generation battery management systems such as individual cell‑scale state‑of‑health diagnostics.

Abstract:
Machine learning (ML) has been increasingly used for topology optimization (TO). However, most existing ML‑based approaches focus on simplified benchmark problems due to their high computational cost, spectral bias, and difficulty in handling complex physics. These limitations become more pronounced in multi‑material, multi‑physics problems whose objective or constraint functions are not self‑adjoint. To address these challenges, we propose a framework based on physics‑informed Gaussian processes (PIGPs). In our approach, the primary, adjoint, and design variables are represented by independent GP priors whose mean functions are parametrized via neural networks whose architectures are particularly beneficial for surrogate modeling of PDE solutions. We estimate all parameters of our model simultaneously by minimizing a loss that is based on the objective function, multi‑physics potential energy functionals, and design‑constraints. We demonstrate the capability of the proposed framework on benchmark TO problems such as compliance minimization, heat conduction optimization, and compliant mechanism design under single‑ and multi‑material settings. Additionally, we leverage thermo‑mechanical TO with single‑ and multi‑material options as a representative multi‑physics problem. We also introduce differentiation and integration schemes that dramatically accelerate the training process. Our results demonstrate that the proposed PIGP framework can effectively solve coupled multi‑physics and design problems simultaneously ‑‑ generating super‑resolution topologies with sharp interfaces and physically interpretable material distributions. We validate these results using open‑source codes and the commercial software package COMSOL.

Abstract:
Physics‑governed models are increasingly paired with machine learning for accelerated predictions, yet most "physics‑‑informed" formulations treat the governing equations as a penalty loss whose scale and meaning are set by heuristic balancing. This blurs operator structure, thereby confounding solution approximation error with governing‑equation enforcement error and making the solving and learning progress hard to interpret and control. Here we introduce the Neural Basis Method, a projection‑based formulation that couples a predefined, physics‑conforming neural basis space with an operator‑induced residual metric to obtain a well‑conditioned deterministic minimization. Stability and reliability then hinge on this metric: the residual is not merely an optimization objective but a computable certificate tied to approximation and enforcement, remaining stable under basis enrichment and yielding reduced coordinates that are learnable across parametric instances. We use advective multiscale Darcian dynamics as a concrete demonstration of this broader point. Our method produce accurate and robust solutions in single solves and enable fast and effective parametric inference with operator learning.

Abstract:
We present a generative modeling framework for synthesizing physically feasible two‑dimensional incompressible flows under arbitrary obstacle geometries and boundary conditions. Whereas existing diffusion‑based flow generators either ignore physical constraints, impose soft penalties that do not guarantee feasibility, or specialize to fixed geometries, our approach integrates three complementary components: (1) a boundary‑conditioned diffusion model operating on velocity fields; (2) a physics‑informed training objective incorporating a divergence penalty; and (3) a projection‑constrained reverse diffusion process that enforces exact incompressibility through a geometry‑aware Helmholtz‑Hodge operator. We derive the method as a discrete approximation to constrained Langevin sampling on the manifold of divergence‑free vector fields, providing a connection between modern diffusion models and geometric constraint enforcement in incompressible flow spaces. Experiments on analytic Navier‑Stokes data and obstacle‑bounded flow configurations demonstrate significantly improved divergence, spectral accuracy, vorticity statistics, and boundary consistency relative to unconstrained, projection‑only, and penalty‑only baselines. Our formulation unifies soft and hard physical structure within diffusion models and provides a foundation for generative modeling of incompressible fields in robotics, graphics, and scientific computing.

Abstract:
We introduce a compact, physics‑driven event representation, RMM‑C46, designed to compress the high‑dimensional rapidity mass matrix (RMM) into a low‑dimensional, interpretable feature set suitable for physics‑informed machine learning (ML) and quantum computing applications. The full RMM encodes detailed pairwise correlations among jets, b‑jets, leptons, photons, and missing transverse energy but contains more than a thousand values per event, making it computationally heavy for large‑scale training and incompatible with current low‑qubit quantum devices. The proposed RMM‑C46 input space for ML preserves the physical block structure of the RMM through aggregated invariant mass, rapidity difference, and transverse energy components, reducing the size of the original RMM by over an order of magnitude while maintaining interpretability. Applied to simulated proton‑proton collisions at centre‑of‑mass energy of 13.6 TeV, these representations match or exceed the discriminative performance of the full RMM in both supervised and unsupervised ML tasks. Their compactness, stability, and physics transparency also make them naturally compatible with near‑term quantum machine learning architectures. RMM‑C46 provides a scalable, efficient, and quantum‑ready alternative to the full RMM for next‑generation collider physics analyses.

Abstract:
Reconstructing complex magnetization textures from nitrogen‑vacancy (NV) magnetometry stray‑field measurements presents a challenging inverse problem. In this work, we introduce a physics‑informed method that addresses this by incorporating the full micromagnetic energy directly into the variational formulation. Built on a PyTorch backend, our forward model integrates an auto‑differentiable finite‑differences micromagnetic framework with FFT‑based stray‑field calculations and Fourier‑space upward continuation. This enables efficient gradient‑based optimization via the adjoint method and allows the sensor‑sample distance to be treated as an optimization parameter. By doing so, we eliminate the experimental uncertainty arising from unknown NV implantation depths and surface oxidation layers. Validation on synthetic data demonstrates high‑fidelity reconstruction of spin textures and precise sensor height estimation. Furthermore, when applied to NV measurements of the van der Waals ferromagnet Fe_3‑xGaTe_2, the method reconstructs the previously unknown NV‑sample distance and physically plausible magnetization textures, which accurately reproduce the experimental observations.

Abstract:
Friction losses in rough pipes are often predicted using semi‑empirical correlations, such as the Colebrook‑White equation (Colebrook,1939), which do not fully replicate Nikuradse's rough‑pipe experiments (1950). This study derives scaling relations for the viscous and turbulent contributions to the streamwise pressure drop through an order‑of‑magnitude analysis of the Reynolds‑averaged Navier‑Stokes equations and the kinetic‑energy transport equations. These relations impose constraints on the local sensitivity of the pressure drop to factors such as mean velocity, roughness, viscosity, and density through exponent envelopes and serve as a physical prior for symbolic regression. By combining Nikuradse's rough‑pipe and smooth‑pipe data of Zagarola and Smits (1998), we aim to derive compact correlations for the friction factor that fit experimental data while adhering to the derived constraints. A modified genetic programming engine (GPTIPS2) optimizes model structure and evaluates it based on fitness, complexity, and constraint violation. This method yields interpretable expressions that accurately reproduce friction factors across various roughness levels and Reynolds numbers, validated up to Re ～ 10^7.

Abstract:
Multi‑agent reinforcement learning (MARL) has made significant progress in recent years, but most algorithms still rely on a discrete‑time Markov Decision Process (MDP) with fixed decision intervals. This formulation is often ill‑suited for complex multi‑agent dynamics, particularly in high‑frequency or irregular time‑interval settings, leading to degraded performance and motivating the development of continuous‑time MARL (CT‑MARL). Existing CT‑MARL methods are mainly built on Hamilton‑Jacobi‑Bellman (HJB) equations. However, they rarely account for safety constraints such as collision penalties, since these introduce discontinuities that make HJB‑based learning difficult. To address this challenge, we propose a continuous‑time constrained MDP (CT‑CMDP) formulation and a novel MARL framework that transforms discrete MDPs into CT‑CMDPs via an epigraph‑based reformulation. We then solve this by proposing a novel physics‑informed neural network (PINN)‑based actor‑critic method that enables stable and efficient optimization in continuous time. We evaluate our approach on continuous‑time safe multi‑particle environments (MPE) and safe multi‑agent MuJoCo benchmarks. Results demonstrate smoother value approximations, more stable training, and improved performance over safe MARL baselines, validating the effectiveness and robustness of our method.

Abstract:
Cosmological analyses are moving past the well understood 2‑point statistics to extract more information from cosmological fields. A natural step in extending inference pipelines to other summary statistics is to include higher order N‑point correlation functions (NPCFs), which are computationally expensive and difficult to model. At the same time it is unclear how many NPCFs one would have to include to reasonably exhaust the cosmological information in the observable fields. An efficient alternative is given by learned and optimized summary statistics, largely driven by overparametrization through neural networks. This, however, largely abandons our physical intuition on the NPCF formalism and information extraction becomes opaque to the practitioner. We design a simulation‑based inference pipeline, that not only benefits from the efficiency of machine learned summaries through optimization, but also holds on to the NPCF program. We employ the heavily constrained Cosmological Correlator Convolutional Neural Network (C3NN) which extracts summary statistics that can be directly linked to a given order NPCF. We present an application of our framework to simulated lensing convergence maps and study the information content of our learned summary at various orders in NPCFs for this idealized example. We view our approach as an exciting new avenue for physics‑informed simulation‑based inference.

Abstract:
The solar dynamo relies on the regeneration of the poloidal magnetic field through processes strongly modulated by nonlinear feedbacks such as tilt quenching (TQ) and latitude quenching (LQ). These mechanisms play a decisive role in regulating the buildup of the Sun's polar field and, in turn, the amplitude of future solar cycles. In this work, we employ Physics‑Informed Neural Networks (PINN) to solve the surface flux transport (SFT) equation, embedding physical constraints directly into the neural network framework. By systematically varying transport parameters, we isolate the relative contributions of TQ and LQ to polar dipole buildup. We use the residual dipole moment as a diagnostic for cycle‑to‑cycle amplification and show that TQ suppression strengthens with increasing diffusivity, while LQ dominates in advection‑dominated regimes. The ratio ΔD_\mathrmLQ/ΔD_\mathrmTQ exhibits a smooth inverse‑square dependence on the dynamo effectivity range, refining previous empirical fits with improved accuracy and reduced scatter. The results further reveal that the need for a decay term is not essential for PINN set‑up due to the training process. Compared with the traditional 1D SFT model, the PINN framework achieves significantly lower error metrics and more robust recovery of nonlinear trends. Our results suggest that the nonlinear interplay between LQ and TQ can naturally produce alternations between weak and strong cycles, providing a physical explanation for the observed even‑odd cycle modulation. These findings demonstrate the potential of PINN as an accurate, efficient, and physically consistent tool for solar cycle prediction.

Abstract:
Kolmogorov‑Arnold Networks (KANs) have recently emerged as a compelling alternative to multilayer perceptrons, offering enhanced interpretability via functional decomposition. However, existing KAN architectures, including spline‑, wavelet‑, radial‑basis variants, etc., suffer from high computational cost and slow convergence, limiting scalability and practical applicability. Here, we introduce Feature‑Enriched Kolmogorov‑Arnold Networks (FEKAN), a simple yet effective extension that preserves all the advantages of KAN while improving computational efficiency and predictive accuracy through feature enrichment, without increasing the number of trainable parameters. By incorporating these additional features, FEKAN accelerates convergence, increases representation capacity, and substantially mitigates the computational overhead characteristic of state‑of‑the‑art KAN architectures. We investigate FEKAN across a comprehensive set of benchmarks, including function‑approximation tasks, physics‑informed formulations for diverse partial differential equations (PDEs), and neural operator settings that map between input and output function spaces. For function approximation, we systematically compare FEKAN against a broad family of KAN variants, FastKAN, WavKAN, ReLUKAN, HRKAN, ChebyshevKAN, RBFKAN, and the original SplineKAN. Across all tasks, FEKAN demonstrates substantially faster convergence and consistently higher approximation accuracy than the underlying baseline architectures. We also establish the theoretical foundations for FEKAN, showing its superior representation capacity compared to KAN, which contributes to improved accuracy and efficiency.

Abstract:
SLOT (Soft Legged Omnidirectional Tetrapod), a tendon‑driven soft quadruped robot with 3D‑printed TPU legs, is presented to study physics‑informed modeling and control of compliant legged locomotion using only four actuators. Each leg is modeled as a deformable continuum using discrete Cosserat rod theory, enabling the capture of large bending deformations, distributed elasticity, tendon actuation, and ground contact interactions. A modular whole‑body modeling framework is introduced, in which compliant leg dynamics are represented through physically consistent reaction forces applied to a rigid torso, providing a scalable interface between continuum soft limbs and rigid‑body locomotion dynamics. This formulation allows efficient whole‑body simulation and real‑time control without sacrificing physical fidelity. The proposed model is embedded into a convex model predictive control framework that optimizes ground reaction forces over a 0.495 s prediction horizon and maps them to tendon actuation through a physics‑informed force‑angle relationship. The resulting controller achieves asymptotic stability under diverse perturbations. The framework is experimentally validated on a physical prototype during crawling and walking gaits, achieving high accuracy with less than 5 mm RMSE in center of mass trajectories. These results demonstrate a generalizable approach for integrating continuum soft legs into model‑based locomotion control, advancing scalable and reusable modeling and control methods for soft quadruped robots.

Abstract:
Accurate estimation of the relative concentrations of chromophores in a spectroscopic photoacoustic (sPA) image can reveal immense structural, functional, and molecular information about physiological processes. However, due to nonlinearities and ill‑posedness inherent to sPA imaging, concentration estimation is intractable. The Spectroscopic Photoacoustic Optical Inversion Autoencoder (SPOI‑AE) aims to address the sPA optical inversion and spectral unmixing problems without assuming linearity. Herein, SPOI‑AE was trained and tested on in vivo mouse lymph node sPA images with unknown ground truth chromophore concentrations. SPOI‑AE better reconstructs input sPA pixels than conventional algorithms while providing biologically coherent estimates for optical parameters, chromophore concentrations, and the percent oxygen saturation of tissue. SPOI‑AE's unmixing accuracy was validated using a simulated mouse lymph node phantom ground truth.

Abstract:
Several complex physical systems are governed by multi‑scale partial differential equations (PDEs) that exhibit both smooth low‑frequency components and localized high‑frequency structures. Existing physics‑informed neural network (PINN) methods typically train with fixed coordinate system inputs, where geometric misalignment with these structures induces gradient stiffness and ill‑conditioning that hinder convergence. To address this issue, we introduce a mapping paradigm that reshapes the input coordinates through differentiable geometric compactification mappings and couples the geometric structure of PDEs with the spectral properties of residual operators. Based on this paradigm, we propose Geometric Compactification (GC)‑PINN, a framework that introduces three mapping strategies for periodic boundaries, far‑field scale expansion, and localized singular structures in the input domain without modifying the underlying PINN architecture. Extensive empirical evaluation demonstrates that this approach yields more uniform residual distributions and higher solution accuracy on representative 1D and 2D PDEs, while improving training stability and convergence speed.

Abstract:
Physics‑informed neural networks and neural operators often suffer from severe optimization difficulties caused by ill‑conditioned gradients, multi‑scale spectral behavior, and stiffness induced by physical constraints. Recently, the Muon optimizer has shown promise by performing orthogonalized updates in the singular‑vector basis of the gradient, thereby improving geometric conditioning. However, its unit‑singular‑value updates may lead to overly aggressive steps and lack explicit stability guarantees when applied to physics‑informed learning. In this work, we propose SpecMuon, a spectral‑aware optimizer that integrates Muon's orthogonalized geometry with a mode‑wise relaxed scalar auxiliary variable (RSAV) mechanism. By decomposing matrix‑valued gradients into singular modes and applying RSAV updates individually along dominant spectral directions, SpecMuon adaptively regulates step sizes according to the global loss energy while preserving Muon's scale‑balancing properties. This formulation interprets optimization as a multi‑mode gradient flow and enables principled control of stiff spectral components. We establish rigorous theoretical properties of SpecMuon, including a modified energy dissipation law, positivity and boundedness of auxiliary variables, and global convergence with a linear rate under the Polyak‑Lojasiewicz condition. Numerical experiments on physics‑informed neural networks, DeepONets, and fractional PINN‑DeepONets demonstrate that SpecMuon achieves faster convergence and improved stability compared with Adam, AdamW, and the original Muon optimizer on benchmark problems such as the one‑dimensional Burgers equation and fractional partial differential equations.

Abstract:
Discovering the unknown governing equations of grid‑connected inverters from external measurements holds significant attraction for analyzing modern inverter‑intensive power systems. However, existing methods struggle to balance the identification of unmodeled nonlinearities with the preservation of physical consistency. To address this, this paper proposes a Physics‑Informed Sparse Machine Learning (PISML) framework. The architecture integrates a sparse symbolic backbone to capture dominant model skeletons with a neural residual branch that compensates for complex nonlinear control logic. Meanwhile, a Jacobian‑regularized physics‑informed training mechanism is introduced to enforce multi‑scale consistency including large/small‑scale behaviors. Furthermore, by performing symbolic regression on the neural residual branch, PISML achieves a tractable mapping from black‑box data to explicit control equations. Experimental results on a high‑fidelity Hardware‑in‑the‑Loop platform demonstrate the framework's superior performance. It not only achieves high‑resolution identification by reducing error by over 340 times compared to baselines but also realizes the compression of heavy neural networks into compact explicit forms. This restores analytical tractability for rigorous stability analysis and reduces computational complexity by orders of magnitude. It also provides a unified pathway to convert structurally inaccessible devices into explicit mathematical models, enabling stability analysis of power systems with unknown inverter governing equations.

Abstract:
In this work, we reformulate the relativistic BDNK (Bemfica‑Disconzi‑Noronha‑Kovtun) diffusion equation in flux‑conservative form, and solve the resulting equations in (1+1)D using both a second‑order Kurganov‑Tadmor finite volume scheme and physics‑informed neural networks (PINNs). In particular, we introduce the SA‑PINN‑ACTO framework, which combines the self‑adaptive PINN technique with an exact enforcement of initial and periodic boundary conditions through an algebraic transform of the network's raw output, allowing the network to focus solely on minimizing the PDE residual. We test both approaches on smooth and discontinuous initial data, for both trivial and dynamically evolving velocity and temperature BDNK backgrounds, and for two characteristic speeds. The SA‑PINN‑ACTO method matches the converged Kurganov‑Tadmor solutions for smooth profiles, while for discontinuous profiles the errors increase, reflecting an expected limitation of PINNs near sharp gradients.

Abstract:
Dense granular flows exhibit nonlocal effects due to stress transmission in microplastic events, especially in quasi‑static or slowly sheared regions. Hence, traditional local rheological models fail to capture spatial cooperativity effects that are prominent in many granular systems. The nonlocal granular fluidity (NGF) model addresses this limitation by introducing a diffusive‑like partial differential equation for a fluidity field, governed by a key material‑dependent parameter: the nonlocal amplitude A. However, determining A from experiments or simulations is known to be difficult and typically requires extensive calibration across multiple geometries. In this work, we present a data‑driven platform based on Physics‑Informed Neural Networks (PINNs) embedded with the NGF model, capable of solving granular flows in a forward or inverse manner. We show that once trained on transient flow fields, these non‑local PINNs can readily infer the material parameters, as well as the pressure and stress fields. These data‑driven frameworks allow for accurate recovery of small variations in the nonlocal amplitude, A, which lead to sharp bifurcation‑like transitions in the flow field. This approach demonstrates the feasibility of data‑driven parameter inference in complex nonlocal models and opens up new possibilities for characterizing granular materials from sparse experimental observations.

Abstract:
Purpose of Review Imaging derived fractional flow reserve (FFR) is rapidly evolving beyond conventional computational fluid dynamics (CFD) based pipelines toward machine learning (ML), deep learning (DL), and physics informed approaches that enable fast, wire free, and scalable functional assessment of coronary artery stenosis. This review synthesizes recent advances in computed tomography (CT)‑ and angiography‑based FFR measurement, with particular emphasis on emerging physics‑informed neural networks and neural operators (PINNs and PINOs), as well as key considerations for their clinical translation. Recent Findings ML/DL approaches have markedly improved automation and computational speed, enabling prediction of pressure and FFR from anatomical descriptors or angiographic contrast dynamics. However, their real‑world performance and generalizability can remain variable and sensitive to domain shift, due to multi‑center heterogeneity, interpretability challenges, and differences in acquisition protocols and image quality. Physics informed learning introduces conservation structure and boundary condition consistency into model training, improving generalizability and reducing dependence on dense supervision while maintaining rapid inference. Recent evaluation trends increasingly highlight deployment oriented metrics, including calibration, uncertainty quantification, and quality control gatekeeping, as essential for safe clinical use. Summary The field is converging toward imaging derived FFR methods that are faster, more automated, and more reliable. While ML/DL offers substantial efficiency gains, physics informed frameworks such as PINNs and PINOs may provide a more robust balance between speed and physical consistency. Prospective multi center validation and standardized evaluation will be critical to support broad and safe clinical adoption.

Abstract:
The reliable discrimination of tectonic earthquakes from anthropogenic quarry blasts and transient noise remains a critical challenge in single station seismic monitoring. In this study, we introduce a novel Physics Informed Convolutional Recurrent Neural Network (PI CRNN) that embeds seismological domain knowledge directly into the feature extraction and learning process. We adapt a multistream architecture with three parallel encoders: (i) Time Domain: SincNet Encoder, (ii) MultiResolution Spectrogram branch, and, (iii) Physics Branch, followed by a fusion and a bidirectionalLSTM module. Evaluated on the Curated Pacific Northwest AI ready Seismic Dataset, the PI CRNN achieves an overall classification accuracy of 97.56 percent on an independent test set. It outperforms a standard CRNN baseline, a classical P to S amplitude ratio method, and a Physics Informed Neural Network (PINN) that enforces physical constraints via the loss function. Furthermore, the model demonstrates perfect precision in noise rejection (100 percent Recall). Interpretability analysis using saliency maps confirms that the architecture successfully learns distinct physical signatures, identifying bimodal P‑ and S‑wave arrivals for earthquakes versus singular impulsive onsets for blasts. This work establishes a scalable, transparent framework for AI‑driven seismology, proving that architectural inductive bias provides an alternative reliable approach compared to purely data‑driven approaches.

Abstract:
Reliable spacecraft attitude control depends on accurate prediction of attitude dynamics, particularly when model‑based strategies such as Model Predictive Control (MPC) are employed, where performance is limited by the quality of the internal system model. For spacecraft with complex dynamics, obtaining accurate physics‑based models can be difficult, time‑consuming, or computationally heavy. Learning‑based system identification presents a compelling alternative; however, models trained exclusively on data frequently exhibit fragile stability properties and limited extrapolation capability. This work explores Physics‑Informed Neural Networks (PINNs) for modeling spacecraft attitude dynamics and contrasts it with a conventional data‑driven approach. A comprehensive dataset is generated using high‑fidelity numerical simulations, and two learning methodologies are investigated: a purely data‑driven pipeline and a physics‑regularized approach that incorporates prior knowledge into the optimization process. The results indicate that embedding physical constraints during training leads to substantial improvements in predictive reliability, achieving a 68.17% decrease in mean relative error relative. When deployed within an MPC architecture, the physics‑informed models yield superior closed‑loop tracking performance and improved robustness to uncertainty. Furthermore, a hybrid control formulation that merges the learned nonlinear dynamics with a nominal linear model enables consistent steady‑state convergence and significantly faster response, reducing settling times by 61.52%‑76.42% under measurement noise and reaction wheel friction.

Abstract:
Physics‑Informed Neural Networks (PINNs) offer a powerful paradigm for flow reconstruction, seamlessly integrating sparse velocity measurements with the governing Navier‑Stokes equations to recover complete velocity and latent pressure fields. However, scaling such models to large spatiotemporal domains is hindered by computational bottlenecks and optimization instabilities. In this work, we propose a robust distributed PINNs framework designed for efficient flow reconstruction via spatiotemporal domain decomposition. A critical challenge in such distributed solvers is pressure indeterminacy, where independent sub‑networks drift into inconsistent local pressure baselines. We address this issue through a reference anchor normalization strategy coupled with decoupled asymmetric weighting. By enforcing a unidirectional information flow from designated master ranks where the anchor point lies to neighboring ranks, our approach eliminates gauge freedom and guarantees global pressure uniqueness while preserving temporal continuity. Furthermore, to mitigate the Python interpreter overhead associated with computing high‑order physics residuals, we implement a high‑performance training pipeline accelerated by CUDA graphs and JIT compilation. Extensive validation on complex flow benchmarks demonstrates that our method achieves near‑linear strong scaling and high‑fidelity reconstruction, establishing a scalable and physically rigorous pathway for flow reconstruction and understanding of complex hydrodynamics.

Abstract:
Restrictive phenomenological assumptions represent a major roadblock for the development of accurate subgrid‑scale models of fluid turbulence. Specifically, these assumptions limit a model's ability to describe key quantities of interest, such as local fluxes of energy and enstrophy, in the presence of diverse coherent structures. This paper introduces a symbolic data‑driven subgrid‑scale model that requires no phenomenological assumptions and has no adjustable parameters, yet it outperforms leading LES models. A combination of a priori and a posteriori benchmarks shows that the model produces accurate predictions of various quantities including local fluxes across a broad range of two‑dimensional turbulent flows. While the model is inferred using LES‑style spatial coarse‑graining, its structure is more similar to RANS models, as it employs an additional field to describe subgrid scales. We find that this field must have a rank‑two tensor structure in order to correctly represent both the components of the subgrid‑scale stress tensor and the various fluxes.

Abstract:
Spatiotemporal flows govern diverse phenomena across physics, biology, and engineering, yet modelling their multiscale dynamics remains a central challenge. Despite major advances in physics‑informed machine learning, existing approaches struggle to simultaneously maintain long‑term temporal evolution and resolve fine‑scale structure across chaotic, turbulent, and physiological regimes. Here, we introduce Uni‑Flow, a unified autoregressive‑diffusion framework that explicitly separates temporal evolution from spatial refinement for modelling complex dynamical systems. The autoregressive component learns low‑resolution latent dynamics that preserve large‑scale structure and ensure stable long‑horizon rollouts, while the diffusion component reconstructs high‑resolution physical fields, recovering fine‑scale features in a small number of denoising steps. We validate Uni‑Flow across canonical benchmarks, including two‑dimensional Kolmogorov flow, three‑dimensional turbulent channel inflow generation with a quantum‑informed autoregressive prior, and patient‑specific simulations of aortic coarctation derived from high‑fidelity lattice Boltzmann hemodynamic solvers. In the cardiovascular setting, Uni‑Flow enables task‑level faster than real‑time inference of pulsatile hemodynamics, reconstructing high‑resolution pressure fields over physiologically relevant time horizons in seconds rather than hours. By transforming high‑fidelity hemodynamic simulation from an offline, HPC‑bound process into a deployable surrogate, Uni‑Flow establishes a pathway to faster‑than‑real‑time modelling of complex multiscale flows, with broad implications for scientific machine learning in flow physics.

Abstract:
This paper develops a tractable analytical channel model for first‑hitting‑time molecular communication (MC) systems under time‑varying drift. While existing studies of nonstationary transport rely primarily on numerical solutions of advection‑diffusion equations or parametric impulse‑response fitting, they do not provide an explicit analytical description of trajectory‑level arrival dynamics at absorbing boundaries. By adopting a change‑of‑measure formulation, we reveal a structural decomposition of the first‑hitting‑time density into a cumulative‑drift displacement term and a stochastic boundary‑flux modulation factor. This leads to a closed‑form Corrected‑Inverse‑Gaussian (C‑IG) density that generalizes the classical IG model to nonstationary drift while preserving O(1) evaluation complexity. Monte Carlo simulations under both smooth pulsatile and abrupt switching drift profiles confirm that the proposed C‑IG model accurately captures complex transport phenomena, including phase modulation, multi‑pulse dispersion, and transient backflow ‑‑ effects that traditionally complicate symbol synchronization and induce severe inter‑symbol interference. The resulting framework provides a physics‑informed, computationally efficient channel model suitable for system‑level analysis and advanced receiver design, such as real‑time maximum likelihood detection, in dynamic biological and MC environments.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful mesh‑free framework for solving ordinary and partial differential equations by embedding the governing physical laws directly into the loss function. However, their classical formulation relies on multilayer perceptrons (MLPs), whose fixed activation functions and global approximation biases limit performance in problems with oscillatory behavior, multiscale dynamics, or sharp gradients. In parallel, Kolmogorov‑Arnold Networks (KANs) have been introduced as a functionally adaptive architecture based on learnable univariate transformations along each edge, providing richer local approximations and improved expressivity. This work presents a systematic and controlled comparison between standard MLP‑based PINNs and their KAN‑based counterparts, Physics‑Informed Kolmogorov‑Arnold Networks (PIKANs), using identical physics‑informed formulations and matched parameter budgets to isolate the architectural effect. Both models are evaluated across a representative collection of ODEs and PDEs, including cases with known analytical solutions that allow direct assessment of gradient reconstruction accuracy. The results show that PIKANs consistently achieve more accurate solutions, converge in fewer iterations, and yield superior gradient estimates, highlighting their advantage for physics‑informed learning. These findings underline the potential of KAN‑based architectures as a next‑generation approach for scientific machine learning and provide rigorous evidence to guide model selection in differential equation solving.

Abstract:
This paper presents a physics‑informed neural network approach for dynamic modeling of saturable synchronous machines, including cases with spatial harmonics. We introduce an architecture that incorporates gradient networks directly into the fundamental machine equations, enabling accurate modeling of the nonlinear and coupled electromagnetic constitutive relationship. By learning the gradient of the magnetic field energy, the model inherently satisfies energy balance (reciprocity conditions). The proposed architecture can universally approximate any physically feasible magnetic behavior and offers several advantages over lookup tables and standard machine learning models: it requires less training data, ensures monotonicity and reliable extrapolation, and produces smooth outputs. These properties further enable robust model inversion and optimal trajectory generation, often needed in control applications. We validate the proposed approach using measured and finite‑element method (FEM) datasets from a 5.6‑kW permanent‑magnet (PM) synchronous reluctance machine. Results demonstrate accurate and physically consistent models, even with limited training data.

Abstract:
High‑resolution fluid simulations for plasma physics and astrophysics rely on Particle in cell (PIC) and hydrodynamic solvers (e.g., FLASH) to resolve shock dominated, multiscale phenomena, but their high computational cost severely limits scalability. This motivates the development of learning based surrogate models, which offer a promising route to accelerate these simulations while preserving physical fidelity. In this work, we study the Sedov Taylor shock propagation problem using a physics informed graph based surrogate model, Physics Informed MeshGraphNet (PhyMGN), designed for grid‑based hydrodynamics. By incorporating weak physics constraints derived from the Euler equations using finite difference method, the model captures the self similar shock evolution and associated flow structures without explicitly solving the full hydrodynamic equations at each timestep. Comparing to the baseline MeshGraphNet model, PhyMGN is able to generalize beyond the training regime with a higher accuracy and preserves differentiability in parameter space while achieving a substantial reduction in computational cost relative to conventional numerical solvers.

Abstract:
We present pseudo‑differential enhanced physics‑informed neural networks (PINNs), an extension of gradient enhancement but in Fourier space. Gradient enhancement of PINNs dictates that the PDE residual is taken to a higher differential order than prescribed by the PDE, added to the objective as an augmented term in order to improve training and overall learning fidelity. We propose the same procedure after application via Fourier transforms, since differentiating in Fourier space is multiplication with the Fourier wavenumber under suitable decay. Our methods are fast and efficient. Our methods oftentimes achieve superior PINN versus numerical error in fewer training iterations, potentially pair well with few samples in collocation, and can on occasion break plateaus in low collocation settings. Moreover, our methods are suitable for fractional derivatives. We establish that our methods, due to the dynamical effects, improve spectral eigenvalue decay of the neural tangent kernel (NTK), and so our methods contribute towards the learning of high frequencies in early training, mitigating the effects of frequency bias up to the polynomial order and possibly greater with smooth activations. Our methods accommodate advanced techniques in PINNs, such as Fourier feature embeddings. A pitfall of discrete Fourier transforms via the Fast Fourier Transform (FFT) is mesh subjugation, and so we demonstrate compatibility of our methods for greater mesh flexibility and invariance on alternative Euclidean and non‑Euclidean domains via Monte Carlo methods and otherwise.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding governing physical laws directly into the training objective. Recent advances in quantum machine learning have motivated hybrid quantum‑classical extensions aimed at enhancing representational capacity while remaining compatible with near‑term quantum hardware. In this work, we investigate trainable embedding strategies within quantum‑assisted PINNs for solving parabolic PDEs, using one‑ and two‑dimensional heat equations as canonical benchmarks. We introduce two quantum‑assisted architectures that differ in their embedding components. In the first approach, a classical feed‑forward neural network generates trainable feature maps for quantum data encoding (FNN‑TE‑QPINN). In the second, the embedding stage is realized entirely by a parameterized quantum circuit (QNN‑TE‑QPINN), yielding a fully quantum feature map. Our findings emphasize the critical role of embedding design and support hybrid quantum‑classical approaches for parabolic PDE modeling in the NISQ era.

Abstract:
Solving coupled systems of differential equations (DEs) is a central problem across scientific computing. While Physics Informed Neural Networks (PINNs) offer a promising, mesh‑free approach, their standard architectures struggle with the multi‑objective optimization conflicts and local optima traps inherent in coupled problems. To address the first issue, we propose a Forked PINN (FPINN) framework designed for coupled systems of DEs. FPINN employs a shared base network with independent branches, isolating gradient pathways to stabilize training. We demonstrate the effectiveness of FPINN in simulating non‑Markovian open quantum dynamics governed by coupled DEs, where multi‑objective conflicts and local optima traps often cause evolutionary stagnation. To overcome this second challenge, we incorporate an evolution regularization loss that guides the model away from trivial solutions and ensures physically meaningful evolution. We demonstrate the effectiveness of FPINN in simulating non‑Markovian open quantum dynamics governed by coupled DEs, where multi‑objective conflicts and local optima traps often cause evolutionary stagnation. For the spin‑boson and XXZ models, FPINN accurately captures hallmark non‑Markovian features, such as quantum coherence revival and information backflow, significantly outperforming standard PINNs. The proposed FPINN architecture offers a general and effective framework for solving coupled systems of equations, which arise across a broad spectrum from classical physics to modern artificial intelligence, including applications in multi‑body rotational dynamics, multi‑asset portfolio optimization, chemical reaction kinetics, and deep representation learning.

Abstract:
Predicting flows that occur both through and around porous bodies is challenging due to coupled physics across fluid and porous regions and the need to generalize across diverse geometries and boundary conditions. We address this problem using two Physics Informed learning approaches: Physics Informed PointNets (PIPN) and Physics Informed Geometry Aware Neural Operator (P‑IGANO). We enforce the incompressible Navier Stokes equations in the free‑flow region and a Darcy Forchheimer extension in the porous region within a unified loss and condition the networks on geometry and material parameters. Datasets are generated with OpenFOAM on 2D ducts containing porous obstacles and on 3D windbreak scenarios with tree canopies and buildings. We first verify the pipeline via the method of manufactured solutions, then assess generalization to unseen shapes, and for PI‑GANO, to variable boundary conditions and parameter settings. The results show consistently low velocity and pressure errors in both seen and unseen cases, with accurate reproduction of the wake structures. Performance degrades primarily near sharp interfaces and in regions with large gradients. Overall, the study provides a first systematic evaluation of PIPN/PI‑GANO for simultaneous through‑and‑around porous flows and shows their potential to accelerate design studies without retraining per geometry.

Abstract:
We present a physics‑informed voiced backend renderer for singing‑voice synthesis. Given synthetic single‑channel audio and a fund‑amental‑‑frequency trajectory, we train a time‑domain Webster model as a physics‑informed neural network to estimate an interpretable vocal‑tract area function and an open‑end radiation coefficient. Training enforces partial differential equation and boundary consistency; a lightweight DDSP path is used only to stabilize learning, while inference is purely physics‑based. On sustained vowels (/a/, /i/, /u/), parameters rendered by an independent finite‑difference time‑domain Webster solver reproduce spectral envelopes competitively with a compact DDSP baseline and remain stable under changes in discretization, moderate source variations, and about ten percent pitch shifts. The in‑graph waveform remains breathier than the reference, motivating periodicity‑aware objectives and explicit glottal priors in future work.

Abstract:
Physics‑Informed Neural Networks present a novel approach in SciML that integrates physical laws in the form of partial differential equations directly into the NN through soft constraints in the loss function. This work studies the application of PINNs to solve a one dimensional coupled electro‑elastodynamic system modeling linear piezoelectricity in stress‑charge form, governed by elastodynamic and electrodynamic equations. Our simulation employs a feedforward architecture, mapping space‑time coordinates to mechanical displacement and electric potential. Our PINN model achieved global relative L2 errors of 2.34 and 4.87 percent for displacement and electric potential respectively. The results validate PINNs as effective mesh free solvers for coupled time‑dependent PDE systems, though challenges remain regarding error accumulation and stiffness in coupled eigenvalue systems.

Abstract:
Insertion tasks in robotic manipulation demand precise, contact‑rich interactions that vision alone cannot resolve. While tactile feedback is intuitively valuable, existing studies have shown that naïve visuo‑tactile fusion often fails to deliver consistent improvements. In this work, we propose a Cross‑Modal Transformer (CMT) for visuo‑tactile fusion that integrates wrist‑camera observations with tactile signals through structured self‑ and cross‑attention. To stabilize tactile embeddings, we further introduce a physics‑informed regularization that encourages bilateral force balance, reflecting principles of human motor control. Experiments on the TacSL benchmark show that CMT with symmetry regularization achieves a 96.59% insertion success rate, surpassing naïve and gated fusion baselines and closely matching the privileged "wrist + contact force" configuration (96.09%). These results highlight two central insights: (i) tactile sensing is indispensable for precise alignment, and (ii) principled multimodal fusion, further strengthened by physics‑informed regularization, unlocks complementary strengths of vision and touch, approaching privileged performance under realistic sensing.

Abstract:
Rapid and reliable assessment of the intrinsic reactivity of amorphous aluminosilicates is critical for their application in alkali‑activated materials (AAMs) and blended cements. Although physics‑informed glass‑structure descriptors have demonstrated strong structure‑reactivity relationships for predominantly amorphous systems, their extension to heterogeneous precursors with mixed crystalline‑amorphous phases has been limited. Here, quantitative X‑ray diffraction combined with bulk compositional analysis was used to reconstruct the effective amorphous compositions of five fly ashes (FAs) and three ground granulated blast‑furnace slags (GGBSs). These compositions served as inputs for molecular dynamics simulations employing a melt‑and‑quench approach to generate atomic‑scale structural models of the glassy phases. Based on these structures, the previously introduced descriptors, i.e., average metal oxygen dissociation energy and average metal oxygen bond strength, were refined to cover a broader compositional space spanning SiO2‑Al2O3‑TiO2‑Fe2O3‑CaO‑MgO‑MnO‑Na2O‑K2O. The refined descriptors exhibit strong inverse correlations with multiple independent reactivity indicators, including cumulative heat release from isothermal calorimetry, bound water content from thermogravimetric analysis, and compressive strength, for both single precursors and binary FA‑GGBS blends activated with NaOH. These results demonstrate that physics‑informed glass‑structure descriptors can be extended from ideal amorphous systems to heterogeneous mixed‑phase precursors and capture relative intrinsic reactivity trends in alkaline solutions. The proposed framework provides a transferable, structure‑informed basis for comparative assessment of precursor reactivity that complements experimental testing and may inform precursor screening and mix designs for AAM and blended cement systems.

Abstract:
Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data‑driven models often require extensive training data and can generalize poorly, especially in small‑data regimes and under unseen (out‑of‑distribution) input functions that are not represented in the training data. To address these limitations, we propose the Physics‑Informed Laplace Neural Operator (PILNO), which enhances the Laplace Neural Operator (LNO) by embedding governing physics into training through PDE, boundary condition, and initial condition residuals. To improve expressivity, we first introduce an Advanced LNO (ALNO) backbone that retains a pole‑residue transient representation while replacing the steady‑state branch with an FNO‑style Fourier multiplier. To make physics‑informed training both data‑efficient and robust, PILNO further leverages (i) virtual inputs: an unlabeled ensemble of input functions spanning a broad spectral range that provides abundant physics‑only supervision and explicitly targets out‑of‑distribution (OOD) regimes; and (ii) temporal‑causality weighting: a time‑decaying reweighting of the physics residual that prioritizes early‑time dynamics and stabilizes optimization for time‑dependent PDEs. Across four representative benchmarks ‑‑ Burgers' equation, Darcy flow, a reaction‑diffusion system, and a forced KdV equation ‑‑ PILNO consistently improves accuracy in small‑data settings (e.g., N_train <= 27), reduces run‑to‑run variability across random seeds, and achieves stronger OOD generalization than purely data‑driven baselines.

Abstract:
This work introduces the Nirenberg Neural Network: a numerical approach to the Nirenberg problem of prescribing Gaussian curvature on S^2 for metrics that are pointwise conformal to the round metric. Our mesh‑free physics‑informed neural network (PINN) approach directly parametrises the conformal factor globally and is trained with a geometry‑aware loss enforcing the curvature equation. Additional consistency checks were performed via the Gauss‑Bonnet theorem, and spherical‑harmonic expansions were fit to the learnt models to provide interpretability. For prescribed curvatures with known realisability, the neural network achieves very low losses (10^‑7 ‑ 10^‑10), while unrealisable curvatures yield significantly higher losses. This distinction enables the assessment of unknown cases, separating likely realisable functions from non‑realisable ones. The current capabilities of the Nirenberg Neural Network demonstrate that neural solvers can serve as exploratory tools in geometric analysis, offering a quantitative computational perspective on longstanding existence questions.

Abstract:
This study introduces Differentiable Graph Neural Network Simulators (Diff‑GNS) as a physics‑informed and automated framework for estimating post‑liquefaction residual strengths (S_r). Traditional approaches to estimate S_r rely on simplified physics, manual iterations, and assumptions about runout development. Diff‑GNS overcomes these limitations by integrating a Graph Neural Network Simulator (GNS) that simulates granular flows, with gradient‑based optimization through automatic differentiation. GNS accelerates forward runout simulations that are otherwise computationally intensive with conventional numerical methods, while gradient‑based optimization automates the inversion to back‑calculate S_r. The GNS is trained on simulations with the material point method on geometries informed by case‑history runout failures, enabling focused learning of realistic runout mechanisms and the ability to simulate slopes across small and large scales. The Diff‑GNS framework is validated using two well‑documented liquefaction‑induced flow failure case histories: the Lower San Fernando dam and La Marquesa dam. In the two cases, the inferred S_r agrees closely with published estimates and reproduces physically consistent runout behaviors. The framework also has the ability to jointly infer multiple interacting parameters, extending beyond single‑parameter back‑analyses. By embedding the physics of runout processes, minimizing manual intervention, and accelerating the inversion process to estimate S_r, Diff‑GNS provides an efficient, reproducible, and physically grounded approach for geotechnical analysis of liquefaction‑induced flow failures.

Abstract:
Standardized laboratory characterizations for absorbing materials rely on idealized sound field assumptions, which deviate largely from real‑life conditions. Consequently, \emphin‑situ acoustic characterization has become essential for accurate diagnosis and virtual prototyping. We propose a physics‑informed neural field that reconstructs local, near‑surface broadband sound fields from sparse pressure samples to directly infer complex surface impedance. A parallel, multi‑frequency architecture enables a broadband impedance retrieval within runtimes on the order of seconds to minutes. To validate the method, we developed a compact microphone array with low hardware complexity. Numerical verifications and laboratory experiments demonstrate accurate impedance retrieval with a small number of sensors under realistic conditions. We further showcase the approach in a vehicle cabin to provide practical guidance on measurement locations that avoid strong interference. Here, we show that this approach offers a robust means of characterizing \emphin‑situ boundary conditions for architectural and automotive acoustics.

Abstract:
This work presents a two‑stage physics‑informed, data‑driven constitutive modeling framework for hyperelastic soft materials undergoing progressive damage and failure. The framework is grounded in the concept of hyperelasticity with energy limiters and employs Gaussian Process Regression (GPR) to separately learn the intact (undamaged) elastic response and damage evolution directly from data. In Stage I, GPR models learn the intact hyperelastic response through volumetric and isochoric response functions (or only the isochoric response under incompressibility), ensuring energetic consistency of the intact response and satisfaction of fundamental principles such as material frame indifference and balance of angular momentum. In Stage II, damage is modeled via a separate GPR model that learns the mapping between the intact strain energy density predicted by Stage I models and a stress‑reduction factor governing damage and failure, with monotonicity, non‑negativity, and complete‑failure constraints enforced through penalty‑based optimization to ensure thermodynamic admissibility. Validation on synthetic datasets, including benchmarking against analytical constitutive models and competing data‑driven approaches, demonstrates high in‑distribution accuracy under uniaxial tension and robust generalization from limited training data to compression and shear modes not used during training. Application to experimental brain tissue data demonstrates the practical applicability of the framework and enables inference of damage evolution and critical failure energy. Overall, the proposed framework combines the physical consistency, interpretability, and generalizability of analytical models with the flexibility, predictive accuracy, and automation of machine learning, offering a powerful approach for modeling failure in soft materials under limited experimental data.

Abstract:
Ambient intelligence, continuously understanding human presence, activity, and physiology in physical spaces, is fundamental to smart environments, health monitoring, and human‑computer interaction. WiFi infrastructure provides a ubiquitous, always‑on, privacy‑preserving substrate for this capability across billions of IoT devices. Yet this potential remains largely untapped, as wireless sensing has typically relied on task‑specific models that require substantial labeled data and limit practical deployment. We present AM‑FM, the first foundation model for ambient intelligence and sensing through WiFi. AM‑FM is pre‑trained on 9.2 million unlabeled Channel State Information (CSI) samples collected over 439 days from 20 commercial device types deployed worldwide, learning general‑purpose representations via contrastive learning, masked reconstruction, and physics‑informed objectives tailored to wireless signals. Evaluated on public benchmarks spanning nine downstream tasks, AM‑FM shows strong cross‑task performance with improved data efficiency, demonstrating that foundation models can enable scalable ambient intelligence using existing wireless infrastructure.

Abstract:
Machine learning techniques have proven to be effective in addressing the structure of atomic nuclei. Physics‑Informed Neural Networks (PINNs) are a promising machine learning technique suitable for solving integro‑differential problems such as the many‑body Schrödinger problem. So far, there has been no demonstration of extracting nuclear eigenstates using such method. Here, we tackle realistic nucleon‑nucleon interaction in momentum space, including models with strong high‑momentum correlations, and demonstrate highly accurate results for the deuteron. We further provide additional benchmarks in coordinate space. We introduce an expression for the variational energy that enters the loss function, which can be evaluated efficiently within the PINNs framework. Results are in excellent agreement with proven numerical methods, with a relative error between the value of the predicted binding energy by the PINN and the numerical benchmark of the order of 10^‑6. Our approach paves the way for the exploitation of PINNs to solve more complex atomic nuclei.

Abstract:
Many biological processes involve numerous coupled degrees of freedom, yet free‑energy estimation is often restricted to one‑dimensional profiles to mitigate the high computational cost of multidimensional sampling. In this work, we extend Fokker‑‑Planck Score Learning (FPSL) to efficiently reconstruct two‑dimensional free‑energy landscapes from non‑equilibrium molecular dynamics simulations using different types of collective variables. We show that explicitly modeling orthogonal degrees of freedom reveals insights hidden in one‑dimensional projections at negligible computational overhead. Additionally, exploiting symmetries in the underlying landscape enhances reconstruction accuracy, while regularization techniques ensure numerical robustness in sparsely sampled regions. We validate our approach on three distinct systems: the conformational dynamics of alanine dipeptide, as well as coarse‑grained and all‑atom models of solute permeation through lipid bilayers. We demonstrate that, because FPSL learns a smooth score function rather than histogram‑based densities, it overcomes the exponential scaling of grid‑based methods, establishing it as a data‑efficient and scalable tool for multidimensional free‑energy estimation.

Abstract:
We study the training and performance of physics‑informed learning for initial and boundary value problems (IBVP) with physics‑informed neural networks (PINNs) from a statistical learning perspective. Specifically, we restrict ourselves to parameterizations with hard initial and boundary condition constraints and reformulate the problem of estimating PINN parameters as a statistical learning problem. From this perspective, the physics penalty on the IBVP residuals can be better understood not as a regularizing term bus as an infinite source of indirect data, and the learning process as fitting the PINN distribution of residuals p(y \mid x, t, w) q(x, t) to the true data‑generating distribution δ(0) q(x, t) by minimizing the Kullback‑Leibler divergence between the true and PINN distributions. Furthermore, this analysis show that physics‑informed learning with PINNs is a singular learning problem, and we employ singular learning theory tools, namely the so‑called Local Learning Coefficient (Lau et al., 2025) to analyze the estimates of PINN parameters obtained via stochastic optimization for a heat equation IBVP. Finally, we discuss implications of this analysis on the quantification of predictive uncertainty of PINNs and the extrapolation capacity of PINNs.

Abstract:
Physics‑informed neural networks (PINNs) have gained significant attention as a surrogate modeling strategy for partial differential equations (PDEs), particularly in regimes where labeled data are scarce and physical constraints can be leveraged to regularize the learning process. In practice, however, PINNs are frequently trained using experimental or numerical data that are not fully consistent with the governing equations due to measurement noise, discretization errors, or modeling assumptions. The implications of such data‑to‑PDE inconsistencies on the accuracy and convergence of PINNs remain insufficiently understood. In this work, we systematically analyze how data inconsistency fundamentally limits the attainable accuracy of PINNs. We introduce the concept of a consistency barrier, defined as an intrinsic lower bound on the error that arises from mismatches between the fidelity of the data and the exact enforcement of the PDE residual. To isolate and quantify this effect, we consider the 1D viscous Burgers equation with a manufactured analytical solution, which enables full control over data fidelity and residual errors. PINNs are trained using datasets of progressively increasing numerical accuracy, as well as perfectly consistent analytical data. Results show that while the inclusion of the PDE residual allows PINNs to partially mitigate low‑fidelity data and recover the dominant physical structure, the training process ultimately saturates at an error level dictated by the data inconsistency. When high‑fidelity numerical data are employed, PINN solutions become indistinguishable from those trained on analytical data, indicating that the consistency barrier is effectively removed. These findings clarify the interplay between data quality and physics enforcement in PINNs providing practical guidance for the construction and interpretation of physics‑informed surrogate models.

Abstract:
Symbolic regression aims to distill mathematical equations from observational data. Recent approaches have successfully leveraged Large Language Models (LLMs) to generate equation hypotheses, capitalizing on their vast pre‑trained scientific priors. However, existing frameworks predominantly treat the LLM as a static generator, relying on prompt‑level guidance to steer exploration. This paradigm fails to update the model's internal representations based on search feedback, often yielding physically inconsistent or mathematically redundant expressions. In this work, we propose PiT‑PO (Physics‑informed Token‑regularized Policy Optimization), a unified framework that evolves the LLM into an adaptive generator via reinforcement learning. Central to PiT‑PO is a dual‑constraint mechanism that rigorously enforces hierarchical physical validity while simultaneously applying fine‑grained, token‑level penalties to suppress redundant structures. Consequently, PiT‑PO aligns LLM to produce equations that are both scientifically consistent and structurally parsimonious. Empirically, PiT‑PO achieves state‑of‑the‑art performance on standard benchmarks and successfully discovers novel turbulence models for challenging fluid dynamics problems. We also demonstrate that PiT‑PO empowers small‑scale models to outperform closed‑source giants, democratizing access to high‑performance scientific discovery.

Abstract:
Many scientific and engineering systems exhibit intrinsically multimodal behavior arising from latent regime switching and non‑unique physical mechanisms. In such settings, learning the full conditional distribution of admissible outcomes in a physically consistent and interpretable manner remains a challenge. While recent advances in machine learning have enabled powerful multimodal generative modeling, their integration with physics‑constrained scientific modeling remains nontrivial, particularly when physical structure must be preserved or data are limited. This work develops a physics‑informed multimodal conditional modeling framework based on mixture density representations. Mixture density networks (MDNs) provide an explicit and interpretable parameterization of multimodal conditional distributions. Physical knowledge is embedded through component‑specific regularization terms that penalize violations of governing equations or physical laws. This formulation naturally accommodates non‑uniqueness and stochasticity while remaining computationally efficient and amenable to conditioning on contextual inputs. The proposed framework is evaluated across a range of scientific problems in which multimodality arises from intrinsic physical mechanisms rather than observational noise, including bifurcation phenomena in nonlinear dynamical systems, stochastic partial differential equations, and atomistic‑scale shock dynamics. In addition, the proposed method is compared with a conditional flow matching (CFM) model, a representative state‑of‑the‑art generative modeling approach, demonstrating that MDNs can achieve competitive performance while offering a simpler and more interpretable formulation.

Abstract:
Transparent liquid manipulation in robotic pouring remains challenging for perception systems: specular/refraction effects and lighting variability degrade visual cues, undermining reliable level estimation. To address this challenge, we introduce RadarEye, a real‑time mmWave radar signal processing pipeline for robust liquid level estimation and tracking during the whole pouring process. RadarEye integrates (i) a high‑resolution range‑angle beamforming module for liquid level sensing and (ii) a physics‑informed mid‑pour tracker that suppresses multipath to maintain lock on the liquid surface despite stream‑induced clutter and source container reflections. The pipeline delivers sub‑millisecond latency. In real‑robot water‑pouring experiments, RadarEye achieves a 0.35 cm median absolute height error at 0.62 ms per update, substantially outperforming vision and ultrasound baselines.

Abstract:
Control pulses that nominally optimize fidelity are sensitive to routine hardware drift and modeling errors. Robust quantum optimal control seeks error‑insensitive control pulses that maintain fidelity thresholds and obey hardware constraints. Distinct numerical approximations to the first‑order error susceptibility include adjoint end‑point and toggling‑frame approaches. Although theoretically equivalent, we provide a novel, systematic study demonstrating important numerical differences between these two approaches. We also introduce a critical discretization correction to the widely‑used toggling‑frame robustness estimator, measurably improving its estimate of first‑order error susceptibility. We accomplish our study by positioning robustness as a first‑class objective within direct, constrained optimal control. Our approach uniquely handles control and fidelity constraints while cleanly isolating robustness for dedicated optimization. In both single‑ and two‑qubit examples under realistic constraints, our approach provides an analytic edge for obtaining precise, physics‑informed robustness.

Abstract:
We investigate the integration of Kolmogorov‑Arnold Networks (KANs) into hard‑constrained recurrent physics‑informed architectures (HRPINN) to evaluate the fidelity of learned residual manifolds in oscillatory systems. Motivated by the Kolmogorov‑Arnold representation theorem and preliminary gray‑box results, we hypothesized that KANs would enable efficient recovery of unknown terms compared to MLPs. Through initial sensitivity analysis on configuration sensitivity, parameter scale, and training paradigm, we found that while small KANs are competitive on univariate polynomial residuals (Duffing), they exhibit severe hyperparameter fragility, instability in deeper configurations, and consistent failure on multiplicative terms (Van der Pol), generally outperformed by standard MLPs. These empirical challenges highlight limitations of the additive inductive bias in the original KAN formulation for state coupling and provide preliminary empirical evidence of inductive bias limitations for future hybrid modeling.

Abstract:
Standard Physics‑Informed Neural Networks (PINNs) often face challenges when modeling parameterized dynamical systems with sharp regime transitions, such as bifurcations. In these scenarios, the continuous mapping from parameters to solutions can result in spectral bias or "mode collapse", where the network averages distinct physical behaviors. We propose a Topology‑Aware PINN (TAPINN) that aims to mitigate this challenge by structuring the latent space via Supervised Metric Regularization. Unlike standard parametric PINNs that map physical parameters directly to solutions, our method conditions the solver on a latent state optimized to reflect the metric‑based separation between regimes, showing ~49% lower physics residual (0.082 vs. 0.160). We train this architecture using a phase‑based Alternating Optimization (AO) schedule to manage gradient conflicts between the metric and physics objectives. Preliminary experiments on the Duffing Oscillator demonstrate that while standard baselines suffer from spectral bias and high‑capacity Hypernetworks overfit (memorizing data while violating physics), our approach achieves stable convergence with 2.18x lower gradient variance than a multi‑output Sobolev Error baseline, and 5x fewer parameters than a hypernetwork‑based alternative.

Abstract:
Accurate modeling of drug release is essential for designing and developing controlled‑release systems. Classical models (Fick, Higuchi, Peppas) rely on simplifying assumptions that limit their accuracy in complex geometries and release mechanisms. Here, we propose a novel approach using Physics‑Informed Neural Networks (PINNs) and Bayesian PINNs (BPINNs) for predicting release from planar, 1D‑wrinkled, and 2D‑crumpled films. This approach uniquely integrates Fick's diffusion law with limited experimental data to enable accurate long‑term predictions from short‑term measurements, and is systematically benchmarked against classical drug release models. We embedded Fick's second law into PINN as loss with 10,000 Latin‑hypercube collocation points and utilized previously published experimental datasets to assess drug release performance through mean absolute error (MAE) and root mean square error (RMSE), considering noisy conditions and limited‑data scenarios. Our approach reduced mean error by up to 40% relative to classical baselines across all film types. The PINN formulation achieved RMSE <0.05 utilizing only the first 6% of the release time data (reducing 94% of release time required for the experiments) for the planar film. For wrinkled and crumpled films, the PINN reached RMSE <0.05 in 33% of the release time data. BPINNs provide tighter and more reliable uncertainty quantification under noise. By combining physical laws with experimental data, the proposed framework yields highly accurate long‑term release predictions from short‑term measurements, offering a practical route for accelerated characterization and more efficient early‑stage drug release system formulation.

Abstract:
We propose a methodology that combines generative latent diffusion models with physics‑informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid‑based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics‑informed constraints and measurement conditions during inference, applying Adam‑based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier‑‑Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion‑based PDE solvers, which are state of the art for sparse observations. Code is available at https://github.com/deeplearningmethods/PISD.

Abstract:
Integrating domain knowledge into neural networks is a central challenge in scientific machine learning. Three paradigms have emerged ‑‑ data‑driven (Neural Ordinary Differential Equations, NODEs), soft‑constrained (Physics‑Informed Neural Networks, PINNs), and hard‑constrained (Differentiable Programming, DP) ‑‑ each encoding physical knowledge at different levels of structural commitment. However, how these strategies impact not only predictive accuracy but also downstream tasks such as control synthesis remains insufficiently understood. This paper presents a comparative study of NODEs, PINNs, and DP for dynamical system modeling, using the Single Machine Infinite Bus power system as a benchmark. We evaluate these paradigms across three tasks: trajectory prediction, parameter identification, and Linear Quadratic Regulator control synthesis. Our results yield three principal findings. First, knowledge representation determines generalization: NODE, which learns the system operator, enables robust extrapolation, whereas PINN, which approximates a solution map, restricts generalization to the training horizon. Second, hard‑constrained formulations (DP) reduce learning to a low‑dimensional physical parameter space, achieving faster and more reliable convergence than soft‑constrained approaches. Third, knowledge fidelity propagates to control performance: DP produces controllers that closely match those obtained from true system parameters, while NODE provides a viable data‑driven alternative by recovering control‑relevant Jacobians with 3‑4% relative error and yielding LQR gains within 0.36% of the ground truth. Based on these findings, we propose a practical decision framework for selecting knowledge integration strategies in neural modeling of dynamical systems.

Abstract:
We propose a physics‑informed consistency modeling framework for solving partial differential equations (PDEs) via fast, few‑step generative inference. We identify a key stability challenge in physics‑constrained consistency training, where PDE residuals can drive the model toward trivial or degenerate solutions, degrading the learned data distribution. To address this, we introduce a structure‑preserving two‑stage training strategy that decouples distribution learning from physics enforcement by freezing the coefficient decoder during physics‑informed fine‑tuning. We further propose a two‑step residual objective that enforces physical consistency on refined, structurally valid generative trajectories rather than noisy single‑step predictions. The resulting framework enables stable, high‑fidelity inference for both unconditional generation and forward problems. We demonstrate that forward solutions can be obtained via a projection‑based zero‑shot inpainting procedure, achieving consistent accuracy of diffusion baselines with orders of magnitude reduction in computational cost.

Abstract:
Physics‑informed neural networks (PINNs) and hybrid quantum‑classical extensions provide a promising framework for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. In this work, we study embedding strategies for trainable embedding quantum physics‑informed neural networks (TE‑QPINNs) in the context of nonlinear reaction‑diffusion (RD) systems. We introduce an extended TE‑QPINN (x‑TE‑QPINN) architecture that supports both classical and fully quantum embeddings, enabling a controlled comparison between feedforward neural network‑based feature maps and parameterized quantum circuit embeddings. The first architecture is the classical embedding feed‑forward neural network‑based TE‑QPINN (FNN‑TE‑QPINN), while the latter variant is a purely quantum one, referred to as quantum embedding neural network‑based TE‑QPINN (QNN‑TE‑QPINN). The proposed framework employs hardware‑efficient variational quantum circuits and species‑specific readout operators to approximate coupled multi‑field dynamics while enforcing governing equations, boundary conditions, and initial conditions through a physics‑informed loss function. By isolating the embedding mechanism while keeping the variational ansatz, loss formulation, and optimization procedure fixed, we analyze the impact of embedding design on gradient structure, parameter scaling, and quantum resource requirements. Numerical experiments on one‑ and two‑dimensional RD equations demonstrate that quantum embeddings can replace classical embeddings without degradation in solution accuracy and, in certain regimes, exhibit improved optimization behavior compared to classical PINNs and hybrid quantum models with fixed embeddings. These results provide architectural insight into hybrid quantum PDE solvers and inform the design of resource‑efficient quantum physics‑informed learning methods.

Abstract:
Physics‑informed (PI) methodologies have surged to become a pillar route to solve Differential Equations (DEs), sustained by the growth of machine learning methods in scientific contexts. The main proposition of PI is to minimise variationally a loss function, formally ensuring that a neural surrogate of the solution has the DE locally satisfied. The nature of such formulation encouraged the exploration of equivalent quantum algorithms, where the surrogate solution is expressed by variational quantum architectures. The locality of typical loss functions emphasises the DE to hold at an ensemble of points sampled in the domain, but encounters issues when generalising beyond such points, or when propagating boundary conditions. Issues which affect classical and quantum PI algorithms alike. The quest to fill this gap in robustness and accuracy against mainstream DE solvers has led to a plethora of proposals in various directions. In particular, classical DE solvers have long employed the weak form ‑ an integral based approach aiming at imposing a global condition on the solution ‑ prioritising a good average behaviour instead of ``overfitting'' select points. Here, we propose and explore to combine contributions from both local and global loss functions in PI routines, to exploit the advantages and mitigate the weaknesses of both. We showcase this intuition in a variety of problems focusing on differentiable quantum architectures, and demonstrating in particular how orchestrating such hybrid loss formulation with domain decomposition can offer a strong advantage over local‑only strategies.

Abstract:
Quadrotor unmanned aerial vehicles (UAVs) are increasingly deployed in complex missions that demand reliable autonomous navigation and robust obstacle avoidance. However, traditional modular pipelines often incur cumulative latency, whereas purely reinforcement learning (RL) approaches typically provide limited formal safety guarantees. To bridge this gap, we propose an end‑to‑end RL framework augmented with model‑based safety mechanisms. We incorporate physical priors in both training and deployment. During training, we design a physics‑informed reward structure that provides global navigational guidance. During deployment, we integrate a real‑time safety filter that projects the policy outputs onto a provably safe set to enforce strict collision‑avoidance constraints. This hybrid architecture reconciles high‑speed flight with robust safety assurances. Benchmark evaluations demonstrate that our method outperforms both traditional planners and recent end‑to‑end obstacle avoidance approaches based on differentiable physics. Extensive experiments demonstrate strong generalization, enabling reliable high‑speed navigation in dense clutter and challenging outdoor forest environments at velocities up to 7.5m/s.

Abstract:
This work investigates the use of shallow physics‑informed neural networks (PINNs) for solving forward and inverse problems of nonlinear partial differential equations (PDEs). By reformulating PINNs as nonlinear systems, the Levenberg‑Marquardt (LM) algorithm is employed to efficiently optimize the network parameters. Analytical expressions for the neural network derivatives with respect to the input variables are derived, enabling accurate and efficient computation of the Jacobian matrix required by LM. The proposed approach is tested on several benchmark problems, including the Burgers, Schrödinger, Allen‑Cahn, and three‑dimensional Bratu equations. Numerical results demonstrate that LM significantly outperforms BFGS in terms of convergence speed, accuracy, and final loss values, even when using shallow network architectures with only two hidden layers. These findings indicate that, for a wide class of PDEs, shallow PINNs combined with efficient second‑order optimization methods can provide accurate and computationally efficient solutions for both forward and inverse problems.

Abstract:
Radial singular fields, such as 1/r, \log r, and crack‑tip profiles, are difficult to model with current coordinate‑separable neural architectures. We formally establish this result: any C^2 function that is both radial and additively separable must be quadratic, establishing a fundamental obstruction for coordinate‑wise power‑law models. Motivated by this result, we introduce Radial Müntz‑Szász Networks (RMN), which represent fields as linear combinations of learnable radial powers r^μ, including negative exponents, together with a limit‑stable log‑primitive for exact \log r behavior. RMN admits closed‑form spatial gradients and Laplacians, enabling physics‑informed learning on punctured domains. Across ten 2D and 3D benchmarks, RMN achieves between 1.5 and 51 times lower RMSE than MLPs and between 10 and 100 times lower RMSE than SIREN, while using only 27 parameters, compared with 33,537 for MLPs and 8,577 for SIREN. We extend RMN to incorporate angular dependence (RMN‑Angular) and to handle multiple sources with learnable centers (RMN‑MC), whose source‑center recovery errors fall below 10^‑4. We also report controlled failures on smooth, strongly non‑radial targets to delineate RMN's operating regime.

Abstract:
With the success of static black‑hole imaging, the next frontier is the dynamic and 3D imaging of black holes. Recovering the dynamic 3D gas near a black hole would reveal previously‑unseen parts of the universe and inform new physics models. However, only sparse radio measurements from a single viewpoint are possible, making the dynamic 3D reconstruction problem significantly ill‑posed. Previously, BH‑NeRF addressed the ill‑posed problem by assuming Keplerian dynamics of the gas, but this assumption breaks down near the black hole, where the strong gravitational pull of the black hole and increased electromagnetic activity complicate fluid dynamics. To overcome the restrictive assumptions of BH‑NeRF, we propose PI‑DEF, a physics‑informed approach that uses differentiable neural rendering to fit a 4D (time + 3D) emissivity field given EHT measurements. Our approach jointly reconstructs the 3D velocity field with the 4D emissivity field and enforces the velocity as a soft constraint on the dynamics of the emissivity. In experiments on simulated data, we find significantly improved reconstruction accuracy over both BH‑NeRF and a physics‑agnostic approach. We demonstrate how our method may be used to estimate other physics parameters of the black hole, such as its spin.

Abstract:
Deep learning‑based garment draping has emerged as a promising alternative to traditional Physics‑Based Simulation (PBS), yet robust collision handling remains a critical bottleneck. Most existing methods enforce physical validity through soft penalties, creating an intrinsic trade‑off between geometric feasibility and physical plausibility: penalizing collisions often distorts mesh structure, while preserving shape leads to interpenetration. To resolve this conflict, we present PhysDrape, a hybrid neural‑physical solver for physically realistic garment draping driven by explicit forces and constraints. Unlike soft‑constrained frameworks, PhysDrape integrates neural inference with explicit geometric solvers in a fully differentiable pipeline. Specifically, we propose a Physics‑Informed Graph Neural Network conditioned on a physics‑enriched graph ‑‑ encoding material parameters and body proximity ‑‑ to predict residual displacements. Crucially, we integrate a differentiable two‑stage solver: first, a learnable Force Solver iteratively resolves unbalanced forces derived from the Saint Venant‑Kirchhoff (StVK) model to ensure quasi‑static equilibrium; second, a Differentiable Projection strictly enforces collision constraints against the body surface. This differentiable design guarantees physical validity through explicit constraints, while enabling end‑to‑end learning to optimize the network for physically consistent predictions. Extensive experiments demonstrate that PhysDrape achieves state‑of‑the‑art performance, ensuring negligible interpenetration with significantly lower strain energy compared to existing baselines, achieving superior physical fidelity and robustness in real‑time.

Abstract:
Low‑rank adaptation (LoRA) has emerged as a widely adopted parameter‑efficient fine‑tuning method in deep transfer learning, due to its reduced number of trainable parameters and lower memory requirements enabled by Burer‑Monteiro factorization on adaptation matrices. However, classical LoRA training methods treat the low‑rank factor matrices individually and optimize them using standard gradient‑based algorithms. Such decoupled optimization schemes are theoretically and empirically suboptimal, as they fail to fully exploit the intrinsic structure of the LoRA parameterization. In this work, we propose a novel continuous‑time optimization dynamic for LoRA factor matrices in the form of an ordinary differential equation (ODE) that emulates the gradient flow of full fine‑tuning on the balanced manifold. We term this approach ODELoRA. To faithfully track the trajectories of ODELoRA, we adopt well‑established and theoretically grounded time‑discretization schemes, including Euler and Runge‑‑Kutta methods. Our framework provides a unified ODE‑based perspective for understanding and designing LoRA training algorithms. We establish linear convergence of the proposed method under strongly convex objectives for certain discretization schemes under mild conditions, and further extend our analysis to the matrix sensing setting. Moreover, we show that ODELoRA achieves stable feature learning, a property that is crucial for training deep neural networks at different scales of problem dimensionality. Empirical results on matrix sensing tasks confirm the derived linear convergence behavior, and experiments on training physics‑informed neural networks further demonstrate the superiority of ODELoRA over existing baselines, especially in the training stability.

Abstract:
Graph neural networks (GNNs) naturally align with sparse operators and unstructured discretizations, making them a promising paradigm for physics‑informed machine learning in computational mechanics. Motivated by discrete physics losses and Hierarchical Deep Learning Neural Network (HiDeNN) constructions, we embed finite‑element (FEM) computations at nodes and Gauss points directly into message‑passing layers and propose a numerically consistent FEM‑Informed Hypergraph Neural Networks (FHGNN). Similar to conventional physics‑informed neural networks (PINNs), training is purely physics‑driven and requires no labeled data: the input is a node element hypergraph whose edges encode mesh connectivity. Guided by empirical results and condition‑number analysis, we adopt an efficient variational loss. Validated on 3D benchmarks, including cyclic loading with isotropic/kinematic hardening, the proposed method delivers substantially improved accuracy and efficiency over recent, competitive PINN variants. By leveraging GPU‑parallel tensor operations and the discrete representation, it scales effectively to large elastoplastic problems and can be competitive with, or faster than, multi‑core FEM implementations at comparable accuracy. This work establishes a foundation for scalable, physics‑embedded learning in nonlinear solid mechanics.

Abstract:
The development of robust and reliable modeling approaches for crystallization processes is often challenging because of non‑idealities in real data arising from various sources of uncertainty. This study investigated the effectiveness of physics‑informed recurrent neural networks (PIRNNs) that integrate the mechanistic population balance model with recurrent neural networks under the presence of systematic and model uncertainties. Such uncertainties are represented by using synthetic data containing controlled noise, solubility shift, and limited sampling. The research demonstrates that PIRNNs achieve strong generalization and physical consistency, maintain stable learning behavior, and accurately recover kinetic parameters despite significant stochastic variations in the training data. In the case of systematic errors in the solubility model, the inclusion of physics regularization improved the test performance by more than an order of magnitude compared to purely data‑driven models, whereas excessive weighting of physics increased error arising due to the model mismatch. The results also show that PIRNNs are able to recover model parameters and replicate crystallization dynamics even in the limit of very low sampling resolution. These findings validate the robustness of physics‑informed machine learning in handling data imperfections and incomplete domain knowledge, providing a potential pathway toward reliable and practical hybrid modeling of crystallization dynamics and industrial process monitoring and control.

Abstract:
The inherently complex‑valued nature of Polarimetric SAR data necessitates using specialized algorithms capable of directly processing complex‑valued representations. However, this aspect remains underexplored in the deep learning community, with many studies opting to convert complex signals into the real domain before applying conventional real‑valued models. In this work, we leverage complex‑valued neural networks and investigate the performance of complex‑valued Convolutional AutoEncoders. We show that these networks can effectively compress and reconstruct fully polarimetric SAR data while preserving essential physical characteristics, as demonstrated through Pauli, Krogager, and Cameron coherent decompositions, as well as the non‑coherent H‑α decomposition. Finally, we highlight the advantages of complex‑valued neural networks over their real‑valued counterparts. These insights pave the way for developing robust, physics‑informed, complex‑valued generative models for SAR data processing.

Abstract:
This study develops a Lagged Backward‑Compatible Physics‑Informed Neural Network (LBC‑PINN) for simulating and inverting one‑dimensional unsaturated soil consolidation under long‑term loading. To address the challenges of coupled air and water pressure dissipation across multi‑scale time domains, the framework integrates logarithmic time segmentation, lagged compatibility loss enforcement, and segment‑wise transfer learning. In forward analysis, the LBC‑PINN with recommended segmentation schemes accurately predicts pore air and pore water pressure evolution. Model predictions are validated against finite element method (FEM) results, with mean absolute errors below 1e‑2 for time durations up to 1e10 seconds. A simplified segmentation strategy based on the characteristic air‑phase dissipation time improves computational efficiency while preserving predictive accuracy. Sensitivity analyses confirm the robustness of the framework across air‑to‑water permeability ratios ranging from 1e‑3 to 1e3.

Abstract:
Modeling distributed dynamical systems governed by hyperbolic partial differential equations (PDEs) remains challenging due to discontinuities and shocks that hinder the convergence of traditional physics‑informed neural networks (PINNs). The recently proposed vanishing stacked residual PINN (VSR‑PINN) embeds a vanishing‑viscosity mechanism within stacked residual refinements to enable a smooth transition from the parabolic to hyperbolic regime. This paper integrates three curriculum‑learning methods as primal‑dual (PD) optimization, causality progression, and adaptive sampling into the VSR‑PINN. The PD strategy balances physics and data losses, the causality scheme unlocks deeper stacks by respecting temporal and gradient evolution, and adaptive sampling targets high residuals. Numerical experiments on traffic reconstruction confirm that enforcing causality systematically reduces the median point‑wise MSE and its variability across runs, yielding improvements of nearly one order of magnitude over non‑causal training in both the baseline and PD variants.

Abstract:
Symbolic Regression (SR) enables the discovery of interpretable mathematical relationships from experimental and simulation data. These relationships are often coined descriptors which are defined as a fundamental materials property that is directly correlated to a desired or undesired functional property of the material. Although established approaches such as Sure Independence Screening and Sparsifying Operator (SISSO) have successfully identified low‑dimensional descriptors within large feature spaces many existing SR tools integrate poorly with modern Python workflows, offer limited control over the symbolic search space, or struggle with the computational demands of large‑scale studies. This paper introduces DISCOVER (Data‑Informed Symbolic Combination of Operators for Variable Equation Regression), an open‑source symbolic regression package developed to address these challenges through a modular, physics‑motivated design. DISCOVER allows users to guide the symbolic search using domain knowledge, constrain the feature space explicitly, and take advantage of optional GPU acceleration to improve computational efficiency in data‑intensive workflows, enabling reproducible and scalable SR workflows. The software is intended for applications in computational physics, computational chemistry, and materials science, where interpretability, physical consistency, and execution time are especially important, and it complements general‑purpose SR frameworks by emphasizing the discovery of physically meaningful models.

Abstract:
Image‑based patient‑specific simulation of left ventricular (LV) mechanics is valuable for understanding cardiac function and supporting clinical intervention planning, but conventional finite‑element analysis (FEA) is computationally intensive. Current graph‑based surrogates do not have full‑cycle prediction capabilities, and physics‑informed neural networks often struggle to converge on complex cardiac geometries. We present CardioGraphFENet (CGFENet), a unified graph‑based surrogate for rapid full‑cycle estimation of LV myocardial biomechanics, supervised by a large FEA simulation dataset. The proposed model integrates (i) a global‑‑local graph encoder to capture mesh features with weak‑form‑inspired global coupling, (ii) a gated recurrent unit‑based temporal encoder conditioned on the target volume‑time signal to model cycle‑coherent dynamics, and (iii) a cycle‑consistent bidirectional formulation for both loading and inverse unloading within a single framework. These strategies enable high fidelity with respect to traditional FEA ground truths and produce physiologically plausible pressure‑volume loops that match FEA results when coupled with a lumped‑parameter model. In particular, the cycle‑consistency strategy enables a significant reduction in FEA supervision with only minimal loss in accuracy.

Abstract:
Training a neural network requires navigating a high‑dimensional, non‑convex loss surface to find parameters that minimize this loss. In many ways, it is surprising that optimizers such as stochastic gradient descent and ADAM can reliably locate minima which perform well on both the training and test data. To understand the success of training, a "loss landscape" community has emerged to study the geometry of the loss function and the dynamics of optimization, often using visualization techniques. However, these loss landscape studies have mostly been limited to machine learning for image classification. In the newer field of physics‑informed machine learning, little work has been conducted to visualize the landscapes of losses defined not by regression to large data sets, but by differential operators acting on state fields discretized by neural networks. In this work, we provide a comprehensive review of the loss landscape literature, as well as a discussion of the few existing physics‑informed works which investigate the loss landscape. We then use a number of the techniques we survey to empirically investigate the landscapes defined by the Deep Ritz and squared residual forms of the physics loss function. We find that the loss landscapes of physics‑informed neural networks have many of the same properties as the data‑driven classification problems studied in the literature. Unexpectedly, we find that the two formulations of the physics loss often give rise to similar landscapes, which appear smooth, well‑conditioned, and convex in the vicinity of the solution. The purpose of this work is to introduce the loss landscape perspective to the scientific machine learning community, compare the Deep Ritz and the strong form losses, and to challenge prevailing intuitions about the complexity of the loss landscapes of physics‑informed networks.

Abstract:
Advanced opinion dynamics modeling is vital for deciphering social behavior, emphasizing its role in mitigating polarization and securing cyberspace. To synergize mechanistic interpretability with data‑driven flexibility, recent studies have explored the integration of Physics‑Informed Neural Networks (PINNs) for opinion modeling. Despite this promise, existing methods are tailored to incomplete priors, lacking a comprehensive physical system to integrate dynamics from local, global, and endogenous levels. Moreover, penalty‑based constraints adopted in existing methods struggle to deeply encode physical priors, leading to optimization pathologies and discrepancy between latent representations and physical transparency. To this end, we offer a physical view to interpret opinion dynamics via Diffusion‑Convection‑Reaction (DCR) system inspired by interacting particle theory. Building upon the Neural ODEs, we define the neural opinion dynamics to coordinate neural networks with physical priors, and further present the OPINN, a physics‑informed neural framework for opinion dynamics modeling. Evaluated on real‑world and synthetic datasets, OPINN achieves state‑of‑the‑art performance in opinion evolution forecasting, offering a promising paradigm for the nexus of cyber, physical, and social systems.

Abstract:
We present a physics‑informed Bayesian optimization (PIBO) with a concise modification to its acquisition function to incorporate the physical prior knowledge. Specifically, this method multiplies the expected improvement (EI) by a weight encoding prior crystal growth physics. When applied to LaAlO3 molecular‑beam epitaxy, the weighting function defines a flat stoichiometric window and penalizes off‑window proposals, thereby steering the optimization toward physically plausible regions while maintaining controlled exploration. In a closed‑loop optimization, relative to the bare EI, which often proposes off‑stoichiometric conditions, the weighted EI constrains the search toward stoichiometric regions while retaining sufficient flexibility to explore neighboring conditions, eventually identifying an optimum slightly beyond the stoichiometric window. Within only 15 growth runs, the lattice constant of the grown LaAlO3 film converged to the bulk value, evidencing efficient and rapid optimization for the ideal stoichiometric growth. Because physics knowledge is incorporated solely through the weighting function, the approach requires only minimal modification to standard BO workflows and is readily applicable to other material systems, offering a general and practical route to AI‑driven materials synthesis.

Abstract:
We present a unified algorithmic framework for the numerical solution, constrained optimization, and physics‑informed learning of PDEs with a variational structure. Our framework is based on a Galerkin discretization of the underlying variational forms, and its high efficiency stems from a novel highly‑optimized and GPU‑compliant TensorGalerkin framework for linear system assembly (stiffness matrices and load vectors). TensorGalerkin operates by tensorizing element‑wise operations within a Python‑level Map stage and then performs global reduction with a sparse matrix multiplication that performs message passing on the mesh‑induced sparsity graph. The Map and Reduce stages are co‑designed inside PyTorch's autograd so that the assembly graph contains O(1) nodes regardless of how the number of elements and local DoFs scale. We validate this O(1)‑graph property by deploying TensorGalerkin downstream as i) a highly‑efficient numerical PDEs solver, ii) an end‑to‑end differentiable framework for PDE‑constrained optimization, and iii) a physics‑informed operator learning algorithm for PDEs. With multiple benchmarks, including 2D and 3D elliptic, parabolic, and hyperbolic PDEs on unstructured meshes, we demonstrate that the proposed framework provides significant computational efficiency and accuracy gains over a variety of baselines in all the targeted downstream applications.

Abstract:
Physics‑informed neural networks (PINNs) provide a mesh‑free framework for solving partial differential equations by embedding governing physics into neural‑network training. Recent studies have shown that parameterized PINNs can learn Navier‑Stokes solutions across Reynolds numbers by treating Reynolds number as an additional network input. However, physics‑only PINNs often lose accuracy in convection‑dominated high‑Reynolds‑number flows because of optimization stiffness and multiscale flow structures. This study presents a sparse‑supervised hybrid parameterized PINNs framework for incompressible Navier‑Stokes flows with regime‑aware learning and localized Reynolds‑number supervision. The approach is demonstrated for two‑dimensional lid‑driven cavity flow and further validated for backward‑facing step flow. At low Reynolds numbers, physics‑only PINNs accurately predict velocity and pressure fields using only governing equations and boundary conditions. At higher Reynolds numbers, sparse CFD supervision combined with transfer learning is introduced to improve predictive accuracy. Although the training range spans (500 < Re < 1000), CFD supervision is applied only within (750 < Re < 850) using just (3%‑20%) of computational points. Results show that approximately (5%) supervised data are sufficient for accurate flow prediction. Comparisons with CFD simulations demonstrate strong agreement in velocity, pressure, vorticity, and reattachment characteristics across interpolation and limited extrapolation regimes. The proposed framework provides a practical and data‑efficient hybrid strategy for incompressible flows across varying Reynolds numbers.

Abstract:
This industrial Ph.D. project, carried out in collaboration between Radiometer Medical ApS and SDU Centre for Photonics Engineering at the University of Southern Denmark, explored the use of digital holographic microscopy (DHM) for the purposes of differential white blood cell counts (dWBCs) in point‑of‑care (PoC) devices for acute care settings. Two DHM prototypes were developed; an initial lens‑based system serving as the foundation for algorithm development, and experimental validation of the approach, achieving 89.6% classification accuracy on a 3‑part differential, and a subsequent lensless system for simplified design and increased field‑of‑view (FoV). Both prototypes employed convolutional neural networks (CNNs) for cell classification. With further optimizations, the lensless system achieved classification accuracies of 92.65% and 89.44% on the 3‑part and 5‑part differential, respectively. With the lensless system, the derivation of the monocyte distribution width (MDW), a biomarker for sepsis, was also demonstrated. Additionally, pixel super‑resolution and multi‑wavelength DHM approaches were investigated to enhance the obtained cell information. Finally, a proof‑of‑principle physics‑informed neural network (PINN) for holographic reconstruction was implemented, demonstrating the potential for machine learning (ML) reconstruction techniques. In summary, this work represents an initial exploration of DHM for dWBC in PoC devices, laying the groundwork for future research.

Abstract:
Physics‑Informed Neural Networks (PINNs), which integrate deep learning with physical prior knowledge, have proven to be a powerful tool for studying the dynamics of high‑dimensional nonlinear systems. The present work utilizes PINNs to analyze the existence and evolution of quantum droplets (QDs) in a binary Bose‑Einstein condensate (BEC), revealing the ability of this technique to accurately predict structural features of the QDs, their multipeak profiles, and dynamical behavior. The stable evolution of multipole QDs is thus demonstrated. Comparing different network architectures, including the training time, loss values, and \mathbbL_2 error, PINNs accurately predict specific dynamical characteristics of QDs. Furthermore, the PINN robustness is evaluated by the application of PINN to parameter‑discovery tasks, considering both clean training data and data contaminated by 1% random noise. The results highlight the efficiency of PINNs in modeling complex quantum systems and extracting reliable parameters under the noisy conditions.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a powerful tool for solving differential equations, yet their application to quantum scattering problems has been hindered by the oscillatory, non‑decaying nature of scattering wave functions. In this work, I demonstrate that exterior complex scaling (ECS) transforms scattering boundary conditions into exponentially decaying waves suitable for neural network solutions, enabling PINNs to solve nuclear scattering problems for the first time. I develop a driven‑equation formulation where the source term is confined to the real axis, avoiding the need to analytically continue nuclear potentials into the complex plane. The method is validated on nucleon‑nucleus scattering (n+^40Ca at E_\textlab=20~MeV) with 21 partial waves, achieving phase shift accuracy of Δδ< 0.1^\circ for most channels when compared to conventional solvers. I further demonstrate the approach on heavy‑ion scattering (^6Li+^208Pb at 40~MeV) with 41 partial waves and strong Coulomb effects. This work establishes the foundation for extending PINNs to inverse problems where end‑to‑end differentiability enables direct fitting of optical potential parameters, coupled‑channel reactions, and few‑body scattering where traditional grid methods face exponential scaling.

Abstract:
We propose the Convolutional Operator Network for Forward and Inverse Problems (FI‑Conv), a framework capable of predicting system evolution and estimating parameters in complex spatio‑temporal dynamics, such as turbulence. FI‑Conv is built on a U‑Net architecture, in which most convolutional layers are replaced by ConvNeXt V2 blocks. This design preserves U‑Net performance on inputs with high‑frequency variations while maintaining low computational complexity. FI‑Conv uses an initial state, PDE parameters, and evolution time as input to predict the system future state. As a representative example of a system exhibiting complex dynamics, we evaluate the performance of FI‑Conv on the task of predicting turbulent plasma fields governed by the Hasegawa‑Wakatani (HW) equations. The HW system models two‑dimensional electrostatic drift‑wave turbulence and exhibits strongly nonlinear behavior, making accurate approximation and long‑term prediction particularly challenging. Using an autoregressive forecasting procedure, FI‑Conv achieves accurate forward prediction of the plasma state evolution over short times (t ~ 3) and captures the statistic properties of derived physical quantities of interest over longer times (t ~ 100). Moreover, we develop a gradient‑descent‑based inverse estimation method that accurately infers PDE parameters from plasma state evolution data, without modifying the trained model weights. Collectively, our results demonstrate that FI‑Conv can be an effective alternative to existing physics‑informed machine learning methods for systems with complex spatio‑temporal dynamics.

Abstract:
We present a comparative benchmark of Quantum Tensor Networks (QTN), the Hydrodynamic Schrödinger Equation (HSE), and Physics‑Informed Neural Networks (PINN) for simulating the 1D Burgers' equation. Evaluating these emerging paradigms against classical GMRES and Spectral baselines, we analyse solution accuracy, runtime scaling, and resource overhead across grid resolutions ranging from N=4 to N=128. Our results reveal a distinct performance hierarchy. The QTN solver achieves superior precision (L_2 ～ 10^‑7) with remarkable near‑constant runtime scaling, effectively leveraging entanglement compression to capture shock fronts. In contrast, while the Finite‑Difference HSE implementation remains robust, the Spectral HSE method suffers catastrophic numerical instability at high resolutions, diverging significantly at N=128. PINNs demonstrate flexibility as mesh‑free solvers but stall at lower accuracy tiers (L_2 ～ 10^‑1), limited by spectral bias compared to grid‑based methods. Ultimately, while quantum methods offer novel representational advantages for low‑resolution fluid dynamics, this study confirms they currently yield no computational advantage over classical solvers without fault tolerance or significant algorithmic breakthroughs in handling non‑linear feedback.

Abstract:
G‑protein‑coupled receptors (GPCRs), primary targets for over one‑third of approved therapeutics, rely on intricate conformational transitions to transduce signals. While Molecular Dynamics (MD) is essential for elucidating this transduction process, particularly within ligand‑bound complexes, conventional all‑atom MD simulation is computationally prohibitive. In this paper, we introduce GPCRLMD, a deep generative framework for efficient all‑atom GPCR‑ligand simulation.GPCRLMD employs a Harmonic‑Prior Variational Autoencoder (HP‑VAE) to first map the complex into a regularized isometric latent space, preserving geometric topology via physics‑informed constraints. Within this latent space, a Residual Latent Flow samples evolution trajectories, which are subsequently decoded back to atomic coordinates. By capturing temporal dynamics via relative displacements anchored to the initial structure, this residual mechanism effectively decouples static topology from dynamic fluctuations. Experimental results demonstrate that GPCRLMD achieves state‑of‑the‑art performance in GPCR‑ligand dynamics simulation, faithfully reproducing thermodynamic observables and critical ligand‑receptor interactions.

Abstract:
We address dynamic manipulation of deformable linear objects by presenting SPiD, a physics‑informed self‑supervised learning framework that couples an accurate deformable object model with an augmented self‑supervised training strategy. On the modeling side, we extend a mass‑spring model to more accurately capture object dynamics while remaining lightweight enough for high‑throughput rollouts during self‑supervised learning. On the learning side, we train a neural controller using a task‑oriented cost, enabling end‑to‑end optimization through interaction with the differentiable object model. In addition, we propose a self‑supervised DAgger variant that detects distribution shift during deployment and performs offline self‑correction to further enhance robustness without expert supervision. We evaluate our method primarily on the rope stabilization task, where a robot must bring a swinging rope to rest as quickly and smoothly as possible. Extensive experiments in both simulation and the real world demonstrate that the proposed controller achieves fast and smooth rope stabilization, generalizing across unseen initial states, rope lengths, masses, non‑uniform mass distributions, and external disturbances. Additionally, we develop an affordable markerless rope perception method and demonstrate that our controller maintains performance with noisy and low‑frequency state updates. Furthermore, we demonstrate the generality of the framework by extending it to the rope trajectory tracking task. Overall, SPiD offers a data‑efficient, robust, and physically grounded framework for dynamic manipulation of deformable linear objects, featuring strong sim‑to‑real generalization.

Abstract:
The electrification of vertical takeoff and landing aircraft demands high‑fidelity battery management systems capable of predicting voltage response under aggressive power dynamics. While data‑driven models offer high accuracy, they often require complex architectures and extensive training data. Conversely, equivalent circuit models (ECMs), such as the second‑order model, offer physical interpretability but struggle with high C‑rate non‑linearities. This paper investigates the impact of integrating physics‑based information into data‑driven surrogate models. Specifically, we evaluate whether physics‑informed features allow for the simplification of neural network architectures without compromising accuracy. Using the open‑source electric vertical takeoff and landing (eVTOL) battery dataset, we compare pure data‑driven models against physics‑informed data models. Results demonstrate that physics‑informed models achieve comparable accuracy to complex pure data‑driven models while using up to 75% fewer trainable parameters, significantly reducing computational overhead for potential on‑board deployment.

Abstract:
The random feature method (RFM) constructs approximation spaces by initializing features from generic distributions, which provides universal approximation properties to solve general partial differential equations. However, such standard initializations lack awareness of the underlying physical laws and geometry, which limits approximation. In this work, we propose the Physics‑Driven Orthogonal Feature Method (PD‑OFM), a framework for constructing feature representations that are explicitly tailored to both the differential operator and the computational domain by pretraining features using physics‑informed objectives together with orthogonality regularization. This pretraining strategy yields nearly orthogonal feature bases. We provide both theoretical and empirical evidence that physics‑informed pretraining improves the approximation capability of the learned feature space. When employed to solve Helmholtz, Poisson, wave, and Navier‑Stokes equations, the proposed method achieves residual errors 2‑3 orders of magnitude lower than those of comparable methods. Furthermore, the orthogonality regularization improves transferability, enabling pretrained features to generalize effectively across different source terms and domain geometries for the same PDE.

Abstract:
Understanding the mechanism of turbulent skin‑friction drag (TSD) generation is of fundamental and practical importance for designing effective drag reduction strategies. However, many previous studies adopted correlation analysis to reveal the causal map between turbulent motions and TSD generation, an approach that is potentially risky as correlation does not necessarily imply causation. In this study, a novel causal inference method called Liang‑Kleeman information flow (LKIF) is utilized for the first time to identify the velocity‑induced causal structures related to TSD generation in a turbulent channel flow. The statistical properties of the causal structures are comprehensively investigated. The positive and negative causal structures, defined by their signs and respectively associated with an increase and decrease in TSD information entropy, promote and suppress the generation of extreme TSD. Particularly, we find that the underlying physics of causal structures is essentially associated with the processes of streamwise streaks and rolls approaching or receding from the extreme events. Results indicate that the physics‑informed LKIF framework can reveal a more explicit and interpretable causal relationship than correlation analysis.

Abstract:
The potential of Quantum Machine Learning (QML) in data‑intensive science is strictly bottlenecked the difficulty of interfacing high‑dimensional, chaotic classical data into resource‑limited, noisy quantum processors. To bridge this gap, we introduce a physics‑informed Koopman‑Quantum hybrid framework, theoretically grounded in a representation‑level structural isomorphism we establish between the Koopman operator, which linearizes nonlinear dynamics, and quantum evolution. Based on this theoretical foundation, we design a realizable NISQ‑friendly pipeline: the Koopman operator functions as a physics‑aware "data distiller," compressing waveforms into compact, "quantum‑ready" features, which are subsequently processed by a modular, parallel quantum neural network. We validated this framework on 4,763 labeled channel sequences from 433 discharges of the tokamak system. The results demonstrate that our model achieves 97.0% accuracy in screening corrupted diagnostic data, matching the performance of state‑of‑the‑art deep classical CNNs while using orders‑of‑magnitude fewer trainable parameters. This work establishes a practical, physics‑grounded paradigm for leveraging quantum processing in constrained environments, offering a scalable path for quantum‑enhanced edge computing.

Abstract:
The advancement of data‑driven materials science is currently constrained by a fundamental bottleneck: the vast majority of historical experimental data remains locked within the unstructured text and rasterized figures of legacy scientific literature. Manual curation of this knowledge is prohibitively labor‑intensive and prone to human error. To address this challenge, we introduce an autonomous, agent‑based framework powered by Large Language Models (LLMs) designed to excavate high‑fidelity datasets from scientific PDFs without human intervention. By deploying a modular "skill‑based" architecture, the agent orchestrates complex cognitive tasks ‑ including semantic filtering, multi‑modal information extraction, and physics‑informed validation. We demonstrate the efficacy of this framework by constructing a physically self‑consistent database for material creep mechanics, a domain characterized by complex graphical trajectories and heterogeneous constitutive models. Applying the pipeline to 243 publications, the agent achieved a verified extraction success rate exceeding 90% for graphical data digitization. Crucially, we introduce a cross‑modal verification protocol, demonstrating that the agent can autonomously align visually extracted data points with textually extracted constitutive parameters (R^2 > 0.99), ensuring the physical self‑consistency of the database. This work not only provides a critical resource for investigating time‑dependent deformation across diverse material systems but also establishes a scalable paradigm for autonomous knowledge acquisition, paving the way for the next generation of self‑driving laboratories.

Abstract:
In this study, we investigate the capability of physics‑informed neural networks (PINNs) to preserve global physical structures by comparing standard PINNs with a Trefftz‑based PINN (Trefftz‑PINN). The target problem is the reproduction of mag‑netic field‑line structures in a helical fusion reactor configuration. Using identical training data sampled from exact solutions, we perform comparisons under matched mean squared error (MSE) levels. Visualization of magnetic field lines reveals that standard PINNs may exhibit structural collapse across magnetic surfaces even when the MSE is sufficiently small, whereas Trefftz‑PINNs successfully preserve the global topology of magnetic field lines. Furthermore, the proposed framework is extended to computational fluid dynamics (CFD) problems, where streamline structures of veloc‑ity fields are analyzed. Similar tendencies are observed, demonstrating that Trefftz‑PINNs provide superior structure preservation compared to standard PINNs. These results indicate that minimizing numerical error alone does not guarantee physical consistency, and that constraining the solution space prior to learning is an effective strategy for physics‑consistent surrogate modeling.

Abstract:
Inverse scattering in optical coherence tomography (OCT) seeks to recover both structural images and intrinsic tissue optical properties, including refractive index, scattering coefficient, and anisotropy. This inverse problem is challenging due to attenuation, speckle noise, and strong coupling among parameters. We propose a regularized end‑to‑end deep learning framework that jointly reconstructs optical parameter maps and speckle‑reduced OCT structural intensity for layer visualization. Trained with Monte Carlo‑simulated ground truth, our network incorporates a physics‑based OCT forward model that generates predicted signals from the estimated parameters, providing physics‑consistent supervision for parameter recovery and artifact suppression. Experiments on the synthetic corneal OCT dataset demonstrate robust optical map recovery under noise, improved resolution, and enhanced structural fidelity. This approach enables quantitative multi‑parameter tissue characterization and highlights the benefit of combining physics‑informed modeling with deep learning for computational OCT.

Abstract:
Foundation models for echocardiography often struggle to disentangle anatomical signal from the stochastic speckle and acquisition artifacts inherent to ultrasound. We present EchoJEPA, a foundation model trained on 18 million echocardiograms across 300K patients, representing the largest pretraining corpus for this modality to date. By leveraging a latent predictive objective, EchoJEPA learns robust anatomical representations that ignore speckle noise. We validate this using a novel multi‑view probing framework with frozen backbones, where EchoJEPA outperforms leading baselines by approximately 20% in left ventricular ejection fraction (LVEF) estimation and 17% in right ventricular systolic pressure (RVSP) estimation. The model also exhibits remarkable sample efficiency, reaching 79% view classification accuracy with only 1% of labeled data versus 42% for the best baseline trained on 100%. Crucially, EchoJEPA demonstrates superior generalization, degrading by only 2% under physics‑informed acoustic perturbations compared to 17% for competitors. Most remarkably, its zero‑shot performance on pediatric patients surpasses fully fine‑tuned baselines, establishing latent prediction as a superior paradigm for robust, generalizable medical AI.

Abstract:
Physics‑Informed Neural Networks (PINNs) are effective methods for solving inverse problems and discovering governing equations from observational data. However, their performance degrades significantly under complex measurement noise and gross outliers. To address this issue, we propose the Noise‑Adaptive Physics‑Informed Neural Network (naPINN), which robustly recovers physical solutions from corrupted measurements without prior knowledge of the noise distribution. naPINN embeds an energy‑based model into the training loop to learn the latent distribution of prediction residuals. Leveraging the learned energy landscape, a trainable reliability gate adaptively filters data points exhibiting high energy, while a rejection cost regularization prevents trivial solutions where valid data are discarded. We demonstrate the efficacy of naPINN on various benchmark partial differential equations corrupted by non‑Gaussian noise and varying rates of outliers. The results show that naPINN significantly outperforms existing robust PINN baselines, successfully isolating outliers and accurately reconstructing the dynamics under severe data corruption.

Abstract:
Solving partial differential equations remains a central challenge in scientific machine learning. Neural operators offer a promising route by learning mappings between function spaces and enabling resolution‑independent inference, yet they typically require supervised data. Physics‑informed neural networks address this limitation through unsupervised training with physical constraints but often suffer from unstable convergence and limited generalization capability. To overcome these issues, we introduce a multi‑stage physics‑informed training strategy that achieves convergence by progressively enforcing boundary conditions in the loss landscape and subsequently incorporating interior residuals. At each stage the optimizer is re‑initialized, acting as a continuation mechanism that restores stability and prevents gradient stagnation. We further propose the Physics‑Informed Spline Fourier Neural Operator (PhIS‑FNO), combining Fourier layers with Hermite spline kernels for smooth residual evaluation. Across canonical benchmarks, PhIS‑FNO attains a level of accuracy comparable to that of supervised learning, using labeled information only along a narrow boundary region, establishing staged, spline‑based optimization as a robust paradigm for physics‑informed operator learning.

Abstract:
Quantum many‑body systems pose a formidable computational challenge due to the exponential growth of their Hilbert space. While machine learning (ML) has shown promise as an alternative paradigm, most applications remain at the proof‑of‑concept stage, focusing narrowly on energy estimation at the lower end of the spectrum. Here, we push ML beyond this frontier by extensively testing HubbardNet, a deep neural network architecture for the Bose‑Hubbard model. Pushing improvements in the optimizer and learning rates, and introducing physics‑informed output activations that can resolve extremely small wave‑function amplitudes, we achieve ground‑state energy errors reduced by orders of magnitude and wave‑function fidelities exceeding 99%. We further assess physical relevance by analysing generalized inverse participation ratios and multifractal dimensions for ground and excited states in one and two dimensions, demonstrating that optimized ML models reproduce localization, delocalization, and multifractality trends across the spectrum. Crucially, these qualitative predictions remain robust across four decades of the interaction strength, e.g. spanning across superfluid, Mott‑insulating, as well as quantum chaotic regimes. Together, these results suggest ML as a viable qualitative predictor of many‑body structure, complementing the quantitative strengths of exact diagonalization and tensor‑network methods.

Abstract:
Physics‑Informed Neural Networks (PINNs) have become a popular and powerful framework for solving partial differential equations (PDEs), leveraging neural networks to approximate solutions while embedding PDE constraints, boundary conditions, and interface jump conditions directly into the loss function. However, most existing PINN approaches are based on multilayer perceptrons (MLPs), which may require large network sizes and extensive training to achieve high accuracy, especially for complex interface problems. In this work, we propose a novel PINN architecture based on Kolmogorov‑Arnold Networks (KANs), which offer greater flexibility in choosing activation functions and can represent functions with fewer parameters. Specifically, we introduce a dual KANs structure that couples two KANs across subdomains and explicitly enforces interface conditions. To further boost training efficiency and convergence, we integrate the RAR‑D adaptive sampling strategy to dynamically refine training points. Numerical experiments on the elliptic interface problems yield more uniform error distributions across the computational domain, which demonstrates that our PINN‑based KANs achieve superior accuracy with significantly smaller network sizes and faster convergence compared to standard PINNs.

Abstract:
Infrared small target detection (IRSTD) is crucial for surveillance and early‑warning, with deployments spanning both single‑frame analysis and video‑mode tracking. A practical solution should leverage vision foundation models (VFMs) to mitigate infrared data scarcity, while adopting a memory‑attention‑based temporal propagation framework that unifies single‑ and multi‑frame inference. However, infrared small targets exhibit weak radiometric signals and limited semantic cues, which differ markedly from visible‑spectrum imagery. This modality gap makes direct use of semantics‑oriented VFMs and appearance‑driven cross‑frame association unreliable for IRSTD: hierarchical feature aggregation can submerge localized target peaks, and appearance‑only memory attention becomes ambiguous, leading to spurious clutter associations. To address these challenges, we propose SPIRIT, a unified and VFM‑compatible framework that adapts VFMs to IRSTD via lightweight physics‑informed plug‑ins. Spatially, PIFR refines features by approximating rank‑sparsity decomposition to suppress structured background components and enhance sparse target‑like signals. Temporally, PGMA injects history‑derived soft spatial priors into memory cross‑attention to constrain cross‑frame association, enabling robust video detection while naturally reverting to single‑frame inference when temporal context is absent. Experiments on multiple IRSTD benchmarks show consistent gains over VFM‑based baselines and SOTA performance.

Abstract:
The prevailing data‑driven machine learning has been plagued by the absence of physics knowledge and the scarcity of data. We implement the physics‑model informed prior into Bayesian machine learning to evaluate the energy dependence of independent fission product yields, which are crucial for advanced nuclear energy applications but only sparse and imperfect experimental data are available. The informative prior is the posterior after learning the generated data from fission models. Furthermore, cumulative fission yields are included as a physical constraint via a conversion matrix to provide augmented energy dependence. Our work demonstrated a truly Bayesian machine learning by incorporating comprehensive physics knowledges as a powerful tool to exploit the sparse but expensive nuclear data.

Abstract:
We introduce a physics‑informed neural framework for modeling static and time‑dependent galactic gravitational potentials. The method combines data‑driven learning with embedded physical constraints to capture complex, small‑scale features while preserving global physical consistency. We quantify predictive uncertainty through a Bayesian framework, and model time evolution using a neural ODE approach. Applied to mock systems of varying complexity, the model achieves reconstruction errors at the sub‑percent level (0.14% mean acceleration error) and improves dynamical consistency compared to analytic baselines. This method complements existing analytic methods, enabling physics‑informed baseline potentials to be combined with neural residual fields to achieve both interpretable and accurate potential models.

Abstract:
Neural operators have emerged as powerful deep learning frameworks for approximating solution operators of parameterized partial differential equations (PDE). However, current methods predominantly rely on multilayer perceptrons (MLPs) for mapping inputs to solutions, which impairs training robustness in physics‑informed settings due to inherent spectral biases and fixed activation functions. To overcome the architectural limitations, we introduce the Physics‑Informed Chebyshev Polynomial Neural Operator (CPNO), a novel mesh‑free framework that leverages a basis transformation to replace unstable monomial expansions with the numerically stable Chebyshev spectral basis. By integrating parameter dependent modulation mechanism to main net, CPNO constructs PDE solutions in a near‑optimal functional space, decoupling the model from MLP‑specific constraints and enhancing multi‑scale representation. Theoretical analysis demonstrates the Chebyshev basis's near‑minimax uniform approximation properties and superior conditioning, with Lebesgue constants growing logarithmically with degree, thereby mitigating spectral bias and ensuring stable gradient flow during optimization. Numerical experiments on benchmark parameterized PDEs show that CPNO achieves superior accuracy, faster convergence, and enhanced robustness to hyperparameters. The experiment of transonic airfoil flow has demonstrated the capability of CPNO in characterizing complex geometric problems.

Abstract:
Semantic segmentation of microscopy images is a critical task for high‑throughput materials characterisation, yet its automation is severely constrained by the prohibitive cost, subjectivity, and scarcity of expert‑annotated data. While physics‑based simulations offer a scalable alternative to manual labelling, models trained on such data historically fail to generalise due to a significant domain gap, lacking the complex textures, noise patterns, and imaging artefacts inherent to experimental data. This paper introduces a novel framework for labour‑free segmentation that successfully bridges this simulation‑to‑reality gap. Our pipeline leverages phase‑field simulations to generate an abundant source of microstructural morphologies with perfect, intrinsically‑derived ground‑truth masks. We then employ a Cycle‑Consistent Generative Adversarial Network (CycleGAN) for unpaired image‑to‑image translation, transforming the clean simulations into a large‑scale dataset of high‑fidelity, realistic SEM images. A U‑Net model, trained exclusively on this synthetic data, demonstrated remarkable generalisation when deployed on unseen experimental images, achieving a mean Boundary F1‑Score of 0.90 and an Intersection over Union (IOU) of 0.88. Comprehensive validation using t‑SNE feature‑space projection and Shannon entropy analysis confirms that our synthetic images are statistically and featurally indistinguishable from the real data manifold. By completely decoupling model training from manual annotation, our generative framework transforms a data‑scarce problem into one of data abundance, providing a robust and fully automated solution to accelerate materials discovery and analysis.

Abstract:
Physics‑informed neural networks (PINNs) offer a promising framework by embedding partial differential equations (PDEs) into the loss function together with measurement data, making them well‑suited for inverse problems. However, standard PINNs face challenges with time‑dependent PDEs due to the high computational cost of space‑time training and the risk of convergence to local minima. These limitations are particularly pronounced in hemodynamic analysis, where 4D‑flow magnetic resonance imaging (4D‑flow MRI) yields temporally sparse velocity snapshots over the cardiac cycle. To address this challenge, we propose a PINN framework that reconstructs instantaneous flow fields from transient velocity snapshots by inferring the acceleration term in the incompressible Navier‑Stokes equations. By designing the network without explicit time as an input, the proposed approach enables physics enforcement using spatial evaluations alone, improving training efficiency while maintaining physical consistency with transient flow characteristics. In addition, we introduce an acceleration‑mismatch loss that penalizes discrepancies between predicted and measured accelerations, which improves prediction accuracy through regularization. Numerical examples on pulsatile flow behind a stenosis using temporally and spatially downsampled synthetic data generated from time‑resolved CFD demonstrate that the proposed framework reliably reconstructs velocity fields even under sparse temporal sampling, and appropriate regularization for acceleration improves predictions of pressure‑gradient and acceleration fields.

Abstract:
Deep Reinforcement Learning (DRL), a subset of machine learning focused on sequential decision‑making, has emerged as a powerful approach for tackling financial trading problems. In finance, DRL is commonly used either to generate discrete trade signals or to determine continuous portfolio allocations. In this work, we propose a novel reinforcement learning framework for portfolio optimization that incorporates Physics‑Informed Kolmogorov‑Arnold Networks (PIKANs) into several DRL algorithms. The approach replaces conventional multilayer perceptrons with Kolmogorov‑Arnold Networks (KANs) in both actor and critic components‑utilizing learnable B‑spline univariate functions to achieve parameter‑efficient and more interpretable function approximation. During actor updates, we introduce a physics‑informed regularization loss that promotes second‑order temporal consistency between observed return dynamics and the action‑induced portfolio adjustments. The proposed framework is evaluated across three equity markets‑China, Vietnam, and the United States, covering both emerging and developed economies. Across all three markets, PIKAN‑based agents consistently deliver higher cumulative and annualized returns, superior Sharpe and Calmar ratios, and more favorable drawdown characteristics compared to both standard DRL baselines and classical online portfolio‑selection methods. This yields more stable training, higher Sharpe ratios, and superior performance compared to traditional DRL counterparts. The approach is particularly valuable in highly dynamic and noisy financial markets, where conventional DRL often suffers from instability and poor generalization.

Abstract:
Electric vehicle (EV) charging infrastructure introduces complex challenges to urban distribution networks, particularly under extreme demand events. A critical barrier to resilience assessment is the scale gap between micro‑level charging physics and city‑scale planning: minute‑resolution deliverability constraints remain invisible in hourly aggregated datasets, causing purely data‑driven models to exhibit non‑physical behavior in high‑stress regimes. This paper develops a five‑stage scientific machine learning framework bridging this gap through physics‑informed knowledge transfer. Stage 1 learns a temperature‑pressure deliverability surface from Swiss DC fast‑charging telemetry with monotonicity constraints. Stage 2 performs cross‑scale injection via anchored quantile mapping. Stage 3 deploys a dual‑head spatio‑temporal graph neural network for joint forecasting of demand and service loss rate. Stage 4 simulates backlog dynamics under stress shocks and evaluates policy interventions. Stage 5 couples service outcomes to distribution‑grid stress via transformer loading analysis. Validation on the Shenzhen UrbanEV dataset demonstrates that physics injection restores monotone stress‑to‑risk response (Spearman correlation coefficient equals +1.0 versus ‑0.8 without injection) and improves forecasting accuracy. Under a representative demand shock, the hybrid policy reduces backlog by 79.1%, restores full service within the study horizon, and limits grid stress to only 2 additional hours. The derived resilience boundary m_crit as a function of epsilon approximately equals 1.7 minus 1.0 times epsilon, providing actionable guidance linking demand flexibility to maximum absorbable stress, enabling risk‑aware emergency planning under extreme events.

Abstract:
Artificial intelligence (AI) is rapidly emerging as a new paradigm of scientific discovery, namely data‑driven science, across nearly all scientific disciplines. In materials science and engineering, AI has already begun to exert a transformative influence, making it both timely and necessary to examine its interaction with materials plasticity. In this study, we present a holistic survey of the convergence between AI and plasticity, highlighting state‑of‑the‑art AI methodologies employed to discover, construct surrogate models for, and emulate the plastic behavior of materials. From a materials science perspective, we examine cause‑and‑effect relationships governing plastic deformation, including microstructural characterization and macroscopic responses described through plasticity constitutive models. From the perspective of AI methodology, we review a broad spectrum of applied approaches, ranging from frequentist techniques such as classical machine learning (ML), deep learning (DL), and physics‑informed models to probabilistic frameworks that incorporate uncertainty quantification and generative AI methods. These data‑driven approaches are discussed in the context of materials characterization and plasticity‑related applications. The primary objective of this survey is to develop a comprehensive and well‑organized taxonomy grounded in AI methodologies, with particular emphasis on distinguishing critical aspects of these techniques, including model architectures, data requirements, and predictive performance within the specific domain of materials plasticity. By doing so, this work aims to provide a clear road map for researchers and practitioners in the materials community, while offering deeper physical insight and intuition into the role of AI in advancing materials plasticity and characterization, an area of growing importance in the emerging AI‑driven era.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, solving high‑fidelity PDEs remains computationally prohibitive, particularly for parametric systems requiring multiple evaluations across varying parameter configurations. This paper presents MF‑BPINN, a novel multi‑fidelity framework that synergistically combines physics‑informed neural networks with Bayesian uncertainty quantification and adaptive residual learning. Our approach leverages abundant low‑fidelity simulations alongside sparse high‑fidelity data through a hierarchical neural architecture that learns nonlinear correlations across fidelity levels. We introduce an adaptive residual network with learnable gating mechanisms that dynamically balances linear and nonlinear fidelity discrepancies. Furthermore, we develop a rigorous Bayesian framework employing Hamiltonian Monte Carlo.

Abstract:
Foundation models have transformed language, vision, and time series data analysis, yet progress on dynamic predictions for physical systems remains limited. Given the complexity of physical constraints, two challenges stand out. (i) Physics‑computation scalability: physics‑informed learning can enforce physical regularization, but its computation (e.g., ODE integration) does not scale to extensive systems. (ii) Knowledge‑sharing efficiency: the attention mechanism is primarily computed within each system, which limits the extraction of shared ODE structures across systems. We show that enforcing ODE consistency does not require expensive nonlinear integration: a token‑wise locally linear ODE representation preserves physical fidelity while scaling to foundation‑model regimes. Thus, we propose novel token representations that respect locally linear ODE evolution. Such linearity substantially accelerates integration while accurately approximating the local data manifold. Second, we introduce a simple yet effective inter‑system attention that augments attention with a common structure hub (CSH) that stores shared tokens and aggregates knowledge across systems. The resulting model, termed LASS‑ODE (\underlineLArge‑\underlineScale \underlineSmall \underlineODE), is pretrained on our 40GB ODE trajectory collections to enable strong in‑domain performance, zero‑shot generalization across diverse ODE systems, and additional improvements through fine‑tuning.

Abstract:
Autonomous driving systems demand trajectory planners that not only model the inherent uncertainty of future motions but also respect complex temporal dependencies and underlying physical laws. While diffusion‑based generative models excel at capturing multi‑modal distributions, they often fail to incorporate long‑term sequential contexts and domain‑specific physical priors. In this work, we bridge these gaps with two key innovations. First, we introduce a Diffusion Mamba Transformer architecture that embeds mamba and attention into the diffusion process, enabling more effective aggregation of sequential input contexts from sensor streams and past motion histories. Second, we design a Port‑Hamiltonian Neural Network module that seamlessly integrates energy‑based physical constraints into the diffusion model, thereby enhancing trajectory predictions with both consistency and interpretability. Extensive evaluations on standard autonomous driving benchmarks demonstrate that our unified framework significantly outperforms state‑of‑the‑art baselines in predictive accuracy, physical plausibility, and robustness, thereby advancing safe and reliable motion planning.

Abstract:
Geomagnetic map interpolation aims to infer unobserved geomagnetic data at spatial points, yielding critical applications in navigation and resource exploration. However, existing methods for scattered data interpolation are not specifically designed for geomagnetic maps, which inevitably leads to suboptimal performance due to detection noise and the laws of physics. Therefore, we propose a Physics‑informed Diffusion Generation framework~(PDG) to interpolate incomplete geomagnetic maps. First, we design a physics‑informed mask strategy to guide the diffusion generation process based on a local receptive field, effectively eliminating noise interference. Second, we impose a physics‑informed constraint on the diffusion generation results following the kriging principle of geomagnetic maps, ensuring strict adherence to the laws of physics. Extensive experiments and in‑depth analyses on four real‑world datasets demonstrate the superiority and effectiveness of each component of PDG.

Abstract:
In reverse osmosis desalination, ultrafiltration (UF) membranes degrade due to fouling, leading to performance loss and costly downtime. Most plants rely on scheduled preventive maintenance, since existing predictive maintenance models, often based on opaque machine learning methods, lack interpretability and operator trust. This study proposes an explainable prognostic framework for UF membrane remaining useful life (RUL) estimation using fuzzy similarity reasoning. A physics‑informed Health Index, derived from transmembrane pressure, flux, and resistance, captures degradation dynamics, which are then fuzzified via Gaussian membership functions. Using a similarity measure, the model identifies historical degradation trajectories resembling the current state and formulates RUL predictions as Takagi‑Sugeno fuzzy rules. Each rule corresponds to a historical exemplar and contributes to a transparent, similarity‑weighted RUL estimate. Tested on 12,528 operational cycles from an industrial‑scale UF system, the framework achieved a mean absolute error of 4.50 cycles, while generating interpretable rule bases consistent with expert understanding.

Abstract:
Unraveling how macroscopic cognitive phenotypes emerge from microscopic neuronal connectivity remains one of the core pursuits of neuroscience. To this end, researchers typically leverage multi‑modal information from structural connectivity (SC) and functional connectivity (FC) to complete downstream tasks. Recent methodologies explore the intricate coupling mechanisms between SC and FC, attempting to fuse their representations at the regional level. However, lacking fundamental neuroscientific insight, these approaches fail to uncover the latent interactions between neural regions underlying these connectomes, and thus cannot explain why SC and FC exhibit dynamic states of both coupling and heterogeneity. In this paper, we formulate multi‑modal fusion through the lens of neural communication dynamics and propose the Adaptive Flow Routing Network (AFR‑Net), a physics‑informed framework that models how structural constraints (SC) give rise to functional communication patterns (FC), enabling interpretable discovery of critical neural pathways. Extensive experiments demonstrate that AFR‑Net significantly outperforms state‑of‑the‑art baselines. The code is available at https://anonymous.4open.science/r/DIAL‑F0D1.

Abstract:
We propose a data‑efficient, physics‑aware generative framework in function space for inverse PDE problems. Existing plug‑and‑play diffusion posterior samplers represent physics implicitly through joint coefficient‑solution modeling, requiring substantial paired supervision. In contrast, our Decoupled Diffusion Inverse Solver (DDIS) employs a decoupled design: an unconditional diffusion learns the coefficient prior, while a neural operator explicitly models the forward PDE for guidance. This decoupling enables superior data efficiency and effective physics‑informed learning, while naturally supporting Decoupled Annealing Posterior Sampling (DAPS) to avoid over‑smoothing in Diffusion Posterior Sampling (DPS). Theoretically, we prove that DDIS avoids the guidance attenuation failure of joint models when training data is scarce. Empirically, DDIS achieves state‑of‑the‑art performance under sparse observation, improving l_2 error by 11% and spectral error by 54% on average; when data is limited to 1%, DDIS maintains accuracy with 40% advantage in l_2 error compared to joint models.

Abstract:
Three‑dimensional integrated circuit (3D IC) pack‑aging and heterogeneous integration have emerged as central pillars of contemporary semiconductor scaling. Yet, the multi‑physics coupling inherent to stacked architectures manifesting as thermal hot spots, warpage‑induced stresses, and interconnect aging demands monitoring and control capabilities that surpass traditional offline metrology. Although Digital Twin (DT) technology provides a principled route to real‑time reliability management, the existing literature remains fragmented and frequently blurs the distinction between static multiphysics simulation workflows and truly dynamic, closed‑loop twins. This critical review distinguishes itself by addressing these deficiencies through three specific contributions. First, we clarify the Digital Twin hierarchy to resolve terminological ambiguity between digital models, shadows, and twins. Second, we synthesize three foundational enabling technologies: (1) physics‑based modeling, emphasizing the shift from computationally intensive finite‑element analysis (FEA) to real‑time surrogate models; (2) data‑driven paradigms, highlighting virtual metrology (VM) for inferring latent metrics; and (3) in‑situ sensing, the nervous system coupling the physical stack to its virtual counterpart. Third, beyond a descriptive survey, we propose a unified hybrid DT architecture that leverages physics‑informed machine learning (e.g., PINNs) to reconcile data scarcity with latency constraints. Finally, we outline a standards‑aligned roadmap incorporating IEEE 1451 and UCIe protocols to accelerate the transition from passive digital shadows to autonomous, self‑optimizing Digital Twins for 3D IC manufacturing and field operation.

Abstract:
Thermal state reconstruction ‑‑ reversing convection to recover the thermal structure of the mantle at an earlier geologic time ‑‑ is an important tool to understand the evolution of mantle convection and its relation to seismic tomographic images and observations at the surface. Thermal state reconstructions are computationally expensive. Here we transformed the basic computational element, numerical solvers, into neural operators, a class of machine learning models for learning mappings between function spaces. Focusing on a specific architecture, Fourier Neural Operators, we demonstrate that they can represent not only a surrogate model like the Stokes system of equations using a purely physics informed approach, but also discover operators without explicit mathematical formulations or even ill‑posedness from data, including the direct mapping between two convecting thermal states separated by a long time interval much larger than the Courant Fredrich Lewy condition and its reversal. These neural operators significantly accelerate forward and inverse convection modeling by transforming forward physical processes into surrogate models with lower complexity while utilizing auto‑differentiation to calculate gradients. With this framework, we demonstrate the strength and weaknesses of four methods for thermal state reconstructions: Reverse buoyancy, reverse convection operator, an inversion with only the terminal thermal state, and a joint inversion with the terminal thermal state and surface velocity evolution. The reverse convection operator is shown to perform poorly in the presence of observational noise, but the joint inversion overcomes this limitation. The joint technique could probably become a solution to large‑scale thermal state inversion problems using seismic tomography and plate tectonic reconstructions.

Abstract:
Physical systems near singularities, interfaces, and critical points exhibit power‑law scaling, yet standard neural networks leave the governing exponents implicit. We introduce physics‑informed M"untz‑Sz'asz Networks (MSN‑PINN), a power‑law basis network that treats scaling exponents as trainable parameters. The model outputs both the solution and its scaling structure. We prove identifiability, or unique recovery, and show that, under these conditions, the squared error between learned and true exponents scales as O(|μ‑ α|^2). Across experiments, MSN‑PINN achieves single‑exponent recovery with 1‑‑5% error under noise and sparse sampling. It recovers corner singularity exponents for the two‑dimensional Laplace equation with 0.009% error, matches the classical result of Kondrat'ev (1967), and recovers forcing‑induced exponents in singular Poisson problems with 0.03% and 0.05% errors. On a 40‑configuration wedge benchmark, it reaches a 100% success rate with 0.022% mean error. Constraint‑aware training encodes physical requirements such as boundary condition compatibility and improves accuracy by three orders of magnitude over naive training. By combining the expressiveness of neural networks with the interpretability of asymptotic analysis, MSN‑PINN produces learned parameters with direct physical meaning.

Abstract:
We study a model for nonperturbative unitarization of the four‑point contact scalar amplitude in four dimensions. It is defined through an infinite sum of planar diagrams, constructed using two‑particle unitarity and crossing symmetry. We reformulate the problem in terms of a set of nonlinear integral equations obeyed by the single and double discontinuities of the amplitude. We then solve them using a neural‑network ansatz trained by minimizing a physics‑informed loss functional. We obtain a one‑parameter family of amplitudes, which exhibit rich structure: sizeable particle production, nontrivial emergent Regge behavior, Landau curves, a logarithmic decay at high energy and fixed angle. Finally, we go beyond the two‑particle‑reducible setup by treating the multi‑particle data ‑‑ supported above the multi‑particle Landau curves due to multi‑particle unitarity ‑‑ as a dynamical variable. We demonstrate that it can be tuned to suppress low‑spin particle production ‑‑ a phenomenon we call Aks screening ‑‑ at the cost of generating larger and oscillatory double spectral density in the multi‑particle region.

Abstract:
Accurate reconstruction of atmospheric wind fields is essential for applications such as weather forecasting, hazard prediction, and wind energy assessment, yet conventional instruments leave spatio‑temporal gaps within the lower atmospheric boundary layer. Unmanned aircraft systems (UAS) provide flexible in situ measurements, but individual platforms sample wind only along their flight trajectories, limiting full wind‑field recovery. This study presents a framework for reconstructing four‑dimensional atmospheric wind fields using measurements obtained from a coordinated UAS swarm. A synthetic turbulence environment and high‑fidelity multirotor simulation are used to generate training and evaluation data. Local wind components are estimated from UAS dynamics using a bidirectional long short‑term memory network (Bi‑LSTM) and assimilated into a physics‑informed neural network (PINN) to reconstruct a continuous wind field in space and time. For local wind estimation, the bidirectional LSTM achieves root‑mean‑square errors (RMSE) of 0.064 and 0.062 m/s for the north and east components in low‑wind conditions, increasing to 0.122 to 0.129 m/s under moderate winds and 0.271 to 0.273 m/s in high‑wind conditions, while the vertical component exhibits higher error, with RMSE values of 0.029 to 0.091 m/s. The physics‑informed reconstruction recovers the dominant spatial and temporal structure of the wind field up to 1000 m altitude while preserving mean flow direction and vertical shear. Under moderate wind conditions, the reconstructed mean wind field achieves an overall RMSE between 0.118 and 0.154 m/s across evaluated UAS configurations, with the lowest error obtained using a five‑UAS swarm. These results demonstrate that coordinated UAS measurements enable accurate and scalable four‑dimensional wind‑field reconstruction without dedicated wind sensors or fixed infrastructure.

Abstract:
This paper summarizes and consolidates fusion power‑plant costing work performed in support of ARPA‑E from 2017 through 2024, and documents the evolution of the associated analysis framework from early capital‑cost‑focused studies to a standards‑aligned, auditable costing capability. Early efforts applied ARIES‑style cost‑scaling relations to generate Nth‑of‑a‑kind (NOAK) estimates and were calibrated through a pilot study with Bechtel and Decysive Systems to benchmark balance‑of‑plant (BOP) costs and validate plant‑level reasonableness from an engineering, procurement, and construction (EPC) perspective. Subsequent work, informed by Lucid Catalyst studies of nuclear cost drivers, expanded the methodology to treat indirect costs explicitly and to evaluate cost‑reduction pathways for non‑fusion‑island systems through design‑for‑cost practices, modularization, centralized manufacturing, and learning. As ARPA‑E's fusion portfolio expanded, these methods were applied across BETHE and GAMOW concepts (and select ALPHA revisits), including enhanced treatment of tritium handling and plant integration supported by Princeton/PPPL expertise. In 2023 the capability was refactored to align with the IAEA‑GEN‑IV EMWG‑EPRI code‑of‑accounts lineage, while key ARIES‑derived scaling relations were replaced by bottom‑up subsystem models for dominant fusion cost drivers (e.g., magnets, lasers, power supplies, and power‑core components) coupled to physics‑informed power balances and engineering‑constrained radial builds. These developments were implemented in the spreadsheet‑based Fusion Economics code (FECONs) and released as an open‑source Python framework (pyFECONs), providing a transparent mapping from subsystem estimates to standardized accounts and a consistent computation of LCOE.

Abstract:
The field of Embodied AI is witnessing a rapid evolution toward general‑purpose robotic systems, fueled by high‑fidelity simulation and large‑scale data collection. However, this scaling capability remains severely bottlenecked by a reliance on labor‑intensive manual oversight from intricate reward shaping to hyperparameter tuning across heterogeneous backends. Inspired by LLMs' success in software automation and science discovery, we introduce \textscEmboCoach‑Bench, a benchmark evaluating the capacity of LLM agents to autonomously engineer embodied policies. Spanning 32 expert‑curated RL and IL tasks, our framework posits executable code as the universal interface. We move beyond static generation to assess a dynamic closed‑loop workflow, where agents leverage environment feedback to iteratively draft, debug, and optimize solutions, spanning improvements from physics‑informed reward design to policy architectures such as diffusion policies. Extensive evaluations yield three critical insights: (1) autonomous agents can qualitatively surpass human‑engineered baselines by 26.5% in average success rate; (2) agentic workflow with environment feedback effectively strengthens policy development and substantially narrows the performance gap between open‑source and proprietary models; and (3) agents exhibit self‑correction capabilities for pathological engineering cases, successfully resurrecting task performance from near‑total failures through iterative simulation‑in‑the‑loop debugging. Ultimately, this work establishes a foundation for self‑evolving embodied intelligence, accelerating the paradigm shift from labor‑intensive manual tuning to scalable, autonomous engineering in embodied AI field.

Abstract:
Monitoring and forecasting fault slip evolution are fundamental for understanding earthquake cycles and assessing future seismic hazards. This study proposes a physics‑based data assimilation framework that integrates geodetic observations with fault mechanics introducing spatial heterogeneity in frictional properties, with a particular focus on short‑term fault slip forecasting. The proposed method employs physics‑informed neural networks (PINNs) to calculate fault slip evolutions and to optimize the spatial distribution of frictional properties and is applied to the 2010 slow slip event beneath the Bungo Channel, southwest Japan, by changing the data period to be assimilated. When only the initial phase of slip acceleration is assimilated, a velocity‑weakening frictional region is inferred beneath southwest Shikoku, corresponding to the initial nucleation are of the slow slip event. Out results demonstrate that the PINN‑based data assimilation framework successfully forecasts slow transient slip even when only slip acceleration data are assimilated, whereas forecasts based on frictionally homogeneous models result in unstable fast slip. This difference can be interpreted as a consequence of introducing frictional heterogeneity, which allows both the characteristic size of the slipping region and the critical nucleation size to be variable, leading to stable slip evolution consistent with observations. When longer observation periods are assimilated, a velocity‑strengthening region emerges around the slip‑weakening patch, progressively restricting the direction of slip propagation. This velocity‑strengthening region is interpreted as a mechanical constraint imposed by fault physics, linking the slip regions required to reproduce the observed geodetic time series. The results highlight the capability of PINN‑based data assimilation incorporating geodetic observations and fault mechanics.

Abstract:
Reinforcement learning (RL) is widely used for humanoid control, with on‑policy methods such as Proximal Policy Optimization (PPO) enabling robust training via large‑scale parallel simulation and, in some cases, zero‑shot deployment to real robots. However, the low sample efficiency of on‑policy algorithms limits safe adaptation to new environments. Although off‑policy RL and model‑based RL have shown improved sample efficiency, the gap between large‑scale pretraining and efficient finetuning on humanoids still exists. In this paper, we find that off‑policy Soft Actor‑Critic (SAC), with large‑batch update and a high Update‑To‑Data (UTD) ratio, reliably supports large‑scale pretraining of humanoid locomotion policies, achieving zero‑shot deployment on real robots. For adaptation, we demonstrate that these SAC‑pretrained policies can be finetuned in new environments and out‑of‑distribution tasks using model‑based methods. Data collection in the new environment executes a deterministic policy while stochastic exploration is instead confined to a physics‑informed world model. This separation mitigates the risks of random exploration during adaptation while preserving exploratory coverage for improvement. Overall, the approach couples the wall‑clock efficiency of large‑scale simulation during pretraining with the sample efficiency of model‑based learning during fine‑tuning. For code and videos, see https://lift‑humanoid.github.io

Abstract:
Diffusion models have emerged as powerful generative tools for modeling complex data distributions, yet their purely data‑driven nature limits applicability in practical engineering and scientific problems where physical laws need to be followed. This paper proposes Physics‑Informed Learning via Diffusion (PILD), a framework that unifies diffusion modeling and first‑principles physical constraints by introducing a virtual residual observation sampled from a Laplace distribution to supervise generation during training. To further integrate physical laws, a conditional embedding module is incorporated to inject physical information into the denoising network at multiple layers, ensuring consistent guidance throughout the diffusion process. The proposed PILD framework is concise, modular, and broadly applicable to problems governed by ordinary differential equations, partial differential equations, as well as algebraic equations or inequality constraints. Extensive experiments across engineering and scientific tasks including estimating vehicle trajectories, tire forces, Darcy flow and plasma dynamics, demonstrate that our PILD substantially improves accuracy, stability, and generalization over existing physics‑informed and diffusion‑based baselines.

Abstract:
Diffusion models provide expressive priors for forecasting trajectories of dynamical systems, but are typically unreliable in the sparse data regime. Physics‑informed machine learning (PIML) improves reliability in such settings; however, most methods require \emphexplicit governing equations during training, which are often only partially known due to complex and nonlinear dynamics. We introduce PHDME, a port‑Hamiltonian diffusion framework designed for \emphsparse observations and \emphincomplete physics. PHDME leverages port‑Hamiltonian structural prior but does not require full knowledge of the closed‑form governing equations. Our approach first trains a Gaussian process distributed Port‑Hamiltonian system (GP‑dPHS) on limited observations to capture an energy‑based representation of the dynamics. The GP‑dPHS is then used to generate a physically consistent artificial dataset for diffusion training, and to inform the diffusion model with a structured physics residual loss. After training, the diffusion model acts as an amortized sampler and forecaster for fast trajectory generation. Finally, we apply split conformal calibration to provide uncertainty statements for the generated predictions. Experiments on PDE benchmarks and a real‑world spring system show improved accuracy and physical consistency under data scarcity.

Abstract:
In this paper, we investigate several techniques for modeling the one‑dimensional advection equation for a specific class of problems with discontinuous initial and boundary conditions using physics‑informed neural networks (PINNs). To mitigate the spectral bias phenomenon, we employ a Fourier feature mapping layer as the input representation, adopt a two‑stage training strategy in which the Fourier feature parameters and the neural network weights are optimized sequentially, and incorporate adaptive loss weighting. To further enhance the approximation accuracy, a median filter is applied to the spatial data, and the predicted solution is constrained through a bounded linear mapping. Moreover, for certain nonlinear problems, we introduce a modified loss function inspired by the upwind numerical scheme to alleviate the excessive smoothing of discontinuous solutions typically observed in neural network approximations.

Abstract:
High‑quality Fourier Transform Infrared (FTIR) imaging usually needs extensive signal averaging to reduce noise and drift which severely limits clinical speed. Deep learning can accelerate imaging by reconstructing spectra from rapid, single‑scan inputs. However, separating noise and baseline drift simultaneously without ground truth is an ill‑posed inverse problem. Standard black‑box architectures often rely on statistical approximations that introduce spectral hallucinations or fail to generalize to unstable atmospheric conditions. To solve these issues we propose a physics‑informed cascade Unet that separates denoising and baseline correction tasks using a new, deterministic Physics Bridge. This architecture forces the network to separate random noise from chemical signals using an embedded SNIP layer to enforce spectroscopic constraints instead of learning statistical approximations. We benchmarked this approach against a standard single Unet and a traditional Savitzky‑Golay/SNIP workflow. We used a dataset of human hypopharyngeal carcinoma cells (FaDu). The cascade model outperformed all other methods, achieving a 51.3% reduction in RMSE compared to raw single‑scan inputs, surpassing both the single Unet (40.2%) and the traditional workflow (33.7%). Peak‑aware metrics show that the cascade architecture eliminates spectral hallucinations found in standard deep learning. It also preserves peak intensity with much higher fidelity than traditional smoothing. These results show that the cascade Unet is a robust solution for diagnostic‑grade FTIR imaging. It enables imaging speeds 32 times faster than current methods.

Abstract:
Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State‑of‑the‑art methods either use spatial statistics or rely on examples of dense fields in the training phase, which often are not available, and thus rely on synthetic data. Here, we present a reconstruction method that generates dense fields from sparse measurements, without assuming availability of the spatial statistics, nor of examples of the dense fields. This is made possible through the introduction of an automatically differentiable numerical simulator into the training phase of the method. The method is shown to have superior results over statistical and neural network based methods on a set of three standard problems from fluid mechanics.

Abstract:
Physics‑informed neural networks (PINNs) solve time‑dependent partial differential equations (PDEs) by learning a mesh‑free, differentiable solution that can be evaluated anywhere in space and time. However, standard space‑‑time PINNs take time as an input but reuse a single network with shared weights across all times, forcing the same features to represent markedly different dynamics. This coupling degrades accuracy and can destabilize training when enforcing PDE, boundary, and initial constraints jointly. We propose Time‑Induced Neural Networks (TINNs), a novel architecture that parameterizes the network weights as a learned function of time, allowing the effective spatial representation to evolve over time while maintaining shared structure. The resulting formulation naturally yields a nonlinear least‑squares problem, which we optimize efficiently using a Levenberg‑‑Marquardt method. Experiments on various time‑dependent PDEs show up to 4× improved accuracy and 10× faster convergence compared to PINNs and strong baselines.

Abstract:
Reliable predictions of many high‑energy strong interaction processes rely heavily on the non‑perturbative parton fragmentation functions (FFs) extracted from existing experimental data. Conventional methods often require parameterized forms of FFs and additional scale evolution according to the Dokshitzer‑Gribov‑Lipatov‑Altarelli‑Parisi (DGLAP) evolution equations. We introduce a novel approach to determining parton FFs using a Physics‑Informed Neural Network (PINN). Unlike traditional methods, our approach does not require prior parameterized forms and directly integrates the DGLAP evolution equations into the neural network architecture, allowing the FFs to automatically satisfy these equations. We present new sets of parton FFs extracted from hadron spectra in electron‑positron annihilation processes at next‑to‑leading order (NLO) in pQCD using this new technique. To validate our approach, we calculate charged hadron spectra in proton‑(anti)proton collisions using the extracted FFs and demonstrate that the results align well with experimental data across a large range of colliding energies (\sqrts = 130, 200, 500, 630, 900, 1800, 2760, 5020, 5440, 7000 GeV). Our findings indicate that the PINN method not only simplifies the extraction process but also enhances the universal applicability of FFs across different energy scales. By eliminating the need for parameterized forms and additional DGLAP evolution, our approach represents a significant step forward toward fast and accurate extractions of non‑perturbative quantities such as parton fragmentations functions and parton distribution functions.

Abstract:
Fault slip modeling, based on laboratory‑derived friction laws, has significantly enhanced our understanding of fault mechanics. Agreement between model predictions and observations supports the hypothesis that observed slip diversity, including fast earthquakes and slow transient slips (Slow Slip Events; SSEs), originates from frictional heterogeneity. However, quantitative assessments of frictional heterogeneity from geodetic observations while fully incorporating fault mechanics are lacking due to the difficulties of high‑dimensional optimization. In this study, we aim to address this gap using Physics‑Informed Neural Networks (PINNs) to link frictional heterogeneity with geodetic observations. PINNs employ a neural network to represent the spatially variable frictional properties, making their estimation feasible. Targeting the 2010 Bungo SSE in southwest Japan, our estimation reveals heterogeneous friction coinciding with localized SSE nucleation in southwest Shikoku, and subsequent westward propagation. The calculated fault slip of SSE successfully reproduces the spatio‑temporal pattern of observed surface displacements. This PINN‑based inversion provides a mechanically consistent fault slip model validated through quantitative comparison with observations. Furthermore, we predict the future fault slip evolution, demonstrating the importance of assimilating observations spanning multiple SSE cycles. Our results demonstrate the potential of PINN for advancing understanding of fault mechanics and enabling physics‑based fault slip forecasting.

Abstract:
The urgent need for building decarbonization calls for a paradigm shift in future autonomous building energy operation, from human‑intensive engineering workflows toward intelligent agents that interact with physics‑grounded digital environments. This study proposes an end‑to‑end agentic AI‑enabled Physics‑Informed Machine Learning (PIML) environment for scalable building energy modeling, simulation, control, and automation. The framework consists of (1) a modular and physics‑consistent PIML digital environment spanning building thermal dynamics, Heating, Ventilation, and Air Conditioning (HVAC), and distributed energy resources (DER) for grid‑interactive energy management; and (2) an agentic AI layer with 11 specialist agents and 72 Model Context Protocol (MCP) tools that enable end‑to‑end execution of multi‑step energy analytics. A representative case study demonstrates multi‑domain, multi‑agent coordination for assessing how system and control upgrades affect energy use, operating cost, thermal comfort, and flexibility. In addition, a large‑scale benchmark (about 4000 runs) systematically evaluates workflow performance in terms of accuracy, token consumption, execution time, and inference cost. The results quantify the impacts of intelligence mode design, model size, task complexity, and orchestrator‑specialist coordination, and provide key lessons for building future agentic AI systems in real‑world building energy applications. This work establishes a scalable, physics‑grounded foundation for deploying agentic AI in decarbonized and grid‑interactive building operations.

Abstract:
Silicon‑based analog neural networks physically embody the ideal neural network model in an approximate way. We show that by retraining the neural network using a physics‑informed hardware‑aware model one can fully recover the inference accuracy of the ideal network model even in the presence of significant non‑idealities. This is way more promising for scalability and integration density than the default option of improving the fidelity of the analog neural network at the cost of significant energy, area, and design overhead, through extensive calibration and conservative analog design. We first present a physics‑informed hardware‑aware model for a time‑domain vector‑matrix multiplier implemented with single‑transistor floating‑gate memory cells that explicitly accounts for two dominant non‑idealities of the physical implementation ‑ capacitive crosstalk and bit‑line voltage drop ‑ and integrates seamlessly with modern deep‑learning workflows. The model discretizes each operation into adaptive time slots, processes activation patterns in parallel, and accumulates their contributions to predict effective multiplier outputs. Using measurements from a 16x16 silicon array, we calibrate the model, show that crosstalk is layout‑dependent and often dominant, and introduce an improved weight‑extraction procedure that doubles signal‑to‑error ratio versus an ideal vector‑matrix multiplier model. Finally, we show that by training silicon‑based analog neural networks using an hardware‑aware model in the forward pass we can recover the accuracy of the ideal software networks across three architectures ‑‑ custom MLP on low‑resolution MNIST, LeNet‑5 on MNIST, and a VGG‑style CNN on CIFAR‑10 ‑ establishing a complete design‑to‑deployment workflow for time‑domain analog neuromorphic chips.

Abstract:
The numerical solution of differential equations using neural networks has become a central topic in scientific computing, with Physics‑Informed Neural Networks (PINNs) emerging as a powerful paradigm for both forward and inverse problems. However, unlike classical numerical methods that offer established convergence guarantees, neural network‑based approximations typically lack rigorous error bounds. Furthermore, the non‑deterministic nature of their optimization makes it difficult to mathematically certify their accuracy. To address these challenges, we propose a "Learn and Verify" framework that provides computable, mathematically rigorous error bounds for the solutions of differential equations. By combining a novel Doubly Smoothed Maximum (DSM) loss for training with interval arithmetic for verification, we compute rigorous a posteriori error bounds as machine‑verifiable proofs. Numerical experiments on nonlinear Ordinary Differential Equations (ODEs), including problems with time‑varying coefficients and finite‑time blow‑up, demonstrate that the proposed framework successfully constructs rigorous enclosures of the true solutions, establishing a foundation for trustworthy scientific machine learning.

Abstract:
Data‑driven turbulence prediction methods often face challenges related to data dependency and lack of physical interpretability. In this paper, we propose a physics‑informed Transformer operator (PITO) and its implicit variant (PIITO) for predicting three‑dimensional (3D) turbulence, which are developed based on the vision Transformer (ViT) architecture with an appropriate patch size. Given the current flow field, the Transformer operator computes its prediction for the next time step. By embedding the large‑eddy simulation (LES) equations into the loss function, PITO and PIITO can learn solution operators without using labeled data. Furthermore, PITO can automatically learn the subgrid scale (SGS) coefficient using a single set of flow data during training. Both PITO and PIITO exhibit excellent stability and accuracy on the predictions of various statistical properties and flow structures for the situation of long‑term extrapolation exceeding 25 times the training horizon in decaying homogeneous isotropic turbulence (HIT), and outperform the physics‑informed Fourier neural operator (PIFNO). Furthermore, PITO exhibits a remarkable accuracy on the predictions of forced HIT where PIFNO fails. Notably, PITO and PIITO reduce GPU memory consumption by 79.5% and 91.3% while requiring only 31.5% and 3.1% of the parameters, respectively, compared to PIFNO. Moreover, both PITO and PIITO models are much faster compared to traditional LES method.

Abstract:
We propose a method for estimating the magnitude distribution of an acoustic field from spatially sparse magnitude measurements. Such a method is useful when phase measurements are unreliable or inaccessible. Physics‑informed neural networks (PINNs) have shown promise for sound field estimation by incorporating constraints derived from governing partial differential equations (PDEs) into neural networks. However, they do not extend to settings where phase measurements are unavailable, as the loss function based on the governing PDE relies on phase information. To remedy this, we propose a phase‑retrieval‑based PINN for magnitude field estimation. By representing the magnitude and phase distributions with separate networks, the PDE loss can be computed based on the reconstructed complex amplitude. We demonstrate the effectiveness of our phase‑retrieval‑based PINN through experimental evaluation.

Abstract:
Traditional e‑commerce recommender systems primarily optimize for user engagement and purchase likelihood, often neglecting the rigid physiological constraints required for human health. Standard collaborative filtering algorithms are structurally blind to these hard limits, frequently suggesting bundles that fail to meet specific total daily energy expenditure and macronutrient balance requirements. To address this disconnect, this paper introduces a Physics‑Informed Neuro‑Symbolic Recommender System that integrates nutritional science directly into the recommendation pipeline via a dual‑layer architecture. The framework begins by constructing a semantic knowledge graph using sentence‑level encoders to strictly align commercial products with authoritative nutritional data. During the training phase, an implicit physics regularizer applies a differentiable thermodynamic loss function, ensuring that learned latent embeddings reflect nutritional plausibility rather than simple popularity. Subsequently, during the inference phase, an explicit physics optimizer employs simulated annealing and elastic quantity optimization to generate discrete grocery bundles that strictly adhere to the user's protein and caloric targets.

Abstract:
This paper introduces wavelet‑physics‑informed residual neural networks (W‑PIRNNs) to study complex fluid flow problems by reconstructing the flow field from highly sparse, supervised data. Our W‑PIRNNs fundamentally integrate ResNet and employ the wavelet W(t) = w_1 \sin(t) + w_2 \cos(t) as an activation function. Due to the vanishing and ballooning gradient problems associated with typical PINNs' deep networks, we implemented residual‑based skip connections. Our W‑PIRNNs, which integrate supervised data with physical principles, demonstrate efficacy even in scenarios of sparse or partial data, enabling the reconstruction of flow fields using merely 0.05% velocity data for training. The wake flow around a circular cylinder served as the test case for our proposed technique, which depends exclusively on velocity data for training. This technique facilitates the precise reconstruction of velocity, pressure, streamlines, and vorticity, requiring fewer epochs and less processing time. Significantly, our proposed W‑PIRNNs effectively resolve PDEs in both forward and inverse contexts. Burger's equation served as a test case for both the forward and inverse problem configurations. Our network calculates the diffusion or viscosity coefficient (λ_2) with an absolute error of 0.065% and the convection coefficient (λ_1) with an absolute error of 0.002%. Furthermore, the Schrödinger equation is examined in the forward setting to assess the framework's ability to handle periodic boundary conditions. To the best of our knowledge, W‑PIRNNs represent the first method capable of flow reconstruction using highly sparse supervised data, as well as reconstructing streamline and vorticity, and they effectively address both forward and inverse problems with high accuracy.

Abstract:
Monte Carlo simulations of optical photon transport are computationally prohibitive for large‑scale optical systems including detector arrays and PET systems, restricting practical use to single‑crystal studies. This work presents an enhanced conditional generative adversarial network (optiGAN) replacing optical simulations at the crystal array level, extending our single‑crystal approach to a 3x3 BGO array. We enhance the Wasserstein‑GAN framework with Fourier feature encoding, a learnable latent mapping network, and a physics‑informed loss enforcing momentum conservation. Training data is reduced eight‑fold by exploiting symmetry. Evaluation employs three studies: a full array evaluation testing generalization from the fundamental domain to the complete geometry, a high‑resolution study probing out‑of‑distribution generalization to untrained positions, and a pencil beam γ‑photon study assessing practical applicability for experimental detector characterization. Performance is benchmarked against GATE10/Geant4 ground truth, using intrinsic fluctuations between independent Monte Carlo runs as baseline. OptiGAN achieves sliced Wasserstein similarity within 3σ‑agreement of the baseline across all conditions, demonstrating successful generalization to the full array. The model transitions from electron‑emission training data to realistic γ‑photon interactions, producing flood maps that reproduce characteristic patterns including photopeak clusters and inter‑crystal scatter lines. This proof‑of‑concept demonstrates that physics‑informed generative models can accurately simulate optical photon transport in segmented scintillator arrays. The reproduction of experimentally relevant flood map features validates optiGAN for PET detector development and establishes a foundation for models generalizing across diverse array configurations.

Abstract:
Inverse design is a central goal in much of science and engineering, including frequency‑selective surfaces (FSS) that are critical to microelectronics for telecommunications and optical metamaterials. Traditional surrogate‑assisted optimization methods using deep learning can accelerate the design process but do not usually incorporate uncertainty quantification, leading to poorer optimization performance due to erroneous predictions in data‑sparse regions. Here, we introduce and validate a fundamentally different paradigm of Physics‑Informed Uncertainty, where the degree to which a model's prediction violates fundamental physical laws serves as a computationally‑cheap and effective proxy for predictive uncertainty. By integrating physics‑informed uncertainty into a multi‑fidelity uncertainty‑aware optimization workflow to design complex frequency‑selective surfaces within the 20 ‑ 30 GHz range, we increase the success rate of finding performant solutions from less than 10% to over 50%, while simultaneously reducing computational cost by an order of magnitude compared to the sole use of a high‑fidelity solver. These results highlight the necessity of incorporating uncertainty quantification in machine‑learning‑driven inverse design for high‑dimensional problems, and establish physics‑informed uncertainty as a viable alternative to quantifying uncertainty in surrogate models for physical systems, thereby setting the stage for autonomous scientific discovery systems that can efficiently and robustly explore and evaluate candidate designs.

Abstract:
Solving time‑dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics‑informed neural networks (PINNs) often suffer from inefficient point allocation that wastes resources on regions already well‑resolved. This paper presents an adaptive sampling framework for PINNs aimed at efficiently solving time‑dependent partial differential equations with pronounced local singularities. The method employs a residual‑driven strategy, where the spatial‑temporal distribution of training points is iteratively updated according to the error field from the previous iteration. This targeted allocation enables the network to concentrate computational effort on regions with significant residuals, achieving higher accuracy with fewer sampling points compared to uniform sampling. Numerical experiments on representative PDE benchmarks demonstrate that the proposed approach improves solution quality.

Abstract:
The integration of high‑penetration renewable energy introduces significant stochasticity and non‑convexity into power system dispatching, challenging the computational limits of classical optimization. While Variational Quantum Algorithms (VQAs) on Noisy Intermediate‑Scale Quantum (NISQ) devices offer a promising path for combinatorial acceleration, existing approaches typically treat the power grid as a "black box", suffering from poor scalability (barren plateaus) and frequent violations of physical constraints. Bridging these gaps, this paper proposes a Physics‑Informed Hybrid Quantum‑Classical Dispatching (PI‑HQCD) framework. We construct a topology‑aware Hamiltonian that explicitly embeds linearized power flow equations, storage dynamics, and multi‑timescale coupling directly into the quantum substrate, significantly reducing the search space dimensionality. We further derive a noise‑adaptive regularization mechanism that theoretically bounds the effective Lipschitz constant of the objective function, guaranteeing convergence stability under realistic quantum measurement noise. Numerical experiments on the IEEE 39‑bus benchmark and a 118‑bus regional grid demonstrate that PI‑HQCD achieves superior economic efficiency and higher renewable utilization compared to stochastic dual dynamic programming (SDDP). Theoretical analysis confirms that this topology‑aware design leads to an O(1/N) gradient variance scaling, effectively mitigating barren plateaus and ensuring scalability for larger networks. This work establishes a rigorous paradigm for embedding engineering physics into quantum computing, paving the way for practical quantum advantage in next‑generation grid operations.

Abstract:
Separating liquid‑liquid dispersions in gravity settlers is critical in chemical, pharmaceutical, and recycling processes. The dense‑packed zone height is an important performance and safety indicator but it is often expensive and impractical to measure due to optical limitations. We propose a framework to estimate phase heights by combining a PINN model with readily available volume flow measurements, without requiring phase height measurements during deployment. To this end, a physics‑informed neural network (PINN) is first pretrained on synthetic data and physics equations derived from a low‑fidelity (approximate) mechanistic model to reduce the need for extensive experimental data. While the mechanistic model is used to generate synthetic training data, only volume balance equations are used in the PINN, as incorporating droplet coalescence and sedimentation submodels would be computationally prohibitive. The pretrained PINN is then fine‑tuned with scarce experimental phase height and flow‑rate data to capture the actual dynamics of the separator. We then deploy the differentiable PINN as a predictive model in an Extended Kalman Filter inspired state estimation framework, enabling the phase heights to be tracked and updated using flow‑rate measurements only. We first test the two‑stage trained PINN by forward simulation from a known initial state against the mechanistic model and a non‑pretrained PINN. We then evaluate phase height estimation performance with the filter, comparing the two‑stage trained PINN with a two‑stage trained purely data‑driven neural network. All model types are trained and evaluated using ensembles to account for model parameter uncertainty. In all evaluations, the two‑stage trained PINN yields the most accurate phase‑height estimates.

Abstract:
Electrochemical devices (batteries, fuel cells, and electrolyzers) are in full development, driven by the green energy transition. Their real‑time control requires ms predictions in order to take critical decisions during fast transients or faults. The physics behind include coupled multi‑physics phenomena that conventional finite element methods cannot solve so fast with the current CPU technology. This paper evaluates the potential of physics‑informed machine learning represented by three frameworks: \acpinn, \acpideeponet, and \acpino by evaluating their training effort, inference speed, and extrapolation capacity. Our analysis reveals valuable performance trade‑offs. \acppinn offer simplicity for fixed problem instances but require retraining for parameter changes. \acpideeponet enables operator learning across varying conditions with mesh‑free geometric flexibility. \acpino delivers superior performance on regular grids, with the strongest extrapolation capabilities due to spectral derivative computation and resolution invariance. \acpideeponet is particularly suited for irregular, unstructured geometries (e.g., porous electrodes or complex flow fields), while \acpino works best for layered, structured‑grid problems (e.g., transport across stacked electrochemical layers) requiring fast inference. Possible future applications include real‑time lithium concentration prediction for safe fast‑charging and micro short circuit detection, water management in fuel cells, and optimal power management in electrolyzers under intermittent renewable inputs. These findings establish physics‑informed operator learning as a transformative approach for next‑generation electrochemical device controller technology.

Abstract:
Reaction‑diffusion (RD) systems provide fundamental models for understanding self‑organized spatiotemporal patterns across natural and engineered settings, yet reliable parameter estimation remains challenging, particularly when observations are sparse, noisy, and restricted to a subset of state variables. We introduce CLIP (Curriculum Learning Identification via PINNs), a physics‑guided framework built on physics‑informed neural networks for joint parameter inference and hidden‑state reconstruction under partial observability. Leveraging the physical separability of RD systems, the CLIP training progresses from reaction‑dominated regimes to full spatiotemporal dynamics using curriculum learning and an anchored widening transfer strategy. Across three canonical reaction‑diffusion benchmarks, CLIP achieves more accurate and robust identification than baseline methods. Furthermore, the CLIP framework is successfully applied to infer the dynamics of the Min system in bacteria, where only membrane‑bound species are observed and key kinetic rates span multiple orders of magnitude. Ablation experiments and loss‑landscape visualizations demonstrate that both the curriculum stages and the anchored transfer are essential for stable convergence.

Abstract:
We introduce NewPINNs, a physics‑informing learning framework that couples neural networks with conventional numerical solvers for solving differential equations. Rather than enforcing governing equations and boundary conditions through residual‑based loss terms, NewPINNs integrates the solver directly into the training loop and defines learning objectives through solver‑consistency. The neural network produces candidate solution states that are advanced by the numerical solver, and training minimizes the discrepancy between the network prediction and the solver‑evolved state. This pull‑push interaction enables the network to learn physically admissible solutions through repeated exposure to the solver's action, without requiring problem‑specific loss engineering or explicit evaluation of differential equation residuals. By delegating the enforcement of physics, boundary conditions, and numerical stability to established numerical solvers, NewPINNs mitigates several well‑known failure modes of standard physics‑informed neural networks, including optimization pathologies, sensitivity to loss weighting, and poor performance in stiff or nonlinear regimes. We demonstrate the effectiveness of the proposed approach across multiple forward and inverse problems involving finite volume, finite element, and spectral solvers.

Abstract:
More than 10 million coronary angiograms are performed globally each year, providing a gold standard for detecting obstructive coronary artery disease. Yet, no obstructive lesions are identified in 70% of patients evaluated for ischemic heart disease. Up to half of these patients have undiagnosed, life‑limiting coronary microvascular dysfunction (CMD), which remains under‑detected due to the limited availability of invasive tools required to measure coronary flow reserve (CFR). Here, we introduce PUNCH, a non‑invasive, uncertainty‑aware framework for estimating CFR directly from standard coronary angiography. PUNCH integrates physics‑informed neural networks with variational inference to infer coronary blood flow from first‑principles models of contrast transport, without requiring ground‑truth flow measurements or population‑level training. The pipeline runs in approximately three minutes per patient on a single GPU. Validated on synthetic angiograms with controlled noise and imaging artifacts, as well as on clinical bolus thermodilution data from 20 patients, PUNCH demonstrates accurate and uncertainty‑calibrated CFR estimation. This approach establishes a new paradigm for CMD diagnosis and illustrates how physics‑informed inference can substantially expand the diagnostic utility of available clinical imaging.

Abstract:
Inverter‑based resources (IBRs) exhibit fast transient dynamics during network disturbances, which often cannot be properly captured by phasor and SCADA measurements. This shortcoming has recently been addressed with the advent of waveform measurement units (WMUs), which provide high‑resolution, time‑synchronized raw voltage and current waveform samples from multiple locations in the power system. However, transient model learning based on synchro‑waveform measurements remains constrained by the scarcity of network disturbances and the complexity of the underlying nonlinear dynamics of IBRs. We propose to address these problems by developing a data‑efficient physics‑informed machine learning (PIML) framework for synchro‑waveform analytics that estimates the IBR terminal current response from only a few network disturbance signatures. Here, the physics of the electrical circuits are used to compensate for limited data availability by constraining the learning process through known circuit relationships. Two cases are considered, with known and unknown circuit parameters. In the latter case, the framework jointly learns the transient dynamics of the IBRs and the parameters of the electrical circuit. Case studies using WMU disturbance data across multiple sampling rates shows consistently lower current estimation error with substantially fewer training events than a purely data‑driven baseline.

Abstract:
While deep learning is transforming data analysis in high‑energy physics, computational challenges limit its potential. We address these challenges in the context of collider physics by introducing EveNet, an event‑level foundation model pretrained on 500 million simulated collision events using a hybrid objective of self‑supervised learning and physics‑informed supervision. By leveraging a shared particle‑cloud representation, EveNet outperforms state‑of‑the‑art baselines across diverse tasks, including searches for heavy resonances and exotic Higgs decays, and demonstrates exceptional data efficiency in low‑statistics regimes. Crucially, we validate the transferability of the model to experimental data by rediscovering the Υ meson in CMS Open Data and show its capacity for precision physics through the robust extraction of quantum correlation observables stable against systematic uncertainties. These results indicate that EveNet can successfully encode the fundamental physical structure of particle interactions, which offers a unified and resource‑efficient framework to accelerate discovery at current and future colliders.

Abstract:
Live video denoising under realistic, multi‑component sensor noise remains challenging for applications such as autofocus, autonomous driving, and surveillance. We propose PocketDVDNet, a lightweight video denoiser developed using our model compression framework that combines sparsity‑guided structured pruning, a physics‑informed noise model, and knowledge distillation to achieve high‑quality restoration with reduced resource demands. Starting from a reference model, we induce sparsity, apply targeted channel pruning, and retrain a teacher on realistic multi‑component noise. The student network learns implicit noise handling, eliminating the need for explicit noise‑map inputs. PocketDVDNet reduces the original model size by 74% while improving denoising quality and processing 5‑frame patches in real‑time. These results demonstrate that aggressive compression, combined with domain‑adapted distillation, can reconcile performance and efficiency for practical, real‑time video denoising.

Abstract:
The process‑independence of the color dipole amplitude is a cornerstone of high‑energy Quantum Chromodynamics (QCD). However, standard phenomenological approaches typically rely on rigid parametric ansatzes and often require ad‑hoc geometric adjustments to reconcile inclusive and diffractive measurements. To resolve this tension, we introduce Physics‑Informed Neural Networks (PINNs) employing a ``Teacher‑‑Student'' strategy. The physics‑based momentum‑space Balitsky‑Kovchegov evolution dynamics act as the ``Teacher,'' constraining the solution manifold, while the network ``Student'' is refined against inclusive HERA F_2 data. This approach extracts a model‑independent dipole amplitude without assuming initial states. Strikingly, we demonstrate that this amplitude ‑‑ without parameter retuning or geometric rescaling ‑‑ successfully predicts the absolute normalization and kinematic dependence of exclusive J/ψ photoproduction cross‑sections. This parameter‑free prediction of the saturation dynamics provides promising evidence for the process‑independence of the gluon saturation scale and establishes PINNs as a transformative paradigm for uncovering non‑perturbative QCD structures.

Abstract:
Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application‑oriented implications in fields such as hydrogeology where system properties exhibit significant (and often uncertain) spatial heterogeneity. We address this by reformulating a Physics‑Informed Neural Network (PINN) as a differentiable solver that learns the continuous solution manifold for steady‑state Darcy flow. Our framework requires only a single training run, circumventing the need for costly re‑training for each new parameter instance. Its versatility is demonstrated through two representations of spatially heterogeneous hydraulic conductivity fields: a direct analytical form and a novel data‑driven formulation resting on an autoencoder to create a low‑dimensional latent encoding. A key innovation is the integration of the differentiable decoder into the physics‑informed loss function, enabling on‑the‑fly reconstruction of complex conductivity fields via automatic differentiation. The approach yields accurate, mass‑conserving flow solutions and supports efficient uncertainty quantification, providing a general methodology for physics‑constrained data‑driven modeling of heterogeneous systems.

Abstract:
Modern buildings are increasingly interconnected with occupancy, heating, ventilation, and air‑conditioning (HVAC) systems, distributed energy resources (DERs), and power grids. Modeling, control, and optimization of such multi‑domain systems play a critical role in achieving building‑sector decarbonization. However, most existing tools lack scalability and physical consistency for addressing these complex, multi‑scale ecosystem problems. To bridge this gap, this study presents BESTOpt, a modular, physics‑informed machine learning (PIML) framework that unifies building applications, including benchmarking, evaluation, diagnostics, control, optimization, and performance simulation. The framework adopts a cluster‑domain‑system/building‑component hierarchy and a standardized state‑action‑disturbance‑observation data typology. By embedding physics priors into data‑driven modules, BESTOpt improves model accuracy and physical consistency under unseen conditions. Case studies on single‑building and cluster scenarios demonstrate its capability for multi‑level centralized and decentralized control. Looking ahead, BESTOpt lays the foundation for an open, extensible platform that accelerates interdisciplinary research toward smart, resilient, and decarbonized building ecosystems.

Abstract:
Seismic wave forward and inverse modeling are fundamental tools for subsurface imaging and geological hazard assessment. Conventional grid‑based numerical methods, such as finite‑difference and finite‑element approaches, often require dense discretization and repeated forward simulations, leading to high computational cost in inverse problems. Although deep learning has shown promise in seismic applications, its performance is commonly limited by the need for large labeled datasets and weak enforcement of physical constraints. In this study, we propose a unified physics‑informed neural network (PINN) framework for forward modeling and parameter inversion of viscoacoustic wave propagation. By embedding the viscoacoustic wave equation into the learning process, the proposed framework accurately reproduces wavefields, attenuation, and phase characteristics, while enabling the simultaneous inversion of velocity and attenuation parameters from temporally sparse observations. Numerical experiments demonstrate that the PINN approach achieves stable and reliable accuracy compared with finite‑difference solutions, while exhibiting reduced sensitivity to spatial discretization. These results highlight the potential of PINNs as a data‑efficient and physically consistent alternative for high‑resolution seismic modeling and inversion in attenuative media.

Abstract:
Deep learning models, particularly Transformers, are often criticized as "black boxes" and lack interpretability. We propose Prism, a white‑box attention‑based architecture derived from the principles of Maximizing Coding Rate Reduction (\textMCR^2). By modeling the attention mechanism as a gradient ascent process on a distinct signal‑noise manifold, we introduce a specific irrational frequency separation (π‑RoPE) to enforce incoherence between signal (semantic) and noise (syntactic) subspaces. We show empirical evidence that these geometric inductive biases can induce unsupervised functional disentanglement alone. Prism spontaneously specializes its attention heads into spectrally distinct regimes: low‑frequency heads capturing long‑range causal dependencies (signal) and high‑frequency heads handling local syntactic constraints and structural artifacts. To provide a theoretical grounding for these spectral phenomena, we draw an analogy between attention mechanism and a Hamiltonian dynamical system and identify that the standard geometric progression of Rotary Positional Embeddings (RoPE) induces dense resonance networks (Arnold Tongues), leading to feature rank collapse. Empirical validation on 124M‑parameter models trained on OpenWebText demonstrates that Prism spontaneously isolates the Attention Sink pathology and maintains isentropic information flow across layers. Further, we suggest a physics‑informed plug‑and‑play intervention KAM‑RoPE for large language models (LLMs). Our results suggest that interpretability and performance can be unified through principled geometric construction, offering a theoretically grounded alternative to heuristic architectural modifications

Abstract:
Granular materials subjected to impact loading exhibit highly heterogeneous spatiotemporal dynamics governed by wave propagation, pore collapse, and grain‑scale rearrangements. Mesoscale hydrocodes resolve these processes but are computationally expensive, limiting their use in parametric studies and uncertainty quantification. In this work, we develop a convolutional Long Short‑Term Memory (ConvLSTM) neural network as a spatiotemporal surrogate for mesoscale simulations of weak shock propagation in granular media. Using two‑dimensional hydrocode simulations as training data, we first consider a simplified "billiard break" problem in which a cue ball impacts a cluster of nine circular balls, all deformable. Sequences of pressure‑field images serve as input‑output pairs for a sequence‑to‑sequence ConvLSTM, which is trained to predict future frames from a short history. We compare several architectures and show that a relatively compact encoder‑decoder ConvLSTM accurately reproduces the propagation of the pressure wave and the resulting particle motion for an unseen combination of cue‑ball position and impact velocity. As a proof‑of‑concept extension, we apply the same ConvLSTM framework to previously published mesoscale simulations of weak shock compaction in a granular ensemble. When evaluated at piston impact speeds that were completely withheld from training, the surrogate captures the position and shape of the compaction front and its dependence on impact speed, while smoothing fine pore‑scale details in the highly compacted region as expected. These results demonstrate that ConvLSTM models can serve as satisfactory surrogates for spatiotemporal mesoscale simulations of granular wave propagation, enabling accelerated exploration of parameter space and laying the groundwork for physics‑informed, mesoscale simulations of granular materials under shock loading.

Abstract:
The design of spacecraft thermal protection systems (TPS) requires accurate knowledge of thermal transport properties across wide ranges of temperature and pressure. For fibrous insulation, conventional measurement techniques in laboratory settings are typically limited to temperatures much lower than what is reached in atmosphere entry scenarios. Moreover, it is often the case that only temperature measurements are available, meaning that the thermal conductivity of the insulation must be indirectly inferred as an inverse problem. We propose a Bayesian framework using information field theory (IFT) to reconstruct the thermal conductivity of high‑temperature fibrous insulation from sparse experimental data. Under IFT, the conductivity is represented as a Gaussian process, and the physics is enforced via a physics‑informed prior over the temperature derived from the heat equation. Bayes's rule produces an infinite‑dimensional posterior distribution that quantifies uncertainty about the conductivity which can be evaluated in extrapolation regimes. We apply the method to Opacified Fibrous Insulation with both synthetic and experimental data to reconstruct the thermal conductivity beyond the experimental regime. The inferred conductivities are validated against reference data and then propagated into high‑fidelity digital twins of flexible TPS performance under Mars and Earth entry trajectories. The results show that IFT yields accurate predictions with quantified uncertainty, enabling robust TPS sizing in regimes inaccessible to direct measurement.

Abstract:
Partial differential equations (PDEs) form the backbone of simulations of many natural phenomena, for example in climate modeling, material science, and even financial markets. The application of physics‑informed neural networks to accelerate the solution of PDEs is promising, but not competitive with numerical solvers yet. Here, we show how quantum computing can improve the ability of physics‑informed neural networks to solve partial differential equations. For this, we develop hybrid networks consisting of quantum circuits combined with classical layers and systematically test them on various non linear PDEs and boundary conditions in comparison with purely classical networks. We demonstrate that the advantage of using quantum networks lies in their ability to achieve an accurate approximation of the solution in substantially fewer training epochs, particularly for more complex problems. These findings provide the basis for targeted developments of hybrid quantum neural networks with the goal to significantly accelerate numerical modeling.

Abstract:
Partial differential equations (PDEs) are central to scientific modeling. Modern workflows increasingly rely on learning‑based components to support model reuse, inference, and integration across large computational processes. Despite the emergence of various physics‑aware data‑driven approaches, the field still lacks a unified perspective to uncover their relationships, limitations, and appropriate roles in scientific workflows. To this end, we propose a unifying perspective to place two dominant paradigms: Physics‑Informed Neural Networks (PINNs) and Neural Operators (NOs), within a shared design space. We organize existing methods from three fundamental dimensions: what is learned, how physical structures are integrated into the learning process, and how the computational load is amortized across problem instances. In this way, many challenges can be best understood as consequences of these structural properties of learning PDEs. By analyzing advances through this unifying view, our survey aims to facilitate the development of reliable learning‑based PDE solvers and catalyze a synthesis of physics and data.

Abstract:
Wildfires are growing in frequency and intensity, devastating ecosystems and communities while causing billions of dollars in suppression costs and economic damage annually in the U.S. Traditional wildfire management is mostly reactive, addressing fires only after they are detected. We introduce FireCastRL, a proactive artificial intelligence (AI) framework that combines wildfire forecasting with intelligent suppression strategies. Our framework first uses a deep spatiotemporal model to predict wildfire ignition. For high‑risk predictions, we deploy a pre‑trained reinforcement learning (RL) agent to execute real‑time suppression tactics with helitack units inside a physics‑informed 3D simulation. The framework generates a threat assessment report to help emergency responders optimize resource allocation and planning. In addition, we are publicly releasing a large‑scale, spatiotemporal dataset containing \mathbf9.5 million samples of environmental variables for wildfire prediction. Our work demonstrates how deep learning and RL can be combined to support both forecasting and tactical wildfire response. More details can be found at https://sites.google.com/view/firecastrl.

Authors: LSST Dark Energy Science Collaboration, Eric Aubourg, Camille Avestruz, Matthew R. Becker, Biswajit Biswas, Rahul Biswas, Boris Bolliet, Adam S. Bolton, Clecio R. Bom, Raphaël Bonnet-Guerrini, Alexandre Boucaud, Jean-Eric Campagne, Chihway Chang, Aleksandra Ćiprijanović, Johann Cohen-Tanugi, Michael W. Coughlin, John Franklin Crenshaw, Juan C. Cuevas-Tello, Juan de Vicente, Seth W. Digel, Steven Dillmann, Mariano Javier de León Dominguez Romero, Alex Drlica-Wagner, Sydney Erickson, Alexander T. Gagliano, Christos Georgiou, Aritra Ghosh, Matthew Grayling, Kirill A. Grishin, Alan Heavens, Lindsay R. House, Mustapha Ishak, Wassim Kabalan, Arun Kannawadi, François Lanusse, C. Danielle Leonard, Pierre-François Léget, Michelle Lochner, Yao-Yuan Mao, Peter Melchior, Grant Merz, Martin Millon, Anais Möller, Gautham Narayan, Yuuki Omori, Hiranya Peiris, Laurence Perreault-Levasseur, Andrés A. Plazas Malagón, Nesar Ramachandra, Benjamin Remy, Cécile Roucelle, Jaime Ruiz-Zapatero, Stefan Schuldt, Ignacio Sevilla-Noarbe, Ved G. Shah, Tjitske Starkenburg, Stephen Thorp, Laura Toribio San Cipriano, Tilman Tröster, Roberto Trotta, Padma Venkatraman, Amanda Wasserman, Tim White, Justine Zeghal, Tianqing Zhang, Yuanyuan Zhang

Abstract:
The Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) will produce unprecedented volumes of heterogeneous astronomical data (images, catalogs, and alerts) that challenge traditional analysis pipelines. The LSST Dark Energy Science Collaboration (DESC) aims to derive robust constraints on dark energy and dark matter from these data, requiring methods that are statistically powerful, scalable, and operationally reliable. Artificial intelligence and machine learning (AI/ML) are already embedded across DESC science workflows, from photometric redshifts and transient classification to weak lensing inference and cosmological simulations. Yet their utility for precision cosmology hinges on trustworthy uncertainty quantification, robustness to covariate shift and model misspecification, and reproducible integration within scientific pipelines. This white paper surveys the current landscape of AI/ML across DESC's primary cosmological probes and cross‑cutting analyses, revealing that the same core methodologies and fundamental challenges recur across disparate science cases. Since progress on these cross‑cutting challenges would benefit multiple probes simultaneously, we identify key methodological research priorities, including Bayesian inference at scale, physics‑informed methods, validation frameworks, and active learning for discovery. With an eye on emerging techniques, we also explore the potential of the latest foundation model methodologies and LLM‑driven agentic AI systems to reshape DESC workflows, provided their deployment is coupled with rigorous evaluation and governance. Finally, we discuss critical software, computing, data infrastructure, and human capital requirements for the successful deployment of these new methodologies, and consider associated risks and opportunities for broader coordination with external actors.

Abstract:
Neural networks are predominantly trained using gradient‑based methods, yet in many applications their final predictions remain far from the accuracy attainable within the model's expressive capacity. We introduce Linearized Subspace Refinement (LSR), a general and architecture‑agnostic framework that exploits the Jacobian‑induced linear residual model at a fixed trained network state. By solving a reduced direct least‑squares problem within this subspace, LSR computes a subspace‑optimal solution of the linearized residual model, yielding a refined linear predictor with substantially improved accuracy over standard gradient‑trained solutions, without modifying network architectures, loss formulations, or training procedures. Across supervised function approximation, data‑driven operator learning, and physics‑informed operator fine‑tuning, we show that gradient‑based training often fails to access this attainable accuracy, even when local linearization yields a convex problem. This observation indicates that loss‑induced numerical ill‑conditioning, rather than nonconvexity or model expressivity, can constitute a dominant practical bottleneck. In contrast, one‑shot LSR systematically exposes accuracy levels not fully exploited by gradient‑based training, frequently achieving order‑of‑magnitude error reductions. For operator‑constrained problems with composite loss structures, we further introduce Iterative LSR, which alternates one‑shot LSR with supervised nonlinear alignment, transforming ill‑conditioned residual minimization into numerically benign fitting steps and yielding accelerated convergence and improved accuracy. By bridging nonlinear neural representations with reduced‑order linear solvers at fixed linearization points, LSR provides a numerically grounded and broadly applicable refinement framework for supervised learning, operator learning, and scientific computing.

Abstract:
We introduce a physics‑informed Generative Adversarial Network framework that recasts quantum resource‑state generation as an inverse‑design task. By embedding task‑specific utility functions into training, the model learns to generate valid two‑qubit states optimized for teleportation and entanglement broadcasting. Comparing decomposition‑based and direct‑generation architectures reveals that structural enforcement of Hermiticity, trace‑one, and positivity yields higher fidelity and training stability than loss‑only approaches. The framework reproduces theoretical resource boundaries for Werner‑like and Bell‑diagonal states with fidelities exceeding ~98%, establishing adversarial learning as a lightweight yet effective method for constraint‑driven quantum‑state discovery. This approach provides a scalable foundation for automated design of tailored quantum resources for information‑processing applications, exemplified with teleportation and broadcasting of entanglement, and it opens up the possibility of using such states in efficient quantum network design.

Abstract:
Distributed multichannel acoustic sensing (DMAS) enables large‑scale sound event classification (SEC), but performance drops when many channels are degraded and when sensor layouts at test time differ from training layouts. We propose a learning‑free, physics‑informed inpainting frontend based on reverse time migration (RTM). In this approach, observed multichannel spectrograms are first back‑propagated on a 3D grid using an analytic Green's function to form a scene‑consistent image, and then forward‑projected to reconstruct inpainted signals before log‑mel feature extraction and Transformer‑based classification. We evaluate the method on ESC‑50 with 50 sensors and three layouts (circular, linear, right‑angle), where per‑channel SNRs are sampled from ‑30 to 0 dB. Compared with an AST baseline, scaling‑sparsemax channel selection, and channel‑swap augmentation, the proposed RTM frontend achieves the best or competitive accuracy across all layouts, improving accuracy by 13.1 points on the right‑angle layout (from 9.7% to 22.8%). Correlation analyses show that spatial weights align more strongly with SNR than with channel‑‑source distance, and that higher SNR‑‑weight correlation corresponds to higher SEC accuracy. These results demonstrate that a reconstruct‑then‑project, physics‑based preprocessing effectively complements learning‑only methods for DMAS under layout‑open configurations and severe channel degradation.

Abstract:
Four‑dimensional (4D) Flow MRI can noninvasively measure cerebrovascular hemodynamics but remains underused clinically because current workflows rely on manual vessel segmentation and yield velocity fields sensitive to noise, artifacts, and phase aliasing. We present VAST (Vascular Flow Analysis and Segmentation), an automated, unsupervised pipeline for intracranial 4D Flow MRI that couples vessel segmentation with physics‑informed velocity reconstruction. VAST derives vessel masks directly from complex 4D Flow data by iteratively fusing magnitude‑ and phase‑based background statistics. It then reconstructs velocities via continuity‑constrained phase unwrapping, outlier correction, and low‑rank denoising to reduce noise and aliasing while promoting mass‑consistent flow fields, with processing completing in minutes per case on a standard CPU. We validate VAST on synthetic data from an internal carotid artery aneurysm model across SNR = 2‑20 and severe phase wrapping (up to five‑fold), on in vitro Poiseuille flow, and on an in vivo internal carotid aneurysm dataset. In synthetic benchmarks, VAST maintains near quarter‑voxel surface accuracy and reduces velocity root‑mean‑square error by up to fourfold under the most degraded conditions. In vitro, it segments the channel within approximately half a voxel of expert annotations and reduces velocity error by 39% (unwrapped) and 77% (aliased). In vivo, VAST closely matches expert time‑of‑flight masks and lowers divergence residuals by about 30%, indicating a more self‑consistent intracranial flow field. By automating processing and enforcing basic flow physics, VAST helps move intracranial 4D Flow MRI toward routine quantitative use in cerebrovascular assessment.

Abstract:
Physics‑Informed Neural Networks (PINNs) provide a learning‑based framework for solving partial differential equations (PDEs) by embedding governing physical laws into neural network training. In practice, however, their performance is often hindered by limited representational capacity and optimization difficulties caused by competing physical constraints and conflicting gradients. In this work, we study PINN training from a unified architecture‑optimization perspective. We first propose a layer‑wise dynamic attention mechanism to enhance representational flexibility, resulting in the Layer‑wise Dynamic Attention PINN (LDA‑PINN). We then reformulate PINN training as a multi‑task learning problem and introduce a conflict‑resolved gradient update strategy to alleviate gradient interference, leading to the Gradient‑Conflict‑Resolved PINN (GC‑PINN). By integrating these two components, we develop the Architecture‑Conflict‑Resolved PINN (ACR‑PINN), which combines attentive representations with conflict‑aware optimization while preserving the standard PINN loss formulation. Extensive experiments on benchmark PDEs, including the Burgers, Helmholtz, Klein‑Gordon, and lid‑driven cavity flow problems, demonstrate that ACR‑PINN achieves faster convergence and significantly lower relative L_2 and L_\infty errors than standard PINNs. These results highlight the effectiveness of architecture‑optimization co‑design for improving the robustness and accuracy of PINN‑based solvers.

Abstract:
The present study investigates the numerical solution of Black‑Scholes partial differential equation (PDE) for option valuation with multiple underlying assets. We develop a physics‑informed (PI) machine learning algorithm based on a radial basis function neural network (RBFNN) that concurrently optimizes the network architecture and predicts the target option price. The physics‑informed radial basis function neural network (PIRBFNN) combines the strengths of the traditional radial basis function collocation method and the physics‑informed neural network machine learning approach to effectively solve PDE problems in the financial context. By employing a PDE residual‑based technique to adaptively refine the distribution of hidden neurons during the training process, the PIRBFNN facilitates accurate and efficient handling of multidimensional option pricing models featuring non‑smooth payoff conditions. The validity of the proposed method is demonstrated through a set of experiments encompassing a single‑asset European put option, a double‑asset exchange option, and a four‑asset basket call option.

Abstract:
In this paper, we develop a novel mesh‑free framework, termed physics‑informed neural networks with invariant measure score matching (PINN‑IMSM), for reconstructing dynamical systems from unlabeled point‑cloud data that capture the system's invariant measure. The invariant density satisfies the steady‑state Fokker‑Planck (FP) equation. We reformulate this equation in terms of its score function (the gradient of the log‑density), which is estimated directly from data via denoising score matching, thereby bypassing explicit density estimation. This learned score is then embedded into a physics‑informed neural network (PINN) to reconstruct the drift velocity field under the resulting score‑based FP equation. The mesh‑free nature of PINNs allows the framework to scale to higher dimensions, avoiding the curse of dimensionality inherent in mesh‑based methods. To address the ill‑posedness of high‑dimensional inverse problems, we recast the problem as a PDE‑constrained optimization that seeks the minimal‑energy velocity field. Under suitable conditions, we prove that this problem admits a unique solution that depends continuously on the score function. The constrained formulation is solved using a stochastic augmented Lagrangian method. Numerical experiments on representative dynamical systems, including the Van der Pol oscillator, an active swimmer in an anharmonic trap, and the chaotic Lorenz‑63 and Lorenz‑96 systems, demonstrate that PINN‑IMSM accurately recovers invariant measures and reconstructs faithful dynamical behavior for problems in up to five dimensions.

Abstract:
Quantum systems governed by time‑dependent Hamiltonians pose significant challenges for the accurate computation of unitary time‑evolution operators, which are essential for predicting quantum state dynamics. In this work, we introduce a physics‑informed deep learning approach based on Physics‑Informed Neural Networks to estimate these operators over the full time domain. By incorporating physical constraints such as unitarity and leveraging the second‑order Magnus expansion on the evolution operator, the proposed framework enables the estimation of unitary matrices at different time intervals. The model is trained using simulated unitary operators and evaluated on quantum systems ranging from 2 to 6 qubits. For larger many‑body systems, specifically those with 7 and 8 qubits, the same methodology is employed to reconstruct an effective time‑dependent Hamiltonian, from which the corresponding time‑evolution operator is computed over the entire temporal domain. The proposed framework achieves fidelities exceeding 0.92 using a limited number of unitary samples, indicating a potential reduction in measurement and data acquisition costs. These results highlight the effectiveness of the approach for data‑driven simulation and identification of quantum dynamical systems, with direct relevance to quantum computing and quantum simulation applications.

Abstract:
Glacial Lake Outburst Floods (GLOFs) pose a serious threat in high mountain regions. They are hazardous to communities, infrastructure, and ecosystems further downstream. The classical methods of GLOF detection and prediction have so far mainly relied on hydrological modeling, threshold‑based lake monitoring, and manual satellite image analysis. These approaches suffer from several drawbacks: slow updates, reliance on manual labor, and losses in accuracy when clouds interfere and/or lack on‑site data. To tackle these challenges, we present IceWatch: a novel deep learning framework for GLOF prediction that incorporates both spatial and temporal perspectives. The vision component, RiskFlow, of IceWatch deals with Sentinel‑2 multispectral satellite imagery using a CNN‑based classifier and predicts GLOF events based on the spatial patterns of snow, ice, and meltwater. Its tabular counterpart confirms this prediction by considering physical dynamics. TerraFlow models glacier velocity from NASA ITS_LIVE time series while TempFlow forecasts near‑surface temperature from MODIS LST records; both are trained on long‑term observational archives and integrated via harmonized preprocessing and synchronization to enable multimodal, physics‑informed GLOF prediction. Both together provide cross‑validation, which will improve the reliability and interpretability of GLOF detection. This system ensures strong predictive performance, rapid data processing for real‑time use, and robustness to noise and missing information. IceWatch paves the way for automatic, scalable GLOF warning systems. It also holds potential for integration with diverse sensor inputs and global glacier monitoring activities.

Abstract:
Attention‑based neural architectures have become central to state‑of‑the‑art methods in real‑time nonlinear control. As these data‑driven models continue to be integrated into increasingly safety‑critical domains, ensuring statistically grounded and provably safe decision‑making becomes essential. This paper introduces a novel reactive control framework for gap‑based navigation using the Attentive Neural Process (AttNP) and a physics‑informed extension, the PI‑AttNP. Both models are evaluated in a simulated F1TENTH‑style Ackermann steering racecar environment, chosen as a fast‑paced proxy for safety‑critical autonomous driving scenarios. The PI‑AttNP augments the AttNP architecture with approximate model‑based priors to inject physical inductive bias, enabling faster convergence and improved prediction accuracy suited for real‑time control. To further ensure safety, we derive and implement a control barrier function (CBF)‑based filtering mechanism that analytically enforces collision avoidance constraints. This CBF formulation is fully compatible with the learned AttNP controller and generalizes across a wide range of racing scenarios, providing a lightweight and certifiable safety layer. Our results demonstrate competitive closed‑loop performance while ensuring real‑time constraint satisfaction.

Abstract:
Accurately and efficiently solving nonlinear differential equations is crucial for modeling dynamic behavior across science and engineering. Physics‑Informed Neural Networks (PINNs) have emerged as a powerful solution that embeds physical laws in training by enforcing equation residuals. However, these struggle to model nonlinear dynamics, suffering from limited generalization across problems and long training times. To address these limitations, we propose a perturbation‑guided transfer learning framework for PINNs (PTL‑PINN), which integrates perturbation theory with transfer learning to efficiently solve nonlinear equations. Unlike gradient‑based transfer learning, PTL‑PINNs solve an approximate linear perturbative system using closed‑form expressions, enabling rapid generalization with the time complexity of matrix‑vector multiplication. We show that PTL‑PINNs achieve accuracy comparable to various Runge‑Kutta methods, with computational speeds up to one order of magnitude faster. To benchmark performance, we solve a broad set of problems, including nonlinear oscillators across various damping regimes, the equilibrium‑centered Lotka‑Volterra system, the KPP‑Fisher and the Wave equation. Since perturbation theory sets the accuracy bound of PTL‑PINNs, we systematically evaluate its practical applicability. This work connects long‑standing perturbation methods with PINNs, demonstrating how perturbation theory can guide foundational models to solve nonlinear systems with speeds comparable to those of classical solvers.

Abstract:
Indoor occupancy classification enables privacy‑preserving monitoring in settings such as remote elder care, where presence information helps triage alarms without cameras or wearables. Radar suits this role by sensing motion through occlusions and in darkness. Modern deep‑learning pipelines are the standard for interpreting radar returns effectively; however, they are often parameter‑heavy and sensitive at low signal‑to‑noise ratios (SNR), motivating compact alternatives like Hybrid Quantum Neural Networks (HQNNs). A two‑qubit HQNN is benchmarked against convolutional neural networks (CNNs) using a physics‑informed 60GHz digital twin and real radar measurements under matched training protocols. In clean conditions, the HQNN achieves high accuracy (99.7% synthetic; 97.0% real) with up to 170x fewer parameters (0.066M). Its parameter efficiency is shown to be structural, as an ablation of the parameterized quantum circuit (PQC) causes sharp performance drops on real data (to 68.5% and 31.5% for the control heads). A domain‑dependent sensitivity emerges under additive‑noise evaluation, where the HQNN begins recovery earlier in synthetic data while CNNs recover more steeply and peak higher on real measurements. In label‑fraction ablations, CNNs prove more sample‑efficient on real Range‑Doppler Maps (RDMs), with the performance gap being most pronounced (at 50% labels, BA 0.89‑0.99 vs. HQNN 0.75). On synthetic data, this gap narrows significantly, largely vanishing by the 50% label mark. Overall, the HQNN's value lies in parameter efficiency and a compact inductive bias that shapes its distinct sensitivity profile; this work establishes a rigorous baseline for hybrid quantum models in privacy‑preserving radar occupancy sensing.

Abstract:
Wildfire monitoring requires high‑resolution atmospheric measurements, yet low‑cost sensors on Unmanned Aerial Vehicles (UAVs) exhibit baseline drift, cross‑sensitivity, and response lag that corrupt concentration estimates. Traditional deep learning denoising approaches demand large datasets impractical to obtain from limited UAV flight campaigns. We present PC^2DAE, a physics‑informed denoising autoencoder that addresses data scarcity by embedding physical constraints directly into the network architecture. Non‑negative concentration estimates are enforced via softplus activations and physically plausible temporal smoothing, ensuring outputs are physically admissible by construction rather than relying on loss function penalties. The architecture employs hierarchical decoder heads for Black Carbon, Gas, and CO_2 sensor families, with two variants: PC^2DAE‑Lean (21k parameters) for edge deployment and PC^2DAE‑Wide (204k parameters) for offline processing. We evaluate on 7,894 synchronized 1 Hz samples collected from UAV flights during prescribed burns in Saskatchewan, Canada (approximately 2.2 hours of flight data), two orders of magnitude below typical deep learning requirements. PC^2DAE‑Lean achieves 67.3% smoothness improvement and 90.7% high‑frequency noise reduction with zero physics violations. Five baselines (LSTM‑AE, U‑Net, Transformer, CBDAE, DeSpaWN) produce 15‑‑23% negative outputs. The lean variant outperforms wide (+5.6% smoothness), suggesting reduced capacity with strong inductive bias prevents overfitting in data‑scarce regimes. Training completes in under 65 seconds on consumer hardware.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful tool for solving differential equations and modeling physical systems by embedding physical laws into the learning process. However, rigorously quantifying how well a PINN captures the complete dynamical behavior of the system, beyond simple trajectory prediction, remains a challenge. This paper proposes a novel experimental framework to address this by employing Fisher information for differentiable dynamical systems, denoted g_F^C. This Fisher information, distinct from its statistical counterpart, measures inherent uncertainties in deterministic systems, such as sensitivity to initial conditions, and is related to the phase space curvature and the net stretching action of the state space evolution. We hypothesize that if a PINN accurately learns the underlying dynamics of a physical system, then the Fisher information landscape derived from the PINN's learned equations of motion will closely match that of the original analytical model. This match would signify that the PINN has achieved comprehensive fidelity capturing not only the state evolution but also crucial geometric and stability properties. We outline an experimental methodology using the dynamical model of a car to compute and compare g_F^C for both the analytical model and a trained PINN. The comparison, based on the Jacobians of the respective system dynamics, provides a quantitative measure of the PINN's fidelity in representing the system's intricate dynamical characteristics.

Abstract:
Physics‑Informed Neural Networks (PINNs) represent a groundbreaking paradigm in scientific computing, seamlessly integrating the robust framework of deep learning with fundamental physical laws. This paper meticulously applies the standard PINN framework to solve the challenging one‑dimensional nonlinear Fisher‑KPP equation, a critical model in reaction‑diffusion dynamics describing phenomena such as population spread and flame propagation. We detail a comprehensive methodology, encompassing the neural network architecture, the physics‑informed loss function, and an in‑depth investigation into retraining strategies aimed at optimizing model performance. Our approach is rigorously validated through a direct comparison of the PINN solution against both the known analytical solution and a numerical solution derived from the Finite Difference Method (FDM). Through this work, we elucidate the intricate balance between model complexity, training efficiency, and accuracy. Results highlight the PINN's remarkable capability in accurately approximating the solution to this complex PDE, while also shedding light on the critical aspects and challenges of model retraining, particularly concerning the optimizer's state. This study provides a thorough quantitative error analysis, demonstrating the efficacy of PINNs as a viable and competitive alternative to traditional numerical methods for solving nonlinear differential equations, and discusses their broader applications across various scientific domains.

Abstract:
Digital twins are virtual replicas of physical entities and are poised to transform personalized medicine through the real‑time simulation and prediction of human physiology. Translating this paradigm from engineering to biomedicine requires overcoming profound challenges, including anatomical variability, multi‑scale biological processes, and the integration of multi‑physics phenomena. This survey systematically reviews methodologies for building digital twins of human organs, structured around a pipeline decoupled into anatomical twinning (capturing patient‑specific geometry and structure) and functional twinning (simulating multi‑scale physiology from cellular to organ‑level function). We categorize approaches both by organ‑specific properties and by technical paradigm, with particular emphasis on multi‑scale and multi‑physics integration. A key focus is the role of artificial intelligence (AI), especially physics‑informed AI, in enhancing model fidelity, scalability, and personalization. Furthermore, we discuss the critical challenges of clinical validation and translational pathways. This study not only charts a roadmap for overcoming current bottlenecks in single‑organ twins but also outlines the promising, albeit ambitious, future of interconnected multi‑organ digital twins for whole‑body precision healthcare.

Abstract:
We present a theoretical framework for quantum key distribution (QKD) using orbital angular momentum (OAM) encoded BB84 on an unmanned aerial vehicle (UAV) platform. A unified channel model captures Kolmogorov turbulence, pointing induced misalignment, and finite aperture clipping, enabling quantitative predictions of inter mode crosstalk and the resulting quantum bit error rate (QBER). Using a weak plus vacuum decoy state formulation, we derive composable finite key lower bounds on the secret key rate that incorporate statistical fluctuations, detector dark counts, efficiency mismatch, and error correction leakage. To stabilize performance under non stationary flight conditions, we introduce a lightweight physics informed learning module that combines physical priors with measured link statistics to classify valid pulses, reject corrupted data, and recommend decoding strategies. We outline a complete evaluation pipeline including UAV system architecture, turbulence driven QBER maps, decoy optimization, finite key scaling, and AI calibration metrics. Simulations indicate that under moderate turbulence and milliradian level pointing jitter, the proposed AI assisted method can improve the secret key rate by 10 percent to 30 percent while preserving composable security.

Abstract:
In this paper, we explore the evolution of plane curves and surfaces governed by the hyperbolic mean curvature flow. We propose a mesh‑free approach based on the physics‑informed neural networks (PINNs), which eliminates the need for discretization and meshing of computational domains, and is efficient in solving partial differential equations involving high dimensions. To the best of our knowledge, this is the first result on the numerical analysis by employing the PINNs for the hyperbolic geometric evolution equations in the literature. The effectiveness of this method is demonstrated through several numerical simulations by selecting diverse initial curves and surfaces, as well as both constant and non‑constant initial velocities.

Abstract:
Physics‑informed extreme learning machines (PIELMs) typically impose boundary and initial conditions through penalty terms, yielding only approximate satisfaction that is sensitive to user‑specified weights and can propagate errors into the interior solution. This work introduces Null‑Space Projected PIELM (NP‑PIELM), achieving exact constraint enforcement through algebraic projection in coefficient space. The method exploits the geometric structure of the admissible coefficient manifold, recognizing that it admits a decomposition through the null space of the boundary operator. By characterizing this manifold via a translation‑invariant representation and projecting onto the kernel component, optimization is restricted to constraint‑preserving directions, transforming the constrained problem into unconstrained least‑squares where boundary conditions are satisfied exactly at discrete collocation points. This eliminates penalty coefficients, dual variables, and problem‑specific constructions while preserving single‑shot training efficiency. Numerical experiments on elliptic and parabolic problems including complex geometries and mixed boundary conditions validate the framework.

Abstract:
Modeling complex dynamical systems under varying conditions is computationally intensive, often rendering high‑fidelity simulations intractable. Although reduced‑order models (ROMs) offer a promising solution, current methods often struggle with stochastic dynamics and fail to quantify prediction uncertainty, limiting their utility in robust decision‑making contexts. To address these challenges, we introduce a data‑driven framework for learning continuous‑time stochastic ROMs that generalize across parameter spaces and forcing conditions. Our approach, based on amortized stochastic variational inference, leverages a reparametrization trick for Markov Gaussian processes to eliminate the need for computationally expensive forward solvers during training. This enables us to jointly learn a probabilistic autoencoder and stochastic differential equations governing the latent dynamics, at a computational cost that is independent of the dataset size and system stiffness. Additionally, our approach offers the flexibility of incorporating physics‑informed priors if available. Numerical studies are presented for three challenging test problems, where we demonstrate excellent generalization to unseen parameter combinations and forcings, and significant efficiency gains compared to existing approaches.

Abstract:
Physics‑Informed Neural Networks (PINNs) provide a mesh‑free approach for solving differential equations by embedding physical constraints into neural network training. However, PINNs tend to overfit within the training domain, leading to poor generalization when extrapolating beyond trained spatiotemporal regions. This work presents SPIKE (Sparse Physics‑Informed Koopman‑Enhanced), a framework that regularizes PINNs with continuous‑time Koopman operators to learn parsimonious dynamics representations. By enforcing linear dynamics dz/dt = Az in a learned observable space, both PIKE (without explicit sparsity) and SPIKE (with L1 regularization on A) learn sparse generator matrices, embodying the parsimony principle that complex dynamics admit low‑dimensional structure. Experiments across parabolic, hyperbolic, dispersive, and stiff PDEs, including fluid dynamics (Navier‑Stokes) and chaotic ODEs (Lorenz), demonstrate consistent improvements in temporal extrapolation, spatial generalization, and long‑term prediction accuracy. The continuous‑time formulation with matrix exponential integration provides unconditional stability for stiff systems while avoiding diagonal dominance issues inherent in discrete‑time Koopman operators.

Abstract:
Optimization is central to both modern machine learning (ML) and scientific machine learning (SciML), yet the structure of the underlying optimization problems differs substantially across these domains. Classical ML typically relies on stochastic, sample‑separable objectives that favor first‑order and adaptive gradient methods. In contrast, SciML often involves physics‑informed or operator‑constrained formulations in which differential operators induce global coupling, stiffness, and strong anisotropy in the loss landscape. As a result, optimization behavior in SciML is governed by the spectral properties of the underlying physical models rather than by data statistics, frequently limiting the effectiveness of standard stochastic methods and motivating deterministic or curvature‑aware approaches. This document provides a unified introduction to optimization methods in ML and SciML, emphasizing how problem structure shapes algorithmic choices. We review first‑ and second‑order optimization techniques in both deterministic and stochastic settings, discuss their adaptation to physics‑constrained and data‑driven SciML models, and illustrate practical strategies through tutorial examples, while highlighting open research directions at the interface of scientific computing and scientific machine learning.

Abstract:
Theoretical developments in relativistic spin hydrodynamics, which describes the macroscopic transport of spin angular momentum alongside other fundamental conserved quantities, have progressed rapidly since the experimental observation of the global spin polarization of Λ hyperons in relativistic heavy‑ion collision experiments. However, numerical simulations of relativistic spin hydrodynamics remain largely unaddressed due to computational challenges, particularly the accurate numerical conservation of total angular momentum. In this work, we propose the use of physics‑informed neural networks (PINNs) for computational relativistic spin hydrodynamics. As a concrete application, we consider a rotating fluid confined within a cylindrical container. We show that angular‑momentum conservation can be accurately achieved in the PINNs‑based numerical framework. Furthermore, we investigate the spin‑orbit conversion induced by the rotational viscous effect, which is the intrinsic dissipative process of relativistic spin hydrodynamics. Our analysis numerically identifies the mismatch between the transverse thermal vorticity and the spin potential as the driving mechanism of the spin‑orbit conversion.

Abstract:
Kolmogorov‑Arnold Networks (KANs) require significantly smaller architectures compared to multilayer perceptron (MLP)‑based approaches, while retaining expressive power through spline‑based activations. Moving boundary problems are ubiquitous in physical systems, whose numerical solutions are quite complex. We propose a shallow KAN framework combined with a Level‑set formulation that directly approximates the temperature distribution T(\mathbfx,t) and the moving interface Γ(t), enforcing the governing PDEs, phase equilibrium, and Stefan condition through physics‑informed residuals. Numerical experiments in one and two dimensions show that the framework achieves accurate reconstructions of both temperature fields and interface dynamics, highlighting the potential of KANs as a compact and efficient alternative for moving boundary PDEs. First, we validate the model with semi‑infinite analytical solutions. Subsequently, the model is extended to 2D using a level‑set based formulation for interface propagation, which is solved within the KAN framework. This work demonstrates that KANs are capable of solving complex moving boundary problems without the need for measurement data.

Abstract:
We propose two globally continuous neural‑based variants of the Neural Approximated Virtual Element Method (NAVEM), termed B‑NAVEM and P‑NAVEM. Both approaches construct local basis functions using pre‑trained fully connected neural networks while ensuring exact continuity across adjacent mesh elements. B‑NAVEM leverages a Physics‑Informed Neural Network to approximately solve the local Laplace problem that defines the virtual element basis functions, whereas P‑NAVEM directly enforces polynomial reproducibility via a tailored loss function, without requiring harmonicity within the element interior. Numerical experiments assess the methods in terms of computational cost, memory usage, and accuracy during both training and testing phases.

Abstract:
We present the Physics‑Informed Optimal Homotopy Analysis Method (PI‑OHAM) for solving nonlinear differential equations. PI‑OHAM, based on classical HAM, employs a physics‑informed residual loss to optimize convergence‑control parameters systematically by combining data, boundary conditions, and governing equations in the manner similar to Physics Informed Neural Networks (PINNs). The combination of the flexibility of PINNs and the analytical transparency of HAM provides the approach with high numerical stability, rapid convergence, and high consistency with traditional numerical solutions. PI‑OHAM has superior accuracy‑time trade‑offs and faster and more accurate convergence than standard HAM and PINNs when applied to the Blasius boundary‑layer problem. It is also very close to numerical standards available in the literature. PI‑OHAM ensures analytical transparency and interpretability by series‑based solutions, unlike purely data‑driven or data‑free PINNs. Significant contributions are a conceptual bridge between decades of homotopy‑based analysis and modern physics‑inspired methods, and a numerically aided but analytically interpretable solver of nonlinear differential equations. PI‑OHAM appears as a computationally efficient, accurate and understandable alternative to nonlinear fluid flow, heat transfer and other industrial problems in cases where robustness and interpretability are important.

Abstract:
Accurate global medium‑range weather forecasting is fundamental to Earth system science. Most existing Transformer‑based forecasting models adopt vision‑centric architectures that neglect the Earth's spherical geometry and zonal periodicity. In addition, conventional autoregressive training is computationally expensive and limits forecast horizons due to error accumulation. To address these challenges, we propose the Shifted Earth Transformer (Searth Transformer), a physics‑informed architecture that incorporates zonal periodicity and meridional boundaries into window‑based self‑attention for physically consistent global information exchange. We further introduce a Relay Autoregressive (RAR) fine‑tuning strategy that enables learning long‑range atmospheric evolution under constrained memory and computational budgets. Based on these methods, we develop YanTian, a global medium‑range weather forecasting model. YanTian achieves higher accuracy than the high‑resolution forecast of the European Centre for Medium‑Range Weather Forecasts and performs competitively with state‑of‑the‑art AI models at one‑degree resolution, while requiring roughly 200 times lower computational cost than standard autoregressive fine‑tuning. Furthermore, YanTian attains a longer skillful forecast lead time for Z500 (10.3 days) than HRES (9 days). Beyond weather forecasting, this work establishes a robust algorithmic foundation for predictive modeling of complex global‑scale geophysical circulation systems, offering new pathways for Earth system science.

Abstract:
A multi‑preconditioned LBFGS (MP‑LBFGS) algorithm is introduced for training finite‑basis physics‑informed neural networks (FBPINNs). The algorithm is motivated by the nonlinear additive Schwarz method and exploits the domain‑decomposition‑inspired additive architecture of FBPINNs, in which local neural networks are defined on subdomains, thereby localizing the network representation. Parallel, subdomain‑local quasi‑Newton corrections are then constructed on the corresponding local parts of the architecture. A key feature is a novel nonlinear multi‑preconditioning mechanism, in which subdomain corrections are optimally combined through the solution of a low‑dimensional subspace minimization problem. Numerical experiments indicate that MP‑LBFGS can improve convergence speed, as well as model accuracy over standard LBFGS while incurring lower communication overhead.

Abstract:
The accurate identification and control of spatiotemporal chaos in reaction‑diffusion systems remains a grand challenge in chemical engineering, particularly when the underlying catalytic surface possesses complex, unknown topography. In the Defect Turbulence regime, system dynamics are governed by topological phase singularities (spiral waves) whose motion couples to manifold curvature via geometric pinning. Conventional Physics‑Informed Neural Networks (PINNs) using ReLU or Tanh activations suffer from fundamental spectral bias, failing to resolve high‑frequency gradients and causing amplitude collapse or phase drift. We propose a Multi‑Scale SIREN‑PINN architecture leveraging periodic sinusoidal activations with frequency‑diverse initialization, embedding the appropriate inductive bias for wave‑like physics directly into the network structure. This enables simultaneous resolution of macroscopic wave envelopes and microscopic defect cores. Validated on the complex Ginzburg‑Landau equation evolving on latent Riemannian manifolds, our architecture achieves relative state prediction error ε_L_2 \approx 1.92 × 10^‑2, outperforming standard baselines by an order of magnitude while preserving topological invariants (|ΔN_defects| < 1). We solve the ill‑posed inverse pinning problem, reconstructing hidden Gaussian curvature fields solely from partial observations of chaotic wave dynamics (Pearson correlation ρ= 0.965). Training dynamics reveal a distinctive Spectral Phase Transition at epoch ～ 2,100, where cooperative minimization of physics and geometry losses drives the solver to Pareto‑optimal solutions. This work establishes a new paradigm for Geometric Catalyst Design, offering a mesh‑free, data‑driven tool for identifying surface heterogeneity and engineering passive control strategies in turbulent chemical reactors.

Abstract:
Traditional 2D hydraulic models face significant computational challenges that limit their applications that are time‑sensitive or require many model evaluations. This study presents a physics‑informed Deep Operator Network (DeepONet) framework for computational hydraulics modeling that learns the solution operator of the 2D shallow water equations (SWEs) to create fast surrogate models. The framework can operate in two modes: a purely data‑driven SWE‑DeepONet that learns from numerical solver such as SRH‑2D, and a physics‑informed PI‑SWE‑DeepONet that additionally incorporates the continuous SWEs as constraints during training. Based on a real‑world case, steady flows in a reach of the Sacramento River in California, it is demonstrated that PI‑SWE‑DeepONet possesses much enhanced prediction capability than SWE‑DeepONet when applied to out‑of‑distribution scenarios. The physics‑informed model is shown to exhibit slower error growth and larger breakdown distances in comparison with SWE‑DeepONet. The gain of the physics‑informed training, however, comes with costs, chief among which are the simulated results have slightly higher errors for in‑distribution cases. It reflects the existence of a tension between the two competing training objectives: fitting the results from the traditional hydraulic model and satisfying the continuous governing equations. In this study, guidelines are developed for selecting the appropriate approach based on a real‑world case: PI‑SWE‑DeepONet is preferred for out‑of‑distribution predictions, uncertain training data, or when physical consistency is a priority, while SWE‑DeepONet is recommended if the modeling objective is to replicate faithfully the traditional hydraulic model results within the training distribution. Other challenges are also discussed, such as the loss weighting approach.

Abstract:
Kolmogorov‑Arnold Networks (KANs) offer a promising framework for approximating complex nonlinear functions, yet the original B‑spline formulation suffers from significant computational overhead due to De Boor algorithm. While recent RBF‑based variants improve efficiency, they often sacrifice the approximation accuracy inherent in the original spline‑based design. To bridge this gap, we propose Free‑RBF‑KAN, an architecture that integrates adaptive learning grids and trainable smoothness parameters to enable expressive, high‑resolution function approximation. Our method utilizes learnable RBF shapes that dynamically align with activation patterns, and we provide the first formal universal approximation proof for the RBF‑KAN family. Empirical evaluations across multiscale regression, physics‑informed PDEs, and operator learning demonstrate that Free‑RBF‑KAN can achieve accuracy comparable to its B‑spline counterparts while delivering significantly faster training and inference. These results establish Free‑RBF‑KAN as an efficient and adaptive alternative for high‑dimensional structured modeling tasks.

Abstract:
We present a physics‑informed Wasserstein GAN with gradient penalty (WGAN‑GP) for solving the inverse Chafee‑‑Infante problem on two‑dimensional domains with Dirichlet boundary conditions. The objective is to reconstruct an unknown initial condition from a near‑equilibrium state obtained after 100 explicit forward Euler iterations of the reaction‑diffusion equation \[ u_t ‑ γΔu + κ\left(u^3 ‑ u\right)=0. \] Because this mapping strongly damps high‑frequency content, the inverse problem is severely ill‑posed and sensitive to noise. Our approach integrates a U‑Net generator, a PatchGAN critic with spectral normalization, Wasserstein loss with gradient penalty, and several physics‑informed auxiliary terms, including Lyapunov energy matching, distributional statistics, and a crucial forward‑simulation penalty. This penalty enforces consistency between the predicted initial condition and its forward evolution under the \emphsame forward Euler discretization used for dataset generation. Earlier experiments employing an Eyre‑type semi‑implicit solver were not compatible with this residual mechanism due to the cost and instability of Newton iterations within batched GPU training. On a dataset of 50k training and 10k testing pairs on 128×128 grids (with natural [‑1,1] amplitude scaling), the best trained model attains a mean absolute error (MAE) of approximately 0.23988159 on the full test set, with a sample‑wise standard deviation of about 0.00266345. The results demonstrate stable inversion, accurate recovery of interfacial structure, and robustness to high‑frequency noise in the initial data.

Abstract:
A new wave of work on covariance cleaning and nonlinear shrinkage has delivered asymptotically optimal analytical solutions for large covariance matrices. The same framework has been generalized to empirical cross‑covariance matrices, whose singular value decomposition identifies canonical comovement modes between two asset sets, with singular values quantifying the strength of each mode and providing natural targets for shrinkage. Existing analytical cross‑covariance cleaners are derived under strong stationarity and large‑sample assumptions, and they typically rely on mesoscopic regularity conditions such as bounded spectra; macroscopic common modes (e.g., a global market factor) violate these conditions. When applied to real equity returns, where dependence structures drift over time and global modes are prominent, we find that these theoretically optimal formulas do not translate into robust out‑of‑sample performance. We address this gap by designing a random‑matrix‑inspired neural architecture that operates in the empirical singular‑vector basis and learns a nonlinear mapping from empirical singular values to their corresponding cleaned values. By construction, the network can recover the analytical solution as a special case, yet it remains flexible enough to adapt to non‑stationary dynamics and mode‑driven distortions. Trained on a long history of equity returns, the proposed method achieves a more favorable bias‑variance trade‑off than purely analytical cleaners and delivers systematically lower out‑of‑sample cross‑covariance prediction errors. Our results demonstrate that combining random‑matrix theory with machine learning makes asymptotic theories practically effective in realistic time‑varying markets.

Abstract:
This paper develops an approach for multi‑step forecasting of dynamical systems by integrating probabilistic input forecasting with physics‑informed output prediction. Accurate multi‑step forecasting of time series systems is important for the automatic control and optimization of physical processes, enabling more precise decision‑making. While mechanistic‑based and data‑driven machine learning (ML) approaches have been employed for time series forecasting, they face significant limitations. Incomplete knowledge of process mathematical models limits mechanistic‑based direct employment, while purely data‑driven ML models struggle with dynamic environments, leading to poor generalization. To address these limitations, this paper proposes a dual‑level strategy for physics‑informed forecasting of dynamical systems. On the first level, input variables are forecast using a hybrid method that integrates a long short‑term memory (LSTM) network into probabilistic state transition models (STMs). On the second level, these stochastically predicted inputs are sequentially fed into a physics‑informed neural network (PINN) to generate multi‑step output predictions. The experimental results of the paper demonstrate that the hybrid input forecasting models achieve a higher log‑likelihood and lower mean squared errors (MSE) compared to conventional STMs. Furthermore, the PINNs driven by the input forecasting models outperform their purely data‑driven counterparts in terms of MSE and log‑likelihood, exhibiting stronger generalization and forecasting performance across multiple test cases.

Abstract:
The surface pressure field of transportation systems, including cars, trains, and aircraft, is critical for aerodynamic analysis and design. In recent years, deep neural networks have emerged as promising and efficient methods for modeling surface pressure field, being alternatives to computationally expensive CFD simulations. Currently, large‑scale public datasets are available for domains such as automotive aerodynamics. However, in many specialized areas, such as high‑speed trains, data scarcity remains a fundamental challenge in aerodynamic modeling, severely limiting the effectiveness of standard neural network approaches. To address this limitation, we propose the Adaptive Field Learning Framework (AdaField), which pre‑trains the model on public large‑scale datasets to improve generalization in sub‑domains with limited data. AdaField comprises two key components. First, we design the Semantic Aggregation Point Transformer (SAPT) as a high‑performance backbone that efficiently handles large‑scale point clouds for surface pressure prediction. Second, regarding the substantial differences in flow conditions and geometric scales across different aerodynamic subdomains, we propose Flow‑Conditioned Adapter (FCA) and Physics‑Informed Data Augmentation (PIDA). FCA enables the model to flexibly adapt to different flow conditions with a small set of trainable parameters, while PIDA expands the training data distribution to better cover variations in object scale and velocity. Our experiments show that AdaField achieves SOTA performance on the DrivAerNet++ dataset and can be effectively transferred to train and aircraft scenarios with minimal fine‑tuning. These results highlight AdaField's potential as a generalizable and transferable solution for surface pressure field modeling, supporting efficient aerodynamic design across a wide range of transportation systems.

Abstract:
We employ Physics‑Informed Neural Networks (PINNs) to solve the diffusion of heavy quarks within the expanding hot QCD medium generated in relativistic heavy‑ion collisions. Due to the strong coupling between heavy quarks and the bulk medium, the evolution of heavy quarks can be effectively characterized by a diffusion equation. This approach assumes the instantaneous kinetic thermalization of heavy quarks following their production in nuclear collisions. The local density of heavy quarks is intrinsically coupled to the velocity profile of the hot QCD medium. By incorporating the fluid velocity profiles provided by a hydrodynamic model directly into the diffusion equation, we utilize the deep neural network (DNN) to efficiently determine the heavy‑quark evolution. Furthermore, this work provides a valuable reference for the application of deep learning techniques to the treatment of non‑thermalized heavy‑quark dynamics. The rapid calculation of heavy‑quark diffusion using DNNs further facilitates the study of heavy‑quark coalescence within a large ensemble of fluctuating hot media.

Abstract:
Physics‑informed neural networks based on automatic differentiation (AD‑PINNs) and their finite‑difference counterparts (FD‑PINNs) are widely used for solving partial differential equations (PDEs), yet their analytical properties remain poorly understood. This work provides a unified mathematical foundation for both formulations. Under mild regularity assumptions on the activation function and for sufficiently wide neural networks of depth at least two, we prove that both the AD‑ and FD‑PINN optimization problems are ill‑posed: whenever a minimizer exists, there are in fact infinitely many, and uniqueness fails regardless of the choice of collocation points or finite‑difference stencil. Nevertheless, we establish two structural properties. First, whenever the underlying PDE or its finite‑difference discretization admits a solution, the corresponding AD‑PINN or FD‑PINN loss also admits a minimizer, realizable by a neural network of finite width. Second, FD‑PINNs are tightly coupled to the underlying finite‑difference scheme: every FD‑PINN minimizer agrees with a finite‑difference minimizer on the grid, and in regimes where the discrete PDE solution is unique, all zero‑loss FD‑PINN minimizers coincide with the discrete PDE solution on the stencil. Numerical experiments illustrate these theoretical insights: FD‑PINNs remain stable in representative forward and inverse problems, including settings where AD‑PINNs may fail to converge. We also include an inverse problem with noisy data, demonstrating that FD‑PINNs retain robustness in this setting as well. Taken together, our results clarify the analytical limitations of AD‑PINNs and explain the structural reasons for the more stable behavior observed in FD‑PINNs.

Abstract:
Machine learning‑based flow field prediction is emerging as a promising alternative to traditional Computational Fluid Dynamics, offering significant computational efficiency advantage. In this work, we propose the Geometry‑Parameterized Dual‑Encoder Physics‑Informed Neural Network (GP‑DE‑PINN) with a dual‑encoder architecture for effective prediction of unsteady flow fields around parameterized geometries. This framework integrates a geometric parameter encoder to map low‑dimensional shape parameters to high‑dimensional latent features, coupled with a spatiotemporal coordinate encoder, and is trained under the Navier‑Stokes equation constraints. Using 2D unsteady flow past petal‑shaped cylinders as an example, we evaluate the model's reconstruction performance, generalization capability, and hyperparameter sensitivity. Results demonstrate that the GP‑DE‑PINN significantly outperforms the PINN with direct geometric input in flow field reconstruction, accurately capturing vortex shedding structures and pressure evolution, while exhibiting superior generalization accuracy on unseen geometric configurations. Furthermore, sensitivity analyses regarding geometric sampling and network width reveal the model's robustness to these hyperparameter variations. These findings illustrate that the proposed framework can serve as a robust and promising framework for predicting unsteady flows around complex geometric obstacles.

Abstract:
Applying Physics‑Informed Gaussian Process Regression to the eigenvalue problem (\mathcalL‑λ)u = 0 poses a fundamental challenge, where the null source term results in a trivial predictive mean and a degenerate marginal likelihood. Drawing inspiration from system identification, we construct a transfer function‑type indicator for the unknown eigenvalue/eigenfunction using the physics‑informed Gaussian Process posterior. We demonstrate that the posterior covariance is only non‑trivial when λ corresponds to an eigenvalue of the partial differential operator \mathcalL, reflecting the existence of a non‑trivial eigenspace, and any sample from the posterior lies in the eigenspace of the linear operator. We demonstrate the effectiveness of the proposed approach through several numerical examples with both linear and non‑linear eigenvalue problems.

Abstract:
High‑dimensional design spaces underpin a wide range of physics‑based modeling and computational design tasks in science and engineering. These problems are commonly formulated as constrained black‑box searches over rugged objective landscapes, where function evaluations are expensive, and gradients are unavailable or unreliable. Conventional global search engines and optimizers struggle in such settings due to the exponential scaling of design spaces, the presence of multiple local basins, and the absence of physical guidance in sampling. We present a physics‑informed Monte Carlo Tree Search (MCTS) framework that extends policy‑driven tree‑based reinforcement concepts to continuous, high‑dimensional scientific optimization. Our method integrates population‑level decision trees with surrogate‑guided directional sampling, reward shaping, and hierarchical switching between global exploration and local exploitation. These ingredients allow efficient traversal of non‑convex, multimodal landscapes where physically meaningful optima are sparse. We benchmark our approach against standard global optimization baselines on a suite of canonical test functions, demonstrating superior or comparable performance in terms of convergence, robustness, and generalization. Beyond synthetic tests, we demonstrate physics‑consistent applicability to (i) crystal structure optimization from clusters to bulk, (ii) fitting of classical interatomic potentials, and (iii) constrained engineering design problems. Across all cases, the method converges with high fidelity and evaluation efficiency while preserving physical constraints. Overall, our work establishes physics‑informed tree search as a scalable and interpretable paradigm for computational design and high‑dimensional scientific optimization, bridging discrete decision‑making frameworks with continuous search in scientific design workflows.

Abstract:
In this work, we embed hard constraints in a physics informed neural network (PINN) which predicts solutions to the 2D incompressible Navier Stokes equations. We extend the hard constraint method introduced by Chen et al. (arXiv:2012.06148) from a linear PDE to a strongly non‑linear PDE. The PINN is used to estimate the stream function and pressure of the fluid, and by differentiating the stream function we can recover an incompressible velocity field. An unlearnable hard constraint projection (HCP) layer projects the predicted velocity and pressure to a hyperplane that admits only exact solutions to a discretised form of the governing equations.

Abstract:
The efficient construction of accurate channel knowledge maps (CKMs) is crucial for unleashing the full potential of environment‑aware wireless networks, yet it remains a difficult ill‑posed problem due to the sparsity of available location‑specific channel knowledge data. Although diffusion‑based methods such as denoising diffusion probabilistic models (DDPMs) have been exploited for CKM construction, they rely on iterative stochastic sampling, rendering them too slow for real‑time wireless applications. To bridge the gap between high fidelity and efficient CKM construction, this letter introduces a novel framework based on linear transport guided flow matching (LT‑GFM). Deviating from the noise‑removal paradigm of diffusion models, our approach models the CKM generation process as a deterministic ordinary differential equation (ODE) that follows linear optimal transport paths, thereby drastically reducing the number of required inference steps. We propose a unified architecture that is applicable to not only the conventional channel gain map (CGM) construction, but also the more challenging spatial correlation map (SCM) construction. To achieve physics‑informed CKM constructions, we integrate environmental semantics (e.g., building masks) for edge recovery and enforce Hermitian symmetry for property of the SCM. Simulation results verify that LT‑GFM achieves superior distributional fidelity with significantly lower Fréchet Inception Distance (FID) and accelerates inference speed by a factor of 25 compared to DDPMs.

Abstract:
This work introduces Adaptive Density Fields (ADF), a geometric attention framework that formulates spatial aggregation as a query‑conditioned, metric‑induced attention operator in continuous space. By reinterpreting spatial influence as geometry‑preserving attention grounded in physical distance, ADF bridges concepts from adaptive kernel methods and attention mechanisms. Scalability is achieved via FAISS‑accelerated inverted file indices, treating approximate nearest‑neighbor search as an intrinsic component of the attention mechanism. We demonstrate the framework through a case study on aircraft trajectory analysis in the Chengdu region, extracting trajectory‑conditioned Zones of Influence (ZOI) to reveal recurrent airspace structures and localized deviations.

Abstract:
In the analysis of complex physical systems, the objective often extends beyond merely computing a numerical solution to capturing the precise crossover between different regimes and extracting parameters containing meaningful information. However, standard numerical solvers and conventional deep learning approaches, such as Physics‑Informed Neural Networks (PINNs), typically operate as black boxes that output solution fields without disentangling the solution into its interpretable constituent parts. In this work, we propose GlueNN, a physics‑informed learning framework that decomposes the global solution into interpretable, patchwise analytic components. Rather than approximating the solution directly, GlueNN promotes the integration constants of local asymptotic expansions to learnable, scale‑dependent coefficient functions. By constraining these coefficients with the differential equation, the network effectively performs regime transition, smoothly interpolating between asymptotic limits without requiring ad hoc boundary matching. We demonstrate that this coefficient‑centric approach reproduces accurate global solutions in various examples and thus directly extracts physical information that is not explicitly available through standard numerical integration.

Abstract:
Combinatorial materials libraries are an efficient route to generate large families of candidate compositions, but their impact is often limited by the speed and depth of characterization and by the difficulty of extracting actionable structure‑property relations from complex characterization data. Here we develop an autonomous scanning probe microscopy (SPM) framework that integrates automated atomic force and magnetic force microscopy (AFM/MFM) to rapidly explore magnetic and structural properties across combinatorial spread libraries. To enable automated exploration of systems without a clear optimization target, we introduce a combination of a static physics‑informed bag‑of‑features (BoF) representation of measured surface morphology and magnetic structure with multi‑objective Bayesian optimization (MOBO) to discover the relative significance and robustness of features. The resulting closed‑loop workflow selectively samples the compositional gradient and reconstructs feature landscapes consistent with dense grid "ground truth" measurements. The resulting Pareto structure reveals where multiple nanoscale objectives are simultaneously optimized, where trade‑offs between roughness, coherence, and magnetic contrast are unavoidable, and how families of compositions cluster into distinct functional regimes, thereby turning multi‑feature imaging data into interpretable maps of competing structure‑property trends. While demonstrated for Au‑Co‑Ni and AFM/MFM, the approach is general and can be extended to other combinatorial systems, imaging modalities, and feature sets, illustrating how feature‑based MOBO and autonomous SPM can transform microscopy images from static data products into active feedback for real‑time, multi‑objective materials discovery.

Abstract:
High‑power impulse magnetron sputtering (HiPIMS) offers considerable control over ion energy and flux, making it invaluable for tailoring the microstructure and properties of advanced functional coatings. However, compared to conventional sputtering techniques, HiPIMS suffers from reduced deposition rates. Many groups have begun to evaluate complex pulsing schemes to improve upon this, leveraging multi‑pulse schemes (e.g. pre‑ionization or bipolar pulses). Unfortunately, the increased complexity of these pulsing schemes has led to high‑dimensionality parameter spaces that are prohibitive to classic design of experi‑ments. In this work we evaluate bipolar HiPIMS pulses for improving deposition rates of Al and Ti sputter tar‑gets. Over 3000 process conditions were collected via autonomous Bayesian sampling over a 6‑dimensional parameter space. These process conditions were then interpreted using Shapley Additive Explanations (SHAP), to deconvolute complex process influences on deposition rates. This allows us to link observed var‑iations in deposition rate to physical mechanisms such as back‑attraction and plasma ignition. Insights gained from this approach were then used to target specific processes where the positive pulse components were expected to have the highest impact on deposition rates. However, in practice, only minimal improve‑ments in deposition rate were achieved. In most cases, the positive pulse appears to be detrimental when placed immediately after the neg. pulse which we hypothesize relates to quenching of the afterglow plasma. The proposed workflow combining autonomous experimentation and interpretable machine learning is broad‑ly applicable to the discovery and optimization of complex plasma processes, paving the way for physics‑informed, data‑driven advancements in coating technologies.

Abstract:
The CMS programme of indirect searches in the top‑quark sector interprets precision measurements in the Standard Model Effective Field Theory (SMEFT) framework. These proceedings summarize recent CMS results highlighted in the TOP2025 talk: a search for CP violation in ttZ and tZq production using CP‑odd observables constructed with physics‑informed machine learning, a measurement that disentangles the flavour structure of electroweak SMEFT couplings in multilepton final states, and a Run 2 combination of complementary top+X measurements. We close with a brief outlook on the expected sensitivity gains at the high‑luminosity LHC.

Abstract:
Radiation hydrodynamics describes the interaction between high‑temperature hypersonic plasmas and the radiation they emit or absorb, a coupling that plays a central role in many astrophysical phenomena related to accretion and ejection processes. The HADES code was developed to model such systems by coupling hydrodynamics with M1‑gray or M1‑multigroup radiative transfer models, which are well suited to optically intermediate media. Despite its accuracy, radiation hydrodynamics simulations remain extremely demanding in terms of computational cost. Two main limitations are responsible for this. First, the M1‑multigroup model relies on a closure relation with no analytic expression, requiring expensive numerical evaluations. Second, the Courant‑Friedrichs‑Lewy condition strongly restricts the time step of the explicit schemes used in HADES. To overcome these difficulties, two complementary Artificial Intelligence based strategies were developed in this thesis. The first approach consists in training a Multi‑Layer Perceptron to approximate the M1‑multigroup closure relation. This method achieves excellent accuracy while reducing the computational cost by a factor of 3000, making it the most efficient approach currently available for this task. This performance gain enables high‑fidelity simulations of radiative shocks, in which radiation directly influences the shock structure. In particular, increasing spectral resolution slows down the shock and enlarges the radiative precursor. The second approach explores the use of Physics‑Informed Neural Networks to directly solve the radiation hydrodynamics equations and extrapolate simulations beyond their initial time range. Tests on purely hydrodynamic shocks show accurate handling of discontinuities, but application to radiative shocks remains challenging and requires further investigation.

Abstract:
We study diffusion‑controlled growth of a spherical droplet with a moving boundary using a physics‑informed neural network (PINN) formulation. The governing diffusion equation is coupled to the interfacial mass balance, with the droplet radius treated as an additional trainable function of time. The PINN accurately reproduces the self‑similar growth law and concentration profiles for a wide range of initial droplet radii, demonstrating convergence toward the asymptotic diffusive regime. The proposed approach provides a flexible and computationally efficient framework for solving moving‑boundary diffusion problems and can be readily extended to include additional physical effects.

Abstract:
Physics‑informed neural networks (PINNs) have recently emerged as a promising alternative for extracting unknown quantities from experimental data. Despite this potential, much of the recent literature has relied on sparse, high‑fidelity data from direct numerical simulations (DNS) rather than experimental sources like particle image velocimetry (PIV), which are not suitable for validating all reconstructed quantities. In the case of PIV, for example, pressure is not directly measured and the data have imperfections such as noise contamination or a limited field of view. To overcome these limitations, we present a novel methodology where PINNs are first trained on a RANS simulation such that it learns all states at every location in the domain. We then apply transfer learning which updates the PINN using sub‑sampled PIV data. The resulting predictions are in significantly better agreement with the full PIV dataset than PINNs which are trained on experimental data only. This work builds on the recent literature by integrating a Spalart‑Allmaras turbulence model and applying hard constraints to the no‑slip wall boundary condition. We apply this new methodology to a two‑dimensional NACA 0012 airfoil inclined at an angle of attack, α = 15 degrees, for two Reynolds numbers of Re = 10,000 and 75,000. The proposed methodology is initially validated using large eddy simulation (LES) data and then demonstrated on experimental PIV data. Our transfer learning approach results in improved predictions and a reduction in training time when compared to using a random network initialisation.

Abstract:
The explicit constraint force method (ECFM) was recently introduced as a novel formulation of the physics‑informed solution reconstruction problem, and was subsequently extended to inverse problems. In both solution reconstruction and inverse problems, model parameters are estimated with the help of measurement data. In practice, experimentalists seek to design experiments such that the acquired data leads to the most robust recovery of the missing parameters in a subsequent inverse problem. While there are well‑established techniques for designing experiments with standard approaches to the inverse problem, optimal experimental design (OED) has yet to be explored with the ECFM formulation. In this work, we investigate OED with a constraint force objective. First, we review traditional approaches to OED based on the Fisher information matrix, and propose an analogous formulation based on constraint forces. Next, we reflect on the different interpretations of the objective from standard and constraint force‑based inverse problems. We then test our method on several example problems. These examples suggest that an experiment which is optimal in the sense of constraint forces tends to position measurements in the stiffest regions of the system. Because the responses ‑‑ and thus the measurements ‑‑ are small in these regions, this strategy is impractical in the presence of measurement noise and/or finite measurement precision. As such, our provisional conclusion is that ECFM is not a viable approach to OED.

Abstract:
Bioelectrical properties of cells such as relative permittivity, conductivity, and characteristic time constants vary significantly between healthy and malignant cells across different frequencies. These distinctions provide a promising foundation for diagnostic and classification applications. This study systematically reviewed 20 scholarly articles to compile 535 datasets of quantitative bioelectric parameters in the kHz‑MHz frequency range and evaluated their utility in predictive modeling. Three supervised machine learning algorithms‑ Random Forest (RF), Support Vector Machine (SVM), and K‑Nearest Neighbor (KNN) were implemented and tuned using key hyperparameters to assess classification performance. In the second stage, a physics informed framework was incorporated to derive additional dielectric descriptors such as imaginary permittivity, loss tangent and charge relaxation time from the measured parameters. Random Forest based feature importance analysis was employed to identify the most discriminative dielectric parameters influencing the classification process. The results indicate that dielectric loss related parameters, particularly imaginary permittivity and conductivity, contribute significantly to the classification of cellular states. While the incorporation of physics‑derived features improves model interpretability and reduces overfitting tendencies, the overall classification accuracy remains comparable to models trained using primary dielectric descriptors. The proposed approach highlights the potential of physics‑informed machine learning for improving the analysis of dielectric spectroscopy data in the biomedical diagnostics.

Abstract:
Physics‑Informed Neural Networks (PINNs) provide a framework for integrating physical laws with data. However, their application to Prognostics and Health Management (PHM) remains constrained by the limited uncertainty quantification (UQ) capabilities. Most existing PINN‑based prognostics approaches are deterministic or account only for epistemic uncertainty, limiting their suitability for risk‑aware decision‑making. This work introduces a heteroscedastic Bayesian Physics‑Informed Neural Network (B‑PINN) framework that jointly models epistemic and aleatoric uncertainty, yielding full predictive posteriors for spatiotemporal insulation material ageing estimation. The approach integrates Bayesian Neural Networks (BNNs) with physics‑based residual enforcement and prior distributions, enabling probabilistic inference within a physics‑informed learning architecture. The framework is evaluated on transformer insulation ageing application, validated with a finite‑element thermal model and field measurements from a solar power plant, and benchmarked against deterministic PINNs, dropout‑based PINNs (d‑PINNs), and alternative B‑PINN variants. Results show that the proposed B‑PINN provides improved predictive accuracy and better‑calibrated uncertainty estimates than competing approaches. A systematic sensitivity study further analyzes the impact of boundary‑condition, initial‑condition, and residual sampling strategies on accuracy, calibration, and generalization. Overall, the findings highlight the potential of Bayesian physics‑informed learning to support uncertainty‑aware prognostics and informed decision‑making in transformer asset management.

Abstract:
Traditional computational fluid dynamics and physics‑informed neural networks (PINNs) often suffer from high computational cost, mesh sensitivity, and reduced accuracy for strongly nonlinear and time‑dependent flows. To address these limitations, we propose PhysicsFormer, a simple and efficient Transformer‑based physics‑informed neural network framework for complex fluid flow simulations. The proposed architecture employs encoder‑‑decoder multi‑head attention to capture long‑range temporal dependencies and enhance spatio‑temporal information propagation. Unlike conventional multilayer perceptron‑based PINNs, PhysicsFormer utilizes pseudo‑sequential spatio‑temporal representations together with a dynamics‑weighted loss formulation to improve convergence, stability, and predictive accuracy. Owing to its lightweight architecture and parallel learning strategy, the proposed framework achieves faster training and lower computational cost than existing Transformer‑based PINN models. The performance of the proposed framework is demonstrated on the convection equation, Burgers' equation, lid‑driven cavity flow at Re=100, and inverse Navier‑‑Stokes and flow reconstruction problems for flow past a circular cylinder at Re=100 and Re=3900. For the inverse Navier‑‑Stokes problem at Re=100, the proposed framework simultaneously reconstructs the flow field and identifies governing equation parameters with nearly 0% absolute error under both clean and noisy data conditions. Furthermore, for the high‑Reynolds‑number case at Re=3900, PhysicsFormer accurately reconstructs the velocity and pressure fields using only 25 spatial measurements per snapshot over 100 temporal snapshots. The obtained results demonstrate that PhysicsFormer provides an accurate, robust, and computationally efficient framework for complex time‑dependent fluid flow problems.

Abstract:
Understanding and modeling the constitutive behavior of concrete is crucial for civil and defense applications, yet widely used phenomenological models such as Karagozian \& Case concrete (KCC) model depend on empirically calibrated failure surfaces that lack flexibility in model form and associated uncertainty quantification. This work develops a physics‑informed framework that retains the modular elastoplastic structure of KCC model while replacing its empirical failure surface with a constrained Gaussian Process Regression (GPR) surrogate that can be learned directly from experimentally accessible observables. Triaxial compression data under varying confinement levels are used for training, and the surrogate is then evaluated at confinement levels not included in the training set to assess its generalization capability. Results show that an unconstrained GPR interpolates well near training conditions but deteriorates and violates essential physical constraints under extrapolation, even when augmented with simulated data. In contrast, a physics‑informed GPR that incorporates derivative‑based constraints aligned with known material behavior yields markedly better accuracy and reliability, including at higher confinement levels beyond the training range. Probabilistic enforcement of these constraints also reduces predictive variance, producing tighter confidence intervals in data‑scarce regimes. Overall, the proposed approach delivers a robust, uncertainty‑aware surrogate that improves generalization and streamlines calibration without sacrificing the interpretability and numerical efficiency of the KCC model, offering a practical path toward an improved constitutive models for concrete.

Abstract:
Accurate aircraft trajectory prediction (TP) in air traffic management systems is confounded by a number of epistemic uncertainties, dominated by uncertain meteorological conditions and operator specific procedures. Handling this uncertainty necessitates the use of probabilistic, machine learned models for generating trajectories. However, the trustworthiness of such models is limited if generated trajectories are not physically plausible. For this reason we propose a physics‑informed approach in which aircraft thrust and airspeed are learned from data and are used to condition the existing Base of Aircraft Data (BADA) model, which is physics‑based and enforces energy‑based constraints on generated trajectories. A set of informative features are identified and used to condition a probabilistic model of aircraft thrust and airspeed, with the proposed scheme demonstrating a 20% improvement in skilfulness across a set of six metrics, compared against a baseline probabilistic model that ignores contextual information such as meteorological conditions.

Abstract:
This paper presents the first probabilistic Digital Twin of operational en route airspace, developed for the London Area Control Centre. The Digital Twin is intended to support the development and rigorous human‑in‑the‑loop evaluation of AI agents for Air Traffic Control (ATC), providing a virtual representation of real‑world airspace that enables safe exploration of higher levels of ATC automation. This paper makes three significant contributions: firstly, we demonstrate how historical and live operational data may be combined with a probabilistic, physics‑informed machine learning model of aircraft performance to reproduce real‑world traffic scenarios, while accurately reflecting the level of uncertainty inherent in ATC. Secondly, we develop a structured assurance case, following the Trustworthy and Ethical Assurance framework, to provide quantitative evidence for the Digital Twin's accuracy and fidelity. This is crucial to building trust in this novel technology within this safety‑critical domain. Thirdly, we describe how the Digital Twin forms a unified environment for agent testing and evaluation. This includes fast‑time execution (up to x200 real‑time), a standardised Python‑based ``gym'' interface that supports a range of AI agent designs, and a suite of quantitative metrics for assessing performance. Crucially, the framework facilitates competency‑based assessment of AI agents by qualified Air Traffic Control Officers through a Human Machine Interface. We also outline further applications and future extensions of the Digital Twin architecture.

Abstract:
We propose a Pretrained Finite Element Method (PFEM),a physics driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM). PFEM consists of a physics informed pretraining stage and an optional finetuning stage. In the pretraining stage, a neural operator based on the Transolver architecture is trained solely from governing partial differential equations, without relying on labeled solution data. The model operates directly on unstructured point clouds, jointly encoding geometric information, material properties, and boundary conditions, and produces physically consistent initial solutions with extremely high computational efficiency. PDE constraints are enforced through explicit finite element, based differentiation, avoiding the overhead associated with automatic differentiation. In the fine‑tuning stage, the pretrained prediction is used as an initial guess for conventional FEM solvers, preserving their accuracy, convergence guarantees, and extrapolation capability while substantially reducing the number of iterations required to reach a prescribed tolerance. PFEM is validated on a broad range of benchmark problems, including linear elasticity and nonlinear hyperelasticity with complex geometries, heterogeneous materials, and arbitrary boundary conditions. Numerical results demonstrate strong generalization in the pretraining stage with relative errors on the order of 1%, and speedups of up to one order of magnitude in the fine‑tuning stage compared to FEM with zero initial guesses.

Abstract:
A fundamental requirement for full autonomy is the ability to sustain accurate navigation in the absence of external data, such as GNSS signals or visual information. In these challenging environments, the platform must rely exclusively on inertial sensors, leading to pure inertial navigation. However, the inherent noise and other error terms of the inertial sensors in such real‑world scenarios will cause the navigation solution to drift over time. Although conventional deep‑learning models have emerged as a possible approach to inertial navigation, they are inherently black‑box in nature. Furthermore, they struggle to learn effectively with limited supervised sensor data and often fail to preserve physical principles. To address these limitations, we propose PiDR, a physics‑informed inertial dead‑reckoning framework for autonomous platforms in situations of pure inertial navigation. PiDR offers transparency by explicitly integrating inertial navigation principles into the network training process through the physics‑informed residual component. PiDR plays a crucial role in mitigating abrupt trajectory deviations even under limited or sparse supervision. We evaluated PiDR on real‑world datasets collected by a mobile robot and an autonomous underwater vehicle. We obtained more than 29% positioning improvement in both datasets, demonstrating the ability of PiDR to generalize different platforms operating in various environments and dynamics. Thus, PiDR offers a robust, lightweight, yet effective architecture and can be deployed on resource‑constrained platforms, enabling real‑time pure inertial navigation in adverse scenarios.

Abstract:
Optimal power flow (OPF) is one of the fundamental tasks for power system operations. While machine learning (ML) approaches such as deep neural networks (DNNs) have been widely studied to enhance OPF solution speed and performance, their practical deployment faces two critical scaling questions: What is the minimum training data volume required for reliable results? How should ML models' complexity balance accuracy with real‑time computational limits? Existing studies evaluate discrete scenarios without quantifying these scaling relationships, leading to trial‑and‑error‑based ML development in real‑world applications. This work presents the first systematic scaling study for ML‑based OPF across two dimensions: data scale (0.1K‑40K training samples) and compute scale (multiple NN architectures with varying FLOPs). Our results reveal consistent power‑law relationships on both DNNs and physics‑informed NNs (PINNs) between each resource dimension and three core performance metrics: prediction error (MAE), constraint violations and speed. We find that for ACOPF, the accuracy metric scales with dataset size and training compute. These scaling laws enable predictable and principled ML pipeline design for OPF. We further identify the divergence between prediction accuracy and constraint feasibility and characterize the compute‑optimal frontier. This work provides quantitative guidance for ML‑OPF design and deployments.

Abstract:
We present GKFieldFlow, a novel three‑dimensional autoregressive deep learning surrogate model for nonlinear gyrokinetic turbulence. Based on the architecture FieldFlow‑Net, this model combines a multi‑resolution 3D U‑Net encoder‑decoder that operates on evolving plasma potential fields. A dilated temporal convolutional network (TCN) learns the nonlinear time evolution of latent turbulence features. GKFieldFlow simultaneously (i) predicts ion and electron energy fluxes, and particle flux directly from CGYRO turbulence, and (ii) predicts future potential fields autoregressively with desired spatial resolution. This enables the model to replicate both instantaneous transport and the underlying spatio‑temporal dynamics that generate it. The architecture is physics‑informed in its design: 3D convolutions preserve the anisotropic geometry and phase structure of gyrokinetic fluctuations, while dilated temporal convolutions capture multiscale dynamical couplings such as turbulence and zonal‑flow interactions, turbulence decorrelation, and intermittent bursty transport. We provide a complete technical description of the data structure, model components, and rationale behind each architectural choice. The model achieves high accuracy across all three transport channels, with multi‑horizon inference maintaining robustness. Autoregressive field rollouts preserve the spectral content, phase coherence, and energy distribution of the CGYRO nonlinear state with strong fidelity, and flux predictions remain consistent with CGYRO within a small fractional error. This work presents GKFieldFlow as a data‑driven reduced model that can jointly learn turbulence dynamics and transport.

Abstract:
Earthquake prediction and seismic hazard assessment remain fundamental challenges in geophysics, with existing machine learning approaches often operating as black boxes that ignore established physical laws. We introduce POSEIDON (Physics‑Optimized Seismic Energy Inference and Detection Operating Network), a physics‑informed energy‑based model for unified multi‑task seismic event prediction, alongside the Poseidon dataset ‑‑ the largest open‑source global earthquake catalog comprising 2.8 million events spanning 30 years. POSEIDON embeds fundamental seismological principles, including the Gutenberg‑Richter magnitude‑frequency relationship and Omori‑Utsu aftershock decay law, as learnable constraints within an energy‑based modeling framework. The architecture simultaneously addresses three interconnected prediction tasks: aftershock sequence identification, tsunami generation potential, and foreshock detection. Extensive experiments demonstrate that POSEIDON achieves state‑of‑the‑art performance across all tasks, outperforming gradient boosting, random forest, and CNN baselines with the highest average F1 score among all compared methods. Crucially, the learned physics parameters converge to scientifically interpretable values ‑‑ Gutenberg‑Richter b‑value of 0.752 and Omori‑Utsu parameters p=0.835, c=0.1948 days ‑‑ falling within established seismological ranges while enhancing rather than compromising predictive accuracy. The Poseidon dataset is publicly available at https://huggingface.co/datasets/BorisKriuk/Poseidon, providing pre‑computed energy features, spatial grid indices, and standardized quality metrics to advance physics‑informed seismic research.

Abstract:
We propose a graph‑based, multi‑fidelity learning framework for the prediction of stationary Navier‑‑Stokes solutions in non‑parametrized two‑dimensional geometries. The method is designed to guide the learning process through successive approximations, starting from reduced‑order and full Stokes models, and progressively approaching the Navier‑‑Stokes solution. To effectively capture both local and long‑range dependencies in the velocity and pressure fields, we combine graph neural networks with Transformer and Mamba architectures. While Transformers achieve the highest accuracy, we show that Mamba can be successfully adapted to graph‑structured data through an unsupervised node‑ordering strategy. The Mamba approach significantly reduces computational cost while maintaining performance. Physical knowledge is embedded directly into the architecture through an encoding‑processing‑physics informed decoding pipeline. Derivatives are computed through algebraic operators constructed via the Weighted Least Squares method. The flexibility of these operators allows us not only to make the output obey the governing equations, but also to constrain selected hidden features to satisfy mass conservation. We introduce additional physical biases through an enriched graph convolution with the same differential operators describing the PDEs. Overall, we successfully guide the learning process by physical knowledge and fluid dynamics insights, leading to more regular and accurate predictions

Abstract:
We propose a neural network‑based model capable of learning the broad landscape of working regimes in quantum dot simulators, and using this knowledge to autotune these devices ‑ based on transport measurements ‑ toward obtaining Majorana modes in the structure. The model is trained in an unsupervised manner on synthetic data in the form of conductance maps, using a physics‑informed loss that incorporates key properties of Majorana zero modes. We show that, with appropriate training, a deep vision‑transformer network can efficiently memorize relation between Hamiltonian parameters and structures on conductance maps and use it to propose parameters update for a quantum dot chain that drive the system toward topological phase. Starting from a broad range of initial detunings in parameter space, a single update step is sufficient to generate nontrivial zero modes. Moreover, by enabling an iterative tuning procedure ‑ where the system acquires updated conductance maps at each step ‑ we demonstrate that the method can address a much larger region of the parameter space.

Abstract:
Orthognathic surgery repositions jaw bones to restore occlusion and enhance facial aesthetics. Accurate simulation of postoperative facial morphology is essential for preoperative planning. However, traditional biomechanical models are computationally expensive, while geometric deep learning approaches often lack interpretability. In this study, we develop and validate a physics‑informed geometric deep learning framework named PhysSFI‑Net for precise prediction of soft tissue deformation following orthognathic surgery. PhysSFI‑Net consists of three components: a hierarchical graph module with craniofacial and surgical plan encoders combined with attention mechanisms to extract skeletal‑facial interaction features; a Long Short‑Term Memory (LSTM)‑based sequential predictor for incremental soft tissue deformation; and a biomechanics‑inspired module for high‑resolution facial surface reconstruction. Model performance was assessed using point cloud shape error (Hausdorff distance), surface deviation error, and landmark localization error (Euclidean distances of craniomaxillofacial landmarks) between predicted facial shapes and corresponding ground truths. A total of 135 patients who underwent combined orthodontic and orthognathic treatment were included for model training and validation. Quantitative analysis demonstrated that PhysSFI‑Net achieved a point cloud shape error of 1.070 +/‑ 0.088 mm, a surface deviation error of 1.296 +/‑ 0.349 mm, and a landmark localization error of 2.445 +/‑ 1.326 mm. Comparative experiments indicated that PhysSFI‑Net outperformed the state‑of‑the‑art method ACMT‑Net in prediction accuracy. In conclusion, PhysSFI‑Net enables interpretable, high‑resolution prediction of postoperative facial morphology with superior accuracy, showing strong potential for clinical application in orthognathic surgical planning and simulation.

Abstract:
Accurate and efficient modeling of radio wave propagation in railway tunnels is is critical for ensuring reliable communication‑based train control (CBTC) systems. Fine‑grid parabolic wave equation (PWE) solvers provide high‑fidelity field predictions but are computationally expensive for large‑scale tunnels, whereas coarse‑grid models lose essential modal and geometric details. To address this challenge, we propose a physics‑informed recurrent back‑projection propagation network (PRBPN) that reconstructs fine‑resolution received‑signal‑strength (RSS) fields from coarse PWE slices. The network integrates multi‑slice temporal fusion with an iterative projection/back‑projection mechanism that enforces physical consistency and avoids any pre‑upsampling stage, resulting in strong data efficiency and improved generalization. Simulations across four tunnel cross‑section geometries and four frequencies show that the proposed PRBPN closely tracks fine‑mesh PWE references. Engineering‑level validation on the Massif Central tunnel in France further confirms robustness in data‑scarce scenarios, trained with only a few paired coarse/fine RSS. These results indicate that the proposed PRBPN can substantially reduce reliance on computationally intensive fine‑grid solvers while maintaining high‑fidelity tunnel propagation predictions.

Abstract:
In this paper, we present a Wachspress‑based transfinite formulation on convex polygonal domains for exact enforcement of Dirichlet boundary conditions in physics‑informed neural networks. This approach leverages prior advances in geometric design such as blending functions and transfinite interpolation over convex domains. For prescribed Dirichlet boundary function \mathcalB, the transfinite interpolant of \mathcalB, g : \bar P \to C^0(\bar P), lifts functions from the boundary of a two‑dimensional polygonal domain to its interior. The transfinite trial function is expressed as the difference between the neural network's output and the extension of its boundary restriction into the interior of the domain, with g added to it. This ensures kinematic admissibility of the trial function in the deep Ritz method. Wachspress coordinates for an n‑gon are used in the transfinite formula, which generalizes bilinear Coons transfinite interpolation on rectangles to convex polygons. Since Wachspress coordinates are smooth, the neural network trial function has a bounded Laplacian, thereby overcoming a limitation in a previous contribution where approximate distance functions were used to exactly enforce Dirichlet boundary conditions. For a point \boldsymbolx \in \barP, Wachspress coordinates, \boldsymbolλ : \bar P \to [0,1]^n, serve as a geometric feature map for the neural network: \boldsymbolλ encodes the boundary edges of the polygonal domain. This offers a framework for solving problems on parametrized convex geometries using neural networks. The accuracy of physics‑informed neural networks is successfully assessed on forward problems (linear and nonlinear), an inverse heat conduction problem, and a parametrized geometric Poisson boundary‑value problem.

Abstract:
Physics‑based models play a key role in battery management, yet face challenges in real‑time applications due to the high computational cost of solving coupled algebraic‑partial differential equations. To accelerate model simulation, this study benchmarks three physics‑informed neural network (PINN) architectures for modeling the battery single particle model, including two conventional PINN architectures and a DeepONet‑based architecture. Both the accuracy and the generalization of these PINNs have been evaluated and compared under various current conditions. Our results highlight the potential of PINNs in modeling battery physics but also reveal limitations of conventional PINN architectures under highly dynamic current conditions. Among them, the Fourier‑enhanced DeepONet achieves superior generalization performance and offers nearly a 10 times speedup compared with numerical solvers. This work provides an example of integrating physics‑based models

Abstract:
We establish the convergence of physics‑informed neural networks (PINNs) for time‑dependent fractional diffusion equations posed on bounded domains. The presence of fractional Laplacian operators introduces nonlocal behavior and regularity constraints, and standard neural network approximations do not naturally enforce the associated spectral boundary conditions. To address this challenge, we introduce a spectrally‑defined mollification strategy that preserves the structure of the nonlocal operator while ensuring boundary compatibility. This enables the derivation of rigorous energy estimates in Sobolev spaces. Our results rely on analytical tools from PDE theory, highlighting the compatibility of PINN approximations with classical energy estimates for nonlocal equations. We prove convergence of the PINN approximation in any space‑time Sobolev norm H^k (with k \in \N). The analysis highlights the role of mollified residuals in enabling theoretical guarantees for neural‑network‑based solvers of nonlocal PDEs.

Abstract:
Digital twins, as precise digital representations of physical systems, have evolved from passive simulation tools into intelligent and autonomous entities through the integration of artificial intelligence technologies. This paper presents a unified four‑stage framework that systematically characterizes AI integration across the digital twin lifecycle, spanning modeling, mirroring, intervention, and autonomous management. By synthesizing existing technologies and practices, we distill a unified four‑stage framework that systematically characterizes how AI methodologies are embedded across the digital twin lifecycle: (1) modeling the physical twin through physics‑based and physics‑informed AI approaches, (2) mirroring the physical system into a digital twin with real‑time synchronization, (3) intervening in the physical twin through predictive modeling, anomaly detection, and optimization strategies, and (4) achieving autonomous management through large language models, foundation models, and intelligent agents. We analyze the synergy between physics‑based modeling and data‑driven learning, highlighting the shift from traditional numerical solvers to physics‑informed and foundation models for physical systems. Furthermore, we examine how generative AI technologies, including large language models and generative world models, transform digital twins into proactive and self‑improving cognitive systems capable of reasoning, communication, and creative scenario generation. Through a cross‑domain review spanning eleven application domains, including healthcare, aerospace, smart manufacturing, robotics, and smart cities, we identify common challenges related to scalability, explainability, and trustworthiness, and outline directions for responsible AI‑driven digital twin systems.

Abstract:
Variational principles are a unifying mathematical framework across many areas of physics, yet their instruction at the undergraduate level remains primarily analytical. This work presents a pedagogically oriented and computationally enhanced approach to variational modeling that integrates contemporary tools including gradient descent, automatic differentiation, and Physics‑Informed Neural Networks (PINNs). Classical variational problems are reformulated as optimization tasks and implemented using open‑source Python libraries such as NumPy, Matplotlib, PyTorch, and JAX. The proposed approach is demonstrated through a progression of problems drawn from standard undergraduate curricula, including the derivation of Snell's law from Fermat's principle, projectile motion with and without viscous drag, simple harmonic motion, nonlinear pendulum with damping, steady‑state heat conduction governed by the Laplace and Poisson equations with nonlinear temperature‑dependent internal heat generation, the double pendulum via the principle of least action, and variational treatments of vibrating strings. In addition, quantum mechanical applications are presented through variational solutions of the hydrogen atom, helium atom, and a schematic nuclear model of the silicon nucleus, illustrating the breadth of the framework across classical, quantum, and nuclear physics. The approach aims to enhance conceptual understanding while simultaneously introducing students to modern computational research methodologies.

Abstract:
Artificial intelligence has revolutionized optical device design, overcoming the efficiency bottlenecks of traditional methods. For holographic metasurfaces, conventional iterative algorithms suffer from time‑consuming iterations and convergence stagnation, especially as the complexity of 3D target fields increases. While recent deep‑learning‑based algorithms have improved the trade‑off between speed and image quality, most existing models remain constrained by predefined physical scenarios (e.g., fixed distances), limiting their adaptability in dynamic practical applications. To address these challenges, we propose a physics‑informed neural network (PINN) based on local polynomial fitting and multi‑plane wave propagation (LM‑PINN) for the rapid design of terahertz 3D holographic metasurfaces. By leveraging a self‑supervised training strategy, LM‑PINN eliminates the need for labeled datasets, enabling direct end‑to‑end mapping from target holographic patterns to the metasurface structures. Both simulated and experimental results demonstrate that LM‑PINN‑designed metasurfaces offer higher imaging quality than traditional iterative algorithms. Crucially, by incorporating a distance encoding process, a single trained LM‑PINN generalizes effectively across diverse physical configurations, including varying diffraction distances and distinct 2D or 3D targets, eliminating the necessity for retraining. Furthermore, the inference process of LM‑PINN typically takes less than 1 second, providing a multifold speed advantage over traditional algorithms. Consequently, this strategy offers a robust and universal framework that paves the way for high‑quality, real‑time, and large‑scale 3D holographic technologies.

Abstract:
Recent advancements in physics‑informed neural networks (PINNs) and their variants have garnered substantial focus from researchers due to their effectiveness in solving both forward and inverse problems governed by differential equations. In this research, a modified Auxiliary physics‑informed neural network (A‑PINN) framework with balanced adaptive optimizers is proposed for the analysis of structural vibration problems. In order to accurately represent structural systems, it is critical for capturing vibration phenomena and ensuring reliable predictive analysis. So, our investigations are crucial for gaining deeper insight into the robustness of scientific machine learning models for solving vibration problems. Further, to rigorously evaluate the performance of A‑PINN, we conducted different numerical simulations to approximate the Euler‑Bernoulli beam equations under the various scenarios. The numerical results substantiate the enhanced performance of our model in terms of both numerical stability and predictive accuracy. Our model shows improvement of at least 40% over the baselines.

Abstract:
Simulating nonlinear reaction‑diffusion dynamics on complex, non‑Euclidean manifolds remains a fundamental challenge in computational morphogenesis, constrained by high‑fidelity mesh generation costs and symplectic drift in discrete time‑stepping schemes. This study introduces the Intrinsic‑Metric Physics‑Informed Neural Network (IM‑PINN), a mesh‑free geometric deep learning framework that solves partial differential equations directly in the continuous parametric domain. By embedding the Riemannian metric tensor into the automatic differentiation graph, our architecture analytically reconstructs the Laplace‑Beltrami operator, decoupling solution complexity from geometric discretization. We validate the framework on a "Stochastic Cloth" manifold with extreme Gaussian curvature fluctuations (K \in [‑2489, 3580]), where traditional adaptive refinement fails to resolve anisotropic Turing instabilities. Using a dual‑stream architecture with Fourier feature embeddings to mitigate spectral bias, the IM‑PINN recovers the "splitting spot" and "labyrinthine" regimes of the Gray‑Scott model. Benchmarking against the Surface Finite Element Method (SFEM) reveals superior physical rigor: the IM‑PINN achieves global mass conservation error of \mathcalE_mass \approx 0.157 versus SFEM's 0.258, acting as a thermodynamically consistent global solver that eliminates mass drift inherent in semi‑implicit integration. The framework offers a memory‑efficient, resolution‑independent paradigm for simulating biological pattern formation on evolving surfaces, bridging differential geometry and physics‑informed machine learning.

Abstract:
Ka‑band low‑Earth‑orbit (LEO) downlinks can suffer second‑scale reliability collapses during flare‑driven ionospheric disturbances, where fixed fade margins and reactive adaptive coding and modulation (ACM) are either overly conservative or too slow. This paper presents a GNSS‑free, link‑internal predictive controller that senses the same downlink via a geometry‑free dual‑carrier phase observable at 10~Hz: a high‑pass filter and template‑based onset detector, followed by a four‑state nearly‑constant‑velocity Kalman filter, estimate ΔVTEC and its rate, and a short look‑ahead (60~s) yields an endpoint outage probability used as a risk gate to trigger one‑step discrete MCS down‑switch and pilot‑time update with hysteresis. Evaluation uses physics‑informed log replay driven by real GOES X‑ray flare morphologies under a disjoint‑day frozen‑calibration protocol, with uncertainty reported via paired moving‑block bootstrap. Across stressed 60~s windows, the controller reduces peak BLER by 25‑‑30% and increases goodput by 0.10‑‑0.15~bps/Hz versus no‑adaptation baselines under a unified link‑level abstraction. The loop runs in \mathcalO(1) per 0.1~s epoch (about 0.042~ms measured), making on‑board implementation feasible, and scope and deployment considerations for dispersion‑dominated events are discussed.

Abstract:
Large Protein Language Models have shown strong potential for generative protein design, yet they frequently produce structural hallucinations, generating sequences with high linguistic likelihood that fold into thermodynamically unstable conformations. Existing alignment approaches such as Direct Preference Optimization are limited in this setting, as they model preferences as binary labels and ignore the continuous structure of the physical energy landscape. We propose Physio‑DPO, a physics informed alignment framework that grounds protein language models in thermodynamic stability. Physio‑DPO introduces a magnitude aware objective that scales optimization updates according to the energy gap between native structures and physics perturbed hard negatives. Experiments show that Physio‑DPO consistently outperforms strong baselines including SFT, PPO, and standard DPO, reducing self consistency RMSD to 1.28 Å and increasing foldability to 92.8%. Qualitative analysis further demonstrates that Physio‑DPO effectively mitigates structural hallucinations by recovering biophysical interactions such as hydrophobic core packing and hydrogen bond networks.

Abstract:
We present a comprehensive Bayesian analysis of the Hubble constant within the framework of Physics‑Informed Neural Networks (PINNs), focusing on the standard ΛCDM model and its dynamical dark energy extensions described by the Chevallier‑Polarski‑Linder (CPL) parametrization, both with and without massive neutrinos. By embedding the cosmological background equations directly into a Bayesian PINN architecture, we reconstruct the Hubble expansion history H(z) in a data‑driven yet physically consistent manner, while rigorously propagating epistemic uncertainties. Our analysis combines late‑time observational probes, including Cosmic Chronometers, Baryon Acoustic Oscillations (BAO DESI DR2), and the Pantheon supernova sample, and quantifies the resulting tension in the inferred Hubble constant with respect to Planck 2018 Cosmic Microwave Background constraints and the SH0ES (R22) local distance ladder measurement. Within ΛCDM, we find that data combinations involving BAO tend to favor lower values of H_0, alleviating the tension with Planck at the expense of increased disagreement with SH0ES. Allowing for a time‑evolving dark energy equation of state in the CPL framework systematically shifts the posterior of H_0 toward higher values, leading to a notable reduction of the SH0ES tension, particularly for combinations including supernova data. The most flexible scenario, CPL with a free total neutrino mass Σm_ν, yields a balanced reconciliation between early‑ and late‑Universe determinations of H_0, with tension levels typically reduced to the ～1‑2σ range relative to both Planck and SH0ES. Our results highlight the nontrivial interplay between dark energy dynamics and neutrino mass in addressing the Hubble tension and demonstrate the efficacy of Bayesian PINNs as a robust and versatile tool for precision cosmology beyond the standard paradigm.

Abstract:
Physics‑informed neural networks have been widely applied to solid mechanics problems. However, balancing the governing partial differential equations and boundary conditions remains challenging, particularly in fracture mechanics, where accurate predictions strongly depend on refined sampling near crack tips. To overcome these limitations, a Kolosov‑Muskhelishvili informed neural network with Williams enrichment is developed in this study. Benefiting from the holomorphic representation, the governing equations are satisfied by construction, and only boundary points are required for training. Across a series of benchmark problems, the Kolosov‑Muskhelishvili informed neural network shows excellent agreement with analytical and finite element method references, achieving average relative errors below 1% and R^2 above 0.99 for both mode I and mode II loadings. Furthermore, three crack propagation criteria (maximum tangential stress, maximum energy release rate, and principle of local symmetry) are integrated into the framework using a transfer learning strategy to predict crack propagation directions. The predicted paths are nearly identical across all criteria, and the transfer learning strategy reduces the required training time by more than 70%. Overall, the developed framework provides a unified, mesh‑free, and physically consistent approach for accurate and efficient crack propagation analysis.

Abstract:
We revisit the analogy between feed‑forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms. A comparative analysis between the numerical/exact solutions of the Burgers' and Eikonal equations, and the same obtained via PINNs is presented. We show that PINN learning provides a different computational pathway compared to standard numerical discretization in approximating essentially the same underlying dynamics of the system. Within this framework, DNNs can be interpreted as discrete dynamical systems whose layer‑wise evolution approaches attractors, and multiple parameter configurations may yield comparable solutions, reflecting the degeneracy of the inverse mapping. In contrast to the structured operators associated with finite‑difference (FD) procedures, PINNs learn dense parameter representations that are not directly associated with classical discretization stencils. This distributed representation generally involves a larger number of parameters, leading to reduced interpretability and increased computational cost. However, the additional flexibility of such representations may offer advantages in high‑dimensional settings where classical grid‑based methods become impractical.

Abstract:
This paper introduces a hybrid computational framework for the multi‑frequency inverse source problem governed by the Helmholtz equation. By integrating a classical Fourier method with a deep convolutional neural network, we address the challenges inherent in sparse and noisy far‑field data. The Fourier method provides a physics‑informed, low‑frequency approximation of the source, which serves as the input to a U‑Net. The network is trained to map this coarse approximation to a high‑fidelity source reconstruction, effectively suppressing truncation artifacts and recovering fine‑scale geometric details. To enhance computational efficiency and robustness, we propose a high‑to‑low noise transfer learning strategy: a model pre‑trained on high‑noise regimes captures global topological features, offering a robust initialization for fine‑tuning on lower‑noise data. Numerical experiments demonstrate that the framework achieves accurate reconstructions with noise levels up to 100%, significantly outperforms traditional spectral methods under sparse measurement constraints, and generalizes well to unseen source geometries.

Abstract:
In aerodynamics, accurately modeling subsonic compressible flow over airfoils is critical for aircraft design. However, solving the governing nonlinear perturbation velocity potential equation presents computational challenges. Traditional approaches often rely on linearized equations or finite, truncated domains, which introduce non‑negligible errors and limit applicability in real‑world scenarios. In this study, we propose a novel framework utilizing Physics‑Informed Neural Networks (PINNs) to solve the full nonlinear compressible potential equation in an unbounded (infinite) domain. We address the unbounded‑domain and convergence challenges inherent in standard PINNs by incorporating a coordinate transformation and embedding physical asymptotic constraints directly into the network architecture. Furthermore, we employ a Multi‑Stage PINN (MS‑PINN) approach to iteratively minimize residuals, achieving solution accuracy approaching machine precision. We validate this framework by simulating flow over circular and elliptical geometries, comparing our results against traditional finite‑domain and linearized solutions. Our findings quantify the noticeable discrepancies introduced by domain truncation and linearization, particularly at higher Mach numbers, and demonstrate that this new framework is a robust, high‑fidelity tool for computational fluid dynamics.

Abstract:
We present Distortion‑Guided Restoration (DGR), a physics‑informed hybrid CNN‑diffusion framework for acquisition‑free correction of severe susceptibility‑induced distortions in prostate single‑shot EPI diffusion‑weighted imaging (DWI). DGR is trained to invert a realistic forward distortion model using large‑scale paired distorted and undistorted data synthesized from distortion‑free prostate DWI and co‑registered T2‑weighted images from 410 multi‑institutional studies, together with 11 measured B0 field maps from metal‑implant cases incorporated into a forward simulator to generate low‑b DWI (b = 50 s per mm squared), high‑b DWI (b = 1400 s per mm squared), and ADC distortions. The network couples a CNN‑based geometric correction module with conditional diffusion refinement under T2‑weighted anatomical guidance. On a held‑out synthetic validation set (n = 34) using ground‑truth simulated distortion fields, DGR achieved higher PSNR and lower NMSE than FSL TOPUP and FUGUE. In 34 real clinical studies with severe distortion, including hip prostheses and marked rectal distension, DGR improved geometric fidelity and increased radiologist‑rated image quality and diagnostic confidence. Overall, learning the inverse of a physically simulated forward process provides a practical alternative to acquisition‑dependent distortion‑correction pipelines for prostate DWI.

Abstract:
Synchronization control in networked dynamical systems requires regulating not only whether coherence is achieved, but also when and to what extent it emerges. We propose a physics‑informed neural network (PINN) framework for continuous‑time synchronization regulation, in which system trajectories and control inputs are jointly parameterized and constrained by the governing dynamics. Macroscopic synchronization objectives are imposed directly at the trajectory level by enforcing persistence conditions on the order parameter after a prescribed target time. This formulation enables simultaneous control of synchronization time and coherence level without assuming any explicit feedback law or solving a strict optimal control problem. Numerical studies on networked Kuramoto oscillators demonstrate smooth synchronization with reduced transient control effort and competitive cumulative cost relative to analytical baselines. The framework remains effective in non‑gradient and frustrated dynamics, highlighting physics‑informed neural control as a flexible trajectory‑level approach to synchronization regulation.

Abstract:
Wearable technologies have the potential to transform ambulatory and at‑home hemodynamic monitoring by providing continuous assessments of cardiovascular health metrics and guiding clinical management. However, existing cuffless wearable devices for blood pressure (BP) monitoring often rely on methods lacking theoretical foundations, such as pulse wave analysis or pulse arrival time, making them vulnerable to physiological and experimental confounders that undermine their accuracy and clinical utility. Here, we developed a smartwatch device with real‑time electrical bioimpedance (BioZ) sensing for cuffless hemodynamic monitoring. We elucidate the biophysical relationship between BioZ and BP via a multiscale analytical and computational modeling framework, and identify physiological, anatomical, and experimental parameters that influence the pulsatile BioZ signal at the wrist. A signal‑tagged physics‑informed neural network incorporating fluid dynamics principles enables calibration‑free estimation of BP and radial and axial blood velocity. We successfully tested our approach with healthy individuals at rest and after physical activity including physical and autonomic challenges, and with patients with hypertension and cardiovascular disease in outpatient and intensive care settings. Our findings demonstrate the feasibility of BioZ technology for cuffless BP and blood velocity monitoring, addressing critical limitations of existing cuffless technologies.

Abstract:
This paper introduces an advanced Computational Analytical Micromechanics (CAM) framework for linear thermoelastic composites (CMs) with periodic microstructures. The approach is based on an exact new Additive General Integral Equation (AGIE), formulated for compactly supported loading conditions, such as body forces and localized thermal effects (for example laser heating). In addition, new general integral equations (GIEs) are established for arbitrary mechanical and thermal loading. A unified iterative scheme is developed for solving the static AGIEs, where the compact support of loading serves as a new fundamental training parameter. At the core of the methodology lies a generalized Representative Volume Element (RVE) concept that extends Hill classical definition of the RVE. Unlike conventional RVEs, this generalized RVE is not fixed geometrically but emerges naturally from the characteristic scale of localized loading, thereby reducing the analysis of an infinite periodic medium to a finite, data‑driven domain. This formulation automatically filters out nonrepresentative subsets of effective parameters while eliminating boundary effects, edge artifacts, and finite‑size sample dependencies. Furthermore, the AGIE‑based CAM framework integrates seamlessly with machine learning (ML) and neural network (NN) architectures, supporting the development of accurate, physics‑informed surrogate nonlocal operators.

Abstract:
Realistic visual simulations are omnipresent, yet their creation requires computing time, rendering, and expert animation knowledge. Open‑vocabulary visual effects generation from text inputs emerges as a promising solution that can unlock immense creative potential. However, current pipelines lack both physical realism and effective language interfaces, requiring slow offline optimization. In contrast, PhysTalk takes a 3D Gaussian Splatting (3DGS) scene as input and translates arbitrary user prompts into real time, physics based, interactive 4D animations. A large language model (LLM) generates executable code that directly modifies 3DGS parameters through lightweight proxies and particle dynamics. Notably, PhysTalk is the first framework to couple 3DGS directly with a physics simulator without relying on time consuming mesh extraction. While remaining open vocabulary, this design enables interactive 3D Gaussian animation via collision aware, physics based manipulation of arbitrary, multi material objects. Finally, PhysTalk is train‑free and computationally lightweight: this makes 4D animation broadly accessible and shifts these workflows from a "render and wait" paradigm toward an interactive dialogue with a modern, physics‑informed pipeline.

Abstract:
Optical computing could reduce the energy cost of artificial intelligence by leveraging the parallelism and propagation speed of light. However, implementing nonlinear activation, essential for machine learning, remains challenging in low‑power optical systems dominated by linear wave physics. Here, we introduce an optical spiking neural network that uses optical rogue‑wave statistics as a programmable firing mechanism. By establishing a homomorphism between free‑space diffraction and neuronal integration, we demonstrate that phase‑engineered caustics enable robust, passive thresholding: sparse spatial spikes emerge when the local intensity exceeds a significant‑intensity rogue‑wave criterion. Using a physics‑informed digital twin, we optimize granular phase masks to deterministically concentrate energy into targeted detector regions, enabling end‑to‑end co‑design of the optical transformation and a lightweight electronic readout. We experimentally validate the approach on BreastMNIST and Olivetti Faces, achieving accuracies of 82.45% and 95.00%, respectively, competitive with standard digital baselines. These results demonstrate that extreme‑wave phenomena, often treated as deleterious fluctuations, can be harnessed as structural nonlinearity for scalable, energy‑efficient neuromorphic photonic inference.

Abstract:
Fault diagnosis of lithium‑ion batteries is critical for system safety. While existing deep learning methods exhibit superior detection accuracy, their "black‑box" nature hinders interpretability. Furthermore, restricted by binary classification paradigms, they struggle to provide root cause analysis and maintenance recommendations. To address these limitations, this paper proposes BatteryAgent, a hierarchical framework that integrates physical knowledge features with the reasoning capabilities of Large Language Models (LLMs). The framework comprises three core modules: (1) A Physical Perception Layer that utilizes 10 mechanism‑based features derived from electrochemical principles, balancing dimensionality reduction with physical fidelity; (2) A Detection and Attribution Layer that employs Gradient Boosting Decision Trees and SHAP to quantify feature contributions; and (3) A Reasoning and Diagnosis Layer that leverages an LLM as the agent core. This layer constructs a "numerical‑semantic" bridge, combining SHAP attributions with a mechanism knowledge base to generate comprehensive reports containing fault types, root cause analysis, and maintenance suggestions. Experimental results demonstrate that BatteryAgent effectively corrects misclassifications on hard boundary samples, achieving an AUROC of 0.986, which significantly outperforms current state‑of‑the‑art methods. Moreover, the framework extends traditional binary detection to multi‑type interpretable diagnosis, offering a new paradigm shift from "passive detection" to "intelligent diagnosis" for battery safety management.

Abstract:
We present a comparative study of classical numerical solvers, such as Petviashvili's method or finite difference with Newton iterations, and neural network‑based methods for computing ground states or profiles of solitary‑wave solutions to the one‑dimensional dispersive PDEs that include the nonlinear Schrödinger, the nonlinear Klein‑Gordon and the generalized KdV equations. We confirm that classical approaches retain high‑order accuracy and strong computational efficiency for single‑instance problems in the one‑dimensional setting. Physics‑informed neural networks (PINNs) are also able to reproduce qualitative solutions but are generally less accurate and less efficient in low dimensions than classical solvers due to expensive training and slow convergence. We also investigate the operator‑learning methods, which, although computationally intensive during training, can be reused across many parameter instances, providing rapid inference after pretraining, making them attractive for applications involving repeated simulations or real‑time predictions. For single‑instance computations, however, the accuracy of operator‑learning methods remains lower than that of classical methods or PINNs, in general.

Abstract:
Scientific machine learning (SciML) offers neural‑network alternatives to numerical workflows in geotechnical engineering. This paper benchmarks multi‑layer perceptrons (MLPs), physics‑informed neural networks (PINNs), deep operator networks (DeepONet), and graph network simulators (GNS) against finite‑difference and particle‑based references on geotechnical benchmarks, and compares PINN inversion with automatic differentiation (AD) through a conventional solver. We evaluate each method for extrapolation, training, and inference cost, transfer across problem instances, and physics accuracy. An MLP trained on two years of Terzaghi consolidation fits the data, but at year ten predicts ~290 mm with ReLU and ~60 mm with tanh or sigmoid, against a reference of 99.3 mm. A PINN on a damped oscillator with a time domain inside [0,1] matches the closed form within that interval but fails outside, since the residual constrains the fit only where it is sampled. For the 1D wave equation, PINN training is ~96,000 times slower than finite‑difference methods and less accurate. DeepONet avoids PINN retraining, yet for the beam on elastic foundation, its training cost equals ~1.8 million finite‑difference solves, and inference is slower per query than the direct solver. GNS improves geometric transfer through local particle interactions, though formulations still need trajectories, large training sets, and substantial memory. In the inverse wave benchmark, AD through the finite‑difference solver recovers the material profile in seconds with ~1% error. The results support a cautious role for SciML. Neural networks suit interpolation and pattern recognition inside validated domains, while inverse analysis should first try differentiable physics‑based solvers when a reliable forward solver exists.

Abstract:
Plasma kinetic equations exhibit pronounced sensitivity to microscopic perturbations in model parameters and data, making reliable and efficient uncertainty quantification (UQ) essential for predictive simulations. However, the cost of uncertainty sampling, the high‑dimensional phase space, and multiscale stiffness pose severe challenges to both computational efficiency and error control in traditional numerical methods. These aspects are further emphasized in presence of collisions where the high‑dimensional nonlocal collision integrations and conservation properties pose severe constraints. To overcome this, we present a variance‑reduced Monte Carlo framework for UQ in the Vlasov‑‑Poisson‑‑Landau (VPL) system, in which neural network surrogates replace the multiple costly evaluations of the Landau collision term. The method couples a high‑fidelity, asymptotic‑preserving VPL solver with inexpensive, strongly correlated surrogates based on the Vlasov‑‑Poisson‑‑Fokker‑‑Planck (VPFP) and Euler‑‑Poisson (EP) equations. For the surrogate models, we introduce a generalization of the separable physics‑informed neural network (SPINN), developing a class of tensor neural networks based on an anisotropic micro‑macro decomposition, to reduce velocity‑moment costs, model complexity, and the curse of dimensionality. To further increase correlation with VPL, we calibrate the VPFP model and design an asymptotic‑preserving SPINN whose small‑ and large‑Knudsen limits recover the EP and VP systems, respectively. Numerical experiments show substantial variance reduction over standard Monte Carlo, accurate statistics with far fewer high‑fidelity samples, and lower wall‑clock time, while maintaining robustness to stochastic dimension.

Abstract:
Early lane‑change intention prediction is essential for autonomous driving and ADAS, but it remains challenging because lane‑changing behavior depends on evolving traffic risk, surrounding‑vehicle interactions, and target‑lane feasibility rather than only instantaneous vehicle states. This study proposes an evolutionary physics‑informed temporal fusion framework for three‑class lane‑change intention prediction, including left lane change, right lane change, and no lane change. Instead of using static physics‑informed variables alone, the proposed method derives temporal descriptors from conventional traffic signals, including risk evolution, gap persistence, counterfactual lane utility, interaction pressure gradient, maneuver feasibility, and intent consistency. These descriptors are fused with temporal embeddings learned from raw trajectory sequences through a sequence encoder, and the fused representation is used for final classification. Experiments are conducted on the highD and exiD datasets under 1\,s, 2\,s, and 3\,s prediction horizons. The proposed model achieves Macro F1‑scores of 0.9514, 0.9256, and 0.8872 on highD, and 0.9386, 0.9070, and 0.8531 on exiD, respectively. The improvement is especially pronounced in exiD ramp‑adjacent scenarios, indicating that temporal physical evolution is particularly useful in interaction‑rich environments. These results demonstrate that combining evolutionary physics‑informed descriptors with learned temporal representations provides a more dynamic and interpretable solution for early lane‑change intention prediction.

Abstract:
Accurate modeling of scrape‑off layer (SOL) and divertor‑edge dynamics is vital for designing plasma‑facing components in fusion devices. High‑fidelity edge fluid/neutral codes such as SOLPS‑ITER capture SOL physics with high accuracy, but their computational cost limits broad parameter scans and long transient studies. We present transformer‑based, autoregressive surrogates for efficient prediction of 2D, time‑dependent plasma edge state fields. Trained on SOLPS‑ITER spatiotemporal data, the surrogates forecast electron temperature, electron density, and radiated power over extended horizons. We evaluate model variants trained with increasing autoregressive horizons (1‑100 steps) on short‑ and long‑horizon prediction tasks. Longer‑horizon training systematically improves rollout stability and mitigates error accumulation, enabling stable predictions over hundreds to thousands of steps and reproducing key dynamical features such as the motion of high‑radiation regions. Measured end‑to‑end wall‑clock times show the surrogate is orders of magnitude faster than SOLPS‑ITER, enabling rapid parameter exploration. Prediction accuracy degrades when the surrogate enters physical regimes not represented in the training dataset, motivating future work on data enrichment and physics‑informed constraints. Overall, this approach provides a fast, accurate surrogate for computationally intensive plasma edge simulations, supporting rapid scenario exploration, control‑oriented studies, and progress toward real‑time applications in fusion devices.

Abstract:
Predicting interfacial thermodynamics across molecular and continuum scales remains a central challenge in computational science. Classical density functional theory (cDFT) provides a first‑principles route to connect microscopic interactions with macroscopic observables, but its predictive accuracy depends on approximate free‑energy functionals that are difficult to generalize. Here we introduce a physics‑informed learning framework that augments cDFT with neural corrections trained directly against molecular‑dynamics data through adjoint optimization. Rather than replacing the theory with a black‑box surrogate, we embed compact neural networks within the Helmholtz free‑energy functional, learning local and nonlocal corrections that preserve thermodynamic consistency while capturing missing correlations. Applied to Lennard‑Jones fluids, the resulting augmented excess free‑energy functional quantitatively reproduces equilibrium density profiles, coexistence curves, and surface tensions across a broad temperature range, and accurately predicts contact angles and droplet shapes far beyond the training regime. This approach combines the interpretability of statistical mechanics with the adaptability of modern machine learning, establishing a general route to learned thermodynamic functionals that bridge molecular simulations and continuum‑scale models.

Abstract:
Advances in data acquisition and computational methods have accelerated the use of differential equation based modelling for complex systems. Such systems are often described by coupled (or more) variables, yet governing equation is typically available for one variable, while the remaining variable can be accessed only through data. This mismatch between known physics and observed data poses a fundamental challenge for existing physics‑informed machine learning approaches, which generally assume either complete knowledge of the governing equations or full data availability across all variables. In this paper, we introduce MUSIC (Multitask Learning Under Sparse and Incomplete Constraints), a sparsity induced multitask neural network framework that integrates partial physical constraints with data‑driven learning to recover full‑dimensional solutions of coupled systems when physics‑constrained and data‑informed variables are mutually exclusive. MUSIC employs mesh‑free (random) sampling of training data and sparsity regularization, yielding highly compressed models with improved training and evaluation efficiency. We demonstrate that MUSIC accurately learns solutions (shock wave solutions, discontinuous solutions, pattern formation solutions) to complex coupled systems under data‑scarce and noisy conditions, consistently outperforming non‑sparse formulations. These results highlight MUSIC as a flexible and effective approach for modeling partially observed systems with incomplete physical knowledge.

Abstract:
The 3D fast silent multi‑parametric mapping sequence with zero echo time (MuPa‑ZTE) is a novel quantitative MRI (qMRI) acquisition that enables nearly silent scanning by using a 3D phyllotaxis sampling scheme. MuPa‑ZTE improves patient comfort and motion robustness, and generates quantitative maps of T1, T2, and proton density using the acquired weighted image series. In this work, we propose a diffusion model‑based qMRI mapping method that leverages both a deep generative model and physics‑based data consistency to further improve the mapping performance. Furthermore, our method enables additional acquisition acceleration, allowing high‑quality qMRI mapping from a fourfold‑accelerated MuPa‑ZTE scan (approximately 1 minute). Specifically, we trained a denoising diffusion probabilistic model (DDPM) to map MuPa‑ZTE image series to qMRI maps, and we incorporated the MuPa‑ZTE forward signal model as an explicit data consistency (DC) constraint during inference. We compared our mapping method against a baseline dictionary matching approach and a purely data‑driven diffusion model. The diffusion models were trained entirely on synthetic data generated from digital brain phantoms, eliminating the need for large real‑scan datasets. We evaluated on synthetic data, a NISM/ISMRM phantom, healthy volunteers, and a patient with brain metastases. The results demonstrated that our method produces 3D qMRI maps with high accuracy, reduced noise and better preservation of structural details. Notably, it generalised well to real scans despite training on synthetic data alone. The combination of the MuPa‑ZTE acquisition and our physics‑informed diffusion model is termed q3‑MuPa, a quick, quiet, and quantitative multi‑parametric mapping framework, and our findings highlight its strong clinical potential.

Abstract:
We present NeuroSPICE, a physics‑informed neural network (PINN) framework for device and circuit simulation. Unlike conventional SPICE, which relies on time‑discretized numerical solvers, NeuroSPICE leverages PINNs to solve circuit differential‑algebraic equations (DAEs) by minimizing the residual of the equations through backpropagation. It models device and circuit waveforms using analytical equations in time domain with exact temporal derivatives. While PINNs do not outperform SPICE in speed or accuracy during training, they offer unique advantages such as surrogate models for design optimization and inverse problems. NeuroSPICE's flexibility enables the simulation of emerging devices, including highly nonlinear systems such as ferroelectric memories.

Abstract:
Physics‑Informed Neural Networks (PINNs) solve physical systems by incorporating governing partial differential equations directly into neural network training. In electromagnetism, where well‑established methodologies such as FDTD and FEM already exist, new methodologies are expected to provide clear advantages to be accepted. Despite their mesh‑free nature and applicability to inverse problems, PINNs can exhibit deficiencies in accuracy and energy metrics compared to FDTD. This study demonstrates that hybrid training strategies can bring PINNs closer to FDTD‑level accuracy and energy consistency. A hybrid methodology addressing common challenges in wave propagation is presented. Causality collapse in time‑dependent PINN training is addressed via time marching and causality‑aware weighting. To mitigate discontinuities introduced by time marching, a two stage interface continuity loss is applied. To suppress cumulative energy drift in electromagnetic waves, a local Poynting‑based regularizer is developed. In the developed PINN model, high field accuracy is achieved with an average 0.09% NRMSE and 1.01% L^2 error over time. Energy conservation is achieved with only a 0.02% relative energy mismatch in the 2D PEC cavity scenario. Training is performed without labeled field data, using only physics‑based residual losses; FDTD is used solely for post‑training evaluation. The results demonstrate that PINNs can achieve competitive results with FDTD in canonical electromagnetic examples and are a viable alternative.

Abstract:
Physics‑Informed Neural Networks with hard constraints (HC‑PINNs) are increasingly favored for their ability to strictly enforce boundary conditions via a trial function ansatz \tildeu = A + B \cdot N, yet the theoretical mechanisms governing their training dynamics have remained unexplored. Unlike soft‑constrained formulations where boundary terms act as additive penalties, this work reveals that the boundary function B introduces a multiplicative spatial modulation that fundamentally alters the learning landscape. A rigorous Neural Tangent Kernel (NTK) framework for HC‑PINNs is established, deriving the explicit kernel composition law. This relationship demonstrates that the boundary function B(\vecx) functions as a spectral filter, reshaping the eigenspectrum of the neural network's native kernel. Through spectral analysis, the effective rank of the residual kernel is identified as a deterministic predictor of training convergence, superior to classical condition numbers. It is shown that widely used boundary functions can inadvertently induce spectral collapse, leading to optimization stagnation despite exact boundary satisfaction. Validated across multi‑dimensional benchmarks, this framework transforms the design of boundary functions from a heuristic choice into a principled spectral optimization problem, providing a solid theoretical foundation for geometric hard constraints in scientific machine learning.

Abstract:
We construct the first physics‑informed neural‑network (PINN) surrogates for relativistic magnetohydrodynamics (RMHD) using a hybrid PDE and data‑driven workflow. Instead of training for the conservative form of the equations, we work with Jacobians or PDE characteristics directly in terms of primitive variables. We further add to the trainable system the divergence‑free condition, without the need of cleaning modes. Using a novel MUON optimizer implementation, we show that a baseline PINN trained on early‑time snapshots can extrapolate RMHD dynamics in one and two spatial dimensions, and that posterior residual‑guided networks can systematically reduce PDE violations.

Abstract:
Partial differential equations (PDEs) govern a wide range of physical systems, and recent multimodal foundation models have shown promise for learning PDE solution operators across diverse equation families. However, existing multi‑operator learning approaches are data‑hungry and neglect physics during training. Here, we propose a physics‑informed multimodal foundation model (PI‑MFM) framework that directly enforces governing equations during pretraining and adaptation. PI‑MFM takes symbolic representations of PDEs as the input, and automatically assembles PDE residual losses from the input expression via a vectorized derivative computation. These designs enable any PDE‑encoding multimodal foundation model to be trained or adapted with unified physics‑informed objectives across equation families. On a benchmark of 13 parametric one‑dimensional time‑dependent PDE families, PI‑MFM consistently outperforms purely data‑driven counterparts, especially with sparse labeled spatiotemporal points, partially observed time domains, or few labeled function pairs. Physics losses further improve robustness against noise, and simple strategies such as resampling collocation points substantially improve accuracy. We also analyze the accuracy, precision, and computational cost of automatic differentiation and finite differences for derivative computation within PI‑MFM. Finally, we demonstrate zero‑shot physics‑informed fine‑tuning to unseen PDE families: starting from a physics‑informed pretrained model, adapting using only PDE residuals and initial/boundary conditions, without any labeled solution data, rapidly reduces test errors to around 1% and clearly outperforms physics‑only training from scratch. These results show that PI‑MFM provides a practical and scalable path toward data‑efficient, transferable PDE solvers.

Abstract:
Millimeter‑wave (mmWave) radar enables privacy‑preserving, illumination‑invariant Human Pose Estimation (HPE). However, current mmWave‑based HPE systems face a signal‑noise dilemma: Heatmaps retain human reflections but embed environmental clutter, while Point Clouds (PC) suppress noise through aggressive thresholding but discard informative human reflections, limiting robustness across environments and radar configurations. To address this intrinsic bottleneck, we introduce Person Parametric Physics‑informed Representation (PPPR), a physics‑informed parametric intermediate representation that replaces purely signal‑level encodings with human‑centric parameterization. PPPR models each human joint as a Gaussian primitive encoding both kinematic properties, which include position, velocity, orientation, and electromagnetic properties, which include scattering intensity and Doppler signature. These parameters enable optimization through a dual‑constraint process: kinematic objectives enforce biomechanical consistency to suppress spatial artifacts, while electromagnetic objectives ensure adherence to mmWave propagation physics, decoupling input representations from non‑human noise. Experiments across three mmWave‑based HPE datasets with four HPE models demonstrate that replacing conventional inputs with PPPR consistently yields substantial accuracy gains. Furthermore, cross‑scenes and cross‑datasets experiments confirm PPPR's noise decoupling capability: models trained with PPPR maintain stable performance across diverse furniture arrangements and different radar chipsets, demonstrating its promising generalization capability in the challenging cross‑dataset settings. Code will be released upon publication.

Abstract:
Variational quantum algorithms are among the most promising approaches for simulating interacting quantum many‑body systems on noisy intermediate‑scale quantum (NISQ) devices. However, the practical success of variational quantum eigensolvers (VQE) critically depends on the structure of the chosen variational ansatz. In this work, we investigate the ground‑state properties of the one‑dimensional antiferromagnetic Heisenberg spin‑1/2 chain using both generic hardware‑efficient ansatz and physics‑informed, symmetry‑preserving variational circuits. We benchmark variational results against exact diagonalization and noiseless simulations, and subsequently validate the approach on real IQM Garnet quantum hardware. Our results demonstrate that incorporating physical symmetries into the circuit design leads to significantly improved energy estimates, enhanced robustness against hardware noise, and clearer convergence behavior when compared to hardware‑efficient ansatz under identical resource constraints. These findings highlight the importance of problem specific ansatz construction for reliable quantum simulations in the NISQ era.

Abstract:
Fluence map prediction is central to automated radiotherapy planning but remains an ill‑posed inverse problem due to the complex relationship between volumetric anatomy and beam‑intensity modulation. Convolutional methods in prior work often struggle to capture long‑range dependencies, which can lead to structurally inconsistent or physically unrealizable plans. We introduce FluenceFormer, a backbone‑agnostic transformer framework for direct, geometry‑aware fluence regression. The model uses a unified two‑stage design: Stage~1 predicts a global dose prior from anatomical inputs, and Stage~2 conditions this prior on explicit beam geometry to regress physically calibrated fluence maps. Central to the approach is the Fluence‑Aware Regression (FAR) loss, a physics‑informed objective that integrates voxel‑level fidelity, gradient smoothness, structural consistency, and beam‑wise energy conservation. We evaluate the generality of the framework across multiple transformer backbones, including Swin UNETR, UNETR, nnFormer, and MedFormer, using a prostate IMRT dataset. FluenceFormer with Swin UNETR achieves the strongest performance among the evaluated models and improves over existing benchmark CNN and single‑stage methods, reducing Energy Error to \mathbf4.5% and yielding statistically significant gains in structural fidelity (p < 0.05).

Abstract:
We present a latent diffusion‑based differentiable inversion method (LD‑DIM) for PDE‑constrained inverse problems involving high‑dimensional spatially distributed coefficients. LD‑DIM couples a pretrained latent diffusion prior with an end‑to‑end differentiable numerical solver to reconstruct unknown heterogeneous parameter fields in a low‑dimensional nonlinear manifold, improving numerical conditioning and enabling stable gradient‑based optimization under sparse observations. The proposed framework integrates a latent diffusion model (LDM), trained in a compact latent space, with a differentiable finite‑volume discretization of the forward PDE. Sensitivities are propagated through the discretization using adjoint‑based gradients combined with reverse‑mode automatic differentiation. Inversion is performed directly in latent space, which implicitly suppresses ill‑conditioned degrees of freedom while preserving dominant structural modes, including sharp material interfaces. The effectiveness of LD‑DIM is demonstrated using a representative inverse problem for flow in porous media, where heterogeneous conductivity fields are reconstructed from spatially sparse hydraulic head measurements. Numerical experiments assess convergence behavior and reconstruction quality for both Gaussian random fields and bimaterial coefficient distributions. The results show that LD‑DIM achieves consistently improved numerical stability and reconstruction accuracy of both parameter fields and corresponding PDE solutions compared with physics‑informed neural networks (PINNs) and physics‑embedded variational autoencoder (VAE) baselines, while maintaining sharp discontinuities and reducing sensitivity to initialization.

Abstract:
The rapid deployment of electric vehicle charging stations (EVCS) within distribution networks necessitates intelligent and adaptive control to maintain the grid's resilience and reliability. In this work, we propose PHANTOM, a physics‑aware adversarial network that is trained and optimized through a multi‑agent reinforcement learning model. PHANTOM integrates a physics‑informed neural network (PINN) enabled by federated learning (FL) that functions as a digital twin of EVCS‑integrated systems, ensuring physically consistent modeling of operational dynamics and constraints. Building on this digital twin, we construct a multi‑agent RL environment that utilizes deep Q‑networks (DQN) and soft actor‑critic (SAC) methods to derive adversarial false data injection (FDI) strategies capable of bypassing conventional detection mechanisms. To examine the broader grid‑level consequences, a transmission and distribution (T and D) dual simulation platform is developed, allowing us to capture cascading interactions between EVCS disturbances at the distribution level and the operations of the bulk transmission system. Results demonstrate how learned attack policies disrupt load balancing and induce voltage instabilities that propagate across T and D boundaries. These findings highlight the critical need for physics‑aware cybersecurity to ensure the resilience of large‑scale vehicle‑grid integration.

Abstract:
PINNs enhance scientific computing by incorporating physical laws into neural network structures, leading to significant advancements in scientific computing. However, PINNs struggle with multi‑scale and high‑frequency problems due to pathological gradient flow and spectral bias, which severely limit their predictive power. By combining an enhanced network architecture with a dynamically adaptive weighting mechanism featuring upper‑bound constraints, we propose the Dynamic Balancing Adaptive Weighting Physics‑Informed Kolmogorov‑Arnold Network (DBAW‑PIKAN). The proposed method effectively mitigates gradient‑related failure modes and overcomes bottlenecks in function representation. Compared to baseline models, the proposed method accelerates the convergence process and improves solution accuracy by at least an order of magnitude without introducing additional computational complexity. Numerical results on the Klein‑Gordon, Burgers, and Helmholtz equations demonstrate that DBAW‑PIKAN achieves superior accuracy and generalization performance.

Abstract:
Power transformers are critical assets in power networks, whose reliability directly impacts grid resilience and stability. Traditional condition monitoring approaches, often rule‑based or purely physics‑based, struggle with uncertainty, limited data availability, and the complexity of modern operating conditions. Recent advances in machine learning (ML) provide powerful tools to complement and extend these methods, enabling more accurate diagnostics, prognostics, and control. In this two‑part series, we examine the role of Neural Networks (NNs) and their extensions in transformer condition monitoring and health management tasks. This first paper introduces the basic concepts of NNs, explores Convolutional Neural Networks (CNNs) for condition monitoring using diverse data modalities, and discusses the integration of NN concepts within the Reinforcement Learning (RL) paradigm for decision‑making and control. Finally, perspectives on emerging research directions are also provided.

Abstract:
The integration of physics‑based knowledge with machine learning models is increasingly shaping the monitoring, diagnostics, and prognostics of electrical transformers. In this two‑part series, the first paper introduced the foundations of Neural Networks (NNs) and their variants for health assessment tasks. This second paper focuses on integrating physics and uncertainty into the learning process. We begin with the fundamentals of Physics‑Informed Neural Networks (PINNs), applied to spatiotemporal thermal modeling and solid insulation ageing. Building on this, we present Bayesian PINNs as a principled framework to quantify epistemic uncertainty and deliver robust predictions under sparse data. Finally, we outline emerging research directions that highlight the potential of physics‑aware and trustworthy machine learning for critical power assets.

Abstract:
Geomagnetic storms (GSTs) driven by solar wind‑magnetosphere coupling can severely disrupt technological systems, motivating the need for improved prediction accuracy and longer warning times. In this study, we develop a physics‑informed Long Short‑Term Memory (LSTM) model that incorporates cosmic‑ray flux modulation as an additional precursor signal. As coronal mass ejection (CME)‑driven disturbances propagate through the heliosphere, enhanced turbulence and magnetic‑field compression reduce galactic cosmic‑ray (GCR) flux measured by ground‑based neutron monitors (Forbush decreases), providing early information that can precede near‑Earth solar‑wind signatures by 1‑‑3 days. We integrate multi‑source space‑weather data, spanning 1995‑2020, including cosmic‑ray observations, solar wind plasma parameters, interplanetary magnetic‑field data, and geomagnetic indices. Based on these data, we construct a 19‑dimensional feature vector that includes flux background levels, decrease‑related indicators, and inter‑station correlation measures as model inputs. Employing a 50‑unit LSTM architecture, the proposed model achieves root‑mean‑square errors (RMSE) of 5.106 nT, 8.315 nT, 10.854 nT, 12.883 nT, and 14.788 nT for 2‑, 6‑, 12‑, 24‑, and 48‑hour predictions, respectively. Incorporating cosmic‑ray information further improves 48‑hour forecast skill by up to 25.84% (from 0.178 to 0.224). These results demonstrate the value of physics‑informed deep learning and cosmic‑ray precursors for advancing space‑weather forecasting.

Abstract:
The Wiener path integral framework is proposed to model military combat dynamics by incorporating the neglected stochastic effects to the Lanchester's square law. This framework is applied to evaluate the empirical 3:1 combat rule, which posits that an attacker requires a threefold force superiority to achieve victory. Specifically, the attacker's winning probability is computed utilizing a semi‑analytical Rayleigh‑Ritz method. Numerical results demonstrate that the validity of the rule critically depends on specific parameter regimes, primarily contingent upon the relative combat effectiveness ratio between the opposing forces and the tolerance for attrition. This work establishes a physics‑informed theoretical bridge between statistical mechanics and military operations research for analyzing uncertain combat systems.

Abstract:
We tackle the challenge of predicting vibrational stability in inorganic semiconductors for high‑throughput screening, an essential attribute for evaluating synthesizability alongside thermodynamic stability, frequently missing in prominent materials databases. We create a physics‑informed neural network (PINN) that incorporates the Born stability requirements directly into its loss function. This integration is a key learning constraint since it only allows the model to make predictions that do not violate fundamental physics. The model shows consistent and improved performance, having been trained on a dataset of 2112 inorganic materials with validated phonon spectra, and getting an F1‑score of 0.83 for both stable and unstable classes. The model shows an AUC‑ROC of 0.82 on a benchmark dataset of 1296 materials. Our PINN surpasses the best models in comparative tests, especially when it comes to accurately identifying unstable materials, which is crucial for a stability filter. This work offers a comprehensive screening tool for identifying materials and a methodology for incorporating domain knowledge to enhance predictive accuracy in materials informatics.

Abstract:
Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network‑based solvers, such as Physics‑Informed Neural Networks (PINNs) and Deep Galerkin Methods, often face challenges in generalization and long‑time prediction efficiency due to their dependence on full space‑time approximations. To address these issues, we propose a novel and scalable framework that significantly enhances the Neural Galerkin Method (NGM) by incorporating the Meta‑Auto‑Decoder (MAD) paradigm. Our approach leverages space‑time decoupling to enable more stable and efficient time integration, while meta‑learning‑driven adaptation allows rapid generalization to unseen parameter configurations with minimal retraining. Furthermore, randomized sparse updates effectively reduce computational costs without compromising accuracy. Together, these advancements enable our method to achieve physically consistent, long‑horizon predictions for complex parameterized evolution equations with significantly lower computational overhead. Numerical experiments on benchmark problems demonstrate that our methods performs comparatively well in terms of accuracy, robustness, and adaptability.

Abstract:
Physics‑Informed Neural Networks (PINNs) have demonstrated considerable success in solving complex fluid dynamics problems. However, their performance often deteriorates in regimes characterized by steep gradients, intricate boundary conditions, and stringent physical constraints, leading to convergence failures and numerical instabilities. To overcome these limitations, we propose a hybrid framework that integrates Long Short‑Term Memory (LSTM) networks into the PINN architecture, enhancing its ability to capture spatial correlations in the steady‑state velocity field of a two‑dimensional charged fluid under an external electric field. Our results demonstrate that the LSTM‑enhanced PINN model significantly outperforms conventional Multilayer Perceptron (MLP)‑based PINNs in terms of convergence rate, numerical stability, and predictive accuracy. This innovative approach offers improved computational efficiency and reliability for modeling electrohydrodynamic flows, providing new insights and strategies for applications in microfluidics and nanofluidics.

Abstract:
The Reference Signal Received Power (RSRP) is a crucial factor that determines communication performance in mobile networks. Accurately predicting the RSRP can help network operators perceive user experiences and maximize throughput by optimizing wireless resources. However, existing research into RSRP prediction has limitations in accuracy and verisimilitude. Theoretical derivations and existing data‑driven methods consider only easily quantifiable Large‑Scale (LS) information, and struggle to effectively capture the intertwined LS and Small‑Scale (SS) signal attenuation characteristics of the wireless channel. Moreover, the lack of prior physical knowledge leads to weak accuracy, interpretability, and transferability. In this paper, we propose a novel RSRP prediction framework, Channel‑Diff. This framework physically models LS and SS attenuation using multimodal conditions and employs physics‑informed conditional diffusion models as the prediction network. Channel‑Diff extracts prior physical information that characterises the signal propagation process from network parameters and multi‑attribute maps of the urban spatial environment. It provides LS physical priors through large‑scale propagation modelling and shadow‑occlusion modelling, and SS physical priors through multipath propagation modelling and urban microenvironment feature extraction. We design a physical‑prior‑guided two‑stage training scheme with a noise prior guidance mechanism, enabling effective fusion of multi‑scale physical knowledge with the diffusion models. Evaluations demonstrate Channel‑Diff exhibits excellent performance on RSRP prediction, achieving at least 25.15%‑37.19% improvement in accuracy relative to baseline methods. Additionally, the model also demonstrated outstanding performance in terms of transferability and training efficiency.

Abstract:
This thesis presents a physics‑informed machine learning framework for solving the Floquet‑Bloch eigenvalue problem associated with particles in two‑dimensional periodic potentials, with a focus on honeycomb lattice geometry, due to its distinctive band topology featuring Dirac points and its relevance to materials such as graphene. By leveraging neural networks to learn complex Bloch functions and their associated eigenvalues (energies) simultaneously, we develop a mesh‑free solver enforcing the governing Schrödinger equation, Bloch periodicity, and normalization constraints through a composite loss function without supervision. The model is trained over the Brillouin zone to recover band structures and Bloch modes, with numerical validation against traditional plane‑wave expansion methods. We further explore transfer learning techniques to adapt the solver from nearly‑free electron potentials to strongly varying potentials, demonstrating its ability to capture changes in band structure topology. This work contributes to the growing field of physics‑informed machine learning for quantum eigenproblems, providing insights into the interplay between symmetry, band structure, and neural architectures.

Abstract:
Regulating grasping force to reduce slippage during dynamic object interaction remains a fundamental challenge in robotic manipulation, especially when objects are manipulated by multiple rolling contacts, have unknown properties (such as mass or surface conditions), and when external sensing is unreliable. In contrast, humans can quickly regulate grasping force by touch, even without visual cues. Inspired by this ability, we aim to enable robotic hands to rapidly explore objects and learn tactile‑driven grasping force control under motion and limited sensing. We propose a physics‑informed energy abstraction that models the object as a virtual energy container. The inconsistency between the fingers' applied power and the object's retained energy provides a physically grounded signal for inferring slip‑aware stability. Building on this abstraction, we employ model‑based learning and planning to efficiently model energy dynamics from tactile sensing and perform real‑time grasping force optimization. Experiments in both simulation and hardware demonstrate that our method can learn grasping force control from scratch within minutes, effectively reduce slippage, and extend grasp duration across diverse motion‑object pairs, all without relying on external sensing or prior object knowledge.

Abstract:
As wildfires increasingly evolve into urban conflagrations, traditional risk models that treat structures as isolated assets fail to capture the non‑linear contagion dynamics characteristic of the wildland urban interface (WUI). This research bridges the gap between mechanistic physics and data driven learning by establishing a novel dual specialist ensemble framework that disentangles vulnerability into two distinct vectors, environmental contagion and structural fragility. The architecture integrates two specialized predictive streams, an environmental specialist, implemented as a graph neural network (GNN) that operationalizes the community as a directed contagion graph weighted by physics informed convection, radiation, and ember probabilities, and enriched with high dimensional Google AlphaEarth Foundation embeddings, and a Structural Specialist, implemented via XGBoost to isolate granular asset level resilience. Applied to the 2025 Eaton Fire, the framework reveals a critical dichotomy in risk drivers. The GNN demonstrates that neighborhood scale environmental pressure overwhelmingly dominates intrinsic structural features in defining propagation pathways, while the XGBoost model identifies eaves as the primary micro scale ingress vector. By synthesizing these divergent signals through logistic stacking, the ensemble achieves robust classification and generates a diagnostic risk topology. This capability empowers decision makers to move beyond binary loss prediction and precisely target mitigation prioritizing vegetation management for high connectivity clusters and structural hardening for architecturally vulnerable nodes thereby operationalizing a proactive, data driven approach to community resilience.

Abstract:
Solar active regions (ARs) do not appear randomly but cluster along longitudinally warped toroidal bands ('toroids') that encode information about magnetic structures in the tachocline, where global‑scale organization likely originates. Global MagnetoHydroDynamic Shallow‑Water Tachocline (MHD‑SWT) models have shown potential to simulate such toroids, matching observations qualitatively. For week‑scale early prediction of flare‑producing AR emergence, forward‑integration of these toroids is necessary. This requires model initialization with a dynamically self‑consistent MHD state‑vector that includes magnetic, flow fields, and shell‑thickness variations. However, synoptic magnetograms provide only geometric shape of toroids, not the state‑vector needed to initialize MHD‑SWT models. To address this challenging task, we develop PINNBARDS, a novel Physics‑Informed Neural Network (PINN)‑Based AR Distribution Simulator, that uses observational toroids and MHD‑SWT equations to derive initial state‑vector. Using Feb‑14‑2024 SDO/HMI synoptic map, we show that PINN converges to physically consistent, predominantly antisymmetric toroids, matching observed ones. Although surface data provides north and south toroids' central latitudes, and their latitudinal widths, they cannot determine tachocline field strengths, connected to AR emergence. We explore here solutions across a broad parameter range, finding hydrodynamically‑dominated structures for weak fields (~2 kG) and overly rigid behavior for strong fields (~100 kG). We obtain best agreement with observations for 20‑30 kG toroidal fields, and ~10 degree bandwidth, consistent with low‑order longitudinal mode excitation. To our knowledge, PINNBARDS serves as the first method for reconstructing state‑vectors for hidden tachocline magnetic structures from surface patterns; potentially leading to weeks ahead prediction of flare‑producing AR‑emergence.

Abstract:
Numerical simulation of time‑dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high‑fidelity solvers are often prohibitively expensive for long‑horizon or time‑critical settings. Neural operator (NO) surrogates offer fast inference across parametric and functional inputs; however, most autoregressive NO frameworks remain vulnerable to compounding errors, and ensemble‑averaged metrics provide limited guarantees for individual inference trajectories. In practice, error accumulation can become unacceptable beyond the training horizon, and existing methods lack mechanisms for online monitoring or correction. To address this gap, we propose ANCHOR (Adaptive Numerical Correction for High‑fidelity Operator Rollouts), an online, instance‑aware hybrid inference framework for stable long‑horizon prediction of nonlinear, time‑dependent PDEs. ANCHOR treats a pretrained NO as the primary inference engine and adaptively couples it with a classical numerical solver using a physics‑informed, residual‑based error estimator. Inspired by adaptive time‑stepping in numerical analysis, ANCHOR monitors an exponential moving average (EMA) of the normalized PDE residual to detect accumulating error and trigger corrective solver interventions without requiring access to ground‑truth solutions. We show that the EMA‑based estimator correlates strongly with the true relative L2 error, enabling data‑free, instance‑aware error control during inference. Evaluations on four canonical PDEs: 1D and 2D Burgers', 2D Allen‑Cahn, and 3D heat conduction, demonstrate that ANCHOR reliably bounds long‑horizon error growth, stabilizes extrapolative rollouts, and significantly improves robustness over standalone neural operators, while remaining substantially more efficient than high‑fidelity numerical solvers.

Abstract:
Coronal mass ejections (CMEs) are key drivers of space weather events, posing risks to both space‑borne and ground‑based systems. Accurate prediction of their arrival time at Earth is critical for impact mitigation. To this end, physics‑informed artificial intelligence (AI) approaches have proven more effective than purely data‑driven or physics‑based methods, generally offering higher accuracy and better explainability than the former and lower computational cost than the latter. In this work, we propose a generalization of the physics‑driven AI framework based on the classical drag‑based model (DBM) by integrating the extended version of the drag‑based model (EDBM). This enhancement allows us to include in the training process CME events whose interplanetary dynamics are incompatible with those assumed by the DBM. We achieve travel‑time prediction accuracy comparable to state‑of‑the‑art methods. We also perform a parametric robustness analysis, highlighting the stability of our approach under small variations in the drag coefficient. Furthermore, we propose a categorization of CMEs into speed regimes defined by the EDBM using a multiclass classification model based on logistic regression, which could be implemented in near‑real‑time operational space weather forecasting systems. The results show that the EDBM framework broadens the applicability of forecasting models while preserving good predictive accuracy.

Abstract:
We investigate the disordered spin‑\frac12Heisenberg model in two dimensions and employ tree tensor networks (TTNs) with a physics‑informed structural optimization of the tree layout, to simulate dynamics in the many‑body localization problem. We find that TTNs are able to capture two‑dimensional entanglement patterns more effectively than matrix product states (MPS) while being more efficient to contract than projected entangled pair states (PEPS) to probe larger systems and longer times. Structural optimization of the trees based on time evolution of the entanglement in the system allows to keep the necessary bond dimensions low and to maximally exploit the increased expressiveness of TTNs over MPS. In this way, we achieve more accurate results in all considered parameter regimes both below and above the ergodicity‑to‑localization crossover at a comparable compute‑time cost.

Abstract:
Axial piston pumps are crucial components in fluid power systems, where reliable fault diagnosis is essential for ensuring operational safety and efficiency. Traditional data‑driven methods require extensive labeled fault data, which is often impractical to obtain, while model‑based approaches suffer from parameter uncertainties. This paper proposes a digital twin (DT)‑driven zero‑shot fault diagnosis framework utilizing fluid‑borne noise (FBN) signals. The framework calibrates a high‑fidelity DT model using only healthy‑state data, generates synthetic fault signals for training deep learning classifiers, and employs a physics‑informed neural network (PINN) as a virtual sensor for flow ripple estimation. Gradient‑weighted class activation mapping (Grad‑CAM) is integrated to visualize the decision‑making process of neural networks, revealing that large kernels matching the subsequence length in time‑domain inputs and small kernels in time‑frequency domain inputs enable higher diagnostic accuracy by focusing on physically meaningful features. Experimental validations demonstrate that training on signals from the calibrated DT model yields diagnostic accuracies exceeding 95% on real‑world benchmarks, while uncalibrated models result in significantly lower performance, highlighting the framework's effectiveness in data‑scarce scenarios.

Abstract:
Solving high‑dimensional Fokker‑Planck (FP) equations is a challenge in computational physics and stochastic dynamics, due to the curse of dimensionality (CoD) and unbounded domains. Existing deep learning approaches, such as Physics‑Informed Neural Networks, face computational challenges as dimensionality increases, driven by the O(d^2) complexity of automatic differentiation for second‑order derivatives. While recent probability flow approaches bypass this by learning score functions or matching velocity fields, they often involve serial operations or depend on sampling efficiency in complex distributions. To address these issues, we propose the Adaptive Probability Flow Residual Minimization (A‑PFRM) method. The second‑order FP equation is reformulated as an equivalent first‑order deterministic Probability Flow ODE (PF‑ODE) constraint, which avoids explicit Hessian computation. Unlike score matching or velocity matching, A‑PFRM solves FP equations by minimizing the residual of the continuity equation induced by the PF‑ODE. By utilizing Continuous Normalizing Flows combined with the Hutchinson Trace Estimator, the training complexity is reduced to a linear scale of O(d), achieving an efficient O(1) wall‑clock time on GPUs. To address data sparsity in high dimensions, a generative adaptive sampling strategy is employed, and we further prove that dynamically aligning collocation points with the evolving probability mass is a necessary condition to bound the approximation error. Experiments on diverse benchmarks ‑‑ ranging from anisotropic Ornstein‑Uhlenbeck (OU) processes and high‑dimensional Brownian motions with time‑varying diffusion terms, to Geometric OU processes featuring non‑Gaussian solutions ‑‑ demonstrate that A‑PFRM effectively mitigates the CoD, maintaining high accuracy and constant temporal cost for problems up to 100 dimensions.

Abstract:
Radiotherapy treatment planning remains a time‑intensive iterative process requiring expert intervention in commercial treatment planning system (TPS). While machine learning approaches have demonstrated promise, most remain depedent on TPS‑based dose calculation or surrogate dose models, preventing direct optimization of deliverable treatment plan parameters. We propose PyDoseRT (PDRT), a physics‑informed, GPU‑accelerated dose engine implemented in PyTorch that computes dose distributions directly from treatment delivery parameters (i.e., MLC leaf positions, jaw positions, gantry angles, and monitor units). The engine preserves gradient information throughout the dose computation pipeline, enabling gradient‑based optimization of hardware‑constrained treatment plans without the reliance on a commercial TPS. PDRT was evaluated on 19 and 162 clinical VMAT prostate cancer plans from two hospitals (with different treatment machines). When recalculating clinical plans, PDRT achieved high 3D gamma pass rates (mean 96.8% for 2%/2 mm and 98.9% for 3%/3 mm, depending on cohort). All optimized plans converged to clinically acceptable solutions and passed deliverability verification when imported into a commercial TPS. This physics‑informed framework eliminates TPS dependency for radiotherapy optimization research by enabling gradient‑based planning while ensuring that delivery parameters remain in the machine‑feasible range. The gradient‑enabled dose engine allows exploration of novel optimization strategies and objective functions while maintaining clinical validity. The proposed approach provides a research platform for investigating real‑time adaptive radiotherapy concepts, automated planning workflows, and TPS‑independent optimization strategies, and democratizing radiotherapy research, by exposing gradient‑enabled, hardware‑aware, open‑source dose computation.

Abstract:
Aleatoric uncertainties ‑ irremovable variability in microstructure morphology, constituent behavior, and processing conditions ‑ pose a major challenge to developing uncertainty‑robust digital twins. We introduce the Variational Deep Material Network (VDMN), a physics‑informed surrogate model that enables efficient and probabilistic forward and inverse predictions of material behavior. The VDMN captures microstructure‑induced variability by embedding variational distributions within its hierarchical, mechanistic architecture. Using an analytic propagation scheme based on Taylor‑series expansion and automatic differentiation, the VDMN efficiently propagates uncertainty through the network during training and prediction. We demonstrate its capabilities in two digital‑twin‑driven applications: (1) as an uncertainty‑aware materials digital twin, it predicts and experimentally validates the nonlinear mechanical variability in additively manufactured polymer composites; and (2) as an inverse calibration engine, it disentangles and quantitatively identifies overlapping sources of uncertainty in constituent properties. Together, these results establish the VDMN as a foundation for uncertainty‑robust materials digital twins.

Abstract:
We propose physics‑informed neural networks (PINNs) as a numerical solver for relativistic spin hydrodynamics and demonstrate that the total angular momentum, i.e., the sum of orbital and spin angular momentum, is accurately conserved throughout the fluid evolution by imposing the conservation law directly in the loss function as a training target. This enables controlled numerical studies of the mutual conversion between spin and orbital angular momentum, a central feature of relativistic spin hydrodynamics driven by the rotational viscous effect. We present two physical scenarios with a rotating fluid confined in a cylindrical container: one case in which initial orbital angular momentum is converted into spin angular momentum in analogy with the Barnett effect, and the opposite case in which initial spin angular momentum is converted into orbital angular momentum in analogy with the Einstein‑de Haas effect. We investigate these conversion processes governed by the rotational viscous effect by analyzing the spacetime profiles of thermal vorticity and spin potential. Our PINNs‑based framework provides the first numerical evidence for spin‑orbit angular momentum conversion with fully nonlinear computational relativistic spin hydrodynamics.

Abstract:
Raman spectroscopy enables non‑destructive, label‑free molecular analysis with high specificity, making it a powerful tool for biomedical diagnostics. However, its application to biological tissues is challenged by inherently weak Raman scattering and strong fluorescence background, which significantly degrade signal quality. In this study, we present a simulation‑driven denoising framework that combines a statistically grounded noise model with deep learning to enhance Raman spectra acquired under fluorescence‑dominated conditions. We comprehensively modeled major noise sources. Based on this model, we generated biologically realistic Raman spectra and used them to train a cascaded deep neural network designed to jointly suppress stochastic detector noise and fluorescence baseline interference. To evaluate the performance of our approach, we simulated human skin spectra derived from real experimental data as a validation case study. Our results demonstrate the potential of physics‑informed learning to improve spectral quality and enable faster, more accurate Raman‑based tissue analysis.

Abstract:
High‑resolution (HR) quantitative MRI (qMRI) relaxometry provides objective tissue characterization but remains clinically underutilized due to lengthy acquisition times. We propose a physics‑informed, self‑supervised framework for qMRI super‑resolution that uses routinely acquired HR weighted MRI (wMRI) scans as guidance, thus, removing the necessity for HR qMRI ground truth during training. We formulate super‑resolution as Bayesian maximum a posteriori inference, minimizing two discrepancies: (1) between HR images synthesized from super‑resolved qMRI maps and acquired wMRI guides via forward signal models, and (2) between acquired LR qMRI and downsampled predictions. This physics‑informed objective allows the models to learn from clinical wMRI without HR qMRI supervision. To validate the concept, we generate training data by synthesizing wMRI guides from HR qMRI using signal equations, then degrading qMRI resolution via k‑space truncation. A deep neural network learns the super‑resolution mapping. Ablation experiments demonstrate that T1‑weighted images primarily enhance T1 maps, T2‑weighted images improve T2 maps, and combined guidance optimally enhances all parameters simultaneously. Validation on independently acquired in‑vivo data from a different qMRI sequence confirms cross‑qMRI sequence generalizability. Models trained on synthetic data can produce super‑resolved maps from a 1‑minute acquisition with quality comparable to a 5‑minute reference scan, leveraging the scanner‑independent nature of relaxometry parameters. By decoupling training from HR qMRI requirement, our framework enables fast qMRI acquisitions enhanced via routine clinical images, offering a practical pathway for integrating quantitative relaxometry into clinical workflows with acceptable additional scan time.

Abstract:
Physics‑informed neural networks (PINNs) have recently emerged as a prominent paradigm for solving partial differential equations (PDEs), yet their training strategies remain underexplored. While hard prioritization methods inspired by finite element methods are widely adopted, recent research suggests that easy prioritization can also be effective. Nevertheless, we find that both approaches exhibit notable trade‑offs and inconsistent performance across PDE types. To address this issue, we develop a hybrid strategy that combines the strengths of hard and easy prioritization through an alternating training algorithm. On PDEs with steep gradients, nonlinearity, and high dimensionality, the proposed method achieves consistently high accuracy, with relative L2 errors mostly in the range of O(10^‑5) to O(10^‑6), significantly surpassing baseline methods. Moreover, it offers greater reliability across diverse problems, whereas compared approaches often suffer from variable accuracy depending on the PDE. This work provides new insights into designing hybrid training strategies to enhance the performance and robustness of PINNs.

Abstract:
We introduce BumpNet, a sparse multilayer perceptron (MLP) framework for PDE numerical solution and operator learning. BumpNet is based on basis function expansion, which makes them superficially similar to radial‑basis function (RBF) networks. However, the basis functions in BumpNet are constructed from ordinary sigmoid activation functions in a sparse multi‑layer framework. This makes BumpNet a MLP, not a RBF neural network, enabling the efficient use of modern training techniques optimized for MLPs. All parameters of the basis functions, including shape, location, and amplitude, are fully trainable. Model parsimony is encouraged through a basis function pruning scheme. BumpNet is a general meshless framework that can be combined with existing neural architectures for learning PDE solutions: here, we propose Bump‑PINNs (BumpNet with physics‑informed neural networks) for solving general PDEs; Bump‑EDNN (BumpNet with evolutionary deep neural networks) to solve time‑evolution PDEs; and Bump‑DeepONet (BumpNet with deep operator networks) for PDE operator learning. We prove that BumpNets and Bump‑DeepONets are universal approximators of continuous functions and continuous operators, respectively. Bump‑PINNs are trained using the same collocation‑based approach used by PINNs; Bump‑EDNN uses a BumpNet only in the spatial domain and uses EDNNs to advance the solution in time; while Bump‑DeepONets employ a BumpNet regression network as the trunk network of a DeepONet. Extensive numerical experiments demonstrate the efficiency and accuracy of BumpNets.

Abstract:
Fine‑grained fire prediction plays a crucial role in emergency response. Infrared images and fire masks provide complementary thermal and boundary information, yet current methods are predominantly limited to binary mask modeling with inherent signal sparsity, failing to capture the complex dynamics of fire. While world models show promise in video generation, their physical inconsistencies pose significant challenges for fire forecasting. This paper introduces PhysFire‑WM, a Physics‑informed World Model for emulating Fire spread dynamics. Our approach internalizes combustion dynamics by encoding structured priors from a Physical Simulator to rectify physical discrepancies, coupled with a Cross‑task Collaborative Training strategy (CC‑Train) that alleviates the issue of limited information in mask‑based modeling. Through parameter sharing and gradient coordination, CC‑Train effectively integrates thermal radiation dynamics and spatial boundary delineation, enhancing both physical realism and geometric accuracy. Extensive experiments on a fine‑grained multimodal fire dataset demonstrate the superior accuracy of PhysFire‑WM in fire spread prediction. Validation underscores the importance of physical priors and cross‑task collaboration, providing new insights for applying physics‑informed world models to disaster prediction.

Abstract:
This paper presents a Multi‑Robot Multi‑Source Term Estimation (MRMSTE) framework that enables teams of mobile robots to collaboratively sample gas concentrations and infer the parameters of an unknown number of airborne releases. The framework is built on a hybrid Bayesian inference scheme that represents the joint multi‑source probability density and incorporates physics‑informed state transitions, including source birth, removal, and merging induced by atmospheric dispersion. A superposition‑based measurement model is naturally accommodated, allowing sparse concentration measurements to be exploited efficiently. To guide robot deployment, we introduce a wind‑aware coverage control (WCC) strategy that integrates the evolving multi‑source belief with local wind information to prioritize regions of high detection likelihood. Unlike conventional coverage control or information‑theoretic planners, WCC explicitly accounts for anisotropic plume transport when modelling sensor performance, leading to more effective sensor placement for multi‑source estimation. Monte Carlo studies demonstrate faster convergence and improved separation of individual source beliefs compared to traditional coverage‑based strategies and small‑scale static sensor networks. Real‑world experiments with CO2 releases using TurtleBot platforms further validate the proposed approach, demonstrating its practicality for scalable multi‑robot gas‑sensing applications.

Abstract:
Short‑term prediction (nowcasting) of low‑visibility and precipitation events is critical for aviation safety and operational efficiency. Current operational approaches rely on computationally intensive numerical weather prediction guidance and human‑issued TAF products, which often exhibit conservative biases and limited temporal resolution. This study presents a lightweight gradient boosting framework (XGBoost) trained exclusively on surface observation data (METAR) and enhanced through physics‑guided feature engineering based on thermodynamic principles. The framework is evaluated across 11 international airports representing distinct climatic regimes (including SCEL, KJFK, KORD, KDEN, SBGR, and VIDP) using historical data from 2000 to 2024. Results suggest that the model successfully captures underlying local physical processes without manual configuration. In a blind comparative evaluation against operational TAF forecasts, the automated model achieved substantially higher detection rates at tactical horizons (3 hours), with a 2.5 to 4.0 times improvement in recall while reducing false alarms. Furthermore, SHAP analysis reveals that the model performs an implicit reconstruction of local physical drivers (advection, radiation, and subsidence), providing actionable explainability for operational situational awareness. Keywords: aviation meteorology; physics‑guided machine learning; explainable artificial intelligence; lightweight machine learning; nowcasting; METAR; TAF verification; edge computing

Abstract:
In this work, we propose a novel physics informed neural network based algorithm for real time plasma boundary reconstruction in tokamak devices. The approach is based on a single Extreme Learning Machine network used to solve the homogeneous Grad Shafranov equation, which is required to identify the plasma boundary. This architecture enables the real time training of the network parameters using the available magnetic sensor data and, consequently, dynamically adapting the network output to the evolving plasma equilibrium. We demonstrate that, the network performs accurate plasma boundary reconstruction for complex configurations, outperforming well established methods, such as the algorithm used for decades at the Joint European Torus, the world's largest tokamak, until it ceased operation in 2023. Indeed, compared to the latter, the proposed solution better generalizes the poloidal flux function, without requiring algorithm retuning across different plasma equilibria. The proposed neural network reconstructor demonstrates also greater robustness with respect to noise on the magnetic measurements. Moreover, this method takes advantage of the generalization power of neural networks but without the need for extensive, time consuming training based on a huge amount of experimental data, making its implementation on existing devices straightforward.

Abstract:
Determining the absolute configuration of gas‑phase molecules in position‑space has long been a fundamental challenge in molecular physics. While strong‑field‑induced Coulomb explosion imaging (CEI) has emerged as a powerful tool for probing molecular stereochemistry in momentum‑space, reconstructing the original three‑dimensional structure of polyatomic molecules remains a long‑standing challenge due to the inherent complexity of multidimensional inversion. Here, we introduce a deep learning framework that bridges this gap by directly recovering position‑space molecular structures from Coulomb explosion momentum patterns. Our approach combines CEI simulations with a neural network trained to establish the mapping between momentum‑space Newton plots and real‑space geometries. The trained model demonstrates high fidelity in reconstructing the structure of CHF_3 from experimental CEI data. This generalizable framework can not only be extended to other molecular systems but also opens avenues for time‑resolved structural analysis of molecular dynamics.

Abstract:
Solving Partial Differential Equations (PDEs) is a cornerstone of engineering and scientific research. Traditional methods for PDE solving are cumbersome, relying on manual setup and domain expertise. While Physics‑Informed Neural Network (PINNs) introduced end‑to‑end neural network‑based solutions, and frameworks like DeepXDE further enhanced automation, these approaches still depend on expert knowledge and lack full autonomy. In this work, we frame PDE solving as tool invocation via LLM‑driven agents and introduce PDE‑Agent, the first toolchain‑augmented multi‑agent collaboration framework, inheriting the reasoning capacity of LLMs and the controllability of external tools and enabling automated PDE solving from natural language descriptions. PDE‑Agent leverages the strengths of multi‑agent and multi‑tool collaboration through two key innovations: (1) A Prog‑Act framework with graph memory for multi‑agent collaboration, which enables effective dynamic planning and error correction via dual‑loop mechanisms (localized fixes and global revisions). (2) A Resource‑Pool integrated with a tool‑parameter separation mechanism for multi‑tool collaboration. This centralizes the management of runtime artifacts and resolves inter‑tool dependency gaps in existing frameworks. To validate and evaluate this new paradigm for PDE solving , we develop PDE‑Bench, a multi‑type PDE Benchmark for agent‑based tool collaborative solving, and propose multi‑level metrics for assessing tool coordination. Evaluations verify that PDE‑Agent exhibits superior applicability and performance in complex multi‑step, cross‑step dependent tasks. This new paradigm of toolchain‑augmented multi‑agent PDE solving will further advance future developments in automated scientific computing. Our source code and dataset will be made publicly available.

Abstract:
Understanding the magnetic field environment around Mars and its response to upstream solar wind conditions provide key insights into the processes driving atmospheric ion escape. To date, global models of Martian induced magnetosphere have been exclusively physics‑based, relying on computationally intensive simulations. For the first time, we develop a data‑driven model of the Martian induced magnetospheric magnetic field using Physics‑Informed Neural Network (PINN) combined with MAVEN observations and physical laws. Trained under varying solar wind conditions, including B_IMF, P_SW, and θ_cone, the data‑driven model accurately reconstructs the three‑dimensional magnetic field configuration and its variability in response to upstream solar wind drivers. Based on the PINN results, we identify key dependencies of magnetic field configuration on solar wind parameters, including the hemispheric asymmetries of the draped field line strength in the Mars‑Solar‑Electric coordinates. These findings demonstrate the capability of PINNs to reconstruct complex magnetic field structures in the Martian induced magnetosphere, thereby offering a promising tool for advancing studies of solar wind‑Mars interactions.

Abstract:
Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high‑dimensional or complex problems. Physics‑Informed Neural Networks (PINNs) have emerged as an efficient alternative by embedding physics‑based constraints into deep learning frameworks, but they face challenges in achieving high accuracy and handling complex boundary conditions. In this work, we extend the Time‑Evolving Natural Gradient (TENG) framework to address Dirichlet boundary conditions, integrating natural gradient optimization with numerical time‑stepping schemes, including Euler and Heun methods, to ensure both stability and accuracy. By incorporating boundary condition penalty terms into the loss function, the proposed approach enables precise enforcement of Dirichlet constraints. Experiments on the heat equation demonstrate the superior accuracy of the Heun method due to its second‑order corrections and the computational efficiency of the Euler method for simpler scenarios. This work establishes a foundation for extending the framework to Neumann and mixed boundary conditions, as well as broader classes of PDEs, advancing the applicability of neural network‑based solvers for real‑world problems.

Abstract:
Inverse design of heterogeneous material microstructures is a fundamentally ill‑posed and famously computationally expensive problem. This is exacerbated by the high‑dimensional design spaces associated with finely resolved images, multimodal input property streams, and a highly nonlinear forward physics. Whilst modern generative models excel at accurately modeling such complex forward behavior, most of them are not intrinsically structured to support fast, stable \emphdeterministic inversion with a physics‑informed bias. This work introduces Janus, a unified generative‑predictive framework to address this problem. Janus couples a deep encoder‑decoder architecture with a predictive KHRONOS head, a separable neural architecture. Topologically speaking, Janus learns a latent manifold simultaneously isometric for generative inversion and pruned for physical prediction; the joint objective inducing \emphdisentanglement of the latent space. Janus is first validated on the MNIST dataset, demonstrating high‑fidelity reconstruction, accurate classification and diverse generative inversion of all ten target classes. It is then applied to the inverse design of heterogeneous microstructures labeled with thermal conductivity. It achieves a forward prediction accuracy R^2=0.98 (2% relative error) and sub‑5% pixelwise reconstruction error. Inverse solutions satisfy target properties to within 1% relative error. Inverting a sweep through properties reveal smooth traversal of the latent manifold, and UMAP visualization confirms the emergence of a low‑dimensional, disentangled manifold. By unifying prediction and generation within a single latent space, Janus enables real‑time, physics‑informed inverse microstructure generation at a lower computational cost typically associated with classical optimization‑based approaches.

Abstract:
In a mathematical model of interacting biological organisms, where external interventions may alter behavior over time, traditional models that assume fixed parameters usually do not capture the evolving dynamics. In oncology, this is further exacerbated by the fact that experimental data are often sparse and sometimes are composed of a few time points of tumor volume. In this paper, we propose to learn time‑varying interactions between cells, such as those of bladder cancer tumors and immune cells, and their response to a combination of anticancer treatments in a limited data scenario. We employ the physics‑informed neural network (PINN) approach to predict possible subpopulation trajectories at time points where no observed data are available. We demonstrate that our approach is consistent with the biological explanation of subpopulation trajectories. Our method provides a framework for learning evolving interactions among biological organisms when external interventions are applied to their environment.

Abstract:
Metasurfaces, typically realized as arrays of nanopillars, transform electromagnetic (EM) fields depending on their geometry and spatial arrangement. For solving the inverse problem of designing new metasurfaces that transform EM fields in a desirable manner, it is often necessary to explore large design spaces through full‑wave simulations that can be computationally demanding. In this work, we demonstrate that neural operators, which are artificial neural network architectures designed to learn operators between function spaces, can effectively approximate the differential operators underlying Maxwell's equations, enabling their use as fast and accurate 3D surrogate models that can predict 3D EM fields transformed by metasurfaces. To calibrate neural operators, we generate synthetic training data consisting of 3D metasurface geometries together with their associated 3D EM fields obtained by numerically solving Maxwell's equations. Using the generated synthetic data, we train physics‑informed neural operators to minimize physical inconsistencies of predicted EM fields by incorporating residuals that capture deviations from Maxwell's equations. We observe that a training dataset consisting of fewer than 5000 examples already suffices to achieve reasonable results. In particular, our experiments show that the resulting 3D surrogate model achieves high predictive performance across a wide range of metasurface geometries, including types of structures not encountered during training. Notably, it predicts diffraction efficiencies with relative errors of 3.9 % and provides a 67‑fold speedup compared to conventional 3D simulations. Overall, once trained, our 3D surrogate model can rapidly predict EM fields for previously unseen metasurface geometries, which can facilitate efficient gradient‑based design of nanostructured materials for EM wave control.

Abstract:
Operator learning has become a powerful tool for accelerating the solution of parameterized partial differential equations (PDEs), enabling rapid prediction of full spatiotemporal fields for new initial conditions or forcing functions. Existing architectures such as DeepONet and the Fourier Neural Operator (FNO) show strong empirical performance but often require large training datasets, lack explicit physical structure, and may suffer from instability in their trunk‑network features, where mode imbalance or collapse can hinder accurate operator approximation. Motivated by the stability and locality of classical partition‑of‑unity (PoU) methods, we investigate PoU‑based regularization techniques for operator learning and develop a revised formulation of the existing POU‑‑PI‑‑DeepONet framework. The resulting \emphPhysics‑\emphinformed \emphPartition \emphPenalty Deep Operator Network (PIP^2 Net) introduces a simplified and more principled partition penalty that improved the coordinated trunk outputs that leads to more expressiveness without sacrificing the flexibility of DeepONet. We evaluate PIP^2 Net on three nonlinear PDEs: the viscous Burgers equation, the Allen‑‑Cahn equation, and a diffusion‑‑reaction system. The results show that it consistently outperforms DeepONet, PI‑DeepONet, and POU‑DeepONet in prediction accuracy and robustness.

Abstract:
Since their advent nearly a decade ago, physics‑informed neural networks (PINNs) have been studied extensively as a novel technique for solving forward and inverse problems in physics and engineering. The neural network discretization of the solution field is naturally adaptive and avoids meshing the computational domain, which can both improve the accuracy of the numerical solution and streamline implementation. However, there have been limited studies of PINNs on complex three‑dimensional geometries, as the lack of mesh and the reliance on the strong form of the partial differential equation (PDE) make boundary condition (BC) enforcement challenging. Techniques to enforce BCs with PINNs have proliferated in the literature, but a comprehensive side‑by‑side comparison of these techniques and a study of their efficacy on geometrically complex three‑dimensional test problems are lacking. In this work, we i) systematically compare BC enforcement techniques for PINNs, ii) propose a general solution framework for arbitrary three‑dimensional geometries, and iii) verify the methodology on three‑dimensional, linear and nonlinear test problems with combinations of Dirichlet, Neumann, and Robin boundaries. Our approach is agnostic to the underlying PDE, the geometry of the computational domain, and the nature of the BCs, while requiring minimal hyperparameter tuning. This work represents a step in the direction of establishing PINNs as a mature numerical method, capable of competing head‑to‑head with incumbents such as the finite element method.

Abstract:
Purpose: Magnetic Resonance Spectroscopic Imaging (MRSI) maps endogenous brain metabolism while suppressing the overwhelming water signal. Water‑unsuppressed MRSI (wu‑MRSI) allows simultaneous imaging of water and metabolites, but large water sidebands cause challenges for metabolic fitting. We developed an end‑to‑end deep‑learning pipeline to overcome these challenges at ultra‑high field. Methods:Fast high‑resolution wu‑MRSI was acquired at 7T with non‑cartesian ECCENTRIC sampling and ultra‑short echo time. A water and lipid removal network (WALINET+) was developed to remove lipids, water signal, and sidebands. MRSI reconstruction was performed by DeepER and a physics‑informed network for metabolite fitting. Water signal was used for absolute metabolite quantification, quantitative susceptibility mapping (QSM), and myelin water fraction imaging (MWF). Results: WALINET+ provided the lowest NRMSE (< 2%) in simulations and in vivo the smallest bias (< 20%) and limits‑of‑agreement (+‑63%) between wu‑MRSI and ws‑MRSI scans. Several metabolites such as creatine and glutamate showed higher SNR in wu‑MRSI. QSM and MWF obtained from wu‑MRSI and GRE showed good agreement with 0 ppm/5.5% bias and +‑0.05 ppm/ +‑ 12.75% limits‑of‑agreement. Conclusion: High‑quality metabolic, QSM, and MWF mapping of the human brain can be obtained simultaneously by ECCENTRIC wu‑MRSI at 7T with 2 mm isotropic resolution in 12 min. WALINET+ robustly removes water sidebands while preserving metabolite signal, eliminating the need for water suppression and separate water acquisitions.

Abstract:
Machine learning (ML) has been increasingly applied in concrete research to optimize performance and mixture design. However, one major challenge in applying ML to cementitious materials is the limited size and diversity of available databases. A promising solution is the development of multi‑modal databases that integrate both numerical and graphical data. Conventional ML frameworks in cement research are typically restricted to a single data modality. Graph neural network (GNN) represents a new generation of neural architectures capable of learning from data structured as graphs, capturing relationships through irregular or topology‑dependent connections rather than fixed spatial coordinates. While GNN is inherently designed for graphical data, they can be adapted to extract correlations from numerical datasets and potentially embed physical laws directly into their architecture, enabling explainable and physics‑informed predictions. This work is among the first few studies to implement GNNs to design concrete, with a particular emphasis on establishing a clear and reproducible pathway for converting tabular data into graph representations using the k‑nearest neighbor (K‑NN) approach. Model hyperparameters and feature selection are systematically optimized to enhance prediction performance. The GNN shows performance comparable to the benchmark random forest, which has been demonstrated by many studies to yield reliable predictions for cementitious materials. Overall, this study provides a foundational roadmap for transitioning from traditional ML to advanced AI architectures. The proposed framework establishes a strong foundation for future multi‑modal and physics‑informed GNN models capable of capturing complex material behaviors and accelerating the design and optimization of cementitious materials.

Abstract:
We develop a self‑adaptive physics‑informed neural network (PINN) framework that reliably solves forward Darcy flow and performs accurate permeability inversion in heterogeneous porous media. In the forward setting, the PINN predicts velocity and pressure for discontinuous, piecewise‑constant permeability; in the inverse setting, it identifies spatially varying permeability directly from indirect flow observations. Both models use a region‑aware permeability parameterization with binary spatial masks, which preserves sharp permeability jumps and avoids the smoothing artifacts common in standard PINNs. To stabilize training, we introduce self‑learned loss weights that automatically balance PDE residuals, boundary constraints, and data mismatch, eliminating manual tuning and improving robustness, particularly for inverse problems. An interleaved AdamW‑L‑BFGS optimization strategy further accelerates and stabilizes convergence. Numerical results demonstrate accurate forward surrogates and reliable inverse permeability recovery, establishing the method as an effective mesh‑free solver and data‑driven inversion tool for porous‑media systems governed by partial differential equations.

Abstract:
Quantum state tomography (QST) faces exponential measurement requirements and noise sensitivity in multi‑qubit systems, bottlenecking practical quantum technologies. We present a physics‑informed neural network (PINN) framework integrating quantum mechanical constraints via adaptive weighting, a residual‑and‑attention‑enhanced architecture, and differentiable Cholesky parameterization for physical validity. Evaluations on 2‑‑5 qubit systems and arbitrary‑dimensional states show PINN consistently outperforms traditional neural networks (TNNs), achieving highest fidelity across all dimensions. PINN outperforms baselines, with marked improvements in moderately high‑dimensional systems, superior noise robustness (slower performance degradation), and consistent dimensional robustness. Theoretical analysis shows physical constraints reduce Rademacher complexity and mitigate the curse of dimensionality via constraint‑induced dimension and sample complexity reduction, effective regardless of qubit number. While experiments are limited to 5‑qubit systems due to computational constraints, our theoretical framework (convergence guarantees, generalization bounds, scalability theorems) justifies PINN's advantages will persist and strengthen in larger systems (6+ qubits), where constraint‑induced dimension reduction benefits grow with system size. Practically, this advances quantum error correction and gate calibration by reducing measurement requirements from O(4^n) to O(2^n) while maintaining high fidelity, enabling faster error correction cycles and accelerated calibration critical for scalable quantum computing.

Abstract:
Physics‑Informed Neural Networks (PINN) are emerging as a promising approach for quantitative parameter estimation of Magnetic Resonance Imaging (MRI). While existing deep learning methods can provide an accurate quantitative estimation of the T2 parameter, they still require large amounts of training data and lack theoretical support and a recognized gold standard. Thus, given the absence of PINN‑based approaches for T2 estimation, we propose embedding the fundamental physics of MRI, the Bloch equation, in the loss of PINN, which is solely based on target scan data and does not require a pre‑defined training database. Furthermore, by deriving rigorous upper bounds for both the T2 estimation error and the generalization error of the Bloch equation solution, we establish a theoretical foundation for evaluating the PINN's quantitative accuracy. Even without access to the ground truth or a gold standard, this theory enables us to estimate the error with respect to the real quantitative parameter T2. The accuracy of T2 mapping and the validity of the theoretical analysis are demonstrated on a numerical cardiac model and a water phantom, where our method exhibits excellent quantitative precision in the myocardial T2 range. Clinical applicability is confirmed in 94 acute myocardial infarction (AMI) patients, achieving low‑error quantitative T2 estimation under the theoretical error bound, highlighting the robustness and potential of PINN.

Abstract:
The Stefan problem is a classical free‑boundary problem that models phase‑change processes and poses computational challenges due to its moving interface and nonlinear temperature‑phase coupling. In this work, we develop a physics‑informed neural network framework for solving two‑phase Stefan problems. The proposed method explicitly tracks the interface motion and enforces the discontinuity in the temperature gradient across the interface while maintaining global consistency of the temperature field. Our approach employs two neural networks: one representing the moving interface and the other for the temperature field. The interface network allows rapid categorization of thermal diffusivity in the spatial domain, which is a crucial step for selecting training points for the temperature network. The temperature network's input is augmented with a modified zero‑level set function to accurately capture the jump in its normal derivative across the interface. Numerical experiments on two‑phase dynamical Stefan problems demonstrate the superior accuracy and effectiveness of our proposed method compared with the ones obtained by other neural network methodology in literature. The results indicate that the proposed framework offers a robust and flexible alternative to traditional numerical methods for solving phase‑change problems governed by moving boundaries. In addition, the proposed method can capture an unstable interface evolution associated with the Mullins‑Sekerka instability.

Abstract:
The interaction structure of a complex dynamical system governs its collective behavior, yet existing reconstruction methods struggle with nonlinear, heterogeneous, and higher‑order couplings, especially when only steady states are observable. We propose a Variational Physics‑Informed Ansatz (VPIA) that infers general interaction operators directly from heterogeneous steady‑state data. VPIA embeds the steady‑state constraints of the dynamics into a differentiable variational representation and reconstructs the underlying couplings by minimizing a physics‑derived steady‑state residual, without requiring temporal trajectories, derivative estimation, or supervision. Residual sampling combined with natural‑gradient optimization enables scalable learning of large and higher‑order networks. Across diverse nonlinear systems, VPIA accurately recovers directed, weighted, and multi‑body structures under substantial noise, providing a unified and robust framework for physics‑constrained inference of complex interaction networks in settings where only snapshot observations are available.

Abstract:
This work introduces Knowledge‑Distilled Physics‑Informed Neural Networks (KD‑PINN), a framework that transfers the predictive accuracy of a high‑capacity teacher model to a compact student through a continuous adaptation of the Kullback‑Leibler divergence. In order to confirm its generality for various dynamics and dimensionalities, the framework is evaluated on a representative set of partial differential equations (PDEs). Across the considered benchmarks, the student model achieves inference speedups ranging from x4.8 (Navier‑Stokes) to x6.9 (Burgers), while preserving accuracy. Accuracy is improved by on the order of 1% when the model is properly tuned. The distillation process also revealed a regularizing effect. With an average inference latency of 5.3 ms on CPU, the distilled models enter the ultra‑low‑latency real‑time regime defined by sub‑10 ms performance. Finally, this study examines how knowledge distillation reduces inference latency in PINNs, to contribute to the development of accurate ultra‑low‑latency neural PDE solvers.

Abstract:
We revisit the problem of physics‑informed regression, and propose a method that directly computes the state at the prediction point, simultaneously with the derivative and curvature information of the existing samples. We frame each prediction as a constrained optimisation problem, leveraging multivariate Taylor series expansions and explicitly enforcing physical laws. Each individual query can be processed with low computational cost without any pre‑ or re‑training, in contrast to global function approximator‑based solutions such as neural networks. Our comparative benchmarks on a reaction‑diffusion system show competitive predictive accuracy relative to a neural network‑based solution, while completely eliminating the need for long training loops, and remaining robust to changes in the sampling layout.

Abstract:
Understanding the dynamical response of quantum materials is central to revealing their microscopic properties, yet access to long‑time and large‑scale dynamics remains severely limited by rapidly growing computational costs and entanglement, particularly in gapless systems. Here we introduce an autoregressive machine‑learning framework that enables the extrapolation of dynamical spin correlations in both time and space beyond the reach of conventional numerical methods. Trained on time‑dependent density matrix renormalization group simulations of the gapless XXZ model, our approach is benchmarked against exact solutions available for this analytically solvable system. Combined with physics‑informed spatial extension, multi‑layer perceptron model using ReLU activation functions has been shown to be superior than convolutional neural networks and linear regressions for longer time extrapolation. Perturbation study of error accumulation further demonstrates that our autoregressive neural network extrapolations are highly robust to perturbations, suggesting stable and reliable predictions. This work establishes a new paradigm for studying the dynamics of gapless quantum many‑body systems, in which machine learning extends and complements the capabilities of state‑of‑the‑art numerical approaches.

Abstract:
Advancing artificial intelligence for physical sciences requires representations that are both interpretable and compatible with the underlying laws of nature. We introduce METASTRINGS, a symbolic language for photonics that expresses nanostructures as textual sequences encoding materials, geometries, and lattice configurations. Analogous to molecular textual representations in chemistry, METASTRINGS provides a framework connecting human interpretability with computational design by capturing the structural hierarchy of photonic metasurfaces. Building on this representation, we develop Meta‑GPT, a foundation transformer model trained on METASTRINGS and finetuned with physics‑informed supervised, reinforcement, and chain‑of‑thought learning. Across various design tasks, the model achieves <3% mean‑squared spectral error and maintains >98% syntactic validity, generating diverse metasurface prototypes whose experimentally measured optical responses match their target spectra. These results demonstrate that Meta‑GPT can learn the compositional rules of light‑matter interactions through METASTRINGS, laying a rigorous foundation for AI‑driven photonics and representing an important step toward a metasurface genome project.

Abstract:
We study optimal trade execution with a hard‑zero terminal inventory constraint, modeled via Hamilton‑Jacobi‑Bellman (HJB) equations. Vanilla PINNs often under‑enforce this constraint and produce unstable controls. We propose a Multi‑Trajectory PINN (MT‑PINN) that adds a rollout‑based trajectory loss and propagates a terminal penalty on terminal inventory via backpropagation‑through‑time, directly enforcing zero terminal inventory. A lightweight lambda‑curriculum is adopted to stabilize training as the state expands from a risk‑neutral reduced HJB to a risk‑averse HJB. On the Gatheral‑Schied single‑asset model, MT‑PINN aligns closely with their derived closed‑form solutions and concentrates terminal inventory tightly around zero while reducing errors along optimal paths. We apply MT‑PINNs on SPY intraday data, matching TWAP when risk‑neutral, and achieving lower exposure and competitive costs, especially in falling windows, for higher risk‑aversion.

Abstract:
Integrating domain knowledge into deep learning has emerged as a promising direction for improving model interpretability, generalization, and data efficiency. In this work, we present a novel knowledge‑guided ViT‑based Masked Autoencoder that embeds scientific domain knowledge within the self‑supervised reconstruction process. Instead of relying solely on data‑driven optimization, our proposed approach incorporates the Linear Spectral Mixing Model (LSMM) as a physical constraint and physically‑based Spectral Angle Mapper (SAM), ensuring that learned representations adhere to known structural relationships between observed signals and their latent components. The framework jointly optimizes LSMM and SAM loss with a conventional Huber loss objective, promoting both numerical accuracy and geometric consistency in the feature space. This knowledge‑guided design enhances reconstruction fidelity, stabilizes training under limited supervision, and yields interpretable latent representations grounded in physical principles. The experimental findings indicate that the proposed model substantially enhances reconstruction quality and improves downstream task performance, highlighting the promise of embedding physics‑informed inductive biases within transformer‑based self‑supervised learning.

Abstract:
Accurate estimation of the State of Charge (SOC) is critical for ensuring the safety, reliability, and performance optimization of lithium‑ion battery systems. Conventional data‑driven neural network models often struggle to fully characterize the inherent complex nonlinearities and memory‑dependent dynamics of electrochemical processes, significantly limiting their predictive accuracy and physical interpretability under dynamic operating conditions. To address this challenge, this study proposes a novel neural architecture termed the Fractional Differential Equation Physics‑Informed Neural Network (FDIFF‑PINN), which integrates fractional calculus with deep learning. The main contributions of this paper include: (1) Based on a fractional‑order equivalent circuit model, a discretized fractional‑order partial differential equation is constructed. (2) Comparative experiments were conducted using a dynamic charge/discharge dataset of Panasonic 18650PF batteries under multi‑temperature conditions (from ‑10^\circC to 20^\circC).

Abstract:
Inverse problems are extensively studied in applied mathematics, with applications ranging from acoustic tomography for medical diagnosis to geophysical exploration. Physics informed neural networks (PINNs) have emerged as a powerful tool for solving such problems, while Physics informed Kolmogorov Arnold networks (PIKANs) represent a recent benchmark that, in certain problems, promises greater interpretability and accuracy compared to PINNs, due to their nature, being constructed as a composition of polynomials. In this work, we develop a methodology for addressing inverse problems in infinite and semi infinite domains. We introduce a novel sampling strategy for the network's training points, using the negative exponential and normal distributions, alongside a dual network architecture that is trained to learn the solution and parameters of an equation with the same loss function. This design enables the solution of inverse problems without explicitly imposing boundary conditions, as long as the solutions tend to stabilize when leaving the domain of interest. The proposed architecture is implemented using both PINNs and PIKANs, and their performance is compared in terms of accuracy with respect to a known solution as well as computational time and response to a noisy environment. Our results demonstrate that, in this setting, PINNs provide a more accurate and computationally efficient solution, solving the inverse problem 1,000 times faster and in the same order of magnitude, yet with a lower relative error than PIKANs.

Abstract:
Model Predictive Control (MPC) is a powerful control strategy for power electronics, but it highly relies on manually‑derived and topology‑specific analytical models, which is labor‑intensive and time‑consuming in practical designs. To overcome this bottleneck, this paper introduces a Digital‑Twin‑based MPC (DT‑MPC) framework for generic power converters that can systematically translate a high‑level circuit into an objective‑aware control policy by leveraging a DT as a high‑fidelity system model. Furthermore, a physics‑informed neural surrogate predictor is proposed to accelerate predictions by DT and enable real‑time operation. A gradient‑free simplex search optimizer is also introduced to efficiently handle complex multi‑objective optimization. The efficacy of the framework has been validated through a cloud‑to‑edge deployment on a 1500 W dual active bridge converter. Experimental results show that the synthesized predictive model achieves an inference speed over 7 times faster than real time, the DT‑MPC controller outperforms several human‑designed counterparts, and the overall framework reduces engineering design time by over 95%, verifying the superiority of DT‑MPC on generalized power converters.

Abstract:
Physics‑Informed Neural Networks (PINNs) and more recently Physics‑Informed Kolmogorov‑Arnold Networks (PIKANs) have emerged as promising approaches for solving partial differential equations (PDEs) without reliance on extensive labeled data. In this work, we propose a novel multi‑resolution Hybrid Wavelet‑Fourier‑Enhanced Physics‑Informed Kolmogorov‑Arnold Network (HWF‑PIKAN) for solving advection problems based on collisionless Boltzmann equation (CBE) with both continuous and discontinuous initial conditions. To validate the effectiveness of the proposed model, we conduct systematic benchmarks on classical advection equations in one and two dimensions. These tests demonstrate the model's ability to accurately capture smooth and abrupt features. We then extend the application of HWF‑PIKAN to the high‑dimensional phase‑space setting by solving the CBE in a continuous‑velocity manner. This leverages the Hamiltonian concept of phase‑space dynamics to model the statistical behavior of particles in a collisionless system, where advection governs the evolution of a probability distribution function or number density. Comparative analysis against Vanilla PINN, Vanilla PIKAN, as well as Fourier‑enhanced and Wavelet‑enhanced PIKAN variants, shows that the proposed hybrid model significantly improves solution accuracy and convergence speed. This study highlights the power of multi‑resolution spectral feature embeddings in advancing physics‑informed deep learning frameworks for complex kinetic equations in both space‑time and phase‑space.

Abstract:
This work presents an interpretable parametric surrogate model motivated by the need to identify a hydrodynamic model for resolving the trajectory of an object in real‑time. The surrogate is formulated as a reduced‑order model for a canonical configuration in which a one‑degree‑of‑freedom heaving sphere operates near a vertical wall. High‑fidelity CFD simulations are used to generate a parametric dataset of heave‑decay responses over varying wall distances (WD) and drop heights (DH). Sparse Identification of Nonlinear Dynamics (SINDy) is then applied to each CFD trajectory to identify a low‑order nonlinear ordinary differential equation (ODE) with polynomial terms representing effective hydrostatic restoring and radiation damping, and the harmonic terms representing the wave‑induced excitation forces. The SINDy identified coefficients are then used as a prior constraint in a neural operator network (ONet) that learns a smooth mapping from wall distance and drop height to the ODE coefficients, yielding a surrogate capable of predicting dynamics at arbitrary points in the input space without rerunning expensive CFD calculations. The resulting surrogate reproduces CFD heave‑decay responses with near‑optimal accuracy given the limiting assumptions while being capable of running in real time. The approach provides a practical pathway toward real‑time, physics‑informed surrogate modelling for launch‑and‑recovery operations.

Abstract:
Classical numerical methods solve partial differential equations (PDEs) efficiently on regular meshes, but many of them become unstable on irregular domains. In practice, multiphysics interactions such as diffusion, damage, and healing often take place on irregular meshes. We develop an operator‑consistent graph neural network (OCGNN‑PINN) that approximates PDE evolution under physics‑informed constraints. It couples node‑edge message passing with a consistency loss enforcing the gradient‑divergence relation through the graph incidence matrix, ensuring that discrete node and edge dynamics remain structurally coupled during temporal rollout. We evaluate the model on diffusion processes over physically driven evolving meshes and real‑world scanned surfaces. The results show improved temporal stability and prediction accuracy compared with graph convolutional and multilayer perceptron baselines, approaching the performance of Crank‑Nicolson solvers on unstructured domains.

Abstract:
Cold dark matter (CDM) evolves as a collisionless fluid under the Vlasov‑Poisson equations, but N‑body simulations approximate this evolution by discretising the distribution function into particles, introducing discreteness effects at small scales. We present a physics‑informed neural network approach that evolves CDM fields without any use of N‑body data or methods, using a Kolmogorov‑Arnold network (KAN) to model the continuous displacement field for one‑dimensional halo collapse. Physical constraints derived from the Vlasov‑Poisson equations are embedded directly into the loss function, enabling accurate evolution beyond the first shell crossing. The trained model achieves sub‑percent errors on the residuals even after seven shell crossings and matches N‑body results while providing a resolution‑free representation of the displacement field. In addition, displacement errors do not grow over time, a very interesting feature with respect to N‑body methods. It can also reconstruct initial conditions through backward evolution when sufficient final‑state information is available. Although current runtimes exceed those of N‑body methods, this framework offers a new route to high‑fidelity CDM evolution without particle discretisation, with prospects for extension to higher dimensions.

Abstract:
Seismic acoustic impedance inversion is one of the most challenging tasks in geophysical exploration. Many studies have proposed the use of deep learning for processing; however, most of them are limited by factors such as seismic wavelets and low‑frequency initial models. Furthermore, self‑supervised frameworks constructed entirely using deep learning models struggle to form direct and effective physical constraints to unlabeled outputs during the multi‑model concatenation, which leads to instability in inversion. In this work, we introduced innovations in both the deep learning framework and training strategy. First, we designed a deep learning framework to perform acoustic impedance inversion and seismic wavelet extraction simultaneously. Building on this foundation, considering the scarcity of well data, we proposed a physics‑informed cross‑learning strategy to impose effective constraints on the framework. We conducted comparative experiments and ablation experiments on both synthetic datasets and field datasets. The results demonstrate that the proposed method achieves a significant improvement compared with semi‑supervised learning methods and can extract seismic wavelets with relatively high accuracy. Finally, to ensure the reproducibility of this work, we have made the code open‑source.

Abstract:
This paper introduces a projected Sobolev natural gradient descent (NGD) method for computing ground states of the Gross‑Pitaevskii equation. By projecting a continuous Riemannian Sobolev gradient flow onto the normalized neural network tangent space, we derive a discrete NGD algorithm that preserves the normalization constraint. The numerical implementation employs variational Monte Carlo with a hybrid sampling strategy to accurately account for the normalization constant arising from nonlinear interaction terms. To enhance computational efficiency, a matrix‑free Nyström‑preconditioned conjugate gradient solver is adopted to approximate the NGD operator without explicit matrix assembly. Numerical experiments demonstrate that the proposed method converges significantly faster than physics‑informed neural network approaches and exhibits linear scalability with respect to spatial dimensions. Moreover, the resulting neural‑network solutions provide high‑quality initial guesses that substantially accelerate subsequent refinement by traditional high‑precision solvers.

Abstract:
Lens flare is a degradation phenomenon caused by strong light sources. Existing researches on flare removal have mainly focused on images, while the spatiotemporal characteristics of video flare remain largely unexplored. Video flare synthesis and removal pose significantly greater challenges than in image, owing to the complex and mutually independent motion of flare, light sources, and scene content. This motion independence further affects restoration performance, often resulting in flicker and artifacts. To address this issue, we propose a physics‑informed dynamic flare synthesis pipeline, which simulates light source motion using optical flow and models the temporal behaviors of both scattering and reflective flares. Meanwhile, we design a video flare removal network that employs an attention module to spatially suppress flare regions and incorporates a Mamba‑based temporal modeling component to capture long range spatio‑temporal dependencies. This motion‑independent spatiotemporal representation effectively eliminates the need for multi‑frame alignment, alleviating temporal aliasing between flares and scene content and thereby improving video flare removal performance. Building upon this, we construct the first video flare dataset to comprehensively evaluate our method, which includes a large set of synthetic paired videos and additional real‑world videos collected from the Internet to assess generalization capability. Extensive experiments demonstrate that our method consistently outperforms existing video‑based restoration and image‑based flare removal methods on both real and synthetic videos, effectively removing dynamic flares while preserving light source integrity and maintaining spatiotemporal consistency of scene.

Abstract:
This work develops a control‑centric framework for a custom 4‑DOF rigid‑body manipulator by coupling a reduced‑order Pontryagin's Maximum Principle (PMP) controller with a physics‑informed Gradient Descent stage. The reduced PMP model provides a closed‑form optimal control law for the joint accelerations, while the Gradient Descent module determines the corresponding time horizons by minimizing a cost functional built directly from the full Rigid‑Body Dynamics. Structural‑mechanics reaction analysis is used only to initialize feasible joint velocities‑most critically the azimuthal component‑ensuring that the optimizer begins in a physically admissible region. The resulting kinematic trajectories and dynamically consistent time horizons are then supplied to the symbolic Euler‑Lagrange model to yield closed‑form inverse‑dynamics inputs. This pipeline preserves a strict control‑theoretic structure while embedding the physical constraints and loading behavior of the manipulator in a computationally efficient way.

Abstract:
Physics‑informed machine learning uses governing ordinary and/or partial differential equations to train neural networks to represent the solution field. Like any machine learning problem, the choice of activation function influences the characteristics and performance of the solution obtained from physics‑informed training. Several studies have compared common activation functions on benchmark differential equations, and have unanimously found that the rectified linear unit (ReLU) is outperformed by competitors such as the sigmoid, hyperbolic tangent, and swish activation functions. In this work, we diagnose the poor performance of ReLU on physics‑informed machine learning problems. While it is well‑known that the piecewise linear form of ReLU prevents it from being used on second‑order differential equations, we show that ReLU fails even on variational problems involving only first derivatives. We identify the cause of this failure as second derivatives of the activation, which are taken not in the formulation of the loss, but in the process of training. Namely, we show that automatic differentiation in PyTorch fails to characterize derivatives of discontinuous fields, which causes the gradient of the physics‑informed loss to be mis‑specified, thus explaining the poor performance of ReLU.

Abstract:
Implicit Neural Representations (INRs) have emerged as a powerful paradigm for parameterizing physical fields, yet they often suffer from spectral bias and the computational expense of non‑convex optimization. We introduce the Vekua Layer (VL), a differentiable spectral method grounded in the classical theory of Generalized Analytic Functions. By restricting the hypothesis space to the kernel of the governing differential operator ‑‑ specifically utilizing Harmonic and Fourier‑Bessel bases ‑‑ the VL transforms the learning task from iterative gradient descent to a strictly convex least‑squares problem solved via linear projection. We evaluate the VL against Sinusoidal Representation Networks (SIRENs) on homogeneous elliptic Partial Differential Equations (PDEs). Our results demonstrate that the VL achieves machine precision (\textMSE \approx 10^‑33) on exact reconstruction tasks and exhibits superior stability in the presence of incoherent sensor noise (\textMSE \approx 0.03), effectively acting as a physics‑informed spectral filter. Furthermore, we show that the VL enables "holographic" extrapolation of global fields from partial boundary data via analytic continuation, a capability absent in standard coordinate‑based approximations.

Abstract:
The DC Optimal Power Flow (DC‑OPF) problem is fundamental to power system operations, requiring rapid solutions for real‑time grid management. While traditional optimization solvers provide optimal solutions, their computational cost becomes prohibitive for large‑scale systems requiring frequent recalculations. Machine learning approaches offer promise for acceleration but often struggle with constraint satisfaction and cost optimality. We present a novel two‑stage learning framework that combines physics‑informed Graph Neural Networks (GNNs) with Continuous Flow Matching (CFM) for solving DC‑OPF problems. Our approach embeds fundamental physical principles‑‑including economic dispatch optimality conditions, Kirchhoff's laws, and Karush‑Kuhn‑Tucker (KKT) complementarity conditions‑‑directly into the training objectives. The first stage trains a GNN to produce feasible initial solutions by learning from physics‑informed losses that encode power system constraints. The second stage employs CFM, a simulation‑free continuous normalizing flow technique, to refine these solutions toward optimality through learned vector field regression. Evaluated on the IEEE 30‑bus system across five load scenarios ranging from 70% to 130% nominal load, our method achieves near‑optimal solutions with cost gaps below 0.1% for nominal loads and below 3% for extreme conditions, while maintaining 100% feasibility. Our framework bridges the gap between fast but approximate neural network predictions and optimal but slow numerical solvers, offering a practical solution for modern power systems with high renewable penetration requiring frequent dispatch updates.

Abstract:
The trade‑off between model fidelity and computational cost remains a central challenge in the computational modeling of extrusion‑based 3D printing, particularly for real time optimization and control. Although high fidelity simulations have advanced considerably for offline analysis, dynamical modeling tailored for online, control‑oriented applications is still significantly underdeveloped. In this study, we propose a reduced order dynamical flow model that captures the transient behavior of extrusion‑based 3D printing. The model is grounded in physics‑based principles derived from the Navier Stokes equations and further simplified through spatial averaging and input dependent parameterization. To assess its performance, the model is identified via a nonlinear least squares approach using Computational Fluid Dynamics (CFD) simulation data spanning a range of printing conditions and subsequently validated across multiple combinations of training and testing scenarios. The results demonstrate strong agreement with the CFD data within the nozzle, the nozzle substrate gap, and the deposited layer regions. Overall, the proposed reduced order model successfully captures the dominant flow dynamics of the process while maintaining a level of simplicity compatible with real time control and optimization.

Abstract:
Radio maps (RMs) provide a spatially continuous description of wireless propagation, enabling cross‑layer optimization and unifying communication and sensing for integrated sensing and communications (ISAC). However, constructing high‑fidelity RMs at operational scales is difficult, since physics‑based solvers are time‑consuming and require precise scene models, while learning methods degrade under incomplete priors and sparse measurements, often smoothing away critical discontinuities. We present RMSup, a physics‑informed super‑resolution framework that functions with uniform sparse sampling and imperfect environment priors. RMSup extracts Helmholtz equation‑informed boundary and singularity prompts from the measurements, fuses them with base‑station side information and coarse scene descriptors as conditional inputs, and employs a boundary‑aware dual‑head network to reconstruct a high‑fidelity RM and recover environmental contours jointly. Experimental results show the proposed RMsup achieves state‑of‑the‑art performance both in RM construction and ISAC‑related environment sensing.

Abstract:
Parabolic trough Concentrating Solar Power (CSP) plants operate large hydraulic networks of collector loops that must deliver a uniform outlet temperature despite spatially heterogeneous optical performance, heat losses, and pressure drops. While loop temperatures are measured, loop‑level mass flows and receiver heat‑loss parameters are unobserved, making it impossible to diagnose hydraulic imbalances or receiver degradation using standard monitoring tools. We present a physics‑informed learning framework that infers (i) loop‑level mass‑flow ratios and (ii) time‑varying receiver heat‑transfer coefficients directly from routine operational data. The method exploits nocturnal homogenization periods ‑‑ when hot oil is circulated through a non‑irradiated field ‑‑ to isolate hydraulic and thermal‑loss effects. A differentiable conjugate heat‑transfer model is discretized and embedded into an end‑to‑end learning pipeline optimized using historical plant data from the 50 MW Andasol 3 solar field. The model accurately reconstructs loop temperatures (RMSE <2^\circC) and produces physically meaningful estimates of loop imbalances and receiver heat losses. Comparison against drone‑based infrared thermography (QScan) shows strong correspondence, correctly identifying all areas with high‑loss receivers. This demonstrates that noisy real‑world CSP operational data contain enough information to recover latent physical parameters when combined with appropriate modeling and differentiable optimization.

Abstract:
Physics‑informed polynomial chaos expansions (PC^2) provide an efficient physically constrained surrogate modeling framework by embedding governing equations and other physical constraints into the standard data‑driven polynomial chaos expansions (PCE) and solving via the Karush‑Kuhn‑Tucker (KKT) conditions. This approach improves the physical interpretability of surrogate models while achieving high computational efficiency and accuracy. However, the performance and efficiency of PC^2 can still be degraded with high‑dimensional parameter spaces, limited data availability, or unrepresentative training data. To address this problem, this study explores two complementary enhancements to the PC^2 framework. First, a numerically efficient constrained optimization solver, straightforward updating of Lagrange multipliers (SULM), is adopted as an alternative to the conventional KKT solver. The SULM method significantly reduces computational cost when solving physically constrained problems with high‑dimensionality and derivative boundary conditions that require a large number of virtual points. Second, a D‑optimal sampling strategy is utilized to select informative virtual points to improve the stability and achieve the balance of accuracy and efficiency of the PC^2. The proposed methods are integrated into the PC^2 framework and evaluated through numerical examples of representative physical systems governed by ordinary or partial differential equations. The results demonstrate that the enhanced PC^2 has better comprehensive capability than standard PC^2, and is well‑suited for high‑dimensional uncertainty quantification tasks.

Abstract:
The simulation of microcirculatory blood flow in realistic vascular architectures poses significant challenges due to the multiscale nature of the problem and the topological complexity of capillary networks. In this work, we propose a novel deep learning‑based reduced‑order modeling strategy, leveraging Graph Neural Networks (GNNs) trained on synthetic microvascular graphs to approximate hemodynamic quantities on anatomically realistic domains. Our method combines algorithms for synthetic vascular generation with a physics‑informed training procedure that integrates graph topological information and local flow dynamics. To ensure the physical reliability of the learned surrogates, we incorporate a physics‑informed loss functional derived from the governing equations, allowing enforcement of mass conservation and rheological constraints. The resulting GNN architecture demonstrates robust generalization capabilities across diverse network configurations. The GNN formulation is validated on benchmark problems with linear and nonlinear rheology, showing accurate pressure and velocity field reconstruction with substantial computational gains over full‑order solvers. The methodology showcases significant generalization capabilities with respect to vascular complexity, as highlighted by tests on data from the mouse cerebral cortex. This work establishes a new class of graph‑based surrogate models for microvascular flow, grounded in physical laws and equipped with inductive biases that mirror mass conservation and rheological models, opening new directions for real‑time inference in vascular modeling and biomedical applications.

Abstract:
Data center (DC) infrastructure serves as the backbone to support the escalating demand for computing capacity. Traditional design methodologies that blend human expertise with specialized simulation tools scale poorly with the increasing system complexity. Recent studies adopt generative artificial intelligence to design plausible human‑centric indoor layouts. However, they do not consider the underlying physics, making them unsuitable for the DC design that sets quantifiable operational objectives and strict physical constraints. To bridge the gap, we propose Phythesis, a novel framework that synergizes large language models (LLMs) and physics‑guided evolutionary optimization to automate simulation‑ready (SimReady) scene synthesis for energy‑efficient DC design. Phythesis employs an iterative bi‑level optimization architecture, where (i) the LLM‑driven optimization level generates physically plausible three‑dimensional layouts and self‑criticizes them to refine the scene topology, and (ii) the physics‑informed optimization level identifies the optimal asset parameters and selects the best asset combination. Experiments on three generation scales show that Phythesis achieves 57.3% generation success rate increase and 11.5% power usage effectiveness (PUE) improvement, compared with the vanilla LLM‑based solution.

Abstract:
Surrogate models provide fast alternatives to costly aerodynamic simulations and are extremely useful in design and optimization applications. This study proposes the use of a recent kernel‑based neural surrogate, KHRONOS. In this work, we blend sparse high‑fidelity (HF) data with low‑fidelity (LF) information to predict aerodynamic fields under varying constraints in computational resources. Unlike traditional approaches, KHRONOS is built upon variational principles, interpolation theory, and tensor decomposition. These elements provide a mathematical basis for heavy pruning compared to dense neural networks. Using the AirfRANS dataset as a high‑fidelity benchmark and NeuralFoil to generate low‑fidelity counterparts, this work compares the performance of KHRONOS with three contemporary model architectures: a multilayer perceptron (MLP), a graph neural network (GNN), and a physics‑informed neural network (PINN). We consider varying levels of high‑fidelity data availability (0%, 10%, and 30%) and increasingly complex geometry parameterizations. These are used to predict the surface pressure coefficient distribution over the airfoil. Results indicate that, whilst all models eventually achieve comparable predictive accuracy, KHRONOS excels in resource‑constrained conditions. In this domain, KHRONOS consistently requires orders of magnitude fewer trainable parameters and delivers much faster training and inference than contemporary dense neural networks at comparable accuracy. These findings highlight the potential of KHRONOS and similar architectures to balance accuracy and efficiency in multi‑fidelity aerodynamic field prediction.

Abstract:
Battery energy storage systems (BESS) have become increasingly vital in three‑phase unbalanced distribution grids for maintaining voltage stability and enabling optimal dispatch. However, existing deep learning approaches often lack explicit three‑phase representation, making it difficult to accurately model phase‑specific dynamics and enforce operational constraints‑‑leading to infeasible dispatch solutions. This paper demonstrates that by embedding detailed three‑phase grid information‑‑including phase voltages, unbalanced loads, and BESS states‑‑into heterogeneous graph nodes, diverse GNN architectures (GCN, GAT, GraphSAGE, GPS) can jointly predict network state variables with high accuracy. Moreover, a physics‑informed loss function incorporates critical battery constraints‑‑SoC and C‑rate limits‑‑via soft penalties during training. Experimental validation on the CIGRE 18‑bus distribution system shows that this embedding‑loss approach achieves low prediction errors, with bus voltage MSEs of 6.92e‑07 (GCN), 1.21e‑06 (GAT), 3.29e‑05 (GPS), and 9.04e‑07 (SAGE). Importantly, the physics‑informed method ensures nearly zero SoC and C‑rate constraint violations, confirming its effectiveness for reliable, constraint‑compliant dispatch.

Abstract:
Physics‑informed neural networks have emerged as a powerful tool in the scientific machine learning community, with applications to both forward and inverse problems. While they have shown considerable empirical success, significant challenges remain ‑‑ particularly regarding training stability and the lack of rigorous theoretical guarantees, especially when compared to classical mesh‑based methods. In this work, we focus on the inverse problem of identifying a spatially varying parameter in a constitutive model of three‑dimensional elasticity, using measurements of the system's state. This setting is especially relevant for non‑invasive diagnosis in cardiac biomechanics, where one must also carefully account for the type of boundary data available. To address this inverse problem, we adopt an all‑at‑once optimisation framework, simultaneously estimating the state and parameter through a least‑squares loss that encodes both available data and the governing physics. For this formulation, we prove stability estimates ensuring that our approach yields a stable approximation of the underlying ground‑truth parameter of the physical system independent of a specific discretisation. We then proceed with a neural network‑based discretisation and compare it to traditional mesh‑based approaches. Our theoretical findings are complemented by illustrative numerical examples.

Abstract:
Environmental variables are increasingly affecting agricultural decision‑making, yet accessible and scalable tools for soil assessment remain limited. This study presents a robust and scalable modeling system for estimating soil properties in croplands, including soil organic carbon (SOC), total nitrogen (N), available phosphorus (P), exchangeable potassium (K), and pH, using remote sensing data and environmental covariates. The system employs a hybrid modeling approach, combining the indirect methods of modeling soil through proxies and drivers with direct spectral modeling. We extend current approaches by using interpretable physics‑informed covariates derived from radiative transfer models (RTMs) and complex, nonlinear embeddings from a foundation model. We validate the system on a harmonized dataset that covers Europes cropland soils across diverse pedoclimatic zones. Evaluation is conducted under a robust validation framework that enforces strict spatial blocking, stratified splits, and statistically distinct train‑test sets, which deliberately make the evaluation harder and produce more realistic error estimates for unseen regions. The models achieved their highest accuracy for SOC and N. This performance held across unseen locations, under both spatial cross‑validation and an independent test set. SOC obtained a MAE of 5.12 g/kg and a CCC of 0.77, and N obtained a MAE of 0.44 g/kg and a CCC of 0.77. We also assess uncertainty through conformal calibration, achieving 90 percent coverage at the target confidence level. This study contributes to the digital advancement of agriculture through the application of scalable, data‑driven soil analysis frameworks that can be extended to related domains requiring quantitative soil evaluation, such as carbon markets.

Abstract:
Industrial cyber physical systems operate under heterogeneous sensing, stochastic dynamics, and shifting process conditions, producing data that are often incomplete, unlabeled, imbalanced, and domain shifted. High‑fidelity datasets remain costly, confidential, and slow to obtain, while edge devices face strict limits on latency, bandwidth, and energy. These factors restrict the practicality of centralized deep learning, hinder the development of reliable digital twins, and increase the risk of error escape in safety‑critical applications. Motivated by these challenges, this dissertation develops an efficiency grounded computational framework that enables data lean, physics‑aware, and deployment ready intelligence for modern manufacturing environments. The research advances methods that collectively address core bottlenecks across multimodal and multiscale industrial scenarios. Generative strategies mitigate data scarcity and imbalance, while semi‑supervised learning integrates unlabeled information to reduce annotation and simulation demands. Physics‑informed representation learning strengthens interpretability and improves condition monitoring under small‑data regimes. Spatially aware graph‑based surrogate modeling provides efficient approximation of complex processes, and an edge cloud collaborative compression scheme supports real‑time signal analytics under resource constraints. The dissertation also extends visual understanding through zero‑shot vision language reasoning augmented by domain specific retrieval, enabling generalizable assessment in previously unseen scenarios. Together, these developments establish a unified paradigm of data efficient and resource aware intelligence that bridges laboratory learning with industrial deployment, supporting reliable decision‑making across diverse manufacturing systems.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving partial differential equations (PDEs) by embedding physical laws into neural network training objectives. However, their deployment on resource‑constrained platforms is hindered by substantial computational and memory overhead, primarily stemming from higher‑order automatic differentiation, intensive tensor operations, and reliance on full‑precision arithmetic. To address these challenges, we present a framework that enables scalable and energy‑efficient PINN training on edge devices. This framework integrates fully quantized training, Stein's estimator (SE)‑based residual loss computation, and tensor‑train (TT) decomposition for weight compression. It contributes three key innovations: (1) a mixed‑precision training method that use a square‑block MX (SMX) format to eliminate data duplication during backpropagation; (2) a difference‑based quantization scheme for the Stein's estimator that mitigates underflow; and (3) a partial‑reconstruction scheme (PRS) for TT‑Layers that reduces quantization‑error accumulation. We further design PINTA, a precision‑scalable hardware accelerator, to fully exploit the performance of the framework. Experiments on the 2‑D Poisson, 20‑D Hamilton‑Jacobi‑Bellman (HJB), and 100‑D Heat equations demonstrate that the proposed framework achieves accuracy comparable to or better than full‑precision, uncompressed baselines while delivering 5.5x to 83.5x speedups and 159.6x to 2324.1x energy savings. This work enables real‑time PDE solving on edge devices and paves the way for energy‑efficient scientific computing at scale.

Abstract:
Compressible flow problems are characterized by highly nonlinear, implicit, and often transcendental governing equations. In undergraduate gas dynamics education, solving these equations traditionally relies on iterative numerical methods or extensive look‑up tables, which can obscure the physical intuition of the solution space. This paper introduces a comprehensive framework using Deep Learning to generate high‑fidelity surrogate models for five canonical problems: Rayleigh flow, Fanno flow, oblique shocks, convergent‑divergent nozzles, and unsteady shock tubes. We detail the specific neural network architectures and physics‑informed feature engineering strategies required for each problem, such as using logarithmic inputs for Fanno friction parameters or geometric anchors for oblique shocks. The resulting models achieve high accuracy and enable instantaneous visualization of complex design spaces, such as thermodynamic T s diagrams and unsteady x t wave interactions. This approach demonstrates how modern data‑driven techniques can be integrated into the physics curriculum to enhance conceptual understanding.

Abstract:
We study online statistical inference for the solutions of stochastic optimization problems with equality and inequality constraints. Such problems are prevalent in statistics and machine learning, encompassing constrained M‑estimation, physics‑informed models, safe reinforcement learning, and algorithmic fairness. We develop a stochastic sequential quadratic programming (SSQP) method to solve these problems, where the step direction is computed by sequentially performing a quadratic approximation of the objective and a linear approximation of the constraints. Despite having access to unbiased estimates of population gradients, a key challenge in constrained stochastic problems lies in dealing with the bias in the step direction. As such, we apply a momentum‑style gradient moving‑average technique within SSQP to debias the step. We show that our method achieves global almost‑sure convergence and exhibits local asymptotic normality with an optimal primal‑dual limiting covariance matrix in the sense of Hájek and Le Cam. In addition, we provide a plug‑in covariance matrix estimator for practical inference. To our knowledge, the proposed SSQP method is the first fully online method that attains primal‑dual asymptotic minimax optimality without relying on projection operators onto the constraint set, which are generally intractable for nonlinear problems. Through extensive experiments on benchmark nonlinear problems, as well as on constrained generalized linear models and portfolio allocation problems using both synthetic and real data, we demonstrate superior performance of our method, showing that the method and its asymptotic behavior not only solve constrained stochastic problems efficiently but also provide valid and practical online inference in real‑world applications.

Abstract:
In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply revised and developed in 2022 with M. Trigiante. Instead of a compact 3‑torus T^3=\mathbbR^3/Λ where Λ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic 1‑forms that, for each energy level, decomposes into six components: two Beltrami, two anti‑Beltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an L^2 space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to an hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such a algorithmic construction that is planned for a future publication.

Abstract:
Physics‑Informed Neural Networks (PINNs) have recently emerged as a promising tool for fluid dynamics, particularly for flow reconstruction and parameter identification. In the context of granular media, accurately estimating rheological parameters remains a major challenge, as it typically requires complex and costly experimental setups. In this work, we propose a PINN‑based approach to identify key rheological parameters of granular materials using a simple experiment: the granular column collapse. A proof of concept is presented using synthetic data, where the PINN is trained to infer the flow fields while simultaneously recovering the rheological parameters. Beyond parameter identification, the method also enables reconstruction of the pressure field, which is difficult to access experimentally. The results highlight the potential of PINNs for data‑driven rheometry of granular materials and open perspectives for future applications with real experimental data.

Abstract:
This work proposes a wavelet‑based physics‑informed quantum neural network framework to efficiently address multiscale partial differential equations that involve sharp gradients, stiffness, rapid local variations, and highly oscillatory behavior. Traditional physics‑informed neural networks (PINNs) have demonstrated substantial potential in solving differential equations, and their quantum counterparts, quantum‑PINNs, exhibit enhanced representational capacity with fewer trainable parameters. However, both approaches face notable challenges in accurately solving multiscale features. Furthermore, their reliance on automatic differentiation for constructing loss functions introduces considerable computational overhead, resulting in longer training times. To overcome these challenges, we developed a wavelet‑accelerated physics‑informed quantum neural network that eliminates the need for automatic differentiation, significantly reducing computational complexity. The proposed framework incorporates the multiresolution property of wavelets within the quantum neural network architecture, thereby enhancing the network's ability to effectively capture both local and global features of multiscale problems. Numerical experiments demonstrate that our proposed method achieves superior accuracy while requiring less than five percent of the trainable parameters compared to classical wavelet‑based PINNs, resulting in faster convergence. Moreover, it offers a speedup of three to five times compared to existing quantum PINNs, highlighting the potential of the proposed approach for efficiently solving challenging multiscale and oscillatory problems.

Abstract:
This paper presents a Spatiotemporal Tube (STT)‑based control framework for general control‑affine MIMO nonlinear pure‑feedback systems with unknown dynamics to satisfy prescribed time reach‑avoid‑stay tasks under external disturbances. The STT is defined as a time‑varying ball, whose center and radius are jointly approximated by a Physics‑Informed Neural Network (PINN). The constraints governing the STT are first formulated as loss functions of the PINN, and a training algorithm is proposed to minimize the overall violation. The PINN being trained on certain collocation points, we propose a Lipschitz‑based validity condition to formally verify that the learned PINN satisfies the conditions over the continuous time horizon. Building on the learned STT representation, an approximation‑free closed‑form controller is defined to guarantee satisfaction of the T‑RAS specification. Finally, the effectiveness and scalability of the framework are validated through two case studies involving a mobile robot and an aerial vehicle navigating through cluttered environments.

Abstract:
Recent studies have demonstrated the success of deep learning in solving forward and inverse problems in engineering and scientific computing domains, such as physics‑informed neural networks (PINNs). Source inversion problems under sparse measurements for parabolic partial differential equations (PDEs) are particularly challenging to solve using PINNs, due to their severe ill‑posedness and the multiple unknowns involved including the source function and the PDE parameters. Although the neural tangent kernel (NTK) of PINNs has been widely used in forward problems involving a single neural network, its extension to inverse problems involving multiple neural networks remains less explored. In this work, we propose a weighted adaptive approach based on the NTK of PINNS including multiple separate networks representing the solution, the unknown source, and the PDE parameters. The key idea behind our methodology is to simultaneously solve the joint recovery of the solution, the source function along with the unknown parameters thereby using the underlying partial differential equation as a constraint that couples multiple unknown functional parameters, leading to more efficient use of the limited information in the measurements. We apply our method on the advection‑diffusion equation and we present various 2D and 3D numerical experiments using different types of measurements data that reflect practical engineering systems. Our proposed method is successful in estimating the unknown source function, the velocity and diffusion parameters as well as recovering the solution of the equation, while remaining robust to additional noise in the measurements.

Abstract:
Predicting relaxed atomic structures of chemically complex materials remains a major computational challenge, particularly for high‑entropy systems where traditional first‑principles methods become prohibitively expensive. We introduce the edge‑aware graph attention model, a physics‑informed graph neural network tailored for predicting relaxed atomic structures of high‑entropy systems. the edge‑aware graph attention model employs chemically and geometrically informed descriptors that capture both atomic properties and local structural environments. To effectively capture atomic interactions, our model integrates a multi‑head self‑attention mechanism that adaptively weighs neighbouring atoms using both node and edge features. This edge‑aware attention framework learn complex chemical and structural relationships independent of global orientation or position. We trained and evaluated the edge‑aware GAT model on a dataset of carbide systems, spanning binary to high‑entropy carbide compositions, and demonstrated its accuracy, convergence efficiency, and transferability. The architecture is lightweight, with a very low computational footprint, making it highly suitable for large‑scale materials screening. By providing invariance to rigid‑body transformations and leveraging domain‑informed attention mechanisms, our model delivers a fast, scalable, and cost‑effective alternative to DFT, enabling accelerated discovery and screening of entropy‑stabilised materials.

Abstract:
Real‑time calibration of stochastic volatility models (SVMs) is computationally bottlenecked by the need to repeatedly solve coupled partial differential equations (PDEs). In this work, we propose DeepSVM, a physics‑informed Deep Operator Network (PI‑DeepONet) designed to learn the solution operator of the Heston model across its entire parameter space. Unlike standard data‑driven deep learning (DL) approaches, DeepSVM requires no labelled training data. Rather, we employ a hard‑constrained ansatz that enforces terminal payoffs and static no‑arbitrage conditions by design. Furthermore, we use Residual‑based Adaptive Refinement (RAR) to stabilize training in difficult regions subject to high gradients. Overall, DeepSVM achieves a final training loss of 10^‑5 and predicts highly accurate option prices across a range of typical market dynamics. While pricing accuracy is high, we find that the model's derivatives (Greeks) exhibit noise in the at‑the‑money (ATM) regime, highlighting the specific need for higher‑order regularization in physics‑informed operator learning.

Abstract:
Quantum centric supercomputing (QCSC) framework, such as sample‑based quantum diagonalization (SQD) holds immense promise toward achieving practical quantum utility to solve challenging problems. QCSC leverages quantum computers to perform the classically intractable task of sampling the dominant fermionic configurations from the Hilbert space that have substantial support to a target state, followed by Hamiltonian diagonalization on a classical processor. However, noisy quantum hardware produces erroneous samples upon measurements, making robust and efficient configuration‑recovery strategies essential for a scalable QCSC pipeline. Toward this, in this work, we introduce PIGen‑SQD, an efficiently designed QCSC workflow that utilizes the capability of generative machine learning (ML) along with physics‑informed configuration screening via implicit low‑rank tensor decompositions for accurate fermionic state reconstruction. The physics‑informed pruning is based on a class of efficient perturbative measures that, in conjunction with hardware samples, provide a substantial overlap with the target state. This distribution induces an anchoring effect on the generative ML models to stochastically explore only the dominant sector of the Hilbert space for effective identification of additional important configurations in a self‑consistent manner. Our numerical experiments performed on IBM Heron R2 quantum processors demonstrate this synergistic workflow produces compact, high‑fidelity subspaces that substantially reduce diagonalization cost while maintaining chemical accuracy under strong electronic correlations. By embedding classical many body intuitions directly into the generative ML model, PIGen‑SQD advances the robustness and scalability of QCSC algorithms, offering a promising pathway toward chemically reliable quantum simulations on utility‑scale quantum hardware.

Abstract:
Applications providing automated coaching for physical training are increasing in popularity, for example physical therapy. These applications rely on accurate and robust pose estimation using monocular video streams. State‑of‑the‑art models like BlazePose excel in real‑time pose tracking, but their lack of anatomical constraints indicates improvement potential by including physical knowledge. We present a real‑time post‑processing algorithm fusing the strengths of BlazePose 3D and 2D estimations using a weighted optimization, penalizing deviations from expected bone length and biomechanical models. Bone length estimations are refined to the individual anatomy using a Kalman filter with adapting measurement trust. Evaluation using the Physio2.2M dataset shows a 10.2 percent reduction in 3D MPJPE and a 16.6 percent decrease in errors of angles between body segments compared to BlazePose 3D estimation. Our method provides a robust, anatomically consistent pose estimation based on a computationally efficient video‑to‑3D pose estimation, suitable for automated physiotherapy, healthcare, and sports coaching on consumer‑level laptops and mobile devices. The refinement runs on the backend with anonymized data only.

Abstract:
Proper initialisation strategy is of primary importance to mitigate gradient explosion or vanishing when training neural networks. Yet, the impact of initialisation parameters still lacks a precise theoretical understanding for several well‑established architectures. Here, we propose a new initialisation for networks with sinusoidal activation functions such as \textttSIREN, focusing on gradients control, their scaling with network depth, their impact on training and on generalization. To achieve this, we identify a closed‑form expression for the initialisation of the parameters, differing from the original \textttSIREN scheme. This expression is derived from fixed points obtained through the convergence of pre‑activation distribution and the variance of Jacobian sequences. Controlling both gradients and targeting vanishing pre‑activation helps preventing the emergence of inappropriate frequencies during estimation, thereby improving generalization. We further show that this initialisation strongly influences training dynamics through the Neural Tangent Kernel framework (NTK). Finally, we benchmark \textttSIREN with the proposed initialisation against the original scheme and other baselines on function fitting and image reconstruction. The new initialisation consistently outperforms state‑of‑the‑art methods across a wide range of reconstruction tasks, including those involving physics‑informed neural networks.

Abstract:
We survey classical, machine learning, and data‑driven system identification approaches to learn control‑relevant and physics‑informed models of dynamical systems. Recently, machine learning approaches have enabled system identification from noisy, high‑dimensional, and complex data. However, their utility is limited by their ability to provide provable guarantees on control‑relevant properties. Meanwhile, control theory has identified several properties that are useful in analysis and control synthesis, such as dissipativity, monotonicity, energy conservation, and symmetry‑preserving structures. We posit that merging system identification with such control‑relevant or physics‑informed properties can provide useful inductive bias, enhance explainability, enable control synthesis with provable guarantees, and improve sample complexity. We formulate system identification as an optimization problem where control‑relevant properties can be enforced through direct parameterization (constraining the model structure to satisfy a desired property by construction), soft constraints (encouraging control‑relevant properties through regularization or penalty terms), and hard constraints (imposing control‑relevant properties as constraints in the optimization problem). Through this lens, we survey methods to learn physics‑informed and control‑relevant models spanning classical linear and nonlinear system identification, machine learning approaches, and direct identification through data‑driven and behavioral representations. We also provide several expository examples that are accompanied by code and brief tutorials on a public Github repository. We also describe challenging directions for future research, including identification in networked, switched, and time‑varying systems, experiment design, and bridging the gaps between data‑driven, learning‑based, and control‑oriented approaches.

Abstract:
Unlike traditional multi‑agent coordination frameworks, which assume a fixed number of agents, UAS traffic management (UTM) requires a platform that enables Uncrewed Aerial Systems (UAS) to freely enter or exit constrained low‑altitude airspace. Consequently, the number of UAS operating in a given region is time‑varying, with vehicles dynamically joining or leaving even in dense, obstacle‑laden environments. The primary goal of this paper is to develop a computationally efficient management system that maximizes airspace usability while ensuring safety and efficiency. To achieve this, we first introduce physics‑informed methods to structure fixed skyroads across multiple altitude layers of urban airspace, with the directionality of each skyroad designed to guarantee full reachability. We then present a novel Continuous UTM (C‑UTM) framework that optimally allocates skyroads to UAS requests while accounting for the time‑varying capacity of the airspace. Collectively, the proposed model addresses the key challenges of low‑altitude UTM by providing a scalable, safe, and efficient solution for urban airspace usability.

Abstract:
Early forecasting of individual cognitive decline in Alzheimer's disease (AD) is central to disease evaluation and management. Despite advances, it is as of yet challenging for existing methodological frameworks to integrate multimodal data for longitudinal personalized forecasting while maintaining interpretability. To address this gap, we present the Neural Koopman Machine (NKM), a new machine learning architecture inspired by dynamical systems and attention mechanisms, designed to forecast multiple cognitive scores simultaneously using multimodal genetic, neuroimaging, proteomic, and demographic data. NKM integrates analytical (α) and biological (β) knowledge to guide feature grouping and control the hierarchical attention mechanisms to extract relevant patterns. By implementing Fusion Group‑Aware Hierarchical Attention within the Koopman operator framework, NKM transforms complex nonlinear trajectories into interpretable linear representations. To demonstrate NKM's efficacy, we applied it to study the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset. Our results suggest that NKM consistently outperforms both traditional machine learning methods and deep learning models in forecasting trajectories of cognitive decline. Specifically, NKM (1) forecasts changes of multiple cognitive scores simultaneously, (2) quantifies differential biomarker contributions to predicting distinctive cognitive scores, and (3) identifies brain regions most predictive of cognitive deterioration. Together, NKM advances personalized, interpretable forecasting of future cognitive decline in AD using past multimodal data through an explainable, explicit system and reveals potential multimodal biological underpinnings of AD progression.

Abstract:
We present a physics‑informed neural network framework for predicting the mechanical performance of elastomers exposed to concurrent thermal and gamma‑radiation exposure, such as elastomers in nuclear cables or space electronics. Our demonstrated approach integrates the dual‑network hypothesis with the microsphere concept to represent soft and brittle sub‑networks, while embedding physical laws directly into the machine learning process. Hard constraints, e.g., incompressibility, bounded network fractions are enforced through network architecture, and soft constraints e.g., monotonicity, polyconvexity, and fading effects are imposed through the loss function. This integration reduces the effective search space, guiding the optimization toward physically admissible solutions and enhancing robustness under sparse data. Validation against published datasets on silicone rubber, ethylene propylene diene monomer, and silica‑reinforced silicone foam shows accurate predictions of stress‑strain behavior and elongation‑at‑break at exposure times not used for training. Results confirm that physics‑informed constraints improve extrapolation, capture synergistic thermal‑radiation effects, and provide a reliable tool for lifetime assessment of nuclear cable insulation and other radiation‑exposed elastomers.

Abstract:
The Archimedes Tube Crystallizer (ATC) employs air‑segmented flow in coiled tubes to achieve narrow residence time distributions for continuous crystallization. Taylor and Dean vortices drive particle suspension in this system. However, one‑way coupled models fail to capture the fluid‑particle feedback that becomes critical at higher loadings. We present a particle‑resolved Direct Numerical Simulation (DNS) framework based on a Finite Element‑Fictitious Boundary Method with hard‑contact modeling of particle interactions. Simulations of L‑alanine suspensions across varying particle sizes, solid contents, and rotational speeds are validated against experimental side‑view imaging. Three quantitative metrics‑axial distribution, radial index, and vertical asymmetry‑are introduced to classify suspension regimes. The DNS results reproduce the experimentally observed flow map zones (green, yellow, red/yellow, red) and resolve subtle transitions such as rear loading and loss of vertical symmetry. This feasibility study demonstrates that DNS can reliably predict dense suspension behavior and provides a mechanistic foundation for crystallizer design.

Abstract:
The ringdown phase of a binary black‑hole merger provides a clean probe of strong‑field gravity, as it can be modeled with minimal assumptions. The quasi‑normal‑mode frequencies encode the mass and spin of the Kerr black‑hole remnant, while the mode excitation depends on the progenitor binary. In this paper, we implement a recently developed amplitude model that captures spin precession in a simulation‑based inference pipeline that specifically targets ringdown signals. We present a critical re‑analysis of GW190521 ‑‑ a short‑duration, merger‑dominated event with conflicting interpretations. Spin‑aligned and precessing analyses at two ringdown start times show that precession induces modest but systematic shifts in inferred parameters and subdominant mode amplitudes, although such ringdown‑only analyses provide no strong evidence for precession. Our results demonstrate the feasibility of physics‑informed precessing ringdown modelling, paving the way for the identification of spin precession in gravitational‑wave events using solely their ringdown stages, where waveform systematics are expected to be substantially less prominent.

Abstract:
This review charts the emerging paradigm of intelligent structured light for high‑field laser‑matter interactions, where the precise spatiotemporal and vectorial control of light is a critical degree of freedom. We outline a transformative framework built upon three synergistic pillars. First, we survey the advanced electromagnetic toolkit, moving beyond conventional spatial light modulators to include robust static optics and the promising frontier of plasma light modulators. Second, we detail the optimization engine for this high‑dimensional design space, focusing on physics‑informed digital twins and AI‑driven inverse design to automate the discovery of optimal light structures. Finally, we explore the groundbreaking applications enabled by this integrated approach, including programmable electron beams, orbital‑angular‑momentum‑carrying γ‑rays, compact THz accelerators, and robust communications. The path forward necessitates overcoming grand challenges in material science, real‑time adaptive control at MHz rates, and the extension of these principles to the quantum realm. This review serves as a call to action for a coordinated, interdisciplinary effort to command, rather than merely observe, light‑matter interactions at the extreme.

Abstract:
A physics‑informed machine learning framework based on holomorphic neural networks is introduced for detecting cracks in two‑dimensional solids from strain or displacement data. Crack detection is formulated as an inverse problem in which the crack size, orientation, and location are treated as unknowns. The problem is solved using genetic optimization, where the fitness function is evaluated by expressing the solution of the corresponding plane elasticity problem in terms of holomorphic potentials, which are then determined through the training of two holomorphic neural networks. As the potentials satisfy equilibrium and traction‑free conditions along the crack faces a priori, the training proceeds quickly based solely on boundary information. Training efficiency is further improved by splitting the genetic search into long‑range and short‑range stages, enabling the use of transfer learning in the latter. The new strategy is tested on three benchmark problems, showing that an optimal number of training epochs exists that provides the best overall performance. A comparison is also made with a popular crack detection approach that uses XFEM to compute the model response. Under the assumption of identical stress‑field representation accuracy, the proposed method is found to be between 7 and 23 times faster than the XFEM‑based approach. Furthermore, the proposed method appears to be less sensitive to noise in the input data. Overall, the present findings demonstrate that combining genetic optimization with holomorphic neural networks and transfer learning offers a promising avenue for developing crack detection strategies with higher efficiency than those currently available.

Abstract:
Fine‑grained air pollution forecasting is crucial for urban management and the development of healthy buildings. Deploying portable sensors on mobile platforms such as cars and buses offers a low‑cost, easy‑to‑maintain, and wide‑coverage data collection solution. However, due to the random and uncontrollable movement patterns of these non‑dedicated mobile platforms, the resulting sensor data are often incomplete and temporally inconsistent. By exploring potential training patterns in the reverse process of diffusion models, we propose Spatio‑Temporal Physics‑Informed Diffusion Models (STeP‑Diff). STeP‑Diff leverages DeepONet to model the spatial sequence of measurements along with a PDE‑informed diffusion model to forecast the spatio‑temporal field from incomplete and time‑varying data. Through a PDE‑constrained regularization framework, the denoising process asymptotically converges to the convection‑diffusion dynamics, ensuring that predictions are both grounded in real‑world measurements and aligned with the fundamental physics governing pollution dispersion. To assess the performance of the system, we deployed 59 self‑designed portable sensing devices in two cities, operating for 14 days to collect air pollution data. Compared to the second‑best performing algorithm, our model achieved improvements of up to 89.12% in MAE, 82.30% in RMSE, and 25.00% in MAPE, with extensive evaluations demonstrating that STeP‑Diff effectively captures the spatio‑temporal dependencies in air pollution fields.

Abstract:
This paper introduces a direct comparative study of Physics‑Informed Neural Networks (PINNs) and Long Short‑Term Memory (LSTM) networks for adaptive steam temperature control in Heat Recovery Steam Generators (HRSGs), particularly under valve leakage faults. Maintaining precise steam temperature in HRSGs is critical for efficiency and safety, yet traditional control strategies struggle with nonlinear, fault‑induced dynamics. Both architectures are designed to adaptively tune the gains of a PI‑plus‑feedforward control law in real‑time. The LSTM controller, a purely data‑driven approach, was trained offline on historical operational data, while the PINN controller integrates fundamental thermodynamic laws directly into its online learning process through a physics‑based loss function. Their performance was evaluated using a model validated with data from a combined cycle power plant, under normal load changes and a challenging valve leakage fault scenario. Results demonstrate that while the LSTM controller offers significant improvement over conventional methods, its performance degrades under the unseen fault. The PINN controller consistently delivered superior robustness and performance, achieving a 54% reduction in integral absolute error compared to the LSTM under fault conditions. This study concludes that embedding physical knowledge into data‑driven control is essential for developing reliable, fault‑tolerant autonomous control systems in complex industrial applications.

Abstract:
Accurately mapping the radio environment (e.g., identifying wireless signal strength at specific frequency bands and geographic locations) is crucial for efficient spectrum sharing, enabling Secondary Users~(SUs) to access underutilized spectrum bands while protecting Primary Users~(PUs). While existing models have made progress, they often degrade in performance when multiple transmitters coexist, due to the compounded effects of shadowing, interference from adjacent transmitters. To address this challenge, we extend our prior work on Physics‑Informed Neural Networks~(PINNs) for single‑transmitter mapping to derive a new multi‑transmitter Partial Differential Equation~(PDE) formulation of the Received Signal Strength Indicator~(RSSI). We then propose \emphReVeal‑MT (Re‑constructor and Visualizer of Spectrum Landscape for Multiple Transmitters), a novel PINN which integrates the multi‑source PDE residual into a neural network loss function, enabling accurate spectrum landscape reconstruction from sparse RF sensor measurements. ReVeal‑MT is validated using real‑world measurements from the ARA wireless living lab across rural and suburban environments, and benchmarked against 3GPP and ITU‑R channel models and a baseline PINN model for a single transmitter use‑case. Results show that ReVeal‑MT achieves substantial accuracy gains in multi‑transmitter scenarios, e.g., achieving an RMSE of only 2.66\,dB with as few as 45 samples over a 370‑square‑kilometer region, while maintaining low computational complexity. These findings demonstrate that ReVeal‑MT significantly advances radio environment mapping under realistic multi‑transmitter conditions, with strong potential for enabling fine‑grained spectrum management and precise coexistence between PUs and SUs.

Abstract:
In this paper, we adapt the Discrete Variable (DV)‑Circuit Quantum‑Classical Physics‑Informed Neural Network (QCPINN) and apply it for the first time to four typical reservoir seepage models. These include the pressure diffusion equation for heterogeneous single‑phase flow, the nonlinear Buckley‑Leverett (BL) equation for simplified two‑phase waterflooding, the convection‑diffusion equation for compositional flow considering adsorption, and the fully coupled pressure‑saturation two‑phase oil‑water seepage equation for heterogeneous reservoirs with exponential permeability distribution. The QCPINN integrates classical preprocessing/postprocessing networks with a DV quantum core, leveraging quantum superposition and entanglement to enhance high‑dimensional feature mapping while embedding physical constraints to ensure solution consistency. We test three quantum circuit topologies (Cascade, Cross‑mesh, Alternate) and demonstrate through four numerical experiments that QCPINNs achieve higher prediction accuracy than classical PINNs. Specifically, the Alternate topology outperforms others in heterogeneous single‑phase flow, BL equation simulations and heterogeneous fully coupled pressure‑saturation two‑phase flow, while the Cascade topology excels in compositional flow with convection‑dispersion‑adsorption coupling. The Cross‑mesh topology shows competitive early‑stage convergence and accuracy across scenarios with balanced performance in coupled two‑phase flow. Our work verifies the feasibility of QCPINN for reservoir engineering applications, bridging the gap between quantum computing research and industrial practice in oil and gas engineering.

Abstract:
This paper presents a novel fault‑tolerant control framework for steam temperature regulation in Heat Recovery Steam Generators (HRSGs) subject to actuator faults. Addressing the critical challenge of valve degradation in superheater spray attemperators, we propose a synergistic architecture comprising three components: (1) a Sliding Mode Observer (SMO) for estimation of unmeasured thermal states, (2) a Physics‑Informed Neural Network (PINN) for estimating multiplicative actuator faults using physical laws as constraints, and (3) a one‑sided Sliding Mode Controller (SMC) that adapts to the estimated faults while minimizing excessive actuation. The key innovation lies in the framework of closed‑loop physics‑awareness, where the PINN continuously informs both the observer and controller about fault severity while preserving thermodynamic consistency. Rigorous uniform ultimate boundedness (UUB) is established via Lyapunov analysis under practical assumptions. Validated on real HRSG operational data, the framework demonstrates effective fault adaptation, reduced temperature overshoot, and maintains steam temperature within 1°C of the setpoint under valve effectiveness loss. This work bridges control theory and physics‑guided machine learning to deliver a practically deployable solution for power plant resilience, with extensions applicable to thermal systems subject to multiplicative faults.

Abstract:
Autonomous Driving (AD) vehicles still struggle to exhibit human‑like behavior in highly dynamic and interactive traffic scenarios. The key challenge lies in AD's limited ability to interact with surrounding vehicles, largely due to a lack of understanding the underlying mechanisms of social interaction. To address this issue, we introduce MPCFormer, an explainable socially‑aware autonomous driving approach with physics‑informed and data‑driven coupled social interaction dynamics. In this model, the dynamics are formulated into a discrete space‑state representation, which embeds physics priors to enhance modeling explainability. The dynamics coefficients are learned from naturalistic driving data via a Transformer‑based encoder‑decoder architecture. To the best of our knowledge, MPCFormer is the first approach to explicitly model the dynamics of multi‑vehicle social interactions. The learned social interaction dynamics enable the planner to generate manifold, human‑like behaviors when interacting with surrounding traffic. By leveraging the MPC framework, the approach mitigates the potential safety risks typically associated with purely learning‑based methods. Open‑looped evaluation on NGSIM dataset demonstrates that MPCFormer achieves superior social interaction awareness, yielding the lowest trajectory prediction errors compared with other state‑of‑the‑art approaches. The prediction achieves an ADE as low as 0.86 m over a long prediction horizon of 5 seconds. Close‑looped experiments in highly intense interaction scenarios, where consecutive lane changes are required to exit an off‑ramp, further validate the effectiveness of MPCFormer. Results show that MPCFormer achieves the highest planning success rate of 94.67%, improves driving efficiency by 15.75%, and reduces the collision rate from 21.25% to 0.5%, outperforming a frontier Reinforcement Learning (RL) based planner.

Abstract:
Bridging the gap between theoretical conceptualization and computational implementation is a major bottleneck in Scientific Computing (SciC) and Scientific Machine Learning (SciML). We introduce ATHENA (Agentic Team for Hierarchical Evolutionary Numerical Algorithms), an agentic framework designed as an Autonomous Lab to manage the end‑to‑end computational research lifecycle. Its core is the HENA loop, a knowledge‑driven diagnostic process framed as a Contextual Bandit problem. Acting as an online learner, the system analyzes prior trials to select structural `actions' (A_n) from combinatorial spaces guided by expert blueprints (e.g., Universal Approximation, Physics‑Informed constraints). These actions are translated into executable code (S_n) to generate scientific rewards (R_n). ATHENA transcends standard automation: in SciC, it autonomously identifies mathematical symmetries for exact analytical solutions or derives stable numerical solvers where foundation models fail. In SciML, it performs deep diagnosis to tackle ill‑posed formulations and combines hybrid symbolic‑numeric workflows (e.g., coupling PINNs with FEM) to resolve multiphysics problems. The framework achieves super‑human performance, reaching validation errors of 10^‑14. Furthermore, collaborative ``human‑in‑the‑loop" intervention allows the system to bridge stability gaps, improving results by an order of magnitude. This paradigm shift focuses from implementation mechanics to methodological innovation, accelerating scientific discovery.

Abstract:
Single‑crystal X‑ray diffraction (SC‑XRD) is the gold standard technique to characterize crystal structures in solid state. Despite significant advances in automation for structure solution, the refinement stage still depends heavily on expert intervention and subjective judgment, limiting accessibility and scalability. Herein, we introduce RefrActor, an end‑to‑end deep learning framework that enables crystal structure determination directly from HKL data. By coupling a physics‑informed reciprocal‑space encoder (ReciEncoder) with a symmetry‑aware diffusion‑based generator (StruDiffuser), RefrActor produces fully refined atomic models without requiring initial structural guesses or manual input. Comprehensive evaluations on the GenRef‑10k benchmark demonstrates that RefrActor achieves low R1‑factors across diverse systems, including low‑symmetry, light‑atom, and heavy‑atom crystals. Case studies further confirm that RefrActor can correctly resolve hydrogen positions, elemental assignments, and moderate disorder. This work establishes a new data‑driven paradigm for autonomous crystallographic analysis, offering a foundation for fully automated, high‑throughput crystal structure determination.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful, mesh‑free paradigm for solving partial differential equations (PDEs). However, they notoriously struggle with stiff, multi‑scale, and nonlinear systems due to the inherent spectral bias of standard multilayer perceptron (MLP) architectures, which prevents them from adequately representing high‑frequency components. In this work, we introduce the Adaptive Spectral Physics‑Enabled Network (ASPEN), a novel architecture designed to overcome this critical limitation. ASPEN integrates an adaptive spectral layer with learnable Fourier features directly into the network's input stage. This mechanism allows the model to dynamically tune its own spectral basis during training, enabling it to efficiently learn and represent the precise frequency content required by the solution. We demonstrate the efficacy of ASPEN by applying it to the complex Ginzburg‑Landau equation (CGLE), a canonical and challenging benchmark for nonlinear, stiff spatio‑temporal dynamics. Our results show that a standard PINN architecture catastrophically fails on this problem, diverging into non‑physical oscillations. In contrast, ASPEN successfully solves the CGLE with exceptional accuracy. The predicted solution is visually indistinguishable from the high‑resolution ground truth, achieving a low median physics residual of 5.10 x 10^‑3. Furthermore, we validate that ASPEN's solution is not only pointwise accurate but also physically consistent, correctly capturing emergent physical properties, including the rapid free energy relaxation and the long‑term stability of the domain wall front. This work demonstrates that by incorporating an adaptive spectral basis, our framework provides a robust and physically‑consistent solver for complex dynamical systems where standard PINNs fail, opening new options for machine learning in challenging physical domains.

Abstract:
Diffusion MRI (dMRI) enables non‑invasive assessment of prostate microstructure but conventional dMRI metrics such as the Apparent Diffusion Coefficient in multiparametric MRI and reflect a mixture of underlying tissues features rather than distinct histologic characteristics. Integrating dMRI with the compartment‑based biophysical VERDICT (Vascular, Extracellular, and Restricted Diffusion for Cytometry in Tumours) framework offers richer microstructural insights, though clinical gradient systems (40‑80 mT/m) often suffer from poor signal‑to‑noise ratio at stronger diffusion weightings due to prolonged echo times. Ultra‑strong gradients (e.g., 300 mT/m) can mitigate these limitations by improving SNR and contrast‑to‑noise ratios. This study investigates whether physics‑informed self‑supervised VERDICT (ssVERDICT) fitting when combined with ultra‑strong gradient data, enhances prostate microstructural characterization relative to current fitting approaches and clinical gradient systems. We developed enhanced ssVERDICT fitting approaches using dense multilayer perceptron and convolutional U‑Net architectures, comparing them against non‑linear least‑squares (NLLS) VERDICT fitting, original ssVERDICT implementation, and Diffusion Kurtosis Imaging across clinical‑ to ultra‑strong gradient systems. For the same ultra‑strong gradient data, Dense ssVERDICT outperformed NLLS VERDICT, boosting median CNR by 47%, cutting inter‑patient Coefficient of Variation by 52%, and reducing pooled f_ic variation by 50%. Overall, Dense ssVERDICT delivered the highest CNR, the most stable parameter estimates, and the clearest tumour‑normal contrast compared with conventional fitting methods and clinical gradient systems. These findings underscore that meaningful gains in non‑invasive prostate cancer characterization arise from the combination of advanced gradient systems and deep learning‑based modelling.

Abstract:
Cardiovascular disease is the leading global cause of mortality, with coronary artery disease (CAD) as its most prevalent form, necessitating early risk prediction. While 3D coronary artery digital twins reconstructed from imaging offer detailed anatomy for personalized assessment, their analysis relies on computationally intensive computational fluid dynamics (CFD), limiting scalability. Data‑driven approaches are hindered by scarce labeled data and lack of physiological priors. To address this, we present PINS‑CAD, a physics‑informed self‑supervised learning framework. It pre‑trains graph neural networks on 200,000 synthetic coronary digital twins to predict pressure and flow, guided by 1D Navier‑Stokes equations and pressure‑drop laws, eliminating the need for CFD or labeled data. When fine‑tuned on clinical data from 635 patients in the multicenter FAME2 study, PINS‑CAD predicts future cardiovascular events with an AUC of 0.73, outperforming clinical risk scores and data‑driven baselines. This demonstrates that physics‑informed pretraining boosts sample efficiency and yields physiologically meaningful representations. Furthermore, PINS‑CAD generates spatially resolved pressure and fractional flow reserve curves, providing interpretable biomarkers. By embedding physical priors into geometric deep learning, PINS‑CAD transforms routine angiography into a simulation‑free, physiology‑aware framework for scalable, preventive cardiology.

Abstract:
Machine learning (ML) has emerged as a powerful tool for accelerating the computational design and production of materials. In materials science, ML has primarily supported large‑scale discovery of novel compounds using first‑principles data and digital twin applications for optimizing manufacturing processes. However, applying general‑purpose ML frameworks to complex industrial materials such as steel remains a challenge. A key obstacle is accurately capturing the intricate relationship between chemical composition, processing parameters, and the resulting microstructure and properties. To address this, we introduce a computational framework that combines physical insights with ML to develop a physics‑informed continuous cooling transformation (CCT) model for steels. Our model, trained on a dataset of 4,100 diagrams, is validated against literature and experimental data. It demonstrates high computational efficiency, generating complete CCT diagrams with 100 cooling curves in under 5 seconds. It also shows strong generalizability across alloy steels, achieving phase classification F1 scores above 88% for all phases. For phase transition temperature regression, it attains mean absolute errors (MAE) below 20 °C across all phases except bainite, which shows a slightly higher MAE of 27 °C. This framework can be extended with additional generic and customized ML models to establish a universal digital twin platform for heat treatment. Integration with complementary simulation tools and targeted experiments will further support accelerated materials design workflows.

Abstract:
This paper presents the modeling of autonomous vehicles with high maneuverability used in an experimental framework for educational purposes. Since standard bicycle models typically neglect wide steering angles, we develop modified planar bicycle models and combine them with both parametric and non‑parametric identification techniques that progressively incorporate physical knowledge. The resulting models are systematically compared to evaluate the tradeoff between model accuracy and computational requirements, showing that physics‑informed neural network models surpass the purely physical baseline in accuracy at lower computational cost.

Abstract:
Heat transfer in semiconductor devices is dominated by chip and substrate assemblies, where heat generated within a finite chip layer dissipates into a semi‑infinite substrate with much higher thermophysical properties. This mismatch produces steep interfacial temperature gradients, making the transient thermal response highly sensitive to the interface. Conventional numerical solvers require excessive discretization to resolve these dynamics, while physics‑informed neural networks (PINNs) often exhibit unstable convergence and loss of physical consistency near the material interface. To address these challenges, we introduce HeatTransFormer, a physics‑guided Transformer architecture for interface‑dominated diffusion problems. The framework integrates physically informed spatiotemporal sampling, a Laplace‑based activation emulating analytical diffusion solutions, and a mask‑free attention mechanism supporting bidirectional spatiotemporal coupling. These components enable the model to resolve steep gradients, maintain physical consistency, and remain stable where PINNs typically fail. HeatTransFormer produces coherent temperature fields across the interface when applied to a finite layer and semi‑infinite substrate configuration. Coupled with a physics‑constrained inverse strategy, it further enables reliable identification of three unknown thermal properties simultaneously using only external measurements. Overall, this work demonstrates that physics‑guided Transformer architectures provide a unified framework for forward and inverse modeling in interface‑dominated thermal systems.

Abstract:
Inverse problems arise across scientific and engineering domains, where the goal is to infer hidden parameters or physical fields from indirect and noisy observations. Classical approaches, such as variational regularization and Bayesian inference, provide well established theoretical foundations for handling ill posedness. However, these methods often become computationally restrictive in high dimensional settings or when the forward model is governed by complex physics. Physics Informed Neural Networks (PINNs) have recently emerged as a promising framework for solving inverse problems by embedding physical laws directly into the training process of neural networks. In this paper, we introduce a new perspective on the Bayesian Physics Informed Neural Network (BPINN) framework, extending classical PINNs by explicitly incorporating training data generation, modeling and measurement uncertainties through Bayesian prior modeling and doing inference with the posterior laws. Also, as we focus on the inverse problems, we call this method BPINN‑IP, and we show that the standard PINN formulation naturally appears as its special case corresponding to the Maximum A Posteriori (MAP) estimate. This unified formulation allows simultaneous exploitation of physical constraints, prior knowledge, and data‑driven inference, while enabling uncertainty quantification through posterior distributions. To demonstrate the effectiveness of the proposed framework, we consider inverse problems arising in infrared image processing, including deconvolution and super‑resolution, and present results on both simulated and real industrial data.

Abstract:
Coordinating emergency responses in extreme environments, such as wildfires, requires resilient and high‑bandwidth communication backbones. While autonomous aerial swarms can establish ad‑hoc networks to provide this connectivity, the high risk of individual node attrition in these settings often leads to network fragmentation and mission‑critical downtime. To overcome this challenge, we introduce and formalize the problem of Robust Task Networking Under Attrition (RTNUA), which extends connectivity maintenance in multi‑robot systems to explicitly address proactive redundancy and attrition recovery. We then introduce Physics‑Informed Robust Employment of Multi‑Agent Networks (ΦIREMAN), a topological algorithm leveraging physics‑inspired potential fields to solve this problem. In our evaluations, ΦIREMAN consistently outperforms baselines, and is able to maintain greater than 99.9% task uptime despite substantial attrition in simulations with up to 100 tasks and 500 drones, demonstrating both effectiveness and scalability.

Abstract:
This chapter opens with a review of classic tools for regression, a subset of machine learning that seeks to find relationships between variables. With the advent of scientific machine learning this field has moved from a purely data‑driven (statistical) formalism to a constrained or ``physics‑informed'' formalism, which integrates physical knowledge and methods from traditional computational engineering. In the first part, we introduce the general concepts and the statistical flavor of regression versus other forms of curve fitting. We then move to an overview of traditional methods from machine learning and their classification and ways to link these to traditional computational science. Finally, we close with a note on methods to combine machine learning and numerical methods for physics

Abstract:
Autonomous ground vehicles (AGVs) must navigate safely in cluttered environments while accounting for complex dynamics and environmental uncertainty. Hamilton‑Jacobi Reachability (HJR) offers formal safety guarantees through the computation of forward and backward reachable sets, but its application is hindered by poor scalability in environments with numerous obstacles. In this paper, we present a novel framework called NeuroHJR that leverages Physics‑Informed Neural Networks (PINNs) to approximate the HJR solution for real‑time obstacle avoidance. By embedding system dynamics and safety constraints directly into the neural network loss function, our method bypasses the need for grid‑based discretization and enables efficient estimation of reachable sets in continuous state spaces. We demonstrate the effectiveness of our approach through simulation results in densely cluttered scenarios, showing that it achieves safety performance comparable to that of classical HJR solvers while significantly reducing the computational cost. This work provides a new step toward real‑time, scalable deployment of reachability‑based obstacle avoidance in robotics.

Abstract:
The requirements of modern sensing are rapidly evolving, driven by increasing demands for data efficiency, real‑time processing, and deployment under limited sensing coverage. Complex physical systems are often characterized through the integration of a limited number of point sensors in combination with scientific computations which approximate the dominant, full‑state dynamics. Simulation models, however, inevitably neglect small‑scale or hidden processes, are sensitive to perturbations, or oversimplify parameter correlations, leading to reconstructions that often diverge from the reality measured by sensors. This creates a critical need for data assimilation, the process of integrating observational data with predictive simulation models to produce coherent and accurate estimates of the full state of complex physical systems. We propose a machine learning framework for Data Assimilation with a SHallow REcurrent Decoder (DA‑SHRED) which bridges the simulation‑to‑real (SIM2REAL) gap between computational modeling and experimental sensor data. For real‑world physics systems modeling high‑dimensional spatiotemporal fields, where the full state cannot be directly observed and must be inferred from sparse sensor measurements, we leverage the latent space learned from a reduced simulation model via SHRED, and update these latent variables using real sensor data to accurately reconstruct the full system state. Furthermore, our algorithm incorporates a sparse identification of nonlinear dynamics based regression model in the latent space to identify functionals corresponding to missing dynamics in the simulation model. We demonstrate that DA‑SHRED successfully closes the SIM2REAL gap and additionally recovers missing dynamics in highly complex systems, demonstrating that the combination of efficient temporal encoding and physics‑informed correction enables robust data assimilation.

Abstract:
Thermal analysis is increasingly critical in modern integrated circuits, where non‑uniform power dissipation and high transistor densities can cause rapid temperature spikes and reliability concerns. Traditional methods, such as FEM‑based simulations offer high accuracy but computationally prohibitive for early‑stage design, often requiring multiple iterative redesign cycles to resolve late‑stage thermal failures. To address these challenges, we propose 'ThermAl', a physics‑informed generative AI framework which effectively identifies heat sources and estimates full‑chip transient and steady‑state thermal distributions directly from input activity profiles. ThermAl employs a hybrid U‑Net architecture enhanced with positional encoding and a Boltzmann regularizer to maintain physical fidelity. Our model is trained on an extensive dataset of heat dissipation maps, ranging from simple logic gates (e.g., inverters, NAND, XOR) to complex designs, generated via COMSOL. Experimental results demonstrate that ThermAl delivers precise temperature mappings for large circuits, with a root mean squared error (RMSE) of only 0.71°C, and outperforms conventional FEM tools by running up to ~200 times faster. We analyze performance across diverse layouts and workloads, and discuss its applicability to large‑scale EDA workflows. While thermal reliability assessments often extend beyond 85°C for post‑layout signoff, our focus here is on early‑stage hotspot detection and thermal pattern learning. To ensure generalization beyond the nominal operating range 25‑55°C, we additionally performed cross‑validation on an extended dataset spanning 25‑95°C maintaining a high accuracy (<2.2% full‑scale RMSE) even under elevated temperature conditions representative of peak power and stress scenarios.

Abstract:
Precipitation nowcasting, key for early warning of disasters, currently relies on computationally expensive and restrictive methods that limit access to many countries. To overcome this challenge, we propose precipitation nowcasting using satellite imagery with physics constraints for improved accuracy and physical consistency. We use a novel physics‑informed dual neural operator (PIANO) structure to enforce the fundamental equation of advection‑diffusion during training to predict satellite imagery using a PINN loss. Then, we use a generative model to convert satellite images to radar images, which are used for precipitation nowcasting. Compared to baseline models, our proposed model shows a notable improvement in moderate (4mm/h) precipitation event prediction alongside short‑term heavy (8mm/h) precipitation event prediction. It also demonstrates low seasonal variability in predictions, indicating robustness for generalization. This study suggests the potential of the PIANO and serves as a good baseline for physics‑informed precipitation nowcasting.

Abstract:
Faults and operational disturbances in Heat Recovery Steam Generators (HRSGs), such as valve leakage, present significant challenges, disrupting steam temperature regulation and potentially causing efficiency losses, safety risks, and unit shutdowns. Traditional PI controllers often struggle due to inherent system delays, nonlinear dynamics, and static gain limitations. This paper introduces a fault‑tolerant temperature control framework by integrating a PI plus feedforward control strategy with Physics‑Informed Neural Networks (PINNs). The feedforward component anticipates disturbances, preemptively adjusting control actions, while the PINN adaptively tunes control gains in real‑time, embedding thermodynamic constraints to manage varying operating conditions and valve leakage faults. A Lyapunov‑based stability analysis confirms the asymptotic convergence of temperature tracking errors under bounded leakage conditions. Simulation results using operational data from the Pareh‑Sar combined cycle power plant demonstrate significantly improved response times, reduced temperature deviations, enhanced fault resilience, and smooth gain adjustments. The proposed adaptive, data‑driven methodology shows strong potential for industrial deployment, ensuring reliable operation, autonomous fault recovery, and enhanced performance in HRSG systems.

Abstract:
Demographic forecasting remains a fundamental challenge for policy planning in rapidly evolving nations such as India, where fertility transitions, policy interventions, and age structured dynamics interact in complex ways. In this study, we present a hybrid modelling framework that integrates policy‑aware fertility functions into a Physics‑Informed Neural Network (PINN) enhanced with Long Short‑Term Memory (LSTM) networks to capture physical constraints and temporal dependencies in population dynamics. The model is applied to India's age structured population from 2024 to 2054 under three fertility‑policy scenarios: continuation of current fertility decline, stricter population control, and relaxed fertility promotion. The governing transport‑reaction partial differential equation is formulated with India‑specific demographic indicators, including age‑specific fertility and mortality rates. PINNs embed the core population equation and policy‑driven fertility changes, while LSTM layers improve long‑term forecasting across decades. Results show that fertility policies substantially shape future age distribution, dependency ratios, and workforce size. Stricter controls intensify ageing and reduce labour force participation, whereas relaxed policies support workforce growth but increase population pressure. Our findings suggest that the hybrid LSTM‑PINN is an effective approach for demographic forecasting, offering accuracy with interpretability. Beyond methodological novelty, this work provides actionable insights for India's demographic policy debates, highlighting the need for balanced fertility interventions to ensure sustainable socio‑economic development.

Abstract:
Analog circuit optimization is typically framed as black‑box search over arbitrary smooth functions, yet device physics constrains performance mappings to structured families: exponential device laws, rational transfer functions, and regime‑dependent dynamics. Off‑the‑shelf Gaussian‑process surrogates impose globally smooth, stationary priors that are misaligned with these regime‑switching primitives and can severely misfit highly nonlinear circuits at realistic sample sizes (50‑‑100 evaluations). We demonstrate that pre‑trained tabular models encoding these primitives enable reliable optimization without per‑circuit engineering. Circuit Prior Network (CPN) combines a tabular foundation model (TabPFN v2) with Direct Expected Improvement (DEI), computing expected improvement exactly under discrete posteriors rather than Gaussian approximations. Across 6 circuits and 25 baselines, structure‑matched priors achieve R^2 \approx 0.99 in small‑sample regimes where GP‑Matérn attains only R^2 = 0.16 on Bandgap, deliver 1.05‑‑3.81× higher FoM with 3.34‑‑11.89× fewer iterations, and suggest a shift from hand‑crafting models as priors toward systematic physics‑informed structure identification. Our code will be made publicly available upon paper acceptance.

Abstract:
We present Relightable Holoported Characters (RHC), a novel person‑specific method for free‑view rendering and relighting of full‑body and highly dynamic humans solely observed from sparse‑view RGB videos at inference. In contrast to classical one‑light‑at‑a‑time (OLAT)‑based human relighting, our transformer‑based RelightNet predicts relit appearance within a single network pass, avoiding costly OLAT‑basis capture and generation. For training such a model, we introduce a new capture strategy and dataset recorded in a multi‑view lightstage, where we alternate frames lit by random environment maps with uniformly lit tracking frames, simultaneously enabling accurate motion tracking and diverse illumination as well as dynamics coverage. Inspired by the rendering equation, we derive physics‑informed features that encode geometry, albedo, shading, and the virtual camera view from a coarse human mesh proxy and the input views. Our RelightNet then takes these features as input and cross‑attends them with a novel lighting condition, and regresses the relit appearance in the form of texel‑aligned 3D Gaussian splats attached to the coarse mesh proxy. Consequently, our RelightNet implicitly learns to efficiently compute the rendering equation for novel lighting conditions within a single feed‑forward pass. Experiments demonstrate our method's superior visual fidelity and lighting reproduction compared to state‑of‑the‑art approaches. Project page: https://vcai.mpi‑inf.mpg.de/projects/RHC/

Abstract:
Inverse heat problems refer to the estimation of material thermophysical properties given observed or known heat diffusion behaviour. Inverse heat problems have wide‑ranging uses, but a critical application lies in quantifying how building facade renovation reduces thermal transmittance, a key determinant of building energy efficiency. However, solving inverse heat problems with non‑invasive data collected in situ is error‑prone due to environmental variability or deviations from theoretically assumed conditions. Hence, current methods for measuring thermal conductivity are either invasive, require lengthy observation periods, or are sensitive to environmental and experimental conditions. Here, we present a PINN‑based iterative framework to estimate the thermal conductivity k of a wall from a set of thermographs; our framework alternates between estimating the forward heat problem with a PINN for a fixed k, and optimizing k by comparing the thermographs and surface temperatures predicted by the PINN, repeating until the estimated k's convergence. Using both environmental data captured by a weather station and data generated from Finite‑Volume‑Method software simulations, we accurately predict k across different environmental conditions and data collection sampling times, given the temperature profile of the wall at dawn is close to steady state. Although violating the steady‑state assumption impacts the accuracy of k's estimation, we show that our proposed framework still only exhibits a maximum MAE of 4.0851. Our work demonstrates the potential of PINN‑based methods for reliable estimation of material properties in situ and under realistic conditions, without lengthy measurement campaigns. Given the lack of research on using machine learning, and more specifically on PINNs, for solving in‑situ inverse problems, we expect our work to be a starting point for more research on the topic.

Abstract:
Physics‑informed neural networks (PINNs) often struggle with multi‑scale PDEs featuring sharp gradients and nontrivial boundary conditions, as the physics residual and boundary enforcement compete during optimization. We present a dual‑network framework that decomposes the solution as u = u_\textD + u_\textB, where u_\textD (domain network) captures interior dynamics and u_\textB (boundary network) handles near‑boundary corrections. Both networks share a unified physics residual while being softly specialized via distance‑weighted priors (w_\textbd = \exp(‑d/τ)) that are cosine‑annealed during training. Boundary conditions are enforced through an augmented Lagrangian method, eliminating manual penalty tuning. Training proceeds in two phases: Phase~1 uses uniform collocation to establish network roles and stabilize boundary satisfaction; Phase~2 employs focused sampling (e.g. ring sampling near \partialΩ) with annealed role weights to efficiently resolve localized features. We evaluate our model on four benchmarks, including the 1D Fokker‑Planck equation, the Laplace equation, the Poisson equation, and the 1D wave equation. Across Laplace and Poisson benchmarks, our method reduces error by 36‑90%, improves boundary satisfaction by 21‑88%, and decreases MAE by 2.2‑9.3× relative to a single‑network PINN. Ablations isolate contributions of (i)~soft boundary‑interior specialization, (ii)~annealed role regularization, and (iii)~the two‑phase curriculum. The method is simple to implement, adds minimal computational overhead, and broadly applies to PDEs with sharp solutions and complex boundary data.

Abstract:
Underwater object tracking is challenging due to wavelength dependent attenuation and scattering, which severely distort appearance across depths and water conditions. Existing trackers trained on terrestrial data fail to generalize to these physics‑driven degradations. We present MANTA, a physics‑informed framework integrating representation learning with tracking design for underwater scenarios. We propose a dual‑positive contrastive learning strategy coupling temporal consistency with Beer‑Lambert augmentations to yield features robust to both temporal and underwater distortions. We further introduce a multi‑stage pipeline augmenting motion‑based tracking with a physics‑informed secondary association algorithm that integrates geometric consistency and appearance similarity for re‑identification under occlusion and drift. To complement standard IoU metrics, we propose Center‑Scale Consistency (CSC) and Geometric Alignment Score (GAS) to assess geometric fidelity. Experiments on four underwater benchmarks (WebUOT‑1M, UOT32, UTB180, UWCOT220) show that MANTA achieves state‑of‑the‑art performance, improving Success AUC by up to 6 percent, while ensuring stable long‑term generalized underwater tracking and efficient runtime.

Abstract:
This paper presents a framework for physics‑informed learning in complex cyber‑physical systems governed by differential equations with both unknown dynamics and algebraic invariants. First, we formalize the Hybrid Recurrent Physics‑Informed Neural Network (HRPINN), a general‑purpose architecture that embeds known physics as a hard structural constraint within a recurrent integrator to learn only residual dynamics. Second, we introduce the Projected HRPINN (PHRPINN), a novel extension that integrates a predict‑project mechanism to strictly enforce algebraic invariants by design. The framework is supported by a theoretical analysis of its representational capacity. We validate HRPINN on a real‑world battery prognostics DAE and evaluate PHRPINN on a suite of standard constrained benchmarks. The results demonstrate the framework's potential for achieving high accuracy and data efficiency, while also highlighting critical trade‑offs between physical consistency, computational cost, and numerical stability, providing practical guidance for its deployment.

Abstract:
Physics‑informed neural networks (PINNs) have been applied to simulate multiphase flows, yet they are limited in modeling phase changes and sharp interfaces due to optimization conflicts in the strongly coupled Allen‑Cahn, Cahn‑Hilliard, and Navier‑Stokes equations and the intrinsic smoothness bias of neural representations near discontinuities. To mitigate these limitations, this study presents a discontinuity‑aware physics‑informed neural network (DPINN) based on the phase‑field method to resolve sharp interfaces and phase changes in multiphase flows. It incorporates a discontinuity‑aware network architecture to mitigate spectral bias and automatically detect and model sharp interfacial dynamics, and a learnable local artificial viscosity term to stabilize the calculation near steep gradients. During optimization, adaptive time‑marching and loss‑balancing strategies are employed to reduce long‑horizon errors and mitigate gradient conflicts, ensuring accurate capture of phase changes. Numerical experiments on two‑phase reversed single‑vortex and bubble‑rising problems demonstrate that DPINN accurately resolves sharp interfacial dynamics, while conventional PINNs fail to converge. The method is further extended and tested in a three‑phase droplet‑icing case, where the viscosity and density ratios between ice, water, and air exceed seven and three orders of magnitude, respectively. The predicted phase change dynamics and sharp pointy‑tip formation show excellent agreement with the reference, highlighting the robustness of the proposed approach.

Abstract:
This paper puts forward a framework to accelerate Electromagnetic Transient (EMT) simulations by replacing individual components with trained Physics‑Informed Neural Networks (PINNs). EMT simulations are considered the cornerstone of transient stability assessment of power systems with high shares of Inverter‑Based Resources (IBRs), and, although accurate, they are notorious for their slow simulation speed. Taking a deeper dive into the EMT simulation algorithms, this paper identifies the most computationally expensive components of the simulation and replaces them with fast and accurate PINNs. The proposed novel PINN formulation enables a modular and scalable integration into the simulation algorithm. Using a type‑4 wind turbine EMT model, we demonstrate a 4‑‑6x simulation speedup by capturing the Phase‑Locked Loop (PLL) with a PINN. We validate all our results with PSCAD software.

Abstract:
Serving as an essential prerequisite for modern power system operation, robust state estimation (RSE) could effectively resist noises and outliers in measurements. The emerging neural network (NN) based end‑to‑end (E2E) learning framework enables real‑time application of RSE but cannot strictly enforce the physical constraints involved, potentially yielding solutions that are statistically accurate yet physically inconsistent. To bridge this gap, this work proposes a novel E2E learning based RSE framework, where the RSE problem is innovatively constructed as an explicit differentiable layer of NN for the first time, ensuring physics alignments with rigors. Also, the measurement weights are treated as learnable parameters of NN to enhance estimation robustness. A hybrid loss function is formulated to pursue accurate and physically consistent solutions. To realize the proposed NN structure, the original non‑convex RSE problem is specially relaxed. Extensive numerical simulations have been carried out to demonstrate that the proposed framework can significantly improve the SE performance while fulfilling physical consistency on six testing systems, in comparisons to the classical E2E learning based approach and the physics‑informed neural network (PINN) approach.

Abstract:
The directed motion of active colloids is governed by spatial variations in surface chemistry and interfacial stress, yet these properties remain extremely difficult to measure directly. We introduce a physics‑informed neural network framework that infers the slip distribution driving propulsion from partial observations of the surrounding flow. By combining sparse fluid velocity measurements with the Stokes equations and boundary constraints, the method reconstructs both the near‑surface slip and the full velocity and pressure fields. Validation against analytical solutions and Boundary Element Method calculations for canonical active colloid models shows quantitative agreement in both unbounded and confined geometries. Crucially, the framework recovers the surface slip even when no flow data are available near the particle, demonstrating that accessible bulk measurements encode the interfacial stresses responsible for active motion. These results establish physics‑informed inference as a powerful tool for characterizing and ultimately controlling interfacially driven transport in colloidal active matter.

Abstract:
We apply physics‑informed machine learning (PIML) to solve inverse problems in holography and classical mechanics, focusing on neural ordinary differential equations (Neural ODEs) and physics‑informed neural networks (PINNs) for solving non‑linear differential equations of motion. First, we introduce holographic inverse problems and demonstrate how PIML can reconstruct bulk spacetime and effective potentials from boundary quantum data. To illustrate this, two case studies are explored: the QCD equation of state in holographic QCD and T‑linear resistivity in holographic strange metals. Additionally, we explicitly show how such holographic problems can be analogized to inverse problems in classical mechanics, modeling frictional forces with neural networks. We also explore Kolmogorov‑Arnold Networks (KANs) as an alternative to traditional neural networks, offering more efficient solutions in certain cases. This manuscript aim to provide a systematic framework for using neural networks in inverse problems, serving as a comprehensive reference for researchers in machine learning for high‑energy physics, with methodologies that also have broader applications in mathematics, engineering, and the natural sciences.

Abstract:
Power Flow (PF) calculation based on machine learning (ML) techniques offer significant computational advantages over traditional numerical methods but often struggle to maintain full physical consistency. This paper introduces a physics‑informed test‑time training (PI‑TTT) framework that enhances the accuracy and feasibility of ML‑based PF surrogates by enforcing AC power flow equalities and operational constraints directly at inference time. The proposed method performs a lightweight self‑supervised refinement of the surrogate outputs through few gradient‑based updates, enabling local adaptation to unseen operating conditions without requiring labeled data. Extensive experiments on the IEEE 14‑, 118‑, and 300‑bus systems and the PEGASE 1354‑bus network show that PI‑TTT reduces power flow residuals and operational constraint violations by one to two orders of magnitude compared with purely ML‑based models, while preserving their computational advantage. The results demonstrate that PI‑TTT provides fast, accurate, and physically reliable predictions, representing a promising direction for scalable and physics‑consistent learning in power system analysis.

Abstract:
We present PDE‑FM, a modular foundation model for physics‑informed machine learning that unifies spatial, spectral, and temporal reasoning across heterogeneous partial differential equation (PDE) systems. PDE‑FM combines spatial‑spectral tokenization, physics‑aware conditioning, and a Mamba‑based state‑space backbone with an operator‑theoretic decoder, enabling scalable and data‑efficient modeling of complex physical dynamics. In contrast to task‑specific neural operators, PDE‑FM is pretrained once on diverse PDE datasets and can be transferred to new physical regimes without architectural or data‑specific modifications. Evaluated on twelve 2D and 3D datasets from The Well benchmark ‑ spanning hydrodynamic, radiative, elastic, and astrophysical phenomena ‑ PDE‑FM achieves state‑of‑the‑art accuracy in six domains, reducing mean VRMSE by 46% relative to prior operator‑learning baselines. The model demonstrates robust cross‑physics generalization, excelling in turbulent and radiative systems while maintaining strong performance in linear and steady‑state regimes. These results suggest that large‑scale pretraining across diverse physical processes can yield transferable representations of dynamics, marking a step toward unified, foundation‑level surrogates for multi‑physics simulation and scientific discovery.

Abstract:
Real‑time, physically‑consistent predictions on low‑power edge devices is critical for the next generation embodied AI systems, yet it remains a major challenge. Physics‑Informed Neural Networks (PINNs) combine data‑driven learning with physics‑based constraints to ensure the model's predictions are with underlying physical principles.However, PINNs are energy‑intensive and struggle to strictly enforce physical conservation laws. Brain‑inspired spiking neural networks (SNNs) have emerged as a promising solution for edge computing and real‑time processing. However, naively converting PINNs to SNNs degrades physical fidelity and fails to address long‑term generalization issues. To this end, this paper introduce a novel Physics‑Informed Spiking Neural Network (PISNN) framework. Importantly, to ensure strict physical conservation, we design the Conservative Leaky Integrate‑and‑Fire (C‑LIF) neuron, whose dynamics structurally guarantee local mass preservation. To achieve robust temporal generalization, we introduce a novel Conservative Flux Quantization (CFQ) strategy, which redefines neural spikes as discrete packets of physical flux. Our CFQ learns a time‑invariant physical evolution operator, enabling the PISNN to become a general‑purpose solver ‑‑ conservative‑by‑construction. Extensive experiments show that our PISNN excels on diverse benchmarks. For both the canonical 1D heat equation and the more challenging 2D Laplace's Equation, it accurately simulates the system dynamics while maintaining perfect mass conservation by design ‑‑ a feat that is challenging for conventional PINNs. This work establishes a robust framework for fusing the rigor of scientific computing with the efficiency of neuromorphic engineering, paving the way for complex, long‑term, and energy‑efficient physics predictions for intelligent systems.

Abstract:
Accurate and efficient seismic response prediction is essential for the design of resilient structures. While the Finite Element Method (FEM) remains the standard for nonlinear seismic analysis, its high computational demands limit its scalability and real‑time applicability. Recent developments in deep learning ‑ particularly Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs), and Long Short‑Term Memory (LSTM) models ‑ have shown promise in reducing the computational cost of the nonlinear seismic analysis of structures. However, these data‑driven models often struggle to generalize and capture the underlying physics, leading to reduced reliability. We propose a novel Physics‑Informed U‑Net‑LSTM framework that integrates physical laws with deep learning to enhance both accuracy and efficiency. The proposed 1D U‑Net captures the underlying latent features of the long‑term input sequences. By embedding domain‑specific constraints into the learning process, the proposed model achieves improved predictive performance over conventional Machine Learning (ML) architectures. This approach bridges the gap between purely data‑driven methods and physics‑based modeling, offering a robust and computationally efficient alternative for predicting the seismic response of structures.

Abstract:
This paper presents a deep learning strategy to simultaneously solve Partial Differential Equations (PDEs) and back‑calculate their parameters in the context of deep tunnel excavation. A Physics‑Informed Neural Network (PINN) model is trained with synthetic data that emulates in situ displacement measurements in the host rock and at the cavity wall, obtained from extensometers and convergence monitoring. As acquiring field observations can be costly, a sequential training approach based on active learning is implemented to determine the most informative locations for new sensors. In particular, Monte Carlo dropout is used to quantify epistemic uncertainty and query measurements in regions where the model is least confident. This approach reduces the amount of required field data and optimizes sensor placement. The PINN is tested to reconstruct the displacement field around a deep tunnel of circular section excavated in transversely isotropic elastic rock and to determine rock constitutive and stress‑field parameters. Results demonstrate excellent performance on small, scattered, and noisy datasets, achieving high precision for the Young's moduli, shear modulus, horizontal‑to‑vertical far‑field stress ratio, and the orientation of the bedding planes. The proposed framework shall ultimately support decision‑making for optimal subsurface monitoring and for adaptive tunnel design and control.

Abstract:
Understanding household behaviour is essential for modelling macroeconomic dynamics and designing effective policy. While heterogeneous agent models offer a more realistic alternative to representative agent frameworks, their implementation poses significant computational challenges, particularly in continuous time. The Aiyagari‑Bewley‑Huggett (ABH) framework, recast as a system of partial differential equations, typically relies on grid‑based solvers that suffer from the curse of dimensionality, high computational cost, and numerical inaccuracies. This paper introduces the ABH‑PINN solver, an approach based on Physics‑Informed Neural Networks (PINNs), which embeds the Hamilton‑Jacobi‑Bellman and Kolmogorov Forward equations directly into the neural network training objective. By replacing grid‑based approximation with mesh‑free, differentiable function learning, the ABH‑PINN solver benefits from the advantages of PINNs of improved scalability, smoother solutions, and computational efficiency. Preliminary results show that the PINN‑based approach is able to obtain economically valid results matching the established finite‑difference solvers.

Abstract:
Forward Collision Warning systems are crucial for vehicle safety and autonomous driving, yet current methods often fail to balance precise multi‑agent interaction modeling with real‑time decision adaptability, evidenced by the high computational cost for edge deployment and the unreliability stemming from simplified interaction models.To overcome these dual challenges‑computational complexity and modeling insufficiency‑along with the high false alarm rates of traditional static‑threshold warnings, this paper introduces an integrated FCW framework that pairs a Hierarchical Spatio‑Temporal Attention Network with a Dynamic Risk Threshold Adjustment algorithm. HSTAN employs a decoupled architecture (Graph Attention Network for spatial, cascaded GRU with self‑attention for temporal) to achieve superior performance and efficiency, requiring only 12.3 ms inference time (73% faster than Transformer methods) and reducing the Average Displacement Error (ADE) to 0.73m (42.2% better than Social_LSTM) on the NGSIM dataset. Furthermore, Conformalized Quantile Regression enhances reliability by generating prediction intervals (91.3% coverage at 90% confidence), which the DTRA module then converts into timely warnings via a physics‑informed risk potential function and an adaptive threshold mechanism inspired by statistical process control.Tested across multi‑scenario datasets, the complete system demonstrates high efficacy, achieving an F1 score of 0.912, a low false alarm rate of 8.2%, and an ample warning lead time of 2.8 seconds, validating the framework's superior performance and practical deployment feasibility in complex environments.

Abstract:
Physics‑Informed Neural Networks (PINNs) encounter significant training difficulties when applied to doubly‑clamped beam problems, and the underlying causes are not fully understood. This study investigates the PINN loss landscape to identify the failure mechanisms of two primary formulations: the high‑order strong formulation and the energy‑based formulation. The results demonstrate that the Strong Formulation suffers from landscape ill‑conditioning driven by the boundary conditions (BCs), leading to convergence issues in the doubly‑clamped case. Conversely, while the energy‑based formulation requires only lower‑order derivatives, its loss functional can become indefinite, causing optimization difficulties near saddle points. Based on strain field benchmarks against Finite Element Method (FEM), it is found that the strong formulation, combined with a BC handling method and the L‑BFGS optimizer, yields the best performance across three classical boundary condition cases. These findings clarify distinct, formulation‑dependent failure modes, offering a diagnostic foundation for developing robust physics‑based surrogate models for complex beam systems.

Abstract:
We introduce a generalizable, physics informed strategy for generating training data that enables a machine learning force field accurate over a broad range of twist angles and stacking layer numbers in moire systems. Applying this to multilayer twisted MoTe2 (tMoTe2), we identify a structural and electronic stratification: the two moire interface (MI) layers retain substantial lattice reconstruction even in thick multilayers, while outer bulk like layers show rapidly attenuated distortions.Surprisingly, this stratification becomes strongest not in the ultra‑small twist angle regime (<~1°), where in plane domain formation is well known, but rather at intermediate angles (2‑5°). Simultaneously, interlayer hybridization across the MI‑bulk boundary is strongly suppressed, leading to electronic isolation. In twisted double bilayer MoTe2, this stratification gives rise to coexisting honeycomb and triangular lattice motifs in the frontier valence bands. We further demonstrate that twist angle and weak gating can create energy shift of bands belonging to the two motifs, producing Chern band reordering and nonlinear electric polarization with modest hole doping. Our approach allows efficient simulation of multilayer moire systems and reveals structural‑electronic separation phenomena absent in bilayer systems.

Abstract:
As artificial intelligence emerges as a transformative enabler for fusion energy commercialization, fast and accurate solvers become increasingly critical. In magnetic confinement nuclear fusion, rapid and accurate solution of the Grad‑Shafranov equation (GSE) is essential for real‑time plasma control and analysis. Traditional numerical solvers achieve high precision but are computationally prohibitive, while data‑driven surrogates infer quickly but fail to enforce physical laws and generalize poorly beyond training distributions. To address this challenge, we present a Physics‑Informed Neural Operator (PINO) that directly learns the GSE solution operator, mapping shape parameters of last closed flux surface to equilibrium solutions for realistic nonlinear current profiles. Comprehensive benchmarking of five neural architectures identifies the novel Transformer‑KAN (Kolmogorov‑Arnold Network) Neural Operator (TKNO) as achieving highest accuracy (0.25% mean L2 relative error) under supervised training (only data‑driven). However, all data‑driven models exhibit large physics residuals, indicating poor physical consistency. Our unsupervised training can reduce the residuals by nearly four orders of magnitude through embedding physics‑based loss terms without labeled data. Critically, semi‑supervised learning‑‑integrating sparse labeled data (100 interior points) with physics constraints‑‑achieves optimal balance: 0.48% interpolation error and the most robust extrapolation performance (4.76% error, 8.9x degradation factor vs 39.8x for supervised models). Accelerated by TensorRT optimization, our models enable millisecond‑level inference, establishing PINO as a promising pathway for next‑generation fusion control systems.

Abstract:
Physics‑informed neural networks (PINNs) have shown promise for solving partial differential equations, yet their success in simulating incompressible flows at high Reynolds numbers remains limited. Existing approaches rely on auxiliary labeled data, supervised pretraining, or reference solutions, and no purely unsupervised method comparable to conventional finite‑difference or finite‑volume solvers has been demonstrated. We attribute this gap to the absence of a mechanism for enforcing the divergence‑free constraint and boundary conditions to strict tolerances. To address this, we adopt the physics‑ and equality‑constrained artificial neural network (PECANN) framework with a conditionally adaptive augmented Lagrangian method (CA‑ALM), and introduce a pressure‑Poisson‑based objective. The residual of the pressure Poisson equation is minimized subject to the momentum and continuity equations and boundary conditions on the primitive variables as equality constraints, with CA‑ALM enforcing all constraints tightly. For advection‑dominated, high‑Reynolds‑number flows, we further propose an adaptive vanishing entropy viscosity that stabilizes early training without influencing the converged solution. A baseline that instead uses the momentum residual as the objective proves ineffective under the same machinery, underscoring the critical role of the pressure‑Poisson objective. The method is assessed on lid‑driven cavity flow up to Re=7,500, three‑dimensional unsteady Beltrami flow, and steady and unsteady flow past a circular cylinder with general inflow‑outflow boundary conditions, including an ablation study identifying admissible outlet conditions ‑‑ all without labeled data or supervised pretraining. Notably, it captures the spontaneous onset of periodic vortex shedding in unsteady cylinder flow without external perturbations, starting from a randomly initialized network.

Abstract:
Physics‑informed neural networks (PINNs) typically minimize average residuals, which can conceal large, localized errors. We propose Residual Risk‑Aware Physics‑Informed Neural Networks PINNs (RRaPINNs), a single‑network framework that optimizes tail‑focused objectives using Conditional Value‑at‑Risk (CVaR), we also introduced a Mean‑Excess (ME) surrogate penalty to directly control worst‑case PDE residuals. This casts PINN training as risk‑sensitive optimization and links it to chance‑constrained formulations. The method is effective and simple to implement. Across several partial differential equations (PDEs) such as Burgers, Heat, Korteweg‑de‑Vries, and Poisson (including a Poisson interface problem with a source jump at x=0.5) equations, RRaPINNs reduce tail residuals while maintaining or improving mean errors compared to vanilla PINNs, Residual‑Based Attention and its variant using convolution weighting; the ME surrogate yields smoother optimization than a direct CVaR hinge. The chance constraint reliability level α acts as a transparent knob trading bulk accuracy (lower α ) for stricter tail control (higher α ). We discuss the framework limitations, including memoryless sampling, global‑only tail budgeting, and residual‑centric risk, and outline remedies via persistent hard‑point replay, local risk budgets, and multi‑objective risk over BC/IC terms. RRaPINNs offer a practical path to reliability‑aware scientific ML for both smooth and discontinuous PDEs.

Abstract:
Autonomous aerial vehicles (AAVs) have played a pivotal role in coverage operations and search missions. Recent advances in large language models (LLMs) offer promising opportunities to augment AAV intelligence. These advances help address complex challenges like area coverage optimization, dynamic path planning, and adaptive decision‑making. However, the absence of physical grounding in LLMs leads to hallucination and reproducibility problems in spatial reasoning and decision‑making. To tackle these issues, we present Skypilot, an LLM‑enhanced two‑stage framework that grounds language models in physical reality by integrating monte carlo tree search (MCTS). In the first stage, we introduce a diversified action space that encompasses generate, regenerate, fine‑tune, and evaluate operations, coupled with physics‑informed reward functions to ensure trajectory feasibility. In the second stage, we fine‑tune Qwen3‑4B on 23,000 MCTS‑generated samples, achieving substantial inference acceleration while maintaining solution quality. Extensive numerical simulations and real‑world flight experiments validate the efficiency and superiority of our proposed approach. Detailed information and experimental results are accessible at https://sky‑pilot.top.

Abstract:
Current control algorithms for aerial robots struggle with robustness in dynamic environments and adverse conditions. Model‑based reinforcement learning (RL) has shown strong potential in handling these challenges while remaining sample‑efficient. Additionally, Dreamer has demonstrated that online model‑based RL can be achieved using a recurrent world model trained on replay buffer data. However, applying Dreamer to aerial systems has been quite challenging due to its sample inefficiency and poor generalization of dynamics models. Our work explores a physics‑informed approach to world model learning and improves policy performance. The world model treats the quadcopter as a free‑body system and predicts the net forces and moments acting on it, which are then passed through a 6‑DOF Runge‑Kutta integrator (RK4) to predict future state rollouts. In this paper, we compare this physics‑informed method to a standard RNN‑based world model. Although both models perform well on the training data, we observed that they fail to generalize to new trajectories, leading to rapid divergence in state rollouts, preventing policy convergence.

Abstract:
Neural PDE solvers used for scientific simulation often violate governing equation constraints. While linear constraints can be projected cheaply, many constraints are nonlinear, complicating projection onto the feasible set. Dynamical PDEs are especially difficult because constraints induce long‑range dependencies in time. In this work, we evaluate two training‑free, post hoc projections of approximate solutions: a nonlinear optimization‑based projection, and a local linearization‑based projection using Jacobian‑vector and vector‑Jacobian products. We analyze constraints across representative PDEs and find that both projections substantially reduce violations and improve accuracy over physics‑informed baselines.

Abstract:
Sweat evaporation is the body's primary cooling mechanism, yet the physical factors governing it are not fully understood. We identify a dueling buoyancy effect in the context of the human body, in which perspiration vapor reduces the near skin air density, counteracting the downward flow driven by cooling of warm air upon contact with the skin. In hot, arid, stagnant environments, this opposing buoyancy suppresses free convection and can reduce sweat evaporation by more than half. As a result, commonly used thermoregulation models can substantially underpredict body temperature (e.g., by 1C after 2 hours of exposure to typical Arizona summer conditions). We develop compact, physics informed models for free convective heat transfer coefficients across wide temperature and humidity ranges, enabling improved thermoregulation modeling and thermal audits. These results enhance understanding of human heat balance and support more accurate heat stress assessment to inform behavioral, infrastructural, and policy decisions for extreme heat adaptations.

Abstract:
Precise pose estimation of optical microrobots is essential for enabling high‑precision object tracking and autonomous biological studies. However, current methods rely heavily on large, high‑quality microscope image datasets, which are difficult and costly to acquire due to the complexity of microrobot fabrication and the labour‑intensive labelling. Digital twin systems offer a promising path for sim‑to‑real data augmentation, yet existing techniques struggle to replicate complex optical microscopy phenomena, such as diffraction artifacts and depth‑dependent imaging.This work proposes a novel physics‑informed deep generative learning framework that, for the first time, integrates wave optics‑based physical rendering and depth alignment into a generative adversarial network (GAN), to synthesise high‑fidelity microscope images for microrobot pose estimation efficiently. Our method improves the structural similarity index (SSIM) by 35.6% compared to purely AI‑driven methods, while maintaining real‑time rendering speeds (0.022 s/frame).The pose estimator (CNN backbone) trained on our synthetic data achieves 93.9%/91.9% (pitch/roll) accuracy, just 5.0%/5.4% (pitch/roll) below that of an estimator trained exclusively on real data. Furthermore, our framework generalises to unseen poses, enabling data augmentation and robust pose estimation for novel microrobot configurations without additional training data.

Abstract:
Model Predictive Control evolved as the state of the art paradigm for safety critical control tasks. Control‑as‑Inference approaches thereof model the constrained optimization problem as a probabilistic inference problem. The constraints have to be implemented into the inference model. A recently introduced physics‑informed Gaussian Process method uses Control‑as‑Inference with a Gaussian likelihood for state constraint modeling, but lacks guarantees of open‑loop constraint satisfaction. We mitigate the lack of guarantees via an additional sampling step using Hamiltonian Monte Carlo sampling in order to obtain safe rollouts of the open‑loop dynamics which are then used to obtain an approximation of the truncated normal distribution which has full probability mass in the safe area. We provide formal guarantees of constraint satisfaction while maintaining the ODE structure of the Gaussian Process on a discretized grid. Moreover, we show that we are able to perform optimization of a quadratic cost function by closed form Gaussian Process computations only and introduce the Matérn kernel into the inference model.

Abstract:
In recent years, there has been an increasing need for Nuclear Power Plants (NPPs) to improve flexibility in order to match the rapid growth of renewable energies. The Operator Assistance Predictive System (OAPS) developed by Framatome addresses this problem through Model Predictive Control (MPC). In this work, we aim to improve MPC methods through data‑driven simulation schemes. Thus, from a set of nonlinear stiff ordinary differential equations (ODEs), this paper introduces two surrogate models acting as alternative simulation schemes to enhance nuclear reactor core simulation. We show that both data‑driven and physics‑informed models can rapidly integrate complex dynamics, with a very low computational time (up to 1000x time reduction).

Abstract:
Accurate and efficient thermal dynamics models of permanent magnet synchronous motors are vital to efficient thermal management strategies. Physics‑informed methods combine model‑based and data‑driven methods, offering greater flexibility than model‑based methods and superior explainability compared to data‑driven methods. Nonetheless, there are still challenges in balancing real‑time performance, estimation accuracy, and explainability. This paper presents a hardware‑efficient complex neural dynamics model achieved through the linear decoupling, diagonalization, and reparameterization of the state‑space model, introducing a novel paradigm for the physics‑informed method that offers high explainability and accuracy in electric motor temperature estimation tasks. We validate this physics‑informed method on an NVIDIA A800 GPU using the JAX machine learning framework, parallel prefix sum algorithm, and Compute Unified Device Architecture (CUDA) platform. We demonstrate its superior estimation accuracy and parallelizable hardware acceleration capabilities through experimental evaluation on a real electric motor.

Abstract:
Reconstructing complete traffic flow time‑space diagrams from vehicle trajectories offer a comprehensive view on traffic dynamics at arterial intersections. However, obtaining full trajectories across networks is costly, and accurately inferring lane‑changing (LC) and car‑following behaviors in multi‑lane environments remains challenging. This study proposes a generative framework for arterial vehicle trajectory reconstruction that jointly models lane‑changing and car‑following behaviors through physics‑informed multi‑task joint learning. The framework consists of a Lane‑Change Generative Adversarial Network (LC‑GAN) and a Trajectory‑GAN. The LC‑GAN models stochastic LC behavior from historical trajectories while considering physical conditions of arterial intersections, such as signal control, geometric configuration, and interactions with surrounding vehicles. The Trajectory‑GAN then incorporates LC information from the LC‑GAN with initial trajectories generated from physics‑based car‑following models, refining them in a data‑driven manner to adapt to dynamic traffic conditions. The proposed framework is designed to reconstruct complete trajectories from only a small subset of connected vehicle (CV) trajectories; for example, even a single observed trajectory per lane, by incorporating partial trajectory information into the generative process. A multi‑task joint learning facilitates synergistic interaction between the LC‑GAN and Trajectory‑GAN, allowing each component to serves as both auxiliary supervision and a physical condition for the other. Validation using two real‑world trajectory datasets demonstrates that the framework outperforms conventional benchmark models in reconstructing complete time‑space diagrams for multi‑lane arterial intersections. This research advances the integration of trajectory‑based sensing from CVs with physics‑informed deep learning.

Abstract:
Underground pipeline leaks and infiltrations pose significant threats to water security and environmental safety. Traditional manual inspection methods provide limited coverage and delayed response, often missing critical anomalies. This paper proposes AquaSentinel, a novel physics‑informed AI system for real‑time anomaly detection in urban underground water pipeline networks. We introduce four key innovations: (1) strategic sparse sensor deployment at high‑centrality nodes combined with physics‑based state augmentation to achieve network‑wide observability from minimal infrastructure; (2) the RTCA (Real‑Time Cumulative Anomaly) detection algorithm, which employs dual‑threshold monitoring with adaptive statistics to distinguish transient fluctuations from genuine anomalies; (3) a Mixture of Experts (MoE) ensemble of spatiotemporal graph neural networks that provides robust predictions by dynamically weighting model contributions; (4) causal flow‑based leak localization that traces anomalies upstream to identify source nodes and affected pipe segments. Our system strategically deploys sensors at critical network junctions and leverages physics‑based modeling to propagate measurements to unmonitored nodes, creating virtual sensors that enhance data availability across the entire network. Experimental evaluation using 110 leak scenarios demonstrates that AquaSentinel achieves 100% detection accuracy. This work advances pipeline monitoring by demonstrating that physics‑informed sparse sensing can match the performance of dense deployments at a fraction of the cost, providing a practical solution for aging urban infrastructure.

Abstract:
We introduce SpectralPINN, a hybrid pseudo‑spectral/physics‑informed neural network (PINN) solver for Kerr quasinormal modes that targets the Teukolsky equation in both the separated (radial/angular) and joint two‑dimensional formulations. The solver replaces standard neural activation functions with Chebyshev polynomials of the first kind and supports both soft ‑‑ via loss penalties ‑‑ and hard ‑‑ enforced by analytic masks ‑‑ implementations of Leaver's normalization. Benchmarking against Leaver's continued‑fraction method shows cumulative (real+imaginary part) relative frequency errors of ～ 0.001% for the separated formulation with hard normalization, ～ 0.1% for both the soft separated and soft joint formulations, and ～ 0.01% for the hard joint case. Exploiting our ability to solve the joint equation, we add a small quadrupolar perturbation to the Teukolsky operator, effectively rendering the problem non‑separable. The resulting perturbed quasinormal modes are compared against the expected precision of the Einstein Telescope, allowing us to constrain the magnitude of the perturbation. These proof‑of‑concept results demonstrate that hybrid spectral‑PINN solvers can provide a flexible pathway to quasinormal spectra in settings where separability, asymptotics, or field content become more intricate and high accuracy is required.

Abstract:
Parameter estimation remains a challenging task across many areas of engineering. Because data acquisition can often be costly, limited, or prone to inaccuracies (noise, uncertainty) it is crucial to identify sensor configurations that provide the maximum amount of information about the unknown parameters, in particular for the case of distributed‑parameter systems, where spatial variations are important. Physics‑Informed Neural Networks (PINNs) have recently emerged as a powerful machine‑learning (ML) tool for parameter estimation, particularly in cases with sparse or noisy measurements, overcoming some of the limitations of traditional optimization‑based and Bayesian approaches. Despite the widespread use of PINNs for solving inverse problems, relatively little attention has been given to how their performance depends on sensor placement. This study addresses this gap by introducing a comprehensive PINN‑based framework that simultaneously tackles optimal sensor placement and parameter estimation. Our approach involves training a PINN model in which the parameters of interest are included as additional inputs. This enables the efficient computation of sensitivity functions through automatic differentiation, which are then used to determine optimal sensor locations exploiting the D‑optimality criterion. The framework is validated on two illustrative distributed‑parameter reaction‑diffusion‑advection problems of increasing complexity. The results demonstrate that our PINNs‑based methodology consistently achieves higher accuracy compared to parameter values estimated from intuitively or randomly selected sensor positions.

Abstract:
Gamma‑ray bursts (GRBs) are challenging to identify due to their transient nature, complex temporal profiles, and limited observational datasets. We address this with a one‑dimensional convolutional neural network integrated with an Adaptive Frequency Feature Enhancement module and physics‑informed data augmentation. Our framework generates 100,000 synthetic GRB samples, expanding training data diversity and volume while preserving physical fidelity‑especially for low‑significance events. The model achieves 97.46% classification accuracy, outperforming all tested variants with conventional enhancement modules, highlighting enhanced domain‑specific feature capture. Feature visualization shows model focuses on deep‑seated morphological features and confirms the capability of extracting physically meaningful burst characteristics. Dimensionality reduction and clustering reveal GRBs with similar morphologies or progenitor origins cluster in the feature space, linking learned features to physical properties. This perhaps offers a novel diagnostic tool for identifying kilonova‑ and supernova‑associated GRB candidates, establishing criteria to enhance multi‑messenger early‑warning systems. The framework aids current time‑domain surveys, generalizes to other rare transients, and advances automated detection in large‑volume observational data.

Abstract:
Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation. However, these methods face significant computational challenges when applied to high‑frequency wave problems in complex two‑dimensional domains. This work investigates Finite Basis Physics‑Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative. These methods leverage domain decomposition, partitioning the computational domain into overlapping sub‑domains, each governed by a local neural network. We assess their accuracy and computational efficiency in solving the Helmholtz equation for the homogeneous case, demonstrating their potential to mitigate the limitations of traditional approaches.

Abstract:
The gravitational collapse of a massless scalar field remains a demanding benchmark for numerical methods in numerical relativity, as it exhibits critical behavior at the boundary between dispersion and black hole formation. In this work we revisit this problem by relying on Physics‑Informed Neural Networks (PINNs) as flexible solvers for partial differential equations, thereby providing a comparative assessment of several recent neural architectures. Building on the Einstein‑massless‑Klein‑Gordon formulation in polar‑areal coordinates, we consider four initial‑value problems encompassing subcritical, critical, and supercritical regimes and use high‑resolution finite‑difference simulations as reference solutions. Our study is primarily comparative: we evaluate several state‑of‑the‑art deep learning architectures, including vanilla and high‑precision PINNs, sinusoidal‑feature and quadratic‑residual variants, and Kolmogorov‑Arnold Networks, all trained under a common loss design that encodes the field equations, boundary conditions, and causal time‑space enforcement, together with a novel adaptive spacetime sampling. Within this framework we also introduce ModPINN, a modest modification of standard PINNs that augments standard multilayer perceptrons with coordinate embeddings, quadratic layers, and other common ingredients in recent literature. This study shows that deep‑learning‑based methods can reproduce finite‑difference solutions for the scalar field and the spacetime metric with competitive accuracy using significantly fewer collocation points than more traditional methodologies. While no single architecture dominates in all regimes, ModPINN achieves particularly stable and accurate solutions near criticality, indicating that suitably designed embeddings and adaptive sampling can enhance the robustness of PINNs for challenging gravitational‑collapse scenarios.

Abstract:
Efficient and sustainable maritime transport increasingly depends on reliable forecasting and adaptive routing, yet operational adoption remains difficult due to forecast latencies and the need for human judgment in rapid decision‑making under changing ocean conditions. We introduce SWR‑Viz, an AI‑assisted visual analytics framework that combines a physics‑informed Fourier Neural Operator wave forecast model with SIMROUTE‑based routing and interactive emissions analytics. The framework generates near‑term forecasts directly from current conditions, supports data assimilation with sparse observations, and enables rapid exploration of what‑if routing scenarios. We evaluate the forecast models and SWR‑Viz framework along key shipping corridors in the Japan Coast and Gulf of Mexico, showing both improved forecast stability and realistic routing outcomes comparable to ground‑truth reanalysis wave products. Expert feedback highlights the usability of SWR‑Viz, its ability to isolate voyage segments with high emission reduction potential, and its value as a practical decision‑support system. More broadly, this work illustrates how lightweight AI forecasting can be integrated with interactive visual analytics to support human‑centered decision‑making in complex geospatial and environmental domains.

Abstract:
Nonlinear physical phenomena often show complex multiscale interactions; motivated by the principles of multiscale modeling in scientific computing, we propose PAS‑Net, a physics‑informed Adaptive‑Scale Deep Operator Network for learning solution operators of nonlinear and singularly perturbed evolution PDEs with small parameters and localized features. Specifically, PAS‑Net augments the trunk input in the physics informed Deep Operator Network (PI‑DeepONet) with a prescribed (or learnable) locally rescaled coordinate transformation centered at reference points. This addition introduces a multiscale feature embedding that acts as an architecture‑independent preconditioner which improves the representation of localized, stiff, and multiscale dynamics. From an optimization perspective, the adaptive‑scale embedding in PAS‑Net modifies the geometry of the Neural Tangent Kernel (NTK) associated with the neural network by increasing its smallest eigenvalue, which in turn improves spectral conditioning and accelerates gradient‑based convergence. We further show that this adaptive‑scale mechanism explicitly accelerates neural network training in approximating functions with steep transitions and strong asymptotic behavior, and we provide a rigorous proof of this function‑approximation result within the finite‑dimensional NTK matrix framework. We test the proposed PAS‑Net on three different problems: (i) the one‑dimensional viscous Burgers equation, (ii) a nonlinear diffusion‑reaction system with sharp spatial gradients, and (iii) a two‑dimensional eikonal equation. The numerical results show that PAS‑Net consistently achieves higher accuracy and faster convergence than the standard DeepONet and PI‑DeepONet models under a similar training cost.

Abstract:
Restoring power distribution systems (PDSs) after large‑scale outages requires sequential switching actions that reconfigure feeder topology and coordinate distributed energy resources (DERs) under nonlinear constraints, including power balance, voltage limits, and thermal ratings. These challenges limit the scalability of conventional optimization and value‑based reinforcement learning (RL) approaches. This paper applies a Heterogeneous‑Agent Reinforcement Learning (HARL) framework via Heterogeneous‑Agent Proximal Policy Optimization (HAPPO) to enable coordinated restoration across interconnected microgrids. Each agent controls a distinct microgrid with different loads, DER capacities, and switch counts. Decentralized actors are trained with a centralized critic for stable on‑policy learning, while a physics‑informed OpenDSS environment enforces electrical feasibility. Experiments on IEEE 123‑bus and 8500‑node feeders show HAPPO outperforms PPO, QMIX, Mean‑Field RL, and other baselines in restored power, convergence stability, and multi‑seed reproducibility. Under a 2400 kW generation cap, the framework restores over 95% of available load on both systems with low‑latency execution, supporting practical real‑time PDS restoration.

Abstract:
In recent years, Physics‑Informed Neural Networks (PINNs) have emerged as a powerful and robust framework for solving nonlinear differential equations across a wide range of scientific and engineering disciplines, including biology, geophysics, astrophysics and fluid dynamics. In the PINN framework, the governing partial differential equations, along with initial and boundary conditions, are encoded directly into the loss function, enabling the network to learn solutions that are consistent with the underlying physics. In this work, we employ the PINN framework to solve the dimensionless Navier‑Stokes equations for three two‑dimensional incompressible, steady, laminar flow problems without using any labeled data. The boundary and initial conditions are enforced in a hard manner, ensuring they are satisfied exactly rather than penalized during training. We validate the PINN predicted velocity profiles, drag coefficients and pressure profiles against the conventional computational fluid dynamics (CFD) simulations for moderate to high values of Reynolds number (Re). It is observed that the PINN predictions show good agreement with the CFD results at lower Re. We also extend our analysis to a transient condition and find that our method is equally capable of simulating complex time‑dependent flow dynamics. To quantitatively assess the accuracy, we compute the L_2 normalized error, which lies in the range \mathcalO(10^‑4) ‑ \mathcalO(10^‑1) for our chosen case studies.

Abstract:
Physics‑informed neural networks (PINNs) represent a new paradigm for solving partial differential equations (PDEs) by integrating physical laws into the learning process of neural networks. However, ensuring that such frameworks fully reflect the physical structure embedded in the governing equations remains an open challenge, particularly for maintaining robustness across diverse scientific problems. In this work, we address this issue by introducing a simple, generalized, yet robust irreversibility‑regularized strategy that enforces hidden physical laws as soft constraints during training, thereby recovering the missing physics associated with irreversible processes in the conventional PINN. This approach ensures that the learned solutions consistently respect the intrinsic one‑way nature of irreversible physical processes. Across a wide range of benchmarks spanning traveling wave propagation, steady combustion, ice melting, corrosion evolution, and crack growth, we observe substantial performance improvements over the conventional PINN, demonstrating that our regularization scheme reduces predictive errors by more than an order of magnitude, while requiring only minimal modification to existing PINN frameworks.

Abstract:
Weather forecasting is fundamentally challenged by the chaotic nature of the atmosphere, necessitating probabilistic approaches to quantify uncertainty. While traditional ensemble prediction (EPS) addresses this through computationally intensive simulations, recent advances in Bayesian Deep Learning (BDL) offer a promising but often disconnected alternative. We bridge these paradigms through a unified hybrid Bayesian Deep Learning framework for ensemble weather forecasting that explicitly decomposes predictive uncertainty into epistemic and aleatoric components, learned via variational inference and a physics‑informed stochastic perturbation scheme modeling flow‑dependent atmospheric dynamics, respectively. We further establish a unified theoretical framework that rigorously connects BDL and EPS, providing formal theorems that decompose total predictive uncertainty into epistemic and aleatoric components under the hybrid BDL framework. We validate our framework on the large‑scale 40‑year ERA5 reanalysis dataset (1979‑2019) with 0.25° spatial resolution. Experimental results show that our method not only improves forecast accuracy and yields better‑calibrated uncertainty quantification but also achieves superior computational efficiency compared to state‑of‑the‑art probabilistic diffusion models. We commit to making our code open‑source upon acceptance of this paper.

Abstract:
This study presents a novel deep learning approach aimed at enhancing stochastic Gust Load Alleviation (GLA) specifically for compliant wings. The approach incorporates the concept of smooth wing camber variation, where the camber of the wing's chord is actively adjusted during flight using a control signal to achieve the desired aerodynamic loading. The proposed method employs a deep learning‑based model predictive controller designed for probability density shaping. This controller effectively solves the probability density evolution equation through a custom Physics‑Informed Neural Network (PINN) and utilizes Automatic Differentiation for Model Predictive Control (MPC) optimization. Comprehensive numerical simulations were conducted on a compliant wing (CW) model, evaluating performance of the proposed approach against stochastic gust profiles. The evaluation involved stochastic aerodynamic loads generated from Band‑Limited White Noise (BLWN) and Dryden gust models. The evaluation were conducted for two distinct Compliant Chord Fractions (CCF). The results demonstrate the effectiveness of the proposed probability density shaping model predictive control in alleviating stochastic gust load and reducing wing tip deflection.

Abstract:
The accurate solution of nonlinear hyperbolic partial differential equations (PDEs) remains challenging due to steep gradients, discontinuities, and multiscale structures that make conventional solvers computationally demanding. Physics‑Informed Neural Networks (PINNs) embed the governing equations into the learning process, enabling mesh‑free solution of PDEs, yet they often struggle to capture steep gradients, discontinuities, and complex nonlinear wave interactions. To address these limitations, we employ the Extended Physics‑Informed Neural Network (XPINN) framework to solve the nonlinear Buckley‑Leverett equation with a nonconvex flux, modeling immiscible two‑phase flow in porous media. The computational domain is dynamically decomposed in space and time into evolving pre‑shock and post‑shock subdomains, allowing localized subnetworks to efficiently learn distinct flow behaviors, with coupling enforced via the Rankine‑Hugoniot jump condition to ensure physically consistent flux continuity. We compare XPINN with standard PINNs and its variants, including PINN with artificial viscosity, PINN with Welge construction, and PINN with the Oleinik entropy condition, and across all cases, XPINN consistently outperforms the other methods, accurately resolving sharp fronts and capturing the correct physical behavior. Importantly, XPINN achieves this using the simpler Adam optimizer, whereas some PINN variants require more complex or higher‑order strategies such as L‑BFGS to reach comparable accuracy, demonstrating that XPINN is a robust and scalable approach for challenging hyperbolic PDEs without artificial diffusion or entropy corrections. The code is available at github.com/saifkhanengr/XPINN‑for‑Buckley‑Leverett.

Abstract:
This work addresses the full‑information output regulation problem for nonlinear systems, assuming the states of both the plant and the exosystem are known. In this setting, perfect tracking or rejection is achieved by constructing a zero‑regulation‑error manifold π(w) and a feedforward input c(w) that render such manifold invariant. The pair (π(w), c(w)) is characterized by the regulator equations, i.e., a system of PDEs with an algebraic constraint. We focus on accurately solving the regulator equations introducing a physics‑informed neural network (PINN) approach that directly approximates π(w) and c(w) by minimizing the residuals under boundary and feasibility conditions, without requiring precomputed trajectories or labeled data. The learned operator maps exosystem states to steady state plant states and inputs, enables real‑time inference and, critically, generalizes across families of the exosystem with varying initial conditions and parameters. The framework is validated on a regulation task that synchronizes a helicopter's vertical dynamics with a harmonically oscillating platform. The resulting PINN‑based solver reconstructs the zero‑error manifold with high fidelity and sustains regulation performance under exosystem variations, highlighting the potential of learning‑enabled solvers for nonlinear output regulation. The proposed approach is broadly applicable to nonlinear systems that admit a solution to the output regulation problem.

Abstract:
Coherent anti‑Stokes Raman scattering (CARS) spectroscopy is a powerful and rapid technique widely used in medicine, material science, and chemical analyses. However, its effectiveness is hindered by the presence of a non‑resonant background that interferes with and distorts the true Raman signal. Deep learning methods have been employed to reconstruct the true Raman spectrum from measured CARS data using labeled datasets. A more recent development integrates the domain knowledge of Kramers‑Kronig relationships and smoothness constraints in the form of physics‑informed loss functions. However, these deterministic models lack the ability to quantify uncertainty, an essential feature for reliable deployment in high‑stakes scientific and biomedical applications. In this work, we evaluate and compare various uncertainty quantification (UQ) techniques within the context of CARS‑to‑Raman signal reconstruction. Furthermore, we demonstrate that incorporating physics‑informed constraints into these models improves their calibration, offering a promising path toward more trustworthy CARS data analysis.

Abstract:
With the development of digital twins and smart manufacturing systems, there is an urgent need for real‑time distortion field prediction to control defects in metal Additive Manufacturing (AM). However, numerical simulation methods suffer from high computational cost, long run‑times that prevent real‑time use, while conventional Machine learning (ML) models struggle to extract spatiotemporal features for long‑horizon prediction and fail to decouple thermo‑mechanical fields. This paper proposes a Physics‑informed Neural Operator (PINO) to predict z and y‑direction distortion for the future 15 s. Our method, Physics‑informed Deep Operator Network‑Recurrent Neural Network (PIDeepONet‑RNN) employs trunk and branch network to process temperature history and encode distortion fields, respectively, enabling decoupling of thermo‑mechanical responses. By incorporating the heat conduction equation as a soft constraint, the model ensures physical consistency and suppresses unphysical artifacts, thereby establishing a more physically consistent mapping between the thermal history and distortion. This is important because such a basis function, grounded in physical laws, provides a robust and interpretable foundation for predictions. The proposed models are trained and tested using datasets generated from experimentally validated Finite Element Method (FEM). Evaluation shows that the model achieves high accuracy, low error accumulation, time efficiency. The max absolute errors in the z and y‑directions are as low as 0.9733 mm and 0.2049 mm, respectively. The error distribution shows high errors in the molten pool but low gradient norms in the deposited and key areas. The performance of PINO surrogate model highlights its potential for real‑time long‑horizon physics field prediction in controlling defects.

Abstract:
The semiconductor industry faces a computational crisis in extreme ultraviolet (EUV) lithography optimization, where traditional methods consume billions of CPU hours while failing to achieve sub‑nanometer precision. We present a physics‑constrained adaptive learning framework that automatically calibrates electromagnetic approximations through learnable parameters \boldsymbolθ = \θ_d, θ_a, θ_b, θ_p, θ_c\ while simultaneously minimizing Edge Placement Error (EPE) between simulated aerial images and target photomasks. The framework integrates differentiable modules for Fresnel diffraction, material absorption, optical point spread function blur, phase‑shift effects, and contrast modulation with direct geometric pattern matching objectives, enabling cross‑geometry generalization with minimal training data. Through physics‑constrained learning on 15 representative patterns spanning current production to future research nodes, we demonstrate consistent sub‑nanometer EPE performance (0.664‑2.536 nm range) using only 50 training samples per pattern. Adaptive physics learning achieves an average improvement of 69.9% over CNN baselines without physics constraints, with a significant inference speedup over rigorous electromagnetic solvers after training completion. This approach requires 90% fewer training samples through cross‑geometry generalization compared to pattern‑specific CNN training approaches. This work establishes physics‑constrained adaptive learning as a foundational methodology for real‑time semiconductor manufacturing optimization, addressing the critical gap between academic physics‑informed neural networks and industrial deployment requirements through joint physics calibration and manufacturing precision objectives.

Abstract:
We introduce a Variational Autoencoder (VAE)‑‑Normalizing Flow (NF) framework for rapid probabilistic inference of galaxy properties and emission line fluxes at z \leq 0.3 from SDSS gri imaging and photometry. Our model probabilistically infers stellar mass, star formation rate (SFR), redshift, gas‑phase metallicity, and central black hole mass for a given galaxy. The model accruacy matches current non‑spectroscopic methods for stellar mass and redshift, surpasses them for SFR and metallicity, and introduces the first probabilistic central black hole mass estimates from imaging + photometry. It also delivers probabilistic estimates of Hα, Hβ, [N~\textscii], and [O~\textsciii] emission line fluxes directly from imaging, enabling SFR, metallicity, dust, and AGN/shock diagnostics without spectroscopy. This approach opens new pathways for scalable, physics‑informed inference in upcoming surveys such as Roman and Rubin LSST.

Abstract:
This paper introduces Quantum Orthogonal Separable Physics‑Informed Neural Networks (QO‑SPINNs), a novel architecture for solving Partial Differential Equations, integrating quantum computing principles to address the computational bottlenecks of classical methods. We leverage a quantum algorithm for accelerating matrix multiplication within each layer, achieving a \mathcal O(d\log d/ε^2) complexity, a significant improvement over the classical \mathcal O(d^2) complexity, where d is the dimension of the matrix, ε the accuracy level. This is accomplished by using a Hamming weight‑preserving quantum circuit and a unary basis for data encoding, with a comprehensive theoretical analysis of the overall architecture provided. We demonstrate the practical utility of our model by applying it to solve both forward and inverse PDE problems. Furthermore, we exploit the inherent orthogonality of our quantum circuits (which guarantees a spectral norm of 1) to develop a novel uncertainty quantification method. Our approach adapts the Spectral Normalized Gaussian Process for SPINNs, eliminating the need for the computationally expensive spectral normalization step. By using a Quantum Orthogonal SPINN architecture based on stacking, we provide a robust and efficient framework for uncertainty quantification (UQ) which, to our knowledge, is the first UQ method specifically designed for Separable PINNs. Numerical results based on classical simulation of the quantum circuits, are presented to validate the theoretical claims and demonstrate the efficacy of the proposed method.

Abstract:
Physics‑informed neural networks (PINN) face significant challenges from spectral bias, which impedes their ability to model high‑frequency phenomena and limits extrapolation performance. To address this, we introduce xLSTM‑PINN, a novel architecture that performs representation‑level spectral remodeling through memory gating and residual micro‑steps. Our method consistently achieves markedly lower spectral error and root mean square error (RMSE) across four diverse partial differential equation (PDE) benchmarks, along withhhh a broader stable learning‑rate window. Frequency‑domain analysis confirms that xLSTM‑PINN elevates high‑frequency kernel weights, shifts the resolvable bandwidth rightward, and shortens the convergence time for high‑wavenumber components. Without modifying automatic differentiation or physics loss constraints, this work provides a robust pathway to suppress spectral bias, thereby improving accuracy, reproducibility, and transferability in physics‑informed learning.

Abstract:
l flows and flat‑plate boundary layers. However, it predicts too low a turbulent kinetic energy. This is a feature it shares with most other two‑equation turbulence models. When comparing the terms in the k equations with DNS data it is found that the production and dissipation terms are well predicted but the turbulent diffusion is not. In the present work the poor modeling of the turbulent diffusion is improved using Physics Informed Neural Network (PINN) and Neural Network (NN).The k equation is turned into an ordinary differential equation for the turbulent viscosity in the k equation, nu_t,PINN, which is solved using PINN. A new turbulent Prandtl number is then computed as sigma_k = nu_t/nu_t,PINN where nu_t = k/omega.To compensate for the new, larger turbulent kinetic energy, three coefficients in the new k‑omega model are computed using three NN models. The new turbulence model, called the k‑omega‑PINN‑NN model, is shown to produce excellent velocity, skin friction and turbulent kinetic profiles in channel flow at Re_tau = 2 000, 5 200 and Re_tau = 10 000 as well as in flat‑plate boundary layer flow (slightly too large a k for the latter case). The k‑omega‑PINN‑NN model is also used for predicting the flow over a periodic hill and the agreement with DNS is very good. At the end of the Conclusions, we give an example on how a NN model can be replaced with a Python symbolic regression (pySR); the latter may conveniently be imported in commercial CFD codes. All Python PINN, NN and pySR scripts as well as the Python CFD code can be downloaded (Davidson, 2025a).

Abstract:
Modeling indoor radio propagation is crucial for wireless network planning and optimization. However, existing approaches often rely on labor‑intensive manual modeling of geometry and material properties, resulting in limited scalability and efficiency. To overcome these challenges, this paper presents SenseRay‑3D, a generalizable and physics‑informed end‑to‑end framework that predicts three‑dimensional (3D) path‑loss heatmaps directly from RGB‑D scans, thereby eliminating the need for explicit geometry reconstruction or material annotation. The proposed framework builds a sensing‑driven voxelized scene representation that jointly encodes occupancy, electromagnetic material characteristics, and transmitter‑receiver geometry, which is processed by a SwinUNETR‑based neural network to infer environmental path‑loss relative to free‑space path‑loss. A comprehensive synthetic indoor propagation dataset is further developed to validate the framework and to serve as a standardized benchmark for future research. Experimental results show that SenseRay‑3D achieves a mean absolute error of 4.27 dB on unseen environments and supports real‑time inference at 217 ms per sample, demonstrating its scalability, efficiency, and physical consistency. SenseRay‑3D paves a new path for sense‑driven, generalizable, and physics‑consistent modeling of indoor propagation, marking a major leap beyond our pioneering EM DeepRay framework.

Abstract:
Monitoring battery health is essential for ensuring safe and efficient operation. However, there is an inherent trade‑off between assessment speed and diagnostic depth‑specifically, between rapid overall health estimation and precise identification of internal degradation states. Capturing detailed internal battery information efficiently remains a major challenge, yet such insights are key to understanding the various degradation mechanisms. To address this, we develop a parameterized physics‑informed neural network (P‑PINNSPM) over the key aging‑related parameter space for a single particle model. The model can accurately predict internal battery variables across the parameter space and identifies internal parameters in about 30 seconds‑achieving a 47x speedup over the finite volume method‑while maintaining high accuracy. These parameters improve the battery state‑of‑health (SOH) estimation accuracy by at least 60.61%, compared to models without parameter incorporation. Moreover, they enable extrapolation to unseen SOH levels and support robust estimation across diverse charging profiles and operating conditions. Our results demonstrate the strong potential of physics‑informed machine learning to advance real‑time, data‑efficient, and physics‑aware battery management systems.

Abstract:
An adjoint formulation of energetic particle confinement in axisymmetric tokamak geometry is derived and evaluated using a physics‑informed neural network (PINN). The PINN estimates the mean escape time of energetic ions by solving an inhomogeneous adjoint of the drift kinetic equation with a Lorentz collision operator, yielding predictions of fast ion loss in tokamak geometry due to direct ion orbit loss and collisional transport. To our knowledge, this is the first time a PINN has been used to solve the drift kinetic equation in tokamak geometry, a challenging problem due to the large time scale separation between the rapid transit time of energetic ions and their slow collisional time scale. It is shown that a careful and intentional design of a PINN is able to learn the mean escape time across the majority of the plasma volume, suggesting a path toward constructing a rapid surrogate for use within a broader optimization framework.

Abstract:
Neural differential equations offer a powerful framework for modeling continuous‑time dynamics, but forecasting stiff biophysical systems remains unreliable. Standard Neural ODEs and physics informed variants often require orders of magnitude more iterations, and even then may converge to suboptimal solutions that fail to preserve oscillatory frequency or amplitude. We introduce PhysicsInformed Neural ODEs with with Scale‑Aware Residuals (PI‑NODE‑SR), a framework that combines a low‑order explicit solver (Heun method) residual normalisation to balance contributions between state variables evolving on disparate timescales. This combination stabilises training under realistic iteration budgets and avoids reliance on computationally expensive implicit solvers. On the Hodgkin‑Huxley equations, PI‑NODE‑SR learns from a single oscillation simulated with a stiff solver (Rodas5P) and extrapolates beyond 100 ms, capturing both oscillation frequency and near‑correct amplitudes. Remarkably, end‑to‑end learning of the vector field enables PI‑NODE‑SR to recover morphological features such as sharp subthreshold curvature in gating variables that are typically reserved for higher‑order solvers, suggesting that neural correction can offset numerical diffusion. While performance remains sensitive to initialisation, PI‑NODE‑SR consistently reduces long‑horizon errors relative to baseline Neural‑ODEs and PINNs, offering a principled route to stable and efficient learning of stiff biological dynamics.

Abstract:
Standard physics‑informed neural network implementations have produced large error rates when using these models to solve the regularized long wave (RLW) equation. Two improved PINN approaches were developed in this research: an adaptive approach with self‑adaptive loss weighting and a conservative approach enforcing explicit conservation laws. Three benchmark tests were used to demonstrate how effective PINN's are as they relate to the type of problem being solved (i.e., time dependent RLW equation). The first was a single soliton traveling along a line (propagation), the second was the interaction between two solitons, and the third was the evolution of an undular bore over the course of t=250. The results demonstrated that the effectiveness of PINNs are problem specific. The adaptive PINN was significantly better than both the conservative PINN and the standard PINN at solving problems involving complex nonlinear interactions such as colliding two solitons. The conservative approach was significantly better at solving problems involving long term behavior of single solitons and undular bores. However, the most important finding from this research is that explicitly enforcing conservation laws may be harmful to optimizing the solution of highly nonlinear systems of equations and therefore requires special training methods. The results from our adaptive and conservative approaches were within O(10^‑5) of established numerical solutions for the same problem, thus demonstrating that PINNs can provide accurate solutions to complex systems of partial differential equations without the need for a discretization of space or time (mesh free). Moreover, the finding from this research challenges the assumptions that conservation enforcement will always improve the performance of a PINN and provides researchers with guidelines for designing PINNs for use on specific types of problems.

Abstract:
Buried pipelines transporting oil and gas across geohazard‑prone regions are exposed to potential ground movement, leading to the risk of significant strain demand and structural failure. Reliability analysis, which determines the probability of failure after accounting for pertinent uncertainties, is essential for ensuring the safety of pipeline systems. However, traditional reliability analysis methods involving computationally intensive numerical models, such as finite element simulations of pipeline subjected to ground movement, have limited applications; this is partly because stochastic sampling approaches require repeated simulations over a large number of samples for the uncertain variables when estimating low probabilities. This study introduces Physics‑Informed Neural Network for Reliability Analysis (PINN‑RA) for buried pipelines subjected to ground movement, which integrates PINN‑based surrogate model with Monte Carlo Simulation (MCS) to achieve efficient reliability assessment. To enable its application under uncertain variables associated with soil properties and ground movement, the PINN‑based surrogate model is extended to solve a parametric differential equation system, namely the governing equation of pipelines embedded in soil with different properties. The findings demonstrate that PINN‑RA significantly reduces the computational effort required and thus accelerates reliability analysis. By eliminating the need for repetitive numerical evaluations of pipeline subjected to permanent ground movement, the proposed approach provides an efficient and scalable tool for pipeline reliability assessment, enabling rapid decision‑making in geohazard‑prone regions.

Abstract:
In this work, we propose a modified Hybrid Parallel Kolmogorov‑‑Arnold Network and Multilayer Perceptron Physics‑Informed Neural Network to overcome the high‑frequency and multiscale challenges inherent in Physics‑Informed Neural Networks. This proposed model features a trainable weighting parameter to optimize the convex combination of outputs from the Kolmogorov‑‑Arnold Network and the Multilayer Perceptron, thus maximizing the networks' capabilities to capture different frequency components. Furthermore, we adopt an overlapping domain decomposition technique to decompose complex problems into subproblems, which alleviates the challenge of global optimization. Benchmark results demonstrate that our method reduces training costs and improves computational efficiency compared with manual hyperparameter tuning in solving high‑frequency multiscale problems.

Abstract:
We present SCULPT (Supervised Clustering and Uncovering Latent Patterns with Training), a comprehensive software platform for analyzing tabulated high‑dimensional multi‑particle coincidence data from Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) experiments. The software addresses critical challenges in modern momentum spectroscopy by integrating advanced machine learning techniques with physics‑informed analysis in an interactive web‑based environment. SCULPT implements Uniform Manifold Approximation and Projection (UMAP) for non‑linear dimensionality reduction to reveal correlations in highly dimensional data. We also discuss potential extensions to deep autoencoders for feature learning, and genetic programming for automated discovery of physically meaningful observables. A novel adaptive confidence scoring system provides quantitative reliability assessments by evaluating user‑selected clustering quality metrics with predefined weights that reflect each metric's robustness. The platform features configurable molecular profiles for different experimental systems, interactive visualization with selection tools, and comprehensive data filtering capabilities. Utilizing a subset of SCULPT's capabilities, we analyze photo double ionization data measured using the COLTRIMS method for 3‑body dissociation of the D2O molecule, revealing distinct fragmentation channels and their correlations with physics parameters. The software's modular architecture and web‑based implementation make it accessible to the broader atomic and molecular physics community, significantly reducing the time required for complex multi‑dimensional analyses. This opens the door to finding and isolating rare events exhibiting non‑linear correlations on the fly during experimental measurements, which can help steer exploration and improve the efficiency of experiments.

Abstract:
We propose a framework for solving nonlinear partial differential equations (PDEs) by combining perturbation theory with one‑shot transfer learning in Physics‑Informed Neural Networks (PINNs). Nonlinear PDEs with polynomial terms are decomposed into a sequence of linear subproblems, which are efficiently solved using a Multi‑Head PINN. Once the latent representation of the linear operator is learned, solutions to new PDE instances with varying perturbations, forcing terms, or boundary/initial conditions can be obtained in closed form without retraining. We validate the method on KPP‑Fisher and wave equations, achieving errors on the order of 1e‑3 while adapting to new problem instances in under 0.2 seconds; comparable accuracy to classical solvers but with faster transfer. Sensitivity analyses show predictable error growth with epsilon and polynomial degree, clarifying the method's effective regime. Our contributions are: (i) extending one‑shot transfer learning from nonlinear ODEs to PDEs, (ii) deriving a closed‑form solution for adapting to new PDE instances, and (iii) demonstrating accuracy and efficiency on canonical nonlinear PDEs. We conclude by outlining extensions to derivative‑dependent nonlinearities and higher‑dimensional PDEs.

Abstract:
Time‑efficient estimation of muscle activations and forces across multi‑joint systems is critical for clinical assessment and assistive device control. However, conventional approaches are computationally expensive and lack a high‑quality labeled dataset for multi‑joint applications. To address these challenges, we propose a physics‑informed deep learning framework that estimates muscle activations and forces directly from kinematics. The framework employs a novel Multi‑Joint Cross‑Attention (MJCA) module with Bidirectional Gated Recurrent Unit (BiGRU) layers to capture inter‑joint coordination, enabling each joint to adaptively integrate motion information from others. By embedding multi‑joint dynamics, inter‑joint coupling, and external force interactions into the loss function, our Physics‑Informed MJCA‑BiGRU (PI‑MJCA‑BiGRU) delivers physiologically consistent predictions without labeled data while enabling time‑efficient inference. Experimental validation on two datasets demonstrates that PI‑MJCA‑BiGRU achieves performance comparable to conventional supervised methods without requiring ground‑truth labels, while the MJCA module significantly enhances inter‑joint coordination modeling compared to other baseline architectures.

Abstract:
We investigate photon‑‑pion discrimination in regimes where electromagnetic showers overlap at the scale of calorimeter granularity. Using full detector simulations with fine‑grained calorimeter segmentation of approximately 0.025×0.025 in (η,ϕ), we benchmark three approaches: boosted decision trees (BDTs) on shower‑shape variables, dense neural networks (DNNs) on the same features, and a ResNet‑based convolutional neural network operating directly on calorimeter cell energies. The ResNet significantly outperformed both baseline methods, achieving further gains when augmented with soft scoring and an auxiliary ΔR regression head. Our results demonstrate that residual convolutional architectures, combined with physics‑informed loss functions, can substantially improve photon identification in high‑luminosity collider environments in which overlapping electromagnetic showers challenge traditional methods.

Abstract:
Accurate modeling of gravitational waveforms from compact binary coalescences remains central to gravitational‑wave (GW) astronomy. Post‑Newtonian (PN) approximations capture the early inspiral dynamics analytically but break down near merger, while numerical relativity (NR) provides the accurate yet computationally expensive waveforms over limited parameter ranges. We develop a physics‑informed neural network (PINN) framework that learns corrections mapping PN dynamics and waveforms to their NR counterparts. As a demonstration of the approach, we use the TaylorT4 PN model as the baseline, and train the network on a remarkably small dataset of only eight hybridized NR surrogate waveforms (NRHybSur3dq8) to learn higher‑order corrections to the orbital dynamics and waveform modes for nonspinning noneccentric systems. Physically motivated loss terms enforce known limits and symmetries, such as vanishing corrections in the Newtonian limit and suppression of odd‑m modes in equal‑mass systems, promoting consistent and reliable extrapolation beyond the training region. We simultaneously incorporate corrections that account for the different meaning of mass parameters in PN and NR descriptions. The learned corrections significantly reduce the phase and amplitude error through the inspiral up to about 200M before the merger. This approach provides a differentiable and computationally efficient bridge between PN and NR, offering a path toward waveform models that generalize more robustly beyond existing NR datasets.

Abstract:
Accurate retrieval of vegetation biophysical variables from satellite imagery is crucial for ecosystem monitoring and agricultural management. In this work, we propose a physics‑informed Transformer‑VAE architecture to invert the PROSAIL radiative transfer model for simultaneous estimation of key canopy parameters from Sentinel‑2 data. Unlike previous hybrid approaches that require real satellite images for self‑supevised training. Our model is trained exclusively on simulated data, yet achieves performance on par with state‑of‑the‑art methods that utilize real imagery. The Transformer‑VAE incorporates the PROSAIL model as a differentiable physical decoder, ensuring that inferred latent variables correspond to physically plausible leaf and canopy properties. We demonstrate retrieval of leaf area index (LAI) and canopy chlorophyll content (CCC) on real‑world field datasets (FRM4Veg and BelSAR) with accuracy comparable to models trained with real Sentinel‑2 data. Our method requires no in‑situ labels or calibration on real images, offering a cost‑effective and self‑supervised solution for global vegetation monitoring. The proposed approach illustrates how integrating physical models with advanced deep networks can improve the inversion of RTMs, opening new prospects for large‑scale, physically‑constrained remote sensing of vegetation traits.

Abstract:
Vessels navigating in confined waters are subject to banking effects, which are hydrodynamic forces and moments arising from pressure differentials between the vessel sides, significantly affecting manoeuvrability and safety. Existing numerical approaches such as computational fluid dynamics (CFD) can accurately capture these effects but are computationally expensive and unsuitable for real‑time control or estimation. This paper presents a closed‑form, first‑principles model of banking effects. The model coefficients are identified using physics‑informed regression on towing tank experiment data for a scaled container vessel. Validation through Shapley value analysis confirms the significance of the banking terms in reproducing the measured forces and moments. Lastly, the derived coefficients are shown to be non‑dimensional, making the model applicable across different scales that preserve vessel geometry.

Abstract:
The discovery of advanced metallic alloys is hindered by vast composition spaces, competing property objectives, and real‑world constraints on manufacturability. Here we introduce MATAI, a generalist machine learning framework for property prediction and inverse design of as‑cast alloys. MATAI integrates a curated alloy database, deep neural network‑based property predictors, a constraint‑aware optimization engine, and an iterative AI‑experiment feedback loop. The framework estimates key mechanical propertie, sincluding density, yield strength, ultimate tensile strength, and elongation, directly from composition, using multi‑task learning and physics‑informed inductive biases. Alloy design is framed as a constrained optimization problem and solved using a bi‑level approach that combines local search with symbolic constraint programming. We demonstrate MATAI's capabilities on the Ti‑based alloy system, a canonical class of lightweight structural materials, where it rapidly identifies candidates that simultaneously achieve lower density (<4.45 g/cm3), higher strength (>1000 MPa) and appreciable ductility (>5%) through only seven iterations. Experimental validation confirms that MATAI‑designed alloys outperform commercial references such as TC4, highlighting the framework's potential to accelerate the discovery of lightweight, high‑performance materials under real‑world design constraints.

Abstract:
Friction modeling plays a crucial role in achieving high‑precision motion control in robotic operating systems. Traditional static friction models (such as the Stribeck model) are widely used due to their simple forms; however, they typically require predefined functional assumptions, which poses significant challenges when dealing with unknown functional structures. To address this issue, this paper proposes a physics‑inspired machine learning approach based on the Kolmogorov Arnold Network (KAN) for static friction modeling of robotic joints. The method integrates spline activation functions with a symbolic regression mechanism, enabling model simplification and physical expression extraction through pruning and attribute scoring, while maintaining both high prediction accuracy and interpretability. We first validate the method's capability to accurately identify key parameters under known functional models, and further demonstrate its robustness and generalization ability under conditions with unknown functional structures and noisy data. Experiments conducted on both synthetic data and real friction data collected from a six‑degree‑of‑freedom industrial manipulator show that the proposed method achieves a coefficient of determination greater than 0.95 across various tasks and successfully extracts concise and physically meaningful friction expressions. This study provides a new perspective for interpretable and data‑driven robotic friction modeling with promising engineering applicability.

Abstract:
Inverse problems in imaging are typically ill‑posed and are usually solved by employing regularized optimization techniques. The usage of appropriate constraints can restrict the solution space, thus making it feasible for a reconstruction algorithm to find a meaningful solution. In recent years, deep network based ideas aimed at learning the end‑to‑end mapping between the raw measurements and the target image have gained popularity. In the learning approach, the functional relationship between the measured raw data and the solution image are learned by training a deep network with prior examples. While this approach allows one to significantly increase the real‑time operational speed, it does not change the nature of the underlying ill‑posed inverse problem. It is well‑known that availability of diverse non‑redundant data via additional measurements can generically improve the robustness of the reconstruction algorithms. The multiple data measurements, however, typically demand additional hardware and complex system setups that are not desirable. In this work, we note that in both incoherent and coherent optical imaging, the irradiance patterns corresponding to two phase diverse measurements associated with the same test object have implicit local correlation which may be learned. A physics informed data augmentation scheme is then described where a trained network is used for generating a phase diverse pseudo‑data based on a ground truth data frame. The true data along with the augmented pesudo‑data are observed to provide high quality inverse solutions with simpler reconstruction algorithms. We validate this approach for both incoherent and coherent optical imaging (or phase retrieval) configurations with vortex phase as a diversity mechanism. Our results may open new avenues for leaner high‑fidelity computational imaging systems across a broad range of applications.

Abstract:
Due to its nonlinearity, bi‑harmonic dissipation, and backward heat‑like term in the absence of a divergence‑free condition, the 2‑D/3‑D Kuramoto‑Sivashinsky equation poses significant challenges for both mathematical analysis and numerical approximation. These difficulties motivate the development of methods that blend classical analysis with numerical approximation approaches embodied in the framework of the physics‑informed neural networks (PINNs). In addition, despite the extensive use of PINN frameworks for various linear and nonlinear PDEs, no study had previously established rigorous error estimates for the Kuramoto‑Sivashinsky equation within a PINN setting. In this work, we overcome the inherent challenges, and establish several global regularity criteria based on space‑time integrability conditions in Besov spaces. We then derive the first rigorous error estimates for the PINNs approximation of the Kuramoto‑Sivashinsky equation and validate our theoretical error bounds through numerical simulations.

Abstract:
We present a Physics‑Informed Neural Network (PINN) framework for reconstructing the redshift‑dependent Hubble parameter \(H(z)\) within the Tsallis Holographic Dark Energy (THDE) model extended by massive neutrinos. In this approach, the modified Friedmann equation is incorporated into the neural network loss function, enabling training on Cosmic Chronometers data up to \(z \leq 2\). The framework allows for the simultaneous estimation of the Hubble constant \(H_0\), the neutrino density parameter \(Ω_ν\), and the Tsallis non‑extensivity index \(δ\). Uncertainty quantification is performed through dropout simulations, resulting in statistically consistent \(1σ\) confidence bands. Our results show that the THDE+ν model, reconstructed via PINN, alleviates the statistical Hubble tension from the canonical \(～ 5σ\) level down to a range of \(0.5σ\leq T \leq 2.2σ\), depending on the redshift sampling. Additionally, we constrain the total neutrino mass to \(Σm_ν< 0.11\,\texteV\). A detailed comparison with the traditional Markov Chain Monte Carlo (MCMC) analysis demonstrates the consistency of both methods, while highlighting the competitiveness of the PINN‑based THDE framework as a robust, data‑driven approach for non‑parametric cosmological inference within generalized thermodynamics.

Abstract:
Can a dynamically robust (aka stable) Q‑ball reproduce the rotation curve of a disk galaxy? In an astrophysical environment, Q‑balls are non‑topological solitons that are transparent and only perceived by their gravitational effects. Traditionally, scalar Q‑balls are modelled with a polynomial potential, but axion‑like periodic potentials are also expected to support such solitonic configurations. In the presence of angular momentum, Q‑balls acquire a toroidal structure with a central density void, qualitatively resembling the axially‑symmetric structure of disk galaxies. Motivated by this similarity, we investigate whether rotating scalar Q‑balls can reproduce the observed rotation curves of disk galaxies. In this work, we use a recently developed hybrid numerical framework that combines a high‑accuracy pseudo‑spectral method with a physics‑informed neural network approach to construct both static and rotating Q‑ball solutions. We assess their ability to act as the dark matter halos in galaxies by fitting the observed rotation curves of a sample of disk galaxies from the SPARC catalogue. Our simplified model provides an overall good agreement with observational data, and a reasonable fit when compared to standard dark matter profiles such as the Navarro‑Frenk‑White; we have further found an average constraint on the scalar field particle's mass m～ 10^‑27 eV, in agreement with similar galactic‑scale soliton solutions.

Abstract:
Solar energetic particle (SEP) events, as one of the most prominent manifestations of solar activity, can generate severe hazardous radiation when accelerated by solar flares or shock waves formed aside from coronal mass ejections (CMEs). However, most existing data‑driven methods used for SEP predictions are operated as black‑box models, making it challenging for solar physicists to interpret the results and understand the underlying physical causes of such events rather than just obtain a prediction. To address this challenge, we propose a novel framework that integrates global explanations and ad‑hoc feature mapping to enhance model transparency and provide deeper insights into the decision‑making process. We validate our approach using a dataset of 341 SEP events, including 244 significant (>=10 MeV) proton events exceeding the Space Weather Prediction Center S1 threshold, spanning solar cycles 22, 23, and 24. Furthermore, we present an explainability‑focused case study of major SEP events, demonstrating how our method improves explainability and facilitates a more physics‑informed understanding of SEP event prediction.

Abstract:
Implementing quantum gates on quantum computers can require the application of carefully shaped pulses for high‑fidelity operations. We explore the use of physics‑informed neural networks (PINNs) for quantum optimal control to assess their usefulness in predicting such pulses. Our PINN is a feedforward neural network that utilizes an unsupervised learning approach, whose loss function includes terms that enforce the equations that govern the evolution of a quantum system, measure how close the learned unitary is to the target unitary operation, and ensure state normalization. We use a sinusoidal activation function and adopt variance‑type weight initialization, tailored to our activation function. By analyzing the model's performance with important machine learning metrics, we demonstrate that the choice of our architecture is well‑suited for this type of problem. We ensure that our network avoids the vanishing and exploding gradients with our relevant choices. We build two different PINNs, one based on the Schrödinger equation and another one based on the Lindblad equation. Our PINNs are able to discover high‑fidelity two‑qubit gate pulses for a variety of quantum operations, demonstrating its flexibility and robustness. We build two different PINNs, one based on the Schrödinger equation and another one based on the Lindblad equation. Our PINNs are able to discover high‑fidelity two‑qubit gate pulses for a variety of quantum operations, demonstrating its flexibility and robustness.

Abstract:
We present a forward prediction material screening framework designed to discover Pd‑Cu alloys with improved B2 phase stability, thereby unlocking simultaneous H_2 generation and utilization. First, we trained CatBoost models with literature‑derived Pd alloy data to predict H_2 permeability from composition and testing conditions. We evaluated fractional, composition‑based, and physics‑informed descriptors, individually and in combination, and showed that sequential Pearson filtering and fold‑wise SHAP‑based recursive feature elimination with cross‑fold aggregation reduced errors while controlling complexity. Guided by the one‑SE rule, a narrower domain‑informed set of 13 features provided the best accuracy parsimony trade‑off (R^2=0.81), only 0.01 below the max. R^2 achievable with 3x the number of features. SHAP analysis indicated that high permeability is promoted by elevated temperature, lattice expansion relative to Pd, atomic size mismatch, and favorable mixing tendencies. Second, the selected model was applied to screen Pd_(100‑x‑y)Cu_xM_y spanning 16 co‑dopants M for B2 stabilization. For each M system, we obtained the Pareto set of compositions that minimize Pd content and Miedema heat of formation and maximize the permeability, then picked three compounds, including that with the highest predicted permeability, the lowest Miedema heat of formation, and the lowest Pd content. With a final filter considering M concentration for single‑phase Pd‑M solution formation, we recommend Pd48.48Cu43.00Y8.52, Pd49.08Cu42.45Sc8.47, Pd56.09Cu33.70La10.21, and Pd52.68Cu40.44Mg6.88 for experimental validation. We predict those alloys to exhibit permeabilities 1.7 to 1.9 higher than B2 Pd60Cu40. Our framework provides plausible experimental targets and a scalable pathway for designing stable, high‑temperature, H2‑selective Pd‑alloy membranes.

Abstract:
Flood mapping is crucial for assessing and mitigating flood impacts, yet traditional methods like numerical modeling and aerial photography face limitations in efficiency and reliability. To address these challenges, we propose PIFF, a physics‑informed, flow‑based generative neural network for near real‑time flood depth estimation. Built on an image‑to‑image generative framework, it efficiently maps Digital Elevation Models (DEM) to flood depth predictions. The model is conditioned on a simplified inundation model (SPM) that embeds hydrodynamic priors into the training process. Additionally, a transformer‑based rainfall encoder captures temporal dependencies in precipitation. Integrating physics‑informed constraints with data‑driven learning, PIFF captures the causal relationships between rainfall, topography, SPM, and flooding, replacing costly simulations with accurate, real‑time flood maps. Using a 26 km study area in Tainan, Taiwan, with 182 rainfall scenarios ranging from 24 mm to 720 mm over 24 hours, our results demonstrate that PIFF offers an effective, data‑driven alternative for flood prediction and response.

Abstract:
We propose a novel projection method that guarantees the conservation of integral quantities in Physics‑Informed Neural Networks (PINNs). While the soft constraint that PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non‑linear optimization problems and found that our PINN modified with the projection, which we call PINN‑Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the PDE solution error. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a tractable solution exists.

Abstract:
In the design and operation of complex dynamical systems, it is essential to ensure that all state trajectories of the dynamical system converge to a desired equilibrium within a guaranteed stability region. Yet, for many practical systems ‑‑ especially in aerospace ‑‑ this region cannot be determined a priori and is often challenging to compute. One of the most common methods for computing the stability region is to identify a Lyapunov function. A Lyapunov function is a positive function whose time derivative along system trajectories is non‑positive, which provides a sufficient condition for stability and characterizes an estimated stability region. However, existing methods of characterizing a stability region via a Lyapunov function often rely on explicit knowledge of the system governing equations. In this work, we present a new physics‑informed machine learning method of characterizing an estimated stability region by inferring a Lyapunov function from system trajectory data that treats the dynamical system as a black box and does not require explicit knowledge of the system governing equations. In our presented Lyapunov function Inference method (LyapInf), we propose a quadratic form for the unknown Lyapunov function and fit the unknown quadratic operator to system trajectory data by minimizing the average residual of the Zubov equation, a first‑order partial differential equation whose solution yields a Lyapunov function. The inferred quadratic Lyapunov function can then characterize an ellipsoidal estimate of the stability region. Numerical results on benchmark examples demonstrate that our physics‑informed stability analysis method successfully characterizes a near‑maximal ellipsoid of the system stability region associated with the inferred Lyapunov function without requiring knowledge of the system governing equations.

Abstract:
Gravitational wave sources with electromagnetic counterparts have highlighted the need for predictive, interpretable models linking the parameters of compact binary systems to post‑merger remnants and mass outflows. In this work, we explore AI‑driven symbolic regression (SR) frameworks to derive updated analytical relations for disk ejecta mass in binary neutron star mergers, trained on state‑of‑the‑art numerical relativity simulations. Our method reveals a set of compact equations that outperform existing fitting formulae across multiple statistical metrics while remaining physically interpretable. Notably, SR also enables alternative predictor sets (e.g., \M_1,M_2,\tildeΛ\) that match or exceed the accuracy of models relying solely on compactness of the lightest neutron star (C_1), enabling new parameter constraints from electromagnetic observations. Unlike traditional black‑box machine learning models, these closed‑form expressions generalize robustly to regions of the parameter space not represented in the training data, offering a physics‑informed tool for multimessenger observations and constraints on the neutron star equation of state.

Abstract:
The Method of Moments (MoM) is constrained by the usage of static, geometry‑defined basis functions, such as the Rao‑Wilton‑Glisson (RWG) basis. This letter reframes electromagnetic modeling around a learnable basis representation rather than solving for the coefficients over a fixed basis. We first show that the RWG basis is essentially a static and piecewise‑linear realization of the Kolmogorov‑Arnold representation theorem. Inspired by this insight, we propose PhyKAN, a physics‑informed Kolmogorov‑Arnold Network (KAN) that generalizes RWG into a learnable and adaptive basis family. Derived from the EFIE, PhyKAN integrates a local KAN branch with a global branch embedded with Green's function priors to preserve physical consistency. It is demonstrated that, across canonical geometries, PhyKAN achieves sub‑0.01 reconstruction errors as well as accurate, unsupervised radar cross section predictions, offering an interpretable, physics‑consistent bridge between classical solvers and modern neural network models for electromagnetic modeling.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a promising framework for solving forward and inverse problems governed by differential equations. However, their reliability when used in ill‑posed inverse problems remains poorly understood. In this study, we explore the fundamental limitations of PINNs using a simple illustrative case: RC low‑pass filters. Showing that while PINNs can accurately predict system dynamics in forward problems, they fail to recover unique physical parameters when solving inverse problems when more than two parameters are approximated. Our findings provide grounds to understand the boundaries of PINNs applicability for parameter discovery in physical systems.

Abstract:
We address the inverse problem of identifying a time‑dependent potential coefficient in a one‑dimensional diffusion equation subject to Dirichlet boundary conditions and a nonlocal integral overdetermination constraint reflecting spatially averaged measurements. After establishing well‑posedness for the forward problem and deriving an a priori estimate that ensures uniqueness and continuous dependence on the data, we prove existence and uniqueness for the inverse problem. To compute numerically the unknown coefficient, we propose and compare three numerical methods: an integration‑based scheme, a Newton‑Raphson iterative solver, and a physics‑informed neural network (PINN). Numerical experiments on both exact and noisy data demonstrate the accuracy, robustness, and efficiency of each approach.

Abstract:
Flexible and accurate interpolation schemes using machine learning could be of great benefit for many use‑cases in numerical simulations and post‑processing, such as temporal upsampling or storage reduction. In this work, we adapt the physics‑informed token transformer (PITT) network for multi‑channel data and couple it with Fourier neural operator (FNO). The resulting PITT FNO network is trained for interpolation tasks on a dataset governed by the Euler equations. We compare the performance of our machine learning model with a linear interpolation baseline and show that it requires ～6‑10 times less data to achieve the same mean square error of the interpolated quantities. Additionally, PITT FNO has excellent mass and energy conservation as a result of its physics‑informed nature. We further discuss the ability of the network to recover fine detail using a spectral analysis. Our results suggest that loss of fine details is related to the decreasing correlation time of the data with increasing Fourier mode which cannot be resolved by simply increasing Fourier mode truncation in FNO.

Abstract:
Various neural network architectures are used in many of the state‑of‑the‑art approaches for real‑time nonlinear state estimation. With the ever‑increasing incorporation of these data‑driven models into the estimation domain, model predictions with reliable margins of error are a requirement ‑‑ especially for safety‑critical applications. This paper discusses the application of a novel real‑time, data‑driven estimation approach based on the multi‑fidelity residual physics‑informed neural process (MFR‑PINP) toward the real‑time state estimation of a robotic system. Specifically, we address the model‑mismatch issue of selecting an accurate kinematic model by tasking the MFR‑PINP to also learn the residuals between simple, low‑fidelity predictions and complex, high‑fidelity ground‑truth dynamics. To account for model uncertainty present in a physical implementation, robust uncertainty guarantees from the split conformal (SC) prediction framework are modeled in the training and inference paradigms. We provide implementation details of our MFR‑PINP‑based estimator for a hybrid online learning setting to validate our model's usage in real‑time applications. Experimental results of our approach's performance in comparison to the state‑of‑the‑art variants of the Kalman filter (i.e. unscented Kalman filter and deep Kalman filter) in estimation scenarios showed promising results for the MFR‑PINP model as a viable option in real‑time estimation tasks.

Abstract:
Multidimensional optimization has consistently been a critical challenge in engineering. However, traditional simulation‑based optimization methods have long been plagued by significant limitations: they are typically capable of optimizing only a single problem at a time and require substantial computational time for meshing and numerical simulation. This paper introduces a novel framework leveraging cutting‑edge Scientific Machine Learning (Sci‑ML) methodologies to overcome these inherent drawbacks of conventional approaches. The proposed method provides instantaneous solutions to a spectrum of complex, multidimensional optimization problems. A micromixer case study is employed to demonstrate this methodology. An agent, operating on a Deep Reinforcement Learning (DRL) architecture, serves as the optimizer to explore the relationships between key problem parameters. This optimizer interacts with an environment constituted by a parametric Physics‑Informed Neural Network (PINN), which responds to the agent's actions at a significantly higher speed than traditional numerical methods. The agent's objective, conditioned on the Schmidt number is to discover the optimal geometric and physical parameters that maximize the micromixer's efficiency. After training the agent across a wide range of Schmidt numbers, we analyzed the resulting optimal designs. Across this entire spectrum, the achieved efficiency was consistently greater than the baseline, normalized value. The maximum efficiency occurred at a Schmidt number of 13.3, demonstrating an improvement of approximately 32%. Finally, a comparative analysis with a Genetic Algorithm was conducted under equivalent conditions to underscore the advantages of the proposed method.

Abstract:
We introduce a Nemytskii neural operator framework for nonlinear model reduction of parametrized steady‑state partial differential equations. The method generalizes reduced basis approaches by replacing linear combinations of basis functions with a structured nonlinear mapping realized through a pointwise Nemytskii operator acting on fixed feature functions. Feature functions are learned offline via nonlinear dimension reduction from high‑fidelity snapshots, and a hypernetwork maps model parameters to a lightweight reconstruction network, which is further refined online using physics‑informed residual minimization. The Nemytskii structure preserves analytical regularity and enables efficient evaluation of spatial and parametric derivatives, leading to fast online adaptation. Numerical experiments demonstrate that the proposed method consistently outperforms linear model reduction techniques, particularly for complex solution manifolds.

Abstract:
Scientific Machine Learning (SciML) integrates data‑driven inference with physical modeling to solve complex problems in science and engineering. However, the design of SciML architectures, loss formulations, and training strategies remains an expert‑driven research process, requiring extensive experimentation and problem‑specific insights. Here we introduce AgenticSciML, a collaborative multi‑agent system in which over 10 specialized AI agents collaborate to propose, critique, and refine SciML solutions through structured reasoning and iterative evolution. The framework integrates structured debate, retrieval‑augmented method memory, and ensemble‑guided evolutionary search, enabling the agents to generate and assess new hypotheses about architectures and optimization procedures. Across physics‑informed learning and operator learning tasks, the framework discovers solution methods that outperform single‑agent and human‑designed baselines by up to four orders of magnitude in error reduction. The agents produce novel strategies ‑‑ including adaptive mixture‑of‑expert architectures, decomposition‑based PINNs, and physics‑informed operator learning models ‑‑ that do not appear explicitly in the curated knowledge base. These results show that collaborative reasoning among AI agents can yield emergent methodological innovation, suggesting a path toward scalable, transparent, and autonomous discovery in scientific computing.

Abstract:
In this work, we introduce the Quantum‑Classical Hybrid Physics‑Informed Neural Network with Multiplicative and Additive Couplings (QPINN‑MAC): a novel hybrid architecture that integrates the framework of Physics‑Informed Neural Networks (PINNs) with that of Quantum Neural Networks (QNNs). Specifically, we prove that through strategic couplings between classical and quantum components, the QPINN‑MAC retains the universal approximation property, ensuring its theoretical capacity to represent complex solutions of ordinary differential equations (ODEs). Simultaneously, we demonstrate that the hybrid QPINN‑MAC architecture actively mitigates the barren plateau problem, regions in parameter space where cost‑function gradients decay exponentially with circuit depth, a fundamental obstacle in QNNs that hinders optimization during training. Furthermore, we prove that these couplings prevent gradient collapse, ensuring trainability even in high‑dimensional regimes. Thus, our results establish a new pathway for constructing quantum‑classical hybrid models with theoretical convergence guarantees, which are essential for the practical application of QPINNs.

Abstract:
Attenuation artifacts remain a significant challenge in cardiac Myocardial Perfusion Imaging (MPI) using Single‑Photon Emission Computed Tomography (SPECT), often compromising diagnostic accuracy and reducing clinical interpretability. While hybrid SPECT/CT systems mitigate these artifacts through CT‑derived attenuation maps, their high cost, limited accessibility, and added radiation exposure hinder widespread clinical adoption. In this study, we propose a novel CT‑free solution to attenuation correction in cardiac SPECT. Specifically, we introduce Physics‑aware Attenuation Correction Diffusion Model (PADM), a diffusion‑based generative method that incorporates explicit physics priors via a teacher‑‑student distillation mechanism. This approach enables attenuation artifact correction using only Non‑Attenuation‑Corrected (NAC) input, while still benefiting from physics‑informed supervision during training. To support this work, we also introduce CardiAC, a comprehensive dataset comprising 424 patient studies with paired NAC and Attenuation‑Corrected (AC) reconstructions, alongside high‑resolution CT‑based attenuation maps. Extensive experiments demonstrate that PADM outperforms state‑of‑the‑art generative models, delivering superior reconstruction fidelity across both quantitative metrics and visual assessment.

Abstract:
The resolving ability of widefield fluorescence microscopy is fundamentally limited by out‑of‑focus background owing to its low axial resolution, particularly for densely labeled biological samples. Although total internal reflection fluorescence (TIRF) microscopy provides strong near‑surface sectioning, they are intrinsically restricted to shallow imaging depths. Here we present computational TIRF (cTIRF), a deep learning‑based imaging modality that generates TIRF‑like sectioned images directly from conventional widefield epifluorescence measurements without any optical modification. By integrating a physics‑informed forward model into network training, cTIRF achieves effective background suppression and axial resolution enhancement while maintaining consistency with the measured widefield data. We demonstrate that cTIRF recovers near‑surface structures with performance comparable to experimental TIRF, and further enables both single‑frame and volumetric sectioned reconstruction in densely labeled samples where conventional TIRF fails. This work establishes cTIRF as a practical and deployable alternative to hardware‑based optical sectioning in fluorescence microscopy, enabled by rapid adaptation to new imaging systems with minimal calibration data.

Abstract:
We propose a Newton‑based scheme, initialized by neural operator predictions, to accelerate the parametric solution of nonlinear problems in computational solid mechanics. First, a physics informed conditional neural field is trained to approximate the nonlinear parametric solutionof the governing equations. This establishes a continuous mapping between the parameter and solution spaces, which can then be evaluated for a given parameter at any spatial resolution. Second, since the neural approximation may not be exact, it is subsequently refined using a Newton‑based correction initialized by the neural output. To evaluate the effectiveness of this hybrid approach, we compare three solution strategies: (i) the standard Newton‑Raphson solver used in NFEM, which is robust and accurate but computationally demanding; (ii) physics‑informed neural operators, which provide rapid inference but may lose accuracy outside the training distribution and resolution; and (iii) the neural‑initialized Newton (NiN) strategy, which combines the efficiency of neural operators with the robustness of NFEM. The results demonstrate that the proposed hybrid approach reduces computational cost while preserving accuracy, highlighting its potential to accelerate large‑scale nonlinear simulations.

Abstract:
Self‑supervised goal‑conditioned reinforcement learning enables robots to autonomously acquire diverse skills without human supervision. However, a central challenge is the goal setting problem: robots must propose feasible and diverse goals that are achievable in their current environment. Existing methods like RIG (Visual Reinforcement Learning with Imagined Goals) use variational autoencoder (VAE) to generate goals in a learned latent space but have the limitation of producing physically implausible goals that hinder learning efficiency. We propose Physics‑Informed RIG (PI‑RIG), which integrates physical constraints directly into the VAE training process through a novel Enhanced Physics‑Informed Variational Autoencoder (Enhanced p3‑VAE), enabling the generation of physically consistent and achievable goals. Our key innovation is the explicit separation of the latent space into physics variables governing object dynamics and environmental factors capturing visual appearance, while enforcing physical consistency through differential equation constraints and conservation laws. This enables the generation of physically consistent and achievable goals that respect fundamental physical principles such as object permanence, collision constraints, and dynamic feasibility. Through extensive experiments, we demonstrate that this physics‑informed goal generation significantly improves the quality of proposed goals, leading to more effective exploration and better skill acquisition in visual robotic manipulation tasks including reaching, pushing, and pick‑and‑place scenarios.

Abstract:
Spiking neural networks (SNNs) offer biologically inspired computation but remain underexplored for continuous regression tasks in scientific machine learning. In this work, we introduce and systematically evaluate Quadratic Integrate‑and‑Fire (QIF) neurons as an alternative to the conventional Leaky Integrate‑and‑Fire (LIF) model in both directly trained SNNs and ANN‑to‑SNN conversion frameworks. The QIF neuron exhibits smooth and differentiable spiking dynamics, enabling gradient‑based training and stable optimization within architectures such as multilayer perceptrons (MLPs), Deep Operator Networks (DeepONets), and Physics‑Informed Neural Networks (PINNs). Across benchmarks on function approximation, operator learning, and partial differential equation (PDE) solving, QIF‑based networks yield smoother, more accurate, and more stable predictions than their LIF counterparts, which suffer from discontinuous time‑step responses and jagged activation surfaces. These results position the QIF neuron as a computational bridge between spiking and continuous‑valued deep learning, advancing the integration of neuroscience‑inspired dynamics into physics‑informed and operator‑learning frameworks.

Abstract:
The convergence of statistical learning and molecular physics is transforming our approach to modeling biomolecular systems. Physics‑informed machine learning (PIML) offers a systematic framework that integrates data‑driven inference with physical constraints, resulting in models that are accurate, mechanistic, generalizable, and able to extrapolate beyond observed domains. This review surveys recent advances in physics‑informed neural networks and operator learning, differentiable molecular simulation, and hybrid physics‑ML potentials, with emphasis on long‑timescale kinetics, rare events, and free‑energy estimation. We frame these approaches as solutions to the "biomolecular closure problem", recovering unresolved interactions beyond classical force fields while preserving thermodynamic consistency and mechanistic interpretability. We examine theoretical foundations, tools and frameworks, computational trade‑offs, and unresolved issues, including model expressiveness and stability. We outline prospective research avenues at the intersection of machine learning, statistical physics, and computational chemistry, contending that future advancements will depend on mechanistic inductive biases, and integrated differentiable physical learning frameworks for biomolecular simulation and discovery.

Abstract:
Traditional statistical optimization‑based state estimation (DSSE) algorithms rely on detailed grid parameters and mathematical assumptions of all possible uncertainties. Furthermore, random data missing due to communication failures, congestion, and cyberattacks, makes these methods easily infeasible. Inspired by recent advances in digital twins (DTs), this paper proposes an interactive attention‑based DSSE model for robust grid monitoring by integrating three core components: physical entities, virtual modeling, and data fusion. To enable robustness against various data missing in heterogeneous measurements, we first propose physics‑informed data augmentation and transfer. Moreover, a state‑of‑the‑art attention‑based spatiotemporal feature learning is proposed, followed by a novel cross‑interaction feature fusion for robust voltage estimation. A case study in a real‑world unbalanced 84‑bus distribution system with raw data validates the accuracy and robustness of the proposed DT model in estimating voltage states, with random locational, arbitrary ratios (up to 40% of total measurements) of data missing.

Abstract:
Recently, 3D Gaussian Splatting (3DGS), an explicit scene representation technique, has shown significant promise for dynamic novel‑view synthesis from monocular video input. However, purely data‑driven 3DGS often struggles to capture the diverse physics‑driven motion patterns in dynamic scenes. To fill this gap, we propose Physics‑Informed Deformable Gaussian Splatting (PIDG), which treats each Gaussian particle as a Lagrangian material point with time‑varying constitutive parameters and is supervised by 2D optical flow via motion projection. Specifically, we adopt static‑dynamic decoupled 4D decomposed hash encoding to reconstruct geometry and motion efficiently. Subsequently, we impose the Cauchy momentum residual as a physics constraint, enabling independent prediction of each particle's velocity and constitutive stress via a time‑evolving material field. Finally, we further supervise data fitting by matching Lagrangian particle flow to camera‑compensated optical flow, which accelerates convergence and improves generalization. Experiments on a custom physics‑driven dataset as well as on standard synthetic and real‑world datasets demonstrate significant gains in physical consistency and monocular dynamic reconstruction quality.

Abstract:
Motion blur, caused by relative movement between camera and scene during exposure, significantly degrades image quality and impairs downstream computer vision tasks such as object detection, tracking, and recognition in dynamic environments. While deep learning‑based motion deblurring methods have achieved remarkable progress, existing approaches face fundamental challenges in capturing the long‑range spatial dependencies inherent in motion blur patterns. Traditional convolutional methods rely on limited receptive fields and require extremely deep networks to model global spatial relationships. These limitations motivate the need for alternative approaches that incorporate physical priors to guide feature evolution during restoration. In this paper, we propose a progressive training framework that integrates physics‑informed PDE dynamics into state‑of‑the‑art restoration architectures. By leveraging advection‑diffusion equations to model feature evolution, our approach naturally captures the directional flow characteristics of motion blur while enabling principled global spatial modeling. Our PDE‑enhanced deblurring models achieve superior restoration quality with minimal overhead, adding only approximately 1% to inference GMACs while providing consistent improvements in perceptual quality across multiple state‑of‑the‑art architectures. Comprehensive experiments on standard motion deblurring benchmarks demonstrate that our physics‑informed approach improves PSNR and SSIM significantly across four diverse architectures, including FFTformer, NAFNet, Restormer, and Stripformer. These results validate that incorporating mathematical physics principles through PDE‑based global layers can enhance deep learning‑based image restoration, establishing a promising direction for physics‑informed neural network design in computer vision applications.

Abstract:
Reliability analysis of engineering systems under uncertainty poses significant computational challenges, particularly for problems involving high‑dimensional stochastic inputs, nonlinear system responses, and multiphysics couplings. Traditional surrogate modeling approaches often incur high energy consumption, which severely limits their scalability and deployability in resource‑constrained environments. We introduce NeuroPOL, the first neuroscience‑inspired physics‑informed operator learning framework for reliability analysis. NeuroPOL incorporates Variable Spiking Neurons into a physics‑informed operator architecture, replacing continuous activations with event‑driven spiking dynamics. This innovation promotes sparse communication, significantly reduces computational load, and enables an energy‑efficient surrogate model. The proposed framework lowers both computational and power demands, supporting real‑time reliability assessment and deployment on edge devices and digital twins. By embedding governing physical laws into operator learning, NeuroPOL builds physics‑consistent surrogates capable of accurate uncertainty propagation and efficient failure probability estimation, even for high‑dimensional problems. We evaluate NeuroPOL on five canonical benchmarks, the Burgers equation, Nagumo equation, two‑dimensional Poisson equation, two‑dimensional Darcy equation, and incompressible Navier‑Stokes equation with energy coupling. Results show that NeuroPOL achieves reliability measures comparable to standard physics‑informed operators, while introducing significant communication sparsity, enabling scalable, distributed, and energy‑efficient deployment.

Abstract:
We introduce NeuroPINNs, a neuroscience‑inspired extension of Physics‑Informed Neural Networks (PINNs) that incorporates biologically motivated spiking neuron models to achieve energy‑efficient PDE solving. Unlike conventional PINNs, which rely on continuously firing activations and therefore incur high computational and energy costs, NeuroPINNs leverage Variable Spiking Neurons (VSNs) to enable sparse, event‑driven communication. This makes them particularly well‑suited for deployment on neuromorphic hardware and for scenarios with constrained computational resources, such as embedded and edge devices. A central challenge, however, lies in reconciling the discontinuous dynamics of spiking neurons with the smooth residual‑based loss formulation required in PINNs. Direct smoothing introduces systematic biases, leading to inaccurate PDE learning. To overcome this, we employ a novel stochastic projection method inspired from upscaled theory that faithfully captures spiking behavior while maintaining compatibility with gradient‑based optimization. Standard surrogate backpropagation is used for parameter updates, ensuring computational tractability. We demonstrate the effectiveness of NeuroPINNs on four representative PDE problems across both regular and irregular domains. Furthermore, application of NeuroPINN for linear elastic micromechnics in three dimensions was also explored. Results show that NeuroPINNs achieve high accuracy while substantially reducing communication and energy demands, marking a step toward scalable, neuromorphic‑ready scientific machine learning.

Abstract:
We propose a consistency model based on the optimal‑transport flow. A physics‑informed design of partially input‑convex neural networks (PICNN) plays a central role in constructing the flow field that emulates the displacement interpolation. During the training stage, we couple the Hamilton‑Jacobi (HJ) residual in the OT formulation with the original flow matching loss function. Our approach avoids inner optimization subproblems that are present in previous one‑step OFM approaches. During the prediction stage, our approach supports both one‑step (Brenier‑map) and multi‑step ODE sampling from the same learned potential, leveraging the straightness of the OT flow. We validate scalability and performance on standard OT benchmarks.

Abstract:
Physics‑Informed Neural Networks (PINNs) have recently emerged as powerful tools for solving partial differential equations (PDEs), with the Deep Energy Method (DEM) proving especially effective in fracture mechanics due to its energy‑based formulation. Despite these advances, existing DEM approaches require dense collocation near cracks, face stability challenges, and typically treat discrete and continuous fracture models separately. To overcome these limitations, we introduce the Extended Deep Energy Method (XDEM), a unified deep learning framework that incorporates both displacement discontinuities and crack‑tip asymptotics in the discrete setting, while flexibly coupling displacement and phase fields in the continuous setting. This integration enables accurate fracture predictions using uniformly distributed, relatively sparse collocation points. Validation across benchmark problems including stress intensity factor evaluation, straight and kinked crack growth, and complex crack initiation demonstrates that XDEM consistently outperforms standard DEM in accuracy and efficiency. By bridging discrete and phase‑field models within a single framework, XDEM establishes a robust foundation for applying AI to fracture mechanics and opens new avenues for predictive modeling in engineering and materials science.

Abstract:
Hydrogen crossover in polymer electrolyte membrane water electrolysis poses a critical safety and efficiency bottleneck for scalable green hydrogen production. While machine learning offers real‑time monitoring capabilities, conventional data‑driven newral networks (Pure NNs) and soft‑constraint physics‑informed neural networks (Standard PINNs) suffer from inherent optimization conflicts and fail catastrophically when extrapolating beyond sparse training conditions. Here, we present a hard‑constraint physics‑residual network (PR‑Net) that embeds analytical transport equations ‑‑ Henry's law, Fick's diffusion, and Faraday's law ‑‑ as a deterministic computational backbone, restricting the neural network to learn only systematic physical deviations. Across 184 experimental points spanning six membrane types and operating conditions of 25‑‑85^\circC, 1‑‑200~bar, and 0.05‑‑5.0 A cm^‑2, this architecture intrinsically resolves gradient conflicts, yielding R^2 = 99.57 \pm 0.16% with a 39‑fold reduction in training variance compared to purely data‑driven models (R^2 = 96.47 \pm 6.20%). Crucially, the PR‑Net breaks the extrapolation barrier, maintaining R^2 > 97% at extreme cathode pressures up to 200~bar ‑‑ a 2.5‑fold extrapolation beyond the training domain where Standard PINN severely degrades (R^2 = 72.2%) and Pure NN collapses (R^2 = 58.7%). Furthermore, the learned residuals autonomously capture temperature‑induced membrane swelling (Spearman's ρ= 0.506, p < 0.001) and identify the non‑linear transport regime transition near 0.23 A cm^‑2, without explicit programming. Delivering millisecond‑level inference on edge hardware, the PR‑Net establishes a highly reliable, generalizable foundation for adaptive safety control and predictive maintenance in high‑pressure electrochemical energy systems.

Abstract:
Sea Surface Temperature (SST) is crucial for understanding upper‑ocean thermal dynamics and ocean‑atmosphere interactions, which have profound economic and social impacts. While data‑driven models show promise in SST prediction, their black‑box nature often limits interpretability and overlooks key physical processes. Recently, physics‑informed neural networks have been gaining momentum but struggle with complex ocean‑atmosphere dynamics due to 1) inadequate characterization of seawater movement (e.g., coastal upwelling) and 2) insufficient integration of external SST drivers (e.g., turbulent heat fluxes). To address these challenges, we propose SSTODE, a physics‑informed Neural Ordinary Differential Equations (Neural ODEs) framework for SST prediction. First, we derive ODEs from fluid transport principles, incorporating both advection and diffusion to model ocean spatiotemporal dynamics. Through variational optimization, we recover a latent velocity field that explicitly governs the temporal dynamics of SST. Building upon ODE, we introduce an Energy Exchanges Integrator (EEI)‑inspired by ocean heat budget equations‑to account for external forcing factors. Thus, the variations in the components of these factors provide deeper insights into SST dynamics. Extensive experiments demonstrate that SSTODE achieves state‑of‑the‑art performances in global and regional SST forecasting benchmarks. Furthermore, SSTODE visually reveals the impact of advection dynamics, thermal diffusion patterns, and diurnal heating‑cooling cycles on SST evolution. These findings demonstrate the model's interpretability and physical consistency.

Abstract:
We present an uncertainty‑aware, physics‑informed neural network (PINN) for option pricing that solves the Black‑‑Scholes (BS) partial differential equation (PDE) as a mesh‑free, global surrogate over (S,t). The model embeds the BS operator and boundary/terminal conditions in a residual‑based objective and requires no labeled prices. For American options, early exercise is handled via an obstacle‑style relaxation while retaining the BS residual in the continuation region. To quantify \emphepistemic uncertainty, we introduce an anchored‑ensemble fine‑tuning stage (AT‑‑PINN) that regularizes each model toward a sampled anchor and yields prediction bands alongside point estimates. On European calls/puts, the approach attains low errors (e.g., MAE ～ 5×10^‑2, RMSE ～ 7×10^‑2, explained variance \approx 0.999 in representative settings) and tracks ground truth closely across strikes and maturities. For American puts, the method remains accurate (MAE/RMSE on the order of 10^‑1 with EV \approx 0.999) and does not exhibit the error accumulation associated with time‑marching schemes. Against data‑driven baselines (ANN, RNN) and a Kolmogorov‑‑Arnold FINN variant (KAN), our PINN matches or outperforms on accuracy while training more stably; anchored ensembles provide uncertainty bands that align with observed error scales. We discuss design choices (loss balancing, sampling near the payoff kink), limitations, and extensions to higher‑dimensional BS settings and alternative dynamics.

Abstract:
Physics‑informed deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and efficiency. In this work, we introduce a hybrid adaptive sampling and weighting method to enhance the performance of physics‑informed neural networks (PINNs). The adaptive sampling component identifies training points in regions where the solution exhibits rapid variation, while the adaptive weighting component balances the convergence rate across training points. Numerical experiments show that applying only adaptive sampling or only adaptive weighting is insufficient to consistently achieve accurate predictions, particularly when training points are scarce. Since each method emphasizes different aspects of the solution, their effectiveness is problem dependent. By combining both strategies, the proposed framework consistently improves prediction accuracy and training efficiency, offering a more robust approach for solving PDEs with PINNs.

Abstract:
The papers included in this Focus Point collection are devoted to the studies on the cosmological tensions and challenges stimulated by the latest observational data. The first results of the LARES‑2 laser ranging satellite on the high precision testing of the frame‑dragging effect predicted by General Relativity are presented. The data on the S‑stars monitoring in the Galactic center obtained by GRAVITY collaboration were analysed within the Physics‑informed neural network (PINN) approach. The results enabled to probe the role of the cosmological constant, of the dark matter, the star cluster in the core of the Galaxy obtaining an upper limit for the star density. The topics include the conversion of high‑frequency relic gravitational waves into photons in cosmological magnetic field, cosmological gravitational waves stochastic background generation through the spontaneous breaking of a global baryon number symmetry, observational predictions of the Starobinsky inflation model and other studies.

Abstract:
Predicting quantum wavefunction probability distributions is crucial for computational chemistry and materials science, yet machine learning (ML) models often face a trade‑off between accuracy and interpretability. This study compares Artificial Neural Networks (ANNs) and Adaptive Neuro‑Fuzzy Inference Systems (ANFIS) in modeling quantum probability distributions for the H_2^+ ion, leveraging data generated via Physics‑Informed Neural Networks (PINNs). While ANN achieved superior accuracy (R^2 = 0.99 vs ANFIS's 0.95 with Gaussian membership functions), it required over 50x more parameters (2,305 vs 39‑45). ANFIS, however, provided unique interpretability: its Gaussian membership functions encoded spatial electron localization near proton positions (μ= 1.2 A), mirroring Born probability densities, while fuzzy rules reflected quantum superposition principles. Rules prioritizing the internuclear direction revealed the system's 1D symmetry, aligning with Linear Combination of Atomic Orbitals theory‑‑a novel data‑driven perspective on orbital hybridization. Membership function variances (σ) further quantified electron delocalization trends, and peak prediction errors highlighted unresolved quantum cusps. The choice of functions critically impacted performance: Gaussian/Generalized Bell outperformed Sigmoid, with errors improving as training data increased, showing scalability. This study underscores the context‑dependent value of ML: ANN for precision and ANFIS for interpretable, parameter‑efficient approximations that link inputs to physical behavior. These findings advocate hybrid approaches in quantum simulations, balancing accuracy with explainability to accelerate discovery. Future work should extend ANFIS to multi‑electron systems and integrate domain‑specific constraints (e.g., kinetic energy terms), bridging data‑driven models and fundamental physics.

Abstract:
Modern power systems require fast and accurate dynamic simulations for stability assessment, digital twins, and real‑time control, but classical ODE solvers are often too slow for large‑scale or online applications. We propose a neural‑operator framework for surrogate modeling of power system components, using Deep Operator Networks (DeepONets) to learn mappings from system states and time‑varying inputs to full trajectories without step‑by‑step integration. To enhance generalization and data efficiency, we introduce Physics‑Informed DeepONets (PI‑DeepONets), which embed the residuals of governing equations into the training loss. Our results show that DeepONets, and especially PI‑DeepONets, achieve accurate predictions under diverse scenarios, providing over 30 times speedup compared to high‑order ODE solvers. Benchmarking against Physics‑Informed Neural Networks (PINNs) highlights superior stability and scalability. Our results demonstrate neural operators as a promising path toward real‑time, physics‑aware simulation of power system dynamics.

Abstract:
In this study, we employ a conventional deep neural network (NN) framework integrated with physics‑based constraints to predict charged hadron multiplicity (N_\textch) in heavy‑ion collisions. The goal is to assess the performance of a purely data‑driven deep neural network in comparison to a physics‑informed neural network (PINN). To accomplish this, we have taken data generated from the HYDJET++ model for testing and training purposes. We train our neural network frameworks using the data of one million individual ^96_40\textZr+^96_40\textZr collision events. Our PINN model successfully extracts the hard‑scattering fraction (x) by learning its underlying relation from the event data. For further testing and comparison with the conventional NN, we take data of ^96_44\textRu+^96_44\textRu (isobar of Zr) and ^197_79\textAu+^197_79\textAu collisions using the same simulation model. We found that the NN model needs more time to train with physics. However, once trained, the PINN model is capable of accurately predicting data that it has not encountered during training, such as Au+Au collision results. Especially in a region of sparse data corresponding to high N_\textch in our study, PINN has a clear advantage over a simple NN.

Abstract:
We study chaotic synchronization in a 5D Hindmarsh‑‑Rose neuron model augmented with electromagnetic induction and a switchable memristive autapse. For two diffusively coupled identical neurons, we derive the transverse error dynamical system and analyze local synchronization via the linearized error system around the synchronization manifold. A quadratic Lyapunov function yields explicit sufficient conditions for (i) asymptotic stability when the memristive switching remains dissipative and (ii) practical stability with an explicit ultimate bound under non‑dissipative switching. We complement this with a Hamiltonian‑based viewpoint: a Helmholtz decomposition of the linearized error vector field provides a closed‑form synchronization Hamiltonian and its rate identity. Numerical simulations corroborate convergence or ultimate boundedness of the synchronization errors and an overall decay of the synchronization Hamiltonian and its instantaneous rate toward zero after transients, and show consistent trends between Lyapunov‑ and Hamiltonian‑based diagnostics across parameters. Finally, we propose the first port‑Hamiltonian physics‑informed neural network (pH‑PINN) that learns this synchronization Hamiltonian and its rate from data while preserving conservative/dissipative structure, achieving close agreement with the analytical expressions.

Abstract:
Human‑in‑the‑loop calibration is often addressed via preference‑based optimization, where algorithms learn from pairwise comparisons rather than explicit cost evaluations. While effective, methods such as Preferential Bayesian Optimization or Global optimization based on active preference learning with radial basis functions (GLISp) treat the system as a black box and ignore informative sensor measurements. In this work, we introduce a sensor‑guided regularized extension of GLISp that integrates measurable descriptors into the preference‑learning loop through a physics‑informed hypothesis function and a least‑squares regularization term. This injects grey‑box structure, combining subjective feedback with quantitative sensor information while preserving the flexibility of preference‑based search. Numerical evaluations on an analytical benchmark and on a human‑in‑the‑loop vehicle suspension tuning task show faster convergence and superior final solutions compared to baseline GLISp.

Abstract:
Robotic manipulation policies are advancing rapidly, but their direct evaluation in the real world remains costly, time‑consuming, and difficult to reproduce, particularly for tasks involving deformable objects. Simulation provides a scalable and systematic alternative, yet existing simulators often fail to capture the coupled visual and physical complexity of soft‑body interactions. We present a real‑to‑sim policy evaluation framework that constructs soft‑body digital twins from real‑world videos and renders robots, objects, and environments with photorealistic fidelity using 3D Gaussian Splatting. We validate our approach on representative deformable manipulation tasks, including plush toy packing, rope routing, and T‑block pushing, demonstrating that simulated rollouts correlate strongly with real‑world execution performance and reveal key behavioral patterns of learned policies. Our results suggest that combining physics‑informed reconstruction with high‑quality rendering enables reproducible, scalable, and accurate evaluation of robotic manipulation policies. Website: https://real2sim‑eval.github.io/

Abstract:
PDEs arise ubiquitously in science and engineering, where solutions depend on parameters (physical properties, boundary conditions, geometry). Traditional numerical methods require re‑solving the PDE for each parameter, making parameter space exploration prohibitively expensive. Recent machine learning advances, particularly physics‑informed neural networks (PINNs) and neural operators, have revolutionized parametric PDE solving by learning solution operators that generalize across parameter spaces. We critically analyze two main paradigms: (1) PINNs, which embed physical laws as soft constraints and excel at inverse problems with sparse data, and (2) neural operators (e.g., DeepONet, Fourier Neural Operator), which learn mappings between infinite‑dimensional function spaces and achieve unprecedented generalization. Through comparisons across fluid dynamics, solid mechanics, heat transfer, and electromagnetics, we show neural operators can achieve computational speedups of 10^3 to 10^5 times faster than traditional solvers for multi‑query scenarios, while maintaining comparable accuracy. We provide practical guidance for method selection, discuss theoretical foundations (universal approximation, convergence), and identify critical open challenges: high‑dimensional parameters, complex geometries, and out‑of‑distribution generalization. This work establishes a unified framework for understanding parametric PDE solvers via operator learning, offering a comprehensive, incrementally updated resource for this rapidly evolving field

Abstract:
Physics‑informed machine learning (PIML) integrates partial differential equations (PDEs) into machine learning models to solve inverse problems, such as estimating coefficient functions (e.g., the Hamiltonian function) that characterize physical systems. This framework enables data‑driven understanding and prediction of complex physical phenomena. While coefficient functions in PIML are typically estimated on the basis of predictive performance, physics as a discipline does not rely solely on prediction accuracy to evaluate models. For example, Kepler's heliocentric model was favored owing to small discrepancies in planetary motion, despite its similar predictive accuracy to the geocentric model. This highlights the inherent uncertainties in data‑driven model inference and the scientific importance of selecting physically meaningful solutions. In this paper, we propose a framework to quantify and analyze such uncertainties in the estimation of coefficient functions in PIML. We apply our framework to reduced model of magnetohydrodynamics and our framework shows that there are uncertainties, and unique identification is possible with geometric constraints. Finally, we confirm that we can estimate the reduced model uniquely by incorporating these constraints.

Abstract:
Training physics informed neural networks PINNs for hyperbolic conservation laws near the inviscid limit presents considerable difficulties because strong form residuals become ill posed at shock discontinuities, while small viscosity regularization introduces narrow boundary layers that exacerbate spectral bias. To address these issues this paper proposes a novel two stage adaptive lifting PINN, a lifting based framework designed to mitigate such challenges without requiring a priori knowledge of the interface geometry. The key idea is to augment the physical coordinates by introducing a learned auxiliary field generated through r adaptive coordinate transformations. Theoretically we first derive an a posteriori L2 error estimate to quantify how training difficulty depends on viscosity. Secondly we provide a statistical interpretation revealing that embedded sampling induces variance reduction analogous to importance sampling. Finally we perform an NTK and gradient flow analysis, demonstrating that input augmentation improves conditioning and accelerates residual decay. Supported by these insights our numerical experiments show accelerated and more stable convergence as well as accurate reconstructions near discontinuities.

Abstract:
Background: Non‑invasive imaging‑based assessment of blood flow plays a critical role in evaluating heart function and structure. Computed Tomography (CT) is a widely‑used imaging modality that can robustly evaluate cardiovascular anatomy and function, but direct methods to estimate blood flow velocity from movies of contrast evolution have not been developed. Purpose: This study evaluates the impact of CT imaging on Physics‑Informed Neural Networks (PINN)‑based flow estimation and proposes an improved framework, SinoFlow, which uses sinogram data directly to estimate blood flow. Methods: We generated pulsatile flow fields in an idealized 2D vessel bifurcation using computational fluid dynamics and simulated CT scans with varying gantry rotation speeds, tube currents, and pulse mode imaging settings. We compared the performance of PINN‑based flow estimation using reconstructed images (ImageFlow) to SinoFlow. Results: SinoFlow significantly improved flow estimation performance by avoiding propagating errors introduced by filtered backprojection. SinoFlow was robust across all tested gantry rotation speeds and consistently produced lower mean squared error and velocity errors than ImageFlow. Additionally, SinoFlow was compatible with pulsed‑mode imaging and maintained higher accuracy with shorter pulse widths. Conclusions: This study demonstrates the potential of SinoFlow for CT‑based flow estimation, providing a more promising approach for non‑invasive blood flow assessment. The findings aim to inform future applications of PINNs to CT images and provide a solution for image‑based estimation, with reasonable acquisition parameters yielding accurate flow estimates.

Abstract:
Modern power systems with high penetration of inverter‑based resources exhibit complex dynamic behaviors that challenge the scalability and generalizability of traditional stability assessment methods. This paper presents a dynamic recurrent adjacency memory network (DRAMN) that combines physics‑informed analysis with deep learning for real‑time power system stability forecasting. The framework employs sliding‑window dynamic mode decomposition to construct time‑varying, multi‑layer adjacency matrices from phasor measurement unit and sensor data to capture system dynamics such as modal participation factors, coupling strengths, phase relationships, and spectral energy distributions. As opposed to processing spatial and temporal dependencies separately, DRAMN integrates graph convolution operations directly within recurrent gating mechanisms, enabling simultaneous modeling of evolving dynamics and temporal dependencies. Extensive validations on modified IEEE 9‑bus, 39‑bus, and a multi‑terminal HVDC network demonstrate high performance, achieving 99.85%, 99.90%, and 99.69% average accuracies, respectively, surpassing all tested benchmarks, including classical machine learning algorithms and recent graph‑based models. The framework identifies optimal combinations of measurements that reduce feature dimensionality by 82% without performance degradation. Correlation analysis between dominant measurements for small‑signal and transient stability events validates generalizability across different stability phenomena. DRAMN achieves state‑of‑the‑art accuracy while providing enhanced interpretability for power system operators, making it suitable for real‑time deployment in modern control centers.

Abstract:
Solving inverse problems governed by partial differential equations (PDEs) is central to science and engineering, yet remains challenging when measurements are sparse, noisy, or when the underlying coefficients are high‑dimensional or discontinuous. Existing deep learning approaches either require extensive labeled datasets or are limited to specific measurement types, often leading to failure in such regimes and restricting their practical applicability. Here, a novel generative neural operator framework, IGNO, is introduced to overcome these limitations. IGNO unifies the solution of inverse problems from both point measurements and operator‑valued data without labeled training pairs. This framework encodes high‑dimensional, potentially discontinuous coefficient fields into a low‑dimensional latent space, which drives neural operator decoders to reconstruct both coefficients and PDE solutions. Training relies purely on physics constraints through PDE residuals, while inversion proceeds via efficient gradient‑based optimization in latent space, accelerated by an a priori normalizing flow model. Across a diverse set of challenging inverse problems, including recovery of discontinuous coefficients from solution‑based measurements and the EIT problem with operator‑based measurements, IGNO consistently achieves accurate, stable, and scalable inversion even under severe noise. It consistently outperforms the state‑of‑the‑art method under varying noise levels and demonstrates strong generalization to out‑of‑distribution targets. These results establish IGNO as a unified and powerful framework for tackling challenging inverse problems across computational science domains.

Abstract:
Computational antibody design holds immense promise for therapeutic discovery, yet existing generative models are fundamentally limited by two core challenges: (i) a lack of dynamical consistency, which yields physically implausible structures, and (ii) poor generalization due to data scarcity and structural bias. We introduce FP‑AbDiff, the first antibody generator to enforce Fokker‑Planck Equation (FPE) physics along the entire generative trajectory. Our method minimizes a novel FPE residual loss over the mixed manifold of CDR geometries (R^3 x SO(3)), compelling locally‑learned denoising scores to assemble into a globally coherent probability flow. This physics‑informed regularizer is synergistically integrated with deep biological priors within a state‑of‑the‑art SE(3)‑equivariant diffusion framework. Rigorous evaluation on the RAbD benchmark confirms that FP‑AbDiff establishes a new state‑of‑the‑art. In de novo CDR‑H3 design, it achieves a mean Root Mean Square Deviation of 0.99 Å when superposing on the variable region, a 25% improvement over the previous state‑of‑the‑art model, AbX, and the highest reported Contact Amino Acid Recovery of 39.91%. This superiority is underscored in the more challenging six‑CDR co‑design task, where our model delivers consistently superior geometric precision, cutting the average full‑chain Root Mean Square Deviation by ~15%, and crucially, achieves the highest full‑chain Amino Acid Recovery on the functionally dominant CDR‑H3 loop (45.67%). By aligning generative dynamics with physical laws, FP‑AbDiff enhances robustness and generalizability, establishing a principled approach for physically faithful and functionally viable antibody design.

Abstract:
We present DropleX, the first system that enables liquid sensing using the capacitive touchscreen of commodity tablets. DropleX detects microliter‑scale liquid samples, and performs non‑invasive, through‑container measurements for liquid analysis. These capabilities are made possible by a physics‑informed mechanism that disables the touchscreen's built‑in adaptive filters, originally designed to reject the effects of liquid drops such as rain, without any hardware modifications. We model the touchscreen's sensing capabilities, limits, and non‑idealities to inform the design of a signal processing and learning‑based pipeline for liquid sensing. Under controlled laboratory conditions, our system achieves 89‑99% accuracy in detecting microliter‑scale adulteration in soda, wine, and milk, 94‑96% accuracy in threshold detection of trace chemical concentrations, and 86‑96% accuracy in through‑container adulterant detection. These exploratory results demonstrate the potential of repurposing commodity touchscreens as a liquid characterization platform for laboratory settings, food and beverage testing, and chemical analysis applications.

Abstract:
Human mobility generation in disaster scenarios plays a vital role in resource allocation, emergency response, and rescue coordination. During disasters such as wildfires and hurricanes, human mobility patterns often deviate from their normal states, which makes the task more challenging. However, existing works usually rely on limited data from a single city or specific disaster, significantly restricting the model's generalization capability in new scenarios. In fact, disasters are highly sudden and unpredictable, and any city may encounter new types of disasters without prior experience. Therefore, we aim to develop a one‑for‑all model for mobility generation that can generalize to new disaster scenarios. However, building a universal framework faces two key challenges: 1) the diversity of disaster types and 2) the heterogeneity among different cities. In this work, we propose a unified model for human mobility generation in natural disasters (named UniDisMob). To enable cross‑disaster generalization, we design physics‑informed prompt and physics‑guided alignment that leverage the underlying common patterns in mobility changes after different disasters to guide the generation process. To achieve cross‑city generalization, we introduce a meta‑learning framework that extracts universal patterns across multiple cities through shared parameters and captures city‑specific features via private parameters. Extensive experiments across multiple cities and disaster scenarios demonstrate that our method significantly outperforms state‑of‑the‑art baselines, achieving an average performance improvement exceeding 13%.

Abstract:
Accurate position reconstruction in noble‑element time‑projection chambers (TPCs) is critical for rare‑event searches in astroparticle physics, yet is systematically limited by electric field distortions arising from charge accumulation on detector surfaces. Conventional data‑driven field corrections suffer from three fundamental limitations: discretization artifacts that break smoothness and differentiability, lack of guaranteed consistency with Maxwell's equations, and statistical requirements of \mathcalO(10^7) calibration events. We introduce a physics‑informed continuous normalizing flow architecture that learns the electric field transformation directly from calibration data while enforcing the constraint of field conservativity through the model structure itself. Applied to simulated ^83\mathrmmKr calibration data in an XLZD‑like dual‑phase xenon TPC, our method achieves superior reconstruction accuracy compared to histogram‑based corrections when trained on identical datasets, demonstrating viable performance with only 6×10^5 events\unicodex2013an order of magnitude reduction in calibration requirements. This approach enables practical monthly field monitoring campaigns, propagation of position uncertainties through differentiable transformations, and enhanced background discrimination in next‑generation rare‑event searches.

Abstract:
Blood flow is sensitive to disease and provides insight into cardiac function, making flow field analysis valuable for diagnosis. However, while safer than radiation‑based imaging and more suitable for patients with medical implants, ultrasound suffers from attenuation with depth, limiting the quality of the image. Despite advances in echocardiographic particle image velocimetry (EchoPIV), accurately measuring blood velocity remains challenging due to the technique's limitations and the complexity of blood flow dynamics. Physics‑informed machine learning can enhance accuracy and robustness, particularly in scenarios where noisy or incomplete data challenge purely data‑driven approaches. We present a physics‑informed neural field model with multi‑scale Fourier Feature encoding for estimating blood flow from sparse and noisy ultrasound data without requiring ground truth supervision. We demonstrate that this model achieves consistently low mean squared error in denoising and inpainting both synthetic and real datasets, verified against reference flow fields and ground truth flow rate measurements. While physics‑informed neural fields have been widely used to reconstruct medical images, applications to medical flow reconstruction are mostly prominent in Flow MRI. In this work, we adapt methods that have proven effective in other imaging modalities to address the specific challenge of ultrasound‑based flow reconstruction.

Abstract:
Energy estimation is critical to impact identification on aerospace composites, where low‑velocity impacts can induce internal damage that is undetectable at the surface. Current methodologies for energy prediction are often constrained by data sparsity, signal noise, complex feature interdependencies, non‑linear dynamics, massive design spaces, and the ill‑posed nature of the inverse problem. This study introduces a physics‑informed framework that embeds domain knowledge into machine learning through a dedicated input space. The approach combines observational biases, which guide the design of physics‑motivated features, with targeted feature selection to retain only the most informative indicators. Features are extracted from time, frequency, and time‑frequency domains to capture complementary aspects of the structural response. A structured feature selection process integrating statistical significance, correlation filtering, dimensionality reduction, and noise robustness ensures physical relevance and interpretability. Exploratory data analysis further reveals domain‑specific trends, yielding a reduced feature set that captures essential dynamic phenomena such as amplitude scaling, spectral redistribution, and transient signal behaviour. Together, these steps produce a compact set of energy‑sensitive indicators with both statistical robustness and physical significance, resulting in impact energy predictions that remain interpretable and traceable to measurable structural responses. Using this optimised input space, a fully‑connected neural network is trained and validated with experimental data from multiple impact scenarios, including pristine and damaged states. The resulting model demonstrates significantly improved impact energy prediction accuracy, reducing errors by a factor of three compared to conventional time‑series techniques and purely data‑driven models.

Abstract:
Refractory Complex Concentrated Alloys (RCCAs) can exhibit exceptional high‑temperature strength, making such alloys promising candidates for high‑temperature structural applications. However, current RCCAs do not possess the high‑temperature oxidation resistance required to survive in oxidizing environments for more than a few hours at or above 1000^\circC, without relying primarily on an environmental barrier coating. Here, we present a machine‑learning framework designed to predict the oxidation‑induced specific mass changes of RCCAs exposed for 24 h at 1000^\circC in air, in order to support the search for oxidation‑resistant alloys over a wide range of compositions. A database was constructed of experimental specific mass change data, upon oxidation at 900‑1000^\circC for 24 h in air, for 77 compositions comprised of simple elements, binary alloys, and higher‑order elemental systems. We then developed a Gaussian Process Regression (GPR) model with physics‑informed descriptors based on oxidation products, capturing the fundamental chemistry of oxide formation and stability. Application of this GPR model to the database yielded a MAE (mean absolute error) test score of 5.78 mg/cm^2, which was a significant improvement in accuracy relative to models only utilizing traditional alloy‑based descriptors. Our model was used to screen over 5,100 quaternary RCCAs, revealing compositions with significantly lower predicted specific mass changes compared to existing literature sources. Overall, this work establishes a versatile and efficient strategy to accelerate the discovery of next‑generation RCCAs with enhanced resistance to extreme environments.

Abstract:
We deal with the solution of the forward problem for high‑dimensional parabolic PDEs with random feature (projection) neural networks (RFNNs). We first prove that there exists a single‑hidden layer neural network with randomized heat‑kernels arising from the fundamental solution (Green's functions) of the heat operator, that we call HEATNET, that provides an unbiased universal approximator to the solution of parabolic PDEs in arbitrary (high) dimensions, with the rate of convergence being analogous to the O(N^‑1/2), where N is the size of HEATNET. Thus, HEATNETs are explainable schemes, based on the analytical framework of parabolic PDEs, exploiting insights from physics‑informed neural networks aided by numerical and functional analysis, and the structure of the corresponding solution operators. Importantly, we show how HEATNETs can be scaled up for the efficient numerical solution of arbitrary high‑dimensional parabolic PDEs using suitable transformations and importance Monte Carlo sampling of the integral representation of the solution, in order to deal with the singularities of the heat kernel around the collocation points. We evaluate the performance of HEATNETs through benchmark linear parabolic problems up to 2,000 dimensions. We show that HEATNETs result in remarkable accuracy with the order of the approximation error ranging from 1.0E‑05 to 1.0E‑07 for problems up to 500 dimensions, and of the order of 1.0E‑04 to 1.0E‑03 for 1,000 to 2,000 dimensions, with a relatively low number (up to 15,000) of features.

Abstract:
The analysis of speech production based on physical models of the vocal folds and vocal tract is essential for studies on vocal‑fold behavior and linguistic research. This paper proposes a speech production analysis method using physics‑informed neural networks (PINNs). The networks are trained directly on the governing equations of vocal‑fold vibration and vocal‑tract acoustics. Vocal‑fold collisions introduce nondifferentiability and vanishing gradients, challenging phenomena for PINNs. We demonstrate, however, that introducing a differentiable approximation function enables the analysis of vocal‑fold vibrations within the PINN framework. The period of self‑excited vocal‑fold vibration is generally unknown. We show that by treating the period as a learnable network parameter, a periodic solution can be obtained. Furthermore, by implementing the coupling between glottal flow and vocal‑tract acoustics as a hard constraint, glottis‑tract interaction is achieved without additional loss terms. We confirmed the method's validity through forward and inverse analyses, demonstrating that the glottal flow rate, vocal‑fold vibratory state, and subglottal pressure can be simultaneously estimated from speech signals. Notably, the same network architecture can be applied to both forward and inverse analyses, highlighting the versatility of this approach. The proposed method inherits the advantages of PINNs, including mesh‑free computation and the natural incorporation of nonlinearities, and thus holds promise for a wide range of applications.

Abstract:
Physics‑Informed Neural Networks (PINNs) offer a flexible framework for solving nonlinear partial differential equations (PDEs), yet conventional implementations often fail to preserve key physical invariants during long‑term integration. This paper introduces a \emphstructure‑preserving PINN framework for the nonlinear Korteweg‑‑de Vries (KdV) equation, a prototypical model for nonlinear and dispersive wave propagation. The proposed method embeds the conservation of mass and Hamiltonian energy directly into the loss function, ensuring physically consistent and energy‑stable evolution throughout training and prediction. Unlike standard \texttttanh‑based PINNs~\citeraissi2019pinn,wang2022modifiedpinn, our approach employs sinusoidal activation functions that enhance spectral expressiveness and accurately capture the oscillatory and dispersive nature of KdV solitons. Through representative case studies ‑‑ including single‑soliton propagation (shape‑preserving translation), two‑soliton interaction (elastic collision with phase shift), and cosine‑pulse initialization (nonlinear dispersive breakup) ‑‑ the model successfully reproduces hallmark behaviors of KdV dynamics while maintaining conserved invariants. Ablation studies demonstrate that combining invariant‑constrained optimization with sinusoidal feature mappings accelerates convergence, improves long‑term stability, and mitigates drift without multi‑stage pretraining. These results highlight that computationally efficient, invariant‑aware regularization coupled with sinusoidal representations yields robust, energy‑consistent PINNs for Hamiltonian partial differential equations such as the KdV equation.

Abstract:
Establishing adaptive particles that sense their state, anticipate their evolution, and compute control inputs onboard has been a major challenge in non‑equilibrium physics. We address this challenge by realizing an autonomous brainbot, building on a recently developed programmable bristlebot. First, we construct a physics‑informed digital twin of the device, based on a kinematic model that reproduces measured trajectory statistics and generates long, statistically faithful synthetic trajectories. The kinematics forms the foundation for implementing onboard model predictive control (MPC), enabling autonomous trajectory tracking, demonstrated by accurate execution of a non‑trivial target path. This provides a proof of principle for a brainbot that senses its state, predicts its evolution, and computes control inputs onboard, unlike conventional active particles with fixed motility, thereby transforming the brainbot into an agentic physical entity. By integrating physical modeling, data‑driven parameter identification, and control into a unified framework, our approach provides a scalable platform for machine‑learning‑enabled multi‑agent studies and lays the groundwork for intelligent, adaptive active matter.

Abstract:
This work presents a novel hybrid approach that integrates Deep Operator Networks (DeepONet) with the Neural Tangent Kernel (NTK) to solve complex inverse problem. The method effectively addresses tasks such as source localization governed by the Navier‑Stokes equations and image reconstruction, overcoming challenges related to nonlinearity, sparsity, and noisy data. By incorporating physics‑informed constraints and task‑specific regularization into the loss function, the framework ensures solutions that are both physically consistent and accurate. Validation on diverse synthetic and real datasets demonstrates its robustness, scalability, and precision, showcasing its broad potential applications in computational physics and imaging sciences.

Abstract:
Recent deep trajectory predictors (e.g., Jiang et al., 2023; Zhou et al., 2022) have achieved strong average accuracy but remain unreliable in complex long‑tail driving scenarios. These limitations reveal the weakness of the prevailing "one‑model‑fits‑all" paradigm, particularly in safety‑critical urban contexts where simpler physics‑based models can occasionally outperform advanced networks (Kalman, 1960). To bridge this gap, we propose a dynamic multi‑expert gating framework that adaptively selects the most reliable trajectory predictor among a physics‑informed LSTM, a Transformer, and a fine‑tuned GameFormer on a per‑sample basis. Our method leverages internal model signals (meta‑features) such as stability and uncertainty (Gal and Ghahramani, 2016), which we demonstrate to be substantially more informative than geometric scene descriptors. To the best of our knowledge, this is the first work to formulate trajectory expert selection as a pairwise‑ranking problem over internal model signals (Burges et al., 2005), directly optimizing decision quality without requiring post‑hoc calibration. Evaluated on the nuPlan‑mini dataset (Caesar et al., 2021) with 1,287 samples, our LLM‑enhanced tri‑expert gate achieves a Final Displacement Error (FDE) of 2.567 m, representing a 9.5 percent reduction over GameFormer (2.835 m), and realizes 57.8 percent of the oracle performance bound. In open‑loop simulations, after trajectory horizon alignment, the same configuration reduces FDE on left‑turn scenarios by approximately 10 percent, demonstrating consistent improvements across both offline validation and open‑loop evaluation. These results indicate that adaptive hybrid systems enhance trajectory reliability in safety‑critical autonomous driving, providing a practical pathway beyond static single‑model paradigms.

Abstract:
The increasing energy demands and carbon footprint of large‑scale AI require intelligent workload management in globally distributed data centers. Yet progress is limited by the absence of benchmarks that realistically capture the interplay of time‑varying environmental factors (grid carbon intensity, electricity prices, weather), detailed data center physics (CPUs, GPUs, memory, HVAC energy), and geo‑distributed network dynamics (latency and transmission costs). To bridge this gap, we present DCcluster‑Opt: an open‑source, high‑fidelity simulation benchmark for sustainable, geo‑temporal task scheduling. DCcluster‑Opt combines curated real‑world datasets, including AI workload traces, grid carbon intensity, electricity markets, weather across 20 global regions, cloud transmission costs, and empirical network delay parameters with physics‑informed models of data center operations, enabling rigorous and reproducible research in sustainable computing. It presents a challenging scheduling problem where a top‑level coordinating agent must dynamically reassign or defer tasks that arrive with resource and service‑level agreement requirements across a configurable cluster of data centers to optimize multiple objectives. The environment also models advanced components such as heat recovery. A modular reward system enables an explicit study of trade‑offs among carbon emissions, energy costs, service level agreements, and water use. It provides a Gymnasium API with baseline controllers, including reinforcement learning and rule‑based strategies, to support reproducible ML research and a fair comparison of diverse algorithms. By offering a realistic, configurable, and accessible testbed, DCcluster‑Opt accelerates the development and validation of next‑generation sustainable computing solutions for geo‑distributed data centers.

Abstract:
In this study, we present and validate the predictive capability of the Physics‑Informed Neural Networks (PINNs) methodology for solving a variety of engineering and biological dynamical systems governed by ordinary differential equations (ODEs). While traditional numerical methods a re effective for many ODEs, they often struggle to achieve convergence in problems involving high stiffness, shocks, irregular domains, singular perturbations, high dimensions, or boundary discontinuities. Alternatively, PINNs offer a powerful approach for handling challenging numerical scenarios. In this study, classical ODE problems are employed as controlled testbeds to systematically evaluate the accuracy, training efficiency, and generalization capability under controlled conditions of the PINNs framework. Although not a universal solution, PINNs can achieve superior results by embedding physical laws directly into the learning process. We first analyze the existence and uniqueness properties of several benchmark problems and subsequently validate the PINNs methodology on these model systems. Our results demonstrate that for complex problems to converge to correct solutions, the loss function components data loss, initial condition loss, and residual loss must be appropriately balanced through careful weighting. We further establish that systematic tuning of hyperparameters, including network depth, layer width, activation functions, learning rate, optimization algorithms, w eight initialization schemes, and collocation point sampling, plays a crucial role in achieving accurate solutions. Additionally, embedding prior knowledge and imposing hard constraints on the network architecture, without loss the generality of the ODE system, significantly enhances the predictive capability of PINNs.

Abstract:
We use elliptic partial differential equations (PDEs) as examples to show various properties and behaviors when shallow neural networks (SNNs) are used to represent the solutions. In particular, we study the numerical ill‑conditioning, frequency bias, and the balance between the differential operator and the shallow network representation for different formulations of the PDEs and with various activation functions. Our study shows that the performance of Physics‑Informed Neural Networks (PINNs) or Deep Ritz Method (DRM) using linear SNNs with power ReLU activation is dominated by their inherent ill‑conditioning and spectral bias against high frequencies. Although this can be alleviated by using non‑homogeneous activation functions with proper scaling, achieving such adaptivity for nonlinear SNNs remains costly due to ill‑conditioning.

Abstract:
This paper presents a physics‑informed machine learning approach for synthesizing optimal feedback control policy for infinite‑horizon optimal control problems by solving the Hamilton‑Jacobi‑Bellman (HJB) partial differential equation(PDE). The optimal control policy is derived analytically for affine dynamical systems with separable and strictly convex control costs, expressed as a function of the gradient of the value function. The resulting HJB‑PDE is then solved by approximating the value function using the Extreme Theory of Functional Connections (X‑TFC) ‑ a hybrid approach that combines the Theory of Functional Connections (TFC) with the Extreme Learning Machine (ELM) algorithm. This approach ensures analytical satisfaction of boundary conditions and significantly reduces training cost compared to traditional Physics‑Informed Neural Networks (PINNs). We benchmark the method on linear and non‑linear systems with known analytical solutions as well as demonstrate its effectiveness on control tasks such as spacecraft optimal de‑tumbling control.

Abstract:
The cosmic microwave background power spectra are a primary window into the early universe. However, achieving interpretable, likelihood‑compatible compression and fast inference under weak model assumptions remains challenging. We propose a parameter‑conditioned variational autoencoder (CVAE) that aligns a data‑driven latent representation with cosmological parameters while remaining compatible with standard likelihood analyses. The model achieves high‑fidelity compression of the D_\ell^TT, D_\ell^EE, and D_\ell^TE spectra into just 5 latent dimensions, with reconstruction accuracy exceeding 99.9% within Planck uncertainties. It reliably reconstructs spectra for beyond‑ΛCDM scenarios, even under parameter extrapolation, and enables rapid inference, reducing the computation time from ～40 hours to ～2 minutes while maintaining posterior consistency. The learned latent space demonstrates a physically meaningful structure, capturing a distributed representation that mirrors known cosmological parameters and their degeneracies. Moreover, it supports highly effective unsupervised discrimination among cosmological models, achieving performance competitive with supervised approaches. Overall, this physics‑informed CVAE enables anomaly detection beyond ΛCDM and points to physically meaningful directions for refinement.

Abstract:
Approximating the solutions of boundary value problems governed by partial differential equations with neural networks is challenging, largely due to the difficult training process. This difficulty can be partly explained by the spectral bias, that is, the slower convergence of high‑frequency components, and can be mitigated by localizing neural networks via (overlapping) domain decomposition. We combine this localization with the Gauss‑Newton method as the optimizer to obtain faster convergence than gradient‑based schemes such as Adam; this comes at the cost of solving an ill‑conditioned linear system in each iteration. Domain decomposition induces a block‑sparse structure in the otherwise dense Gauss‑Newton system, reducing the computational cost per iteration. Our numerical results indicate that combining localization and Gauss‑Newton optimization is promising for neural network‑based solvers for partial differential equations.

Abstract:
Integrated energy systems (IES) are complex heterogeneous architectures that typically encompass power sources, hydrogen electrolyzers, energy storage, and heat exchangers. This integration is achieved through operating control strategy optimization. However, the lack of physical understanding as to how these systems evolve over time introduces uncertainties that hinder reliable application thereof. Techniques that can accommodate such uncertainties are fundamental for ensuring proper operation of these systems. Unfortunately, no unifying methodology exists for accommodating uncertainties in this regard. That being said, adaptive control (AC) is a discipline that may allow for accommodating such uncertainties in real‑time. In the present work, we derive an AC formulation for linear systems in which all states are observable and apply it to the control of a glycol heat exchanger (GHX) in an IES. Based on prior research in which we quantified the uncertainties of the GHXs system dynamics, we introduced an error of 50% on four terms of the nominal model. In the case where a linear quadratic regulator is used as the nominal control for the reference system, we found that employing AC can reduce the mean absolute error and integral time absolute error by a factor of 30%‑75%. This reduction is achieved with minimal computing overhead and control infrastructure, thus underscoring the strength of AC. However, the control effort induced is significant, therefore warranting further study in order to estimate its impact on a physical system. To address further challenges, including partially observable and non‑linear dynamics, enhancements of the linear formulation are currently being developed.

Abstract:
We present an application of Physics‑Informed Neural Networks (PINNs) to the study of \mathrmSU(N_c) lattice gauge theories. Our method enables the learning of eigenfunctions and eigenvalues at arbitrary gauge couplings, smoothly moving from the analytically known strong‑coupling regime towards weaker couplings. By encoding the Schrödinger equation and the symmetries of the eigenstates directly into the loss function, the network performs an unsupervised exploration of the spectrum. We validate the approach on the single‑plaquette \mathrmU(1) and \mathrmSU(2) pure‑gauge theories, showing that the PINNs successfully reproduce the hierarchy of energy levels and their corresponding wavefunctions.

Abstract:
We construct a generative network for Monte‑Carlo sampling in lattice field theories and beyond, for which the learning of layerwise propagation is done and optimised independently on each layer. The architecture uses physics‑informed renormalisation group flows that provide access to the layerwise propagation step from one layer to the next in terms of a simple first order partial differential equation for the respective renormalisation group kernel through a given layer. Thus, it transforms the generative task into that of solving once the set of independent and linear differential equations for the kernels of the transformation. As these equations are analytically known, the kernels can be refined iteratively. This allows us to structurally tackle out‑of‑domain problems generally encountered in generative models and opens the path to further optimisation. We illustrate the practical feasibility of the architecture within simulations in scalar field theories.

Abstract:
In this study, we leverage a mixture model learning approach to identify defects in laser‑based Additive Manufacturing (AM) processes. By incorporating physics based principles, we also ensure that the model is sensitive to meaningful physical parameter variations. The empirical evaluation was conducted by analyzing real‑world data from two AM processes: Directed Energy Deposition and Laser Powder Bed Fusion. In addition, we also studied the performance of the developed framework over public datasets with different alloy type and experimental parameter information. The results show the potential of physics‑guided mixture models to examine the underlying physical behavior of an AM system.

Abstract:
In this paper, local Hölder regularization is incorporated into a physics‑informed neural networks (PINNs) framework for solving elliptic partial differential equations (PDEs). Motivated by the interior regularity properties of linear elliptic PDEs, a modified loss function is constructed by introducing local Hölder regularization term. To approximate this term effectively, a variable‑distance discrete sampling strategy is developed. Error estimates are established to assess the generalization performance of the proposed method. Numerical experiments on a range of elliptic problems demonstrate notable improvements in both prediction accuracy and robustness compared to standard physics‑informed neural networks.

Abstract:
Physics‑informed machine learning (PIML) integrates prior physical information, often in the form of differential equation constraints, into the process of fitting machine learning models to physical data. Popular PIML approaches, including neural operators, physics‑informed neural networks, neural ordinary differential equations, and neural discrete equilibria, are typically fit to objectives that simultaneously include both data and physical constraints. However, the multi‑objective nature of this approach creates ambiguity in the measurement of model quality. This is related to a poor understanding of epistemic uncertainty, and it can lead to surprising failure modes, even when existing statistical metrics suggest strong fits. Working within a Gaussian process regression framework, we introduce the Physics‑Informed Log Evidence (PILE) score. Bypassing the ambiguities of test losses, the PILE score is a single, uncertainty‑aware metric that provides a selection principle for hyperparameters of a PIML model. We show that PILE minimization yields excellent choices for a wide variety of model parameters, including kernel bandwidth, least squares regularization weights, and even kernel function selection. We also show that, even prior to data acquisition, a special 'data‑free' case of the PILE score identifies a priori kernel choices that are 'well‑adapted' to a given PDE. Beyond the kernel setting, we anticipate that the PILE score can be extended to PIML at large, and we outline approaches to do so.

Abstract:
This paper systematically reviews the research progress and application prospects of machine learning technologies in the field of polymer materials. Currently, machine learning methods are developing rapidly in polymer material research; although they have significantly accelerated material prediction and design, their complexity has also caused difficulties in understanding and application for researchers in traditional fields. In response to the above issues, this paper first analyzes the inherent challenges in the research and development of polymer materials, including structural complexity and the limitations of traditional trial‑and‑error methods. To address these problems, it focuses on introducing key basic technologies such as molecular descriptors and feature representation, data standardization and cleaning, and records a number of high‑quality polymer databases. Subsequently, it elaborates on the key role of machine learning in polymer property prediction and material design, covering the specific applications of algorithms such as traditional machine learning, deep learning, and transfer learning; further, it deeply expounds on data‑driven design strategies, such as reverse design, high‑throughput virtual screening, and multi‑objective optimization. The paper also systematically introduces the complete process of constructing high‑reliability machine learning models and summarizes effective experimental verification, model evaluation, and optimization methods. Finally, it summarizes the current technical challenges in research, such as data quality and model generalization ability, and looks forward to future development trends including multi‑scale modeling, physics‑informed machine learning, standardized data sharing, and interpretable machine learning.

Abstract:
Multiparametric mapping MRI has become a viable tool for myocardial tissue characterization. However, misalignment between multiparametric maps makes pixel‑wise analysis challenging. To address this challenge, we developed a generalizable physics‑informed deep‑learning model using test‑time adaptation to enable group image registration across contrast weighted images acquired from multiple physical models (e.g., a T1 mapping model and T2 mapping model). The physics‑informed adaptation utilized the synthetic images from specific physics model as registration reference, allows for transductive learning for various tissue contrast. We validated the model in healthy volunteers with various MRI sequences, demonstrating its improvement for multi‑modal registration with a wide range of image contrast variability.

Abstract:
In this study, the capabilities of the Physics‑Informed Neural Network (PINN) method are investigated for three major tasks: modeling, simulation, and optimization in the context of the heat conduction problem. In the modeling phase, the governing equation of heat transfer by conduction is reconstructed through equation discovery using fractional‑order derivatives, enabling the identification of the fractional derivative order that best describes the physical behavior. In the simulation phase, the thermal conductivity is treated as a physical parameter, and a parametric simulation is performed to analyze its influence on the temperature field. In the optimization phase, the focus is placed on the inverse problem, where the goal is to infer unknown physical properties from observed data. The effectiveness of the PINN approach is evaluated across these three fundamental engineering problem types and compared against conventional numerical methods. The results demonstrate that although PINNs may not yet outperform traditional numerical solvers in terms of speed and accuracy for forward problems, they offer a powerful and flexible framework for parametric simulation, optimization, and equation discovery, making them highly valuable for inverse and data‑driven modeling applications.

Abstract:
Scanning tunnelling microscopy (STM) enables atomic‑resolution imaging and atom manipulation, but its utility is often limited by tip degradation and slow serial data acquisition. Fabrication adds another layer of complexity since the tip is often subjected to large voltages, which may alter the shape of its apex, requiring it to be conditioned. Here, we propose a machine learning (ML) approach for image repair and super‑resolution to alleviate both challenges. Using a dataset of only 36 pristine experimental images of Si(001):H, we demonstrate that a physics‑informed synthetic data generation pipeline can be used to train several state‑of‑the‑art flow‑matching and diffusion models. Quantitative evaluation with metrics such as the CLIP Maximum Mean Discrepancy (CMMD) score and structural similarity demonstrates that our models are able to effectively restore images and offer a two‑ to fourfold reduction in image acquisition time by accurately reconstructing images from sparsely sampled data. Our framework has the potential to significantly increase STM experimental throughput by offering a route to reducing the frequency of tip‑conditioning procedures and to enhancing frame rates in existing high‑speed STM systems.

Abstract:
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long‑standing challenge. Here we introduce a method that parameterizes the solution using spectral bases on arbitrary spatiotemporal domains, whereby the basis is defined on a hyperrectangle containing the true domain. We find the coefficients of the basis expansion by solving an optimization problem whereby both the equations, the boundary conditions and any optimization targets are enforced by a loss function, building on a key idea from Physics‑Informed Neural Networks (PINNs). Since the representation of the function natively has exponential convergence, so does the solution of the optimization problem, as long as it can be solved efficiently. We find empirically that the optimization protocols developed for machine learning find solutions with exponential convergence on a wide range of equations. The method naturally allows for the incorporation of data assimilation by including additional terms in the loss function, and for the efficient solution of optimization problems over the PDE solutions.

Abstract:
We introduce a method for efficiently solving initial‑boundary value problems (IBVPs) that uses Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction. By leveraging symmetry transformations, the model inherently incorporates the physical laws and learns solutions from initial and boundary data. As a result, the loss directly measures the model's accuracy, leading to improved convergence. Moreover, for well‑posed IBVPs, our method enables rigorous error estimation. The approach yields compact models, facilitating an efficient optimization. We implement LieSolver and demonstrate its application to linear homogeneous PDEs with a range of initial conditions, showing that it is faster and more accurate than physics‑informed neural networks (PINNs). Overall, our method improves both computational efficiency and the reliability of predictions for PDE‑constrained problems.

Abstract:
Estimating subsurface dielectric properties is essential for applications ranging from environmental surveys of soils to nondestructive evaluation of concrete in infrastructure. Conventional wave inversion methods typically assume few discrete homogeneous layers and require dense measurements or strong prior knowledge of material boundaries, limiting scalability and accuracy in realistic settings where properties vary continuously. We present a physics informed machine learning framework that reconstructs subsurface permittivity as a fully neural, continuous function of depth, trained to satisfy both measurement data and Maxwells equations. We validate the framework with both simulations and custom built radar experiments on multilayered natural materials. Results show close agreement with in‑situ permittivity measurements (R^2=0.93), with sensitivity to even subtle variations (Delta eps_r=2). Parametric analysis reveals that accurate profiles can be recovered with as few as three strategically placed sensors in two layer systems. This approach reframes subsurface inversion from boundary‑driven to continuous property estimation, enabling accurate characterization of smooth permittivity variations and advancing electromagnetic imaging using low cost radar systems.

Abstract:
The accurate prediction of oceanographic variables is crucial for understanding climate change, managing marine resources, and optimizing maritime activities. Traditional ocean forecasting relies on numerical models; however, these approaches face limitations in terms of computational cost and scalability. In this study, we adapt Aurora, a foundational deep learning model originally designed for atmospheric forecasting, to predict sea surface temperature (SST) in the Canary Upwelling System. By fine‑tuning this model with high‑resolution oceanographic reanalysis data, we demonstrate its ability to capture complex spatiotemporal patterns while reducing computational demands. Our methodology involves a staged fine‑tuning process, incorporating latitude‑weighted error metrics and optimizing hyperparameters for efficient learning. The experimental results show that the model achieves a low RMSE of 0.119K, maintaining high anomaly correlation coefficients (ACC \approx 0.997). The model successfully reproduces large‑scale SST structures but faces challenges in capturing finer details in coastal regions. This work contributes to the field of data‑driven ocean forecasting by demonstrating the feasibility of using deep learning models pre‑trained in different domains for oceanic applications. Future improvements include integrating additional oceanographic variables, increasing spatial resolution, and exploring physics‑informed neural networks to enhance interpretability and understanding. These advancements can improve climate modeling and ocean prediction accuracy, supporting decision‑making in environmental and economic sectors.

Abstract:
Soil physics models have long relied on simplifying assumptions to represent complex processes, yet such assumptions can strongly bias model predictions. Here, we propose a paradigm‑shifting differentiable hybrid modeling (DHM) framework that instead of simplifying the unknown, learns it from data. As a proof of concept, we apply the hybrid approach to the challenge of partitioning the soil water retention curve (SWRC) into capillary and adsorbed water components, a problem where traditional assumptions have led to divergent results. The hybrid framework derives this partitioning directly from data while remaining guided by a few parsimonious and universally accepted physical constraints. Using basic soil physical properties as inputs, the hybrid model couples an analytical formula for the dry end of the SWRC with data‑driven physics‑informed neural networks that learn the wet end, the transition between the two ends, and key soil‑specific parameters. The model was trained on a SWRC dataset from 482 undisturbed soil samples from Central Europe, spanning a broad range of soil texture classes and organic carbon contents. The hybrid model successfully learned both the overall shape and the capillary and adsorbed components of the SWRC. Notably, the model revealed physically meaningful pore‑scale features without relying on explicit geometrical assumptions about soil pore shape or its distribution. Moreover, the model revealed a distinctly nonlinear transition between capillary and adsorbed domains, challenging the linear assumptions invoked in previous studies. The methodology introduced here provides a blueprint for learning other soil processes where high‑quality datasets are available but mechanistic understanding is incomplete.

Abstract:
Physics‑Informed Machine Learning (PIML) has successfully integrated mechanistic understanding into machine learning, particularly in domains governed by well‑known physical laws. This success has motivated efforts to apply PIML to biology, a field rich in dynamical systems but shaped by different constraints. Biological modeling, however, presents unique challenges: multi‑faceted and uncertain prior knowledge, heterogeneous and noisy data, partial observability, and complex, high‑dimensional networks. In this position paper, we argue that these challenges should not be seen as obstacles to PIML, but as catalysts for its evolution. We propose Biology‑Informed Machine Learning (BIML): a principled extension of PIML that retains its structural grounding while adapting to the practical realities of biology. Rather than replacing PIML, BIML retools its methods to operate under softer, probabilistic forms of prior knowledge. We outline four foundational pillars as a roadmap for this transition: uncertainty quantification, contextualization, constrained latent structure inference, and scalability. Foundation Models and Large Language Models will be key enablers, bridging human expertise with computational modeling. We conclude with concrete recommendations to build the BIML ecosystem and channel PIML‑inspired innovation toward challenges of high scientific and societal relevance.

Abstract:
Modeling complex spatiotemporal dynamics, particularly in far‑from‑equilibrium systems, remains a grand challenge in science. The governing partial differential equations (PDEs) for these systems are often intractable to derive from first principles, due to their inherent complexity, characterized by high‑order derivatives and strong nonlinearities, coupled with incomplete physical knowledge. This has spurred the development of data‑driven methods, yet these approaches face limitations: Purely data‑driven models are often physically inconsistent and data‑intensive, while existing physics‑informed methods lack the structural capacity to represent complex operators or systematically integrate partial physical knowledge. Here, we propose a hierarchical physics‑embedded learning framework that fundamentally advances both the forward spatiotemporal prediction and inverse discovery of physical laws from sparse and noisy data. The key innovation is a two‑level architecture that mirrors the process of scientific discovery: the first level learns fundamental symbolic components of a PDE, while the second learns their governing combinations. This hierarchical decomposition not only reduces learning complexity but, more importantly, enables a structural integration of prior knowledge. Known physical laws are directly embedded into the models computational graph, guaranteeing physical consistency and improving data efficiency. By building the framework upon adaptive Fourier Neural Operators, we can effectively capture the non‑local dependencies and high‑order operators characteristic of dynamical systems. Additionally, by structurally decoupling known and unknown terms, the framework further enables interpretable discovery of underlying governing equations through symbolic regression, without presupposing functional forms.

Abstract:
A Minkowskian physics‑informed neural network approach (M‑‑PINN) is formulated to solve the Dyson‑‑Schwinger integral equations (DSE) of quantum electrodynamics (QED) directly in Minkowski spacetime. Our novel strategy merges two complementary approaches: (i) a dispersive solver based on Lehmann representations and subtracted dispersion relations, and (ii) a M‑‑PINN that learns the fermion mass function B(p^2), under the same truncation and renormalization configuration (quenched, rainbow, Landau gauge) with the loss integrating the DSE residual with multi‑‑scale regularization, and monotonicity/smoothing penalties in the spacelike branch in the same way as in our previous work in Euclidean space. The benchmarks show quantitative agreement from the infrared (IR) to the ultraviolet (UV) scales in both on‑shell and momentum‑subtraction schemes. In this controlled setting, our M‑‑PINN reproduces the dispersive solution whilst remaining computationally compact and differentiable, paving the way for extensions with realistic vertices, unquenching effects, and uncertainty‑aware variants.

Abstract:
We are delighted to see the recent development of physics‑informed extreme learning machine (PIELM) for its higher computational efficiency and accuracy compared to other physics‑informed machine learning (PIML) paradigms. Since a comprehensive summary or review of PIELM is currently unavailable, we would like to take this opportunity to share our perspectives and experiences on this promising research direction. We can see that many efforts have been made to solve ordinary/partial differential equations (ODEs/PDEs) characterized by sharp gradients, nonlinearities, high‑frequency behavior, hard constraints, uncertainty, multiphysics coupling, and interpretability. Despite these encouraging successes, many pressing challenges remain to be tackled, which also provides opportunities to develop more robust, interpretable, and generalizable PIELM frameworks for scientific and engineering applications.

Abstract:
Machine‑learning based methods like physics‑informed neural networks and physics‑informed neural operators are becoming increasingly adept at solving even complex systems of partial differential equations. Boundary conditions can be enforced either weakly by penalizing deviations in the loss function or strongly by training a solution structure that inherently matches the prescribed values and derivatives. The former approach is easy to implement but the latter can provide benefits with respect to accuracy and training times. However, previous approaches to strongly enforcing Neumann or Robin boundary conditions require a domain with a fully C^1 boundary and, as we demonstrate, can lead to instability if those boundary conditions are posed on a segment of the boundary that is piecewise C^1 but only C^0 globally. We introduce a generalization of the approach by Sukumar \& Srivastava (doi: 10.1016/j.cma.2021.114333), and a new approach based on orthogonal projections that overcome this limitation. The performance of these new techniques is compared against weakly and semi‑weakly enforced boundary conditions for the scalar Darcy flow equation and the stationary Navier‑Stokes equations.

Abstract:
The Multifaceted Asymmetric Radiation From the Edge (MARFE) is a critical plasma instability that often precedes density‑limit disruptions in tokamaks, posing a significant risk to machine integrity and operational efficiency. Early and reliable alert of MARFE formation is therefore essential for developing effective disruption mitigation strategies, particularly for next‑generation devices like ITER. This paper presents a novel, physics‑informed indicator for early MARFE prediction and disruption warning developed for the HL‑3 tokamak. Our framework integrates two core innovations: (1) a high‑fidelity label refinement pipeline that employs a physics‑scored, weighted Expectation‑Maximization (EM) algorithm to systematically correct noise and artifacts in raw visual data from cameras, and (2) a continuous‑time, physics‑constrained Neural Ordinary Differential Equation (Neural ODE) model that predicts the short‑horizon ``worsening" of a MARFE. By conditioning the model's dynamics on key plasma parameters such as normalized density (f_G, derived from core electron density) and core electron temperature (T_e), the predictor achieves superior performance in the low‑false‑alarm regime crucial for control. On a large experimental dataset from HL‑3, our model demonstrates high predictive accuracy, achieving an Area Under the Curve (AUC) of 0.969 for 40ms‑ahead prediction. The indicator has been successfully deployed for real‑time operation with updates every 1 ms. This work lays a very foundation for future proactive MARFE mitigation.

Abstract:
The accuracy of Physics‑Informed Neural Networks (PINNs) critically depends on the placement of collocation points, as the PDE loss is approximated through sampling over the solution domain. Global sampling ensures stability by covering the entire domain but requires many samples and is computationally expensive, whereas local sampling improves efficiency by focusing on high‑residual regions but may neglect well‑learned areas, reducing robustness. We propose a Global‑Local Fusion (GLF) Sampling Strategy that combines the strengths of both approaches. Specifically, new collocation points are generated by perturbing training points with Gaussian noise scaled inversely to the residual, thereby concentrating samples in difficult regions while preserving exploration. To further reduce computational overhead, a lightweight linear surrogate is introduced to approximate the global residual‑based distribution, achieving similar effectiveness at a fraction of the cost. Together, these components, residual‑adaptive sampling and residual‑based approximation, preserve the stability of global methods while retaining the efficiency of local refinement. Extensive experiments on benchmark PDEs demonstrate that GLF consistently improves both accuracy and efficiency compared with global and local sampling strategies. This study provides a practical and scalable framework for enhancing the reliability and efficiency of PINNs in solving complex and high‑dimensional PDEs.

Abstract:
We propose an adaptive sampling method for the training of Physics Informed Neural Networks (PINNs) which allows for sampling based on an arbitrary problem‑specific heuristic which may depend on the network and its gradients. In particular we focus our analysis on the Allen‑Cahn equations, attempting to accurately resolve the characteristic interfacial regions using a PINN without any post‑hoc resampling. In experiments, we show the effectiveness of these methods over residual‑adaptive frameworks.

Abstract:
The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input‑output function pairs), often requiring substantial amounts of high‑fidelity data, which can be challenging to obtain in some real‑world engineering applications. These datasets may be unevenly discretized from one realization to another, with the grid resolution varying across samples. In this study, we introduce a physics‑informed operator learning approach by extending the Resolution Independent Neural Operator (RINO) framework to a fully data‑free setup, addressing both challenges simultaneously. Here, the arbitrarily (but sufficiently finely) discretized input functions are projected onto a latent embedding space (i.e., a vector space of finite dimensions), using pre‑trained basis functions. The operator associated with the underlying partial differential equations (PDEs) is then approximated by a simple multi‑layer perceptron (MLP), which takes as input a latent code along with spatiotemporal coordinates to produce the solution in the physical space. The PDEs are enforced via a finite difference solver in the physical space. The validation and performance of the proposed method are benchmarked on several numerical examples with multi‑resolution data, where input functions are sampled at varying resolutions, including both coarse and fine discretizations.

Abstract:
Accurately separating tectonic, anthropogenic, and geomorphologic seismic sources is essential for Pacific Northwest (PNW) monitoring but remains difficult as networks densify and signals overlap. Prior work largely treats binary discrimination and seldom compares classic ML (feature‑engineered) and deep learning (end‑to‑end) approaches under a common, multi‑class setting with operational constraints. We evaluate methods and features for four‑way source discrimination ‑ earthquakes, explosions, surface events, and noise ‑ and identify models that are both accurate and deployable. Using ~200k three‑component waveforms from >70k events in an AI‑curated PNW dataset, we test random‑forest classifiers on TSFEL, physics‑informed, and scattering features, and CNNs that ingest time series (1D) or spectrograms (2D); we benchmark on a balanced common test set, a 10k event network dataset, and out‑of‑domain data (global surface events; near‑field blasts). CNNs taking spectrograms lead with accuracy performance over 92% for within‑domain (as a short‑and‑fat CNN SeismicCNN 2D) and out‑of‑domain (as a long and skinny CNN QuakeXNet 2D), versus 89% for the best random forest; performance remains strong at low SNR and longer distances, and generalizes to independent network and global datasets. QuakeXNet‑2D is lightweight (~70k parameters; ~1.2 MB), implemented into seisbench, scans a full day of 100 Hz, three‑component data in ~9 s on commodity hardware, with released checkpoints. These results show spectrogram‑based CNNs provide state‑of‑the‑art accuracy, efficiency, and robustness for real‑time PNW operations and transferable surface‑event monitoring.

Abstract:
Since their introduction, Kolmogorov‑Arnold Networks (KANs) have been successfully applied across several domains, with physics‑informed machine learning (PIML) emerging as one of the areas where they have thrived. In the PIML setting, Chebyshev‑based physics‑informed KANs (cPIKANs) have become the standard due to their computational efficiency. However, like their multilayer perceptron‑based counterparts, cPIKANs face significant challenges when scaled to depth, leading to training instabilities that limit their applicability to several PDE problems. To address this, we propose a basis‑agnostic, Glorot‑like initialization scheme that preserves activation variance and yields substantial improvements in stability and accuracy over the default initialization of cPIKANs. Inspired by the PirateNet architecture, we further introduce Residual‑Gated Adaptive KANs (RGA KANs), designed to mitigate divergence in deep cPIKANs where initialization alone is not sufficient. Through empirical tests and information bottleneck analysis, we show that RGA KANs successfully traverse all training phases, unlike baseline cPIKANs, which stagnate in the diffusion phase in specific PDE settings. Evaluations on nine standard forward PDE benchmarks under a fixed training pipeline with adaptive components demonstrate that RGA KANs consistently outperform parameter‑matched cPIKANs and PirateNets ‑ often by several orders of magnitude ‑ while remaining stable in settings where the others diverge.

Abstract:
Polycrystalline metal failure often begins with stress concentration at grain boundaries. Identifying which microstructural features trigger these events is important but challenging because these extreme damage events are rare and the failure mechanisms involve multiple complex processes across scales. Most existing inference methods focus on average behavior rather than rare events, whereas standard sample‑based methods are computationally expensive for high‑dimensional complex systems. In this paper, we develop a new variational inference framework that integrates a recently developed computationally efficient physics‑informed statistical model with extreme value statistics to significantly facilitate the identification of material failure attributions. First, we reformulate the objective to emphasize observed exceedances by incorporating extreme‑value theory into the likelihood, thereby highlighting tail behavior. Second, we constrain inference via a physics‑informed statistical model that characterizes microstructure‑stress relationships, which uniquely provides physically consistent predictions for these rare events. Third, mixture models in a reduced latent space are developed to capture the non‑Gaussian characteristics of microstructural features, allowing the identification of multiple underlying mechanisms. In both controlled and realistic experimental tests for the bicrystal configuration, the framework achieves reliable extreme‑event prediction and reveals the microstructural features associated with material failure, providing physical insights for material design with uncertainty quantification.

Abstract:
Discovering materials with new structural chemistry is key to achieving transformative functionality. Generative artificial intelligence offers a scalable route to propose candidate crystal structures. We introduce a reliable low‑cost proxy for structural novelty as a conditioning property to steer generation towards novel yet physically plausible structures. We then develop a physics‑informed diffusion model that embeds this descriptor of local environment diversity together with compactness as a stability metric to balance physical plausibility with structural novelty. Conditioning on these metrics improves generative performance across diffusion models, shifting generation away from structural motifs that dominate the training data. A chemically grounded validation protocol isolates those candidates that combine plausibility with structural novelty for physics‑based calculation of energetic stability. Both the stability and the novelty of candidates emerging from this workflow can however change when the full potential energy surface at a candidate composition is evaluated with crystal structure prediction (CSP). This suggests a practical generative‑CSP synergy for discovery‑oriented exploration, where AI targets physically viable yet structurally distinct regions of chemical space for detailed physics‑based assessment of novelty and stability.

Abstract:
Physics‑informed statistical learning (PISL) integrates empirical data with physical knowledge to enhance the statistical performance of estimators. While PISL methods are widely used in practice, a comprehensive theoretical understanding of how informed regularization affects statistical properties is still missing. Specifically, two fundamental questions have yet to be fully addressed: (1) what is the trade‑off between considering soft penalties versus hard constraints, and (2) what is the statistical gain of incorporating physical knowledge compared to purely data‑driven empirical error minimisation. In this paper, we address these questions for PISL in convex classes of functions under physical knowledge expressed as linear equations by developing appropriate complexity dependent error rates based on the small‑ball method. We show that, under suitable assumptions, (1) the error rates of physics‑informed estimators are comparable to those of hard constrained empirical error minimisers, differing only by constant terms, and that (2) informed penalization can effectively reduce model complexity, akin to dimensionality reduction, thereby improving learning performance. This work establishes a theoretical framework for evaluating the statistical properties of physics‑informed estimators in convex classes of functions, contributing to closing the gap between statistical theory and practical PISL, with potential applications to cases not yet explored in the literature.

Abstract:
Tracer kinetic modeling serves a vital role in diagnosis, treatment planning, tracer development and oncology, but burdens practitioners with complex and invasive arterial input function estimation (AIF). We adopt a physics‑informed CycleGAN showing promise in DCE‑MRI quantification to dynamic PET quantification. Our experiments demonstrate sound AIF predictions and parameter maps closely resembling the reference.

Abstract:
Accurate prediction of remaining useful life (RUL) is essential to enhance system reliability and reduce maintenance risk. Yet many strong contemporary models are fragile around fault onset and opaque to engineers: short, high‑energy spikes are smoothed away or misread, fixed thresholds blunt sensitivity, and physics‑based explanations are scarce. To remedy this, we introduce SARNet (Spike‑Aware Consecutive Validation Framework), which builds on a Modern Temporal Convolutional Network (ModernTCN) and adds spike‑aware detection to provide physics‑informed interpretability. ModernTCN forecasts degradation‑sensitive indicators; an adaptive consecutive threshold validates true spikes while suppressing noise. Failure‑prone segments then receive targeted feature engineering (spectral slopes, statistical derivatives, energy ratios), and the final RUL is produced by a stacked RF‑‑LGBM regressor. Across benchmark‑ported datasets under an event‑triggered protocol, SARNet consistently lowers error compared to recent baselines (RMSE 0.0365, MAE 0.0204) while remaining lightweight, robust, and easy to deploy.

Abstract:
Compounding climate hazards, such as wildfire‑induced outages and urban heatwaves, challenge the stability and equity of cities. We present a Hazard‑Responsive Digital Twin (H‑RDT) that combines physics‑informed neural network modeling, multimodal data fusion, and equity‑aware risk analytics for urban‑scale response. In a synthetic district with diverse building archetypes and populations, a simulated wildfire‑outage‑heatwave cascade shows that H‑RDT maintains stable indoor temperature predictions (approximately 31 to 33 C) under partial sensor loss, reproducing outage‑driven surges and recovery. The reinforcement learning based fusion module adaptively reweights IoT, UAV, and satellite inputs to sustain spatiotemporal coverage, while the equity‑adjusted mapping isolates high‑vulnerability clusters (schools, clinics, low‑income housing). Prospective interventions, such as preemptive cooling‑center activation and microgrid sharing, reduce population‑weighted thermal risk by 11 to 13 percent, shrink the 95th‑percentile (tail) risk by 7 to 17 percent, and cut overheating hours by up to 9 percent. Beyond the synthetic demonstration, the framework establishes a transferable foundation for real‑city implementation, linking physical hazard modeling with social equity and decision intelligence. The H‑RDT advances digital urban resilience toward adaptive, learning‑based, and equity‑centered decision support for climate adaptation.

Abstract:
Self‑induced stochastic resonance (SISR) is the emergence of coherent oscillations in slow‑fast excitable systems driven solely by noise, without external periodic forcing or proximity to a bifurcation. This work presents a physics‑informed machine learning framework for modeling and predicting SISR in the stochastic FitzHugh‑Nagumo neuron. We embed the governing stochastic differential equations and SISR‑asymptotic timescale‑matching constraints directly into a Physics‑Informed Neural Network (PINN) based on a Noise‑Augmented State Predictor architecture. The composite loss integrates data fidelity, dynamical residuals, and barrier‑based physical constraints derived from Kramers' escape theory. The trained PINN accurately predicts the dependence of spike‑train coherence on noise intensity, excitability, and timescale separation, matching results from direct stochastic simulations with substantial improvements in accuracy and generalization compared with purely data‑driven methods, while requiring significantly less computation. The framework provides a data‑efficient and interpretable surrogate model for simulating and analyzing noise‑induced coherence in multiscale stochastic systems.

Abstract:
Ground‑based imaging systems struggle to achieve diffraction‑limited resolution when atmospheric turbulence and photon scarcity act simultaneously. In this regime, conventional adaptive optics, speckle imaging, and blind deconvolution lack sufficient information diversity to reliably estimate either the scene or the turbulence. We present Turbulence Aware Poisson Blind Deconvolution (TAP‑BD), a framework designed for robust image recovery in these extreme conditions. TAP‑BD extracts more information from coded‑detection through phase diversity and decodes it with a physics‑informed optimization that incorporates low photon Poisson statistics. Experiments show that TAP‑BD provides reliable reconstructions of both scene and turbulence using only a few tens of measurements, even under strong aberrations and photon‑starved conditions where existing methods fail. This capability enables photon‑efficient, turbulence resilient imaging for applications such as space situational awareness and long‑range remote sensing.

Abstract:
Simulating hybrid magnonic quantum systems remains a challenge due to the large disparity between the timescales of the two systems. We present a massively parallel GPU‑based simulation framework that enables fully coupled, large‑scale modeling of on‑chip magnon‑photon circuits. Our approach resolves the dynamic interaction between ferromagnetic and electromagnetic fields with high spatiotemporal fidelity. To accelerate design workflows, we develop a physics‑informed machine learning surrogate trained on the simulation data, reducing computational cost while maintaining accuracy. This combined approach reveals real‑time energy exchange dynamics and reproduces key phenomena such as anti‑crossing behavior and the suppression of ferromagnetic resonance under strong electromagnetic fields. By addressing the multiscale and multiphysics challenges in magnon‑photon modeling, our framework enables scalable simulation and rapid prototyping of next‑generation quantum and spintronic devices.

Abstract:
Conventional AC Power Flow (ACPF) solvers like Newton‑Raphson (NR) face significant computational and convergence challenges in modern, large‑scale power systems. This paper proposes a novel, two‑stage hybrid method that integrates a Physics‑Informed Graph Neural Network (GNN) with a robust, iterative Linear State Estimation (LSE) refinement step to produce fast and physically‑consistent solutions. The GNN, trained with a physics‑informed loss function featuring an efficient dynamic weighting scheme, rapidly predicts a high‑quality initial system state. This prediction is then refined using an iterative, direct linear solver inspired by state estimation techniques. This LSE refinement step solves a series of linear equations to enforce physical laws, effectively bypassing the non‑linearities and convergence issues of traditional solvers. The proposed GNN‑LSE framework is comprehensively validated on systems ranging from small radial distribution networks (IEEE 33‑bus, 69‑bus) to a large, meshed transmission system (IEEE 118‑bus). Results show that our GNN variants are up to 8.4 × 10^3 times faster than NR. The LSE refinement provides a fast route to a physically‑consistent solution, while heavy‑loading stress tests (120%‑150% of nominal) and N‑1 contingencies demonstrate the method's reliability and generalization. This work presents a powerful and flexible framework for bridging fast, data‑driven models with the rigorous constraints of power system physics, offering a practical tool for real‑time operations and analysis.

Abstract:
Unmanned aerial vehicles (UAVs) operating in dynamic wind fields must generate safe and energy‑efficient trajectories under physical and environmental constraints. Traditional planners, such as A and kinodynamic RRT, often yield suboptimal or non‑smooth paths due to discretization and sampling limitations. This paper presents a physics‑informed neural network (PINN) framework that embeds UAV dynamics, wind disturbances, and obstacle avoidance directly into the learning process. Without requiring supervised data, the PINN learns dynamically feasible and collision‑free trajectories by minimizing physical residuals and risk‑aware objectives. Comparative simulations show that the proposed method outperforms A and Kino‑RRT in control energy, smoothness, and safety margin, while maintaining similar flight efficiency. The results highlight the potential of physics‑informed learning to unify model‑based and data‑driven planning, providing a scalable and physically consistent framework for UAV trajectory optimization.

Abstract:
Short‑term forecasting of airport fog (visibility < 1.0 km) presents challenges in geographic generalization because many machine learning models rely on location‑specific features and fail to transfer across sites. This study investigates whether fundamental thermodynamic and radiative processes can be encoded in a coordinate‑free (location‑independent) feature set to enable geographic transferability. A gradient boosting classifier (XGBoost) trained on Santiago, Chile (SCEL, 33S) data from 2002‑2009 was evaluated on a 2010‑2012 holdout set and under strict zero‑shot tests at Puerto Montt (SCTE), San Francisco (KSFO), and London (EGLL). The model achieved AUC values of 0.923‑0.947 across distances up to 11,650 km and different fog regimes (radiative, advective, marine). Consistent SHAP feature rankings show that visibility persistence, solar angle, and thermal gradients dominate predictions, suggesting the model learned transferable physical relationships rather than site‑specific patterns. Results suggest that physics‑informed, coordinate‑free feature engineering can yield geographically transferable atmospheric forecasting tools.

Abstract:
The Madden‑Julian Oscillation (MJO) is an important driver of global weather and climate extremes, but its prediction in operational dynamical models remains challenging, with skillful forecasts typically limited to 3‑4 weeks. Here, we introduce a novel deep learning framework, the Physics‑guided Cascaded Corrector for MJO (PCC‑MJO), which acts as a universal post‑processor to correct MJO forecasts from dynamical models. This two‑stage model first employs a physics‑informed 3D U‑Net to correct spatial‑temporal field errors, then refines the MJO's RMM index using an LSTM optimized for forecast skill. When applied to three different operational forecasts from CMA, ECMWF and NCEP, our unified framework consistently extends the skillful forecast range (bivariate correlation > 0.5) by 2‑8 days. Crucially, the model effectively mitigates the "Maritime Continent barrier", enabling more realistic eastward propagation and amplitude. Explainable AI analysis quantitatively confirms that the model's decision‑making is spatially congruent with observed MJO dynamics (correlation > 0.93), demonstrating that it learns physically meaningful features rather than statistical fittings. Our work provides a promising physically consistent, computationally efficient, and highly generalizable pathway to break through longstanding barriers in subseasonal forecasting.

Abstract:
This study introduces a novel control framework for adaptive cruise control (ACC) in automated driving, leveraging Long Short‑Term Memory (LSTM) networks and physics‑informed constraints. As automated vehicles (AVs) adopt advanced features like ACC, transportation systems are becoming increasingly intelligent and efficient. However, existing AV control strategies primarily focus on optimizing the performance of individual vehicles or platoons, often neglecting their interactions with human‑driven vehicles (HVs) and the broader impact on traffic flow. This oversight can exacerbate congestion and reduce overall system efficiency. To address this critical research gap, we propose a neural network‑based, socially compliant AV control framework that incorporates social value orientation (SVO). This framework enables AVs to account for their influence on HVs and traffic dynamics. By leveraging AVs as mobile traffic regulators, the proposed approach promotes adaptive driving behaviors that reduce congestion, improve traffic efficiency, and lower energy consumption. Within this framework, we define utility functions for both AVs and HVs, which are optimized based on the SVO of each AV to balance its own control objectives with broader traffic flow considerations. Numerical results demonstrate the effectiveness of the proposed method in adapting to varying traffic conditions, thereby enhancing system‑wide efficiency. Specifically, when the AV's control mode shifts from prioritizing energy consumption to optimizing traffic flow efficiency, vehicles in the following platoon experience at least a 58.99% increase in individual energy consumption alongside at least a 38.39% improvement in individual average speed, indicating significant enhancements in traffic dynamics.

Abstract:
The inverse Stefan problem, as a typical phase‑change problem with moving boundaries, finds extensive applications in science and engineering. Recent years have seen the applications of physics‑informed neural networks (PINNs) to solving Stefan problems, yet they still exhibit shortcomings in hyperparameter dependency, training efficiency, and prediction accuracy. To address this, this paper develops a physics‑informed extreme learning machine (PIELM), a rapid physics‑informed learning method framework for inverse Stefan problems. PIELM replaces conventional deep neural networks with an extreme learning machine network. The input weights are fixed in the PIELM framework, and the output weights are determined by optimizing a loss vector of physical laws composed by initial and boundary conditions and governing partial differential equations (PDEs). Then, solving inverse Stefan problems is transformed into finding the Moore‑Penrose generalized inverse by the least squares method. Case studies show that the PIELM can increase the prediction accuracy by 3‑7 order of magnitude in terms of the relative L2 error, and meanwhile saving more than 94% training time, compared to conventional PINNs.

Abstract:
Kinetic modeling enables in vivo quantification of tracer uptake and glucose metabolism in [^18F]Fluorodeoxyglucose ([^18F]FDG) dynamic positron emission tomography (dPET) imaging of mice. However, kinetic modeling requires the accurate determination of the arterial input function (AIF) during imaging, which is time‑consuming and invasive. Recent studies have shown the efficacy of using deep learning to directly predict the input function, surpassing established methods such as the image‑derived input function (IDIF). In this work, we trained a physics‑informed deep learning‑based input function prediction model (PIDLIF) to estimate the AIF directly from the PET images, incorporating a kinetic modeling loss during training. The proposed method uses a two‑tissue compartment model over two regions, the myocardium and brain of the mice, and is trained on a dataset of 70 [^18F]FDG dPET images of mice accompanied by the measured AIF during imaging. The proposed method had comparable performance to the network without a physics‑informed loss, and when sudden movement causing blurring in the images was simulated, the PIDLIF model maintained high performance in severe cases of image degradation. The proposed physics‑informed method exhibits an improved robustness that is promoted by physically constraining the problem, enforcing consistency for out‑of‑distribution samples. In conclusion, the PIDLIF model offers insight into the effects of leveraging physiological distribution mechanics in mice to guide a deep learning‑based AIF prediction network in images with severe degradation as a result of blurring due to movement during imaging.

Abstract:
Recent advances in Scientific Machine Learning have shown that second‑order methods can enhance the training of Physics‑Informed Neural Networks (PINNs), making them a suitable alternative to traditional numerical methods for Partial Differential Equations (PDEs). However, second‑order methods induce large memory requirements, making them scale poorly with the model size. In this paper, we define a local Mixture of Experts (MoE) combining the parameter‑efficiency of ensemble models and sparse coding to enable the use of second‑order training. Our model ‑‑ \textscPINN Balls ‑‑ also features a fully learnable domain decomposition structure, achieved through the use of Adversarial Adaptive Sampling (AAS), which adapts the DD to the PDE and its domain. \textscPINN Balls achieves better accuracy than the state‑of‑the‑art in scientific machine learning, while maintaining invaluable scalability properties and drawing from a sound theoretical background.

Abstract:
As communication networks evolve towards greater complexity (e.g., 6G and beyond), a deep understanding of the wireless environment becomes increasingly crucial. When explicit knowledge of the environment is unavailable, geometry‑aware feature extraction from channel state information (CSI) emerges as a pivotal methodology to bridge physical‑layer measurements with network intelligence. This paper proposes to explore the received signal strength (RSS) data, without explicit 3D environment knowledge, to jointly construct the radio beam map and environmental geometry for a multiple‑input multiple‑output (MIMO) system. Unlike existing methods that only learn blockage structures, we propose an oriented virtual obstacle model that captures the geometric features of both blockage and reflection. Reflective zones are formulated to identify relevant reflected paths according to the geometry relation of the environment. We derive an analytical expression for the reflective zone and further analyze its geometric characteristics to develop a reformulation that is more compatible with deep learning representations. A physics‑informed deep learning framework that incorporates the reflective‑zone‑based geometry model is proposed to learn the blockage, reflection, and scattering components, along with the beam pattern, which leverages physics prior knowledge to enhance network transferability. Numerical experiments demonstrate that, in addition to reconstructing the blockage and reflection geometry, the proposed model can construct a more accurate MIMO beam map with a 32%‑48% accuracy improvement.

Abstract:
How to integrate and verify spatial intelligence in foundation models remains an open challenge. Current practice often proxies Visual‑Spatial Intelligence (VSI) with purely textual prompts and VQA‑style scoring, which obscures geometry, invites linguistic shortcuts, and weakens attribution to genuinely spatial skills. We introduce Spatial Intelligence Grid (SIG): a structured, grid‑based schema that explicitly encodes object layouts, inter‑object relations, and physically grounded priors. As a complementary channel to text, SIG provides a faithful, compositional representation of scene structure for foundation‑model reasoning. Building on SIG, we derive SIG‑informed evaluation metrics that quantify a model's intrinsic VSI, which separates spatial capability from language priors. In few‑shot in‑context learning with state‑of‑the‑art multimodal LLMs (e.g. GPT‑ and Gemini‑family models), SIG yields consistently larger, more stable, and more comprehensive gains across all VSI metrics compared to VQA‑only representations, indicating its promise as a data‑labeling and training schema for learning VSI. We also release SIGBench, a benchmark of 1.4K driving frames annotated with ground‑truth SIG labels and human gaze traces, supporting both grid‑based machine VSI tasks and attention‑driven, human‑like VSI tasks in autonomous‑driving scenarios.

Abstract:
Deep neural networks (DNNs) are powerful black‑box function approximators which have been shown to yield improved performance compared to traditional neural network (NN) architectures. However, black‑box algorithms do not incorporate known physics of the system and can yield results which are physically implausible. Physics‑informed neural networks (PINNs) have grown in popularity due to their ability to leverage known physical principles in the learning process which has been empirically shown to improve performance compared to traditional black‑box methods. This paper introduces the first physics‑informed DNN controller for an Euler‑Lagrange dynamic system where the adaptation laws are designed using a Lyapunov‑based stability analysis to account for the skew‑symmetry property of the inertia matrix and centripetal‑Coriolis matrix. A Lyapunov‑based stability analysis is provided to guarantee asymptotic convergence of the tracking error and the skew‑symmetric prediction error. Simulations indicate that the developed update law demonstrates improvement in individual and overall function approximation capabilities when compared to a physics‑informed adaptation law which does not incorporate knowledge of system symmetries.

Abstract:
Fatigue behaviors of metal components by laser fusion suffer from scattering due to random geometrical defects (e.g., porosity, lack of fusion). Monitoring fatigue crack initiation and growth is critical, especially for laser‑fused components with significant inherent fatigue scattering. Conventional statistics‑based curve‑fitting fatigue models have difficulty incorporating significant scattering in their fatigue life due to the random geometrical defects. A scattering‑informed predictive method is needed for laser‑fused materials' crack size and growth. Current data‑driven machine learning could circumvent the issue of deterministic modeling, but results in a black‑box function that lacks interpretability. To address these challenges, this study explores a novel nondimensionalized physics‑informed machine learning (PIML) model to predict fatigue crack growth of laser‑fused SS‑316L by integrating fatigue laws and constraints with small data to ensure a realistic and interpretable prediction. Resonance process signature data were leveraged with Paris's law to train the PIML model without experimental crack growth data. The results show that Paris's law constants can be learned with good similarity to comparable data from the literature, and the crack growth rate can be predicted to compute crack sizes.

Abstract:
Movable antenna (MA) technology provides a promising avenue for actively shaping wireless channels through dynamic antenna positioning, thereby enabling electromagnetic radiation reconstruction to enhance physical layer security (PLS). However, its practical deployment is hindered by two major challenges: the high computational complexity of real time optimization and a critical temporal mismatch between slow mechanical movement and rapid channel variations. Although data driven methods have been introduced to alleviate online optimization burdens, they are still constrained by suboptimal training labels derived from conventional solvers or high sample complexity in reinforcement learning. More importantly, existing learning based approaches often overlook communication‑specific domain knowledge, particularly the asymmetric roles and adversarial interactions between legitimate users and eavesdroppers, which are fundamental to PLS. To address these issues, this paper reformulates the MA positioning problem as a predictive task and introduces RoleAware‑MAPP, a novel Transformer based framework that incorporates domain knowledge through three key components: role‑aware embeddings that model user specific intentions, physics‑informed semantic features that encapsulate channel propagation characteristics, and a composite loss function that strategically prioritizes secrecy performance over mere geometric accuracy. Extensive simulations under 3GPP‑compliant scenarios show that RoleAware‑MAPP achieves an average secrecy rate of 0.3569 bps/Hz and a strictly positive secrecy capacity of 81.52%, outperforming the strongest baseline by 48.4% and 5.39 percentage points, respectively, while maintaining robust performance across diverse user velocities and noise conditions.

Abstract:
Atlantic Canada faces significant hurricane threats from damaging winds and coastal flooding that are projected to intensify under climate change. This study adopts a two‑stage framework. First, the evolution of wind and coastal‑flood hazards is quantified from a historical baseline (1979‑2014) to two future periods: a near future (2024‑2059) and a far future (2060‑2095). Hazard fields are constructed from large ensembles of physics‑informed synthetic hurricane tracks, and changes are evaluated in return‑period wind speeds and in inundation depth and extent, with sea‑level rise included for flood projections. The second stage estimates hurricane risk using wind as an operational proxy for total loss, combining the simulated wind fields with exposure data and a vulnerability relationship to compute expected damages. This design clarifies how physical drivers change and how those shifts translate into loss potential without requiring fully coupled compound‑loss modeling. Results indicate an intensification of wind extremes and a substantial amplification of coastal inundation, yielding higher wind‑proxy risk for many coastal communities. Spatial patterns show a heterogeneous escalation of risk concentrated along exposed shorelines and urban corridors. This comprehensive analysis of both hazard evolution and proxy risk provides decision‑ready evidence on where and by how much hurricane losses are likely to grow. The approach clarifies the link between physical drivers and loss potential, ensuring compatibility with standard wind‑centric workflows used in engineering and insurance practice.

Abstract:
Spectral bias, the tendency of neural networks to learn low‑frequency features first, is a well‑known issue with many training algorithms for physics‑informed neural networks (PINNs). To overcome this issue, we propose IFeF‑PINN, an algorithm for iterative training of PINNs with Fourier‑enhanced features. The key idea is to enrich the latent space using high‑frequency components through Random Fourier Features. This creates a two‑stage training problem: (i) estimate a basis in the feature space, and (ii) perform regression to determine the coefficients of the enhanced basis functions. For an underlying linear model, it is shown that the latter problem is convex, and we prove that the iterative training scheme converges. Furthermore, we empirically establish that Random Fourier Features enhance the expressive capacity of the network, enabling accurate approximation of high‑frequency PDEs. Through extensive numerical evaluation on classical benchmark problems, the superior performance of our method over state‑of‑the‑art algorithms is shown, and the improved approximation across the frequency domain is illustrated.

Abstract:
Uncertainty‑aware robot motion prediction is crucial for downstream traversability estimation and safe autonomous navigation in unstructured, off‑road environments, where terrain is heterogeneous and perceptual uncertainty is high. Most existing methods assume deterministic or spatially independent terrain uncertainties, ignoring the inherent local correlations of 3D spatial data and often producing unreliable predictions. In this work, we introduce an efficient probabilistic framework that explicitly models spatially correlated aleatoric uncertainty over terrain parameters as a probabilistic world model and propagates this uncertainty through a differentiable physics engine for probabilistic trajectory forecasting. By leveraging structured convolutional operators, our approach provides high‑resolution multivariate predictions at manageable computational cost. Experimental evaluation on a publicly available dataset shows significantly improved uncertainty estimation and trajectory prediction accuracy over aleatoric uncertainty estimation baselines.

Abstract:
Hamilton‑Jacobi (HJ) Reachability offers a framework for generating safe value functions and policies in the face of adversarial disturbance, but is limited by the curse of dimensionality. Physics‑informed deep learning is able to overcome this infeasibility, but itself suffers from slow and inaccurate convergence, primarily due to weak PDE gradients and the complexity of self‑supervised learning. A few works, recently, have demonstrated that enriching the self‑supervision process with regular supervision (based on the nature of the optimal control problem), greatly accelerates convergence and solution quality, however, these have been limited to single player problems and simple games. In this work, we introduce MADR: MPC‑guided Adversarial DeepReach, a general framework to robustly approximate the two‑player, zero‑sum differential game value function. In doing so, MADR yields the corresponding optimal strategies for both players in zero‑sum games as well as safe policies for worst‑case robustness. We test MADR on a multitude of high‑dimensional simulated and real robotic agents with varying dynamics and games, finding that our approach significantly out‑performs state‑of‑the‑art baselines in simulation and produces impressive results in hardware.

Abstract:
Water is essential for agricultural productivity. Assessing water shortages and reduced yield potential is a critical factor in decision‑making for ensuring agricultural productivity and food security. Crop simulation models, which align with physical processes, offer intrinsic explainability but often perform poorly. Conversely, machine learning models for crop yield modeling are powerful and scalable, yet they commonly operate as black boxes and lack adherence to the physical principles of crop growth. This study bridges this gap by coupling the advantages of both worlds. We postulate that the crop yield is inherently defined by the water availability. Therefore, we formulate crop yield as a function of temporal water scarcity and predict both the crop drought stress and the sensitivity to water scarcity at fine‑scale resolution. Sequentially modeling the crop yield response to water enables accurate yield prediction. To enforce physical consistency, a novel physics‑informed loss function is proposed. We leverage multispectral satellite imagery, meteorological data, and fine‑scale yield data. Further, to account for the uncertainty within the model, we build upon a deep ensemble approach. Our method surpasses state‑of‑the‑art models like LSTM and Transformers in crop yield prediction with a coefficient of determination (R^2‑score) of up to 0.82 while offering high explainability. This method offers decision support for industry, policymakers, and farmers in building a more resilient agriculture in times of changing climate conditions.

Abstract:
In millimeter wave (mmWave) communications, beam alignment and tracking are crucial to combat the significant path loss. As scanning the entire directional space is inefficient, designing an efficient and robust method to identify the optimal beam directions is essential. Since traditional bandit algorithms require a long time horizon to converge under large beam spaces, many existing works propose efficient bandit algorithms for beam alignment by relying on unimodality or multimodality assumptions on the reward function's structure. However, such assumptions often do not hold (or cannot be strictly satisfied) in practice, which causes such algorithms to converge to choosing suboptimal beams. In this work, we propose two physics‑informed bandit algorithms pretc and prgreedy that exploit the sparse multipath property of mmWave channels ‑ a generic but realistic assumption ‑ which is connected to the Phase Retrieval Bandit problem. Our algorithms treat the parameters of each path as black boxes and maintain optimal estimates of them based on sampled historical rewards. pretc starts with a random exploration phase and then commits to the optimal beam under the estimated reward function. prgreedy performs such estimation in an online manner and chooses the best beam under current estimates. Our algorithms can also be easily adapted to beam tracking in the mobile setting. Through experiments using both the synthetic DeepMIMO dataset and the real‑world DeepSense6G dataset, we demonstrate that both algorithms outperform existing approaches in a wide range of scenarios across diverse channel environments, showing their generalizability and robustness.

Abstract:
We introduce the Stable Physics‑Informed Kernel Evolution (SPIKE) method for numerical computation of inviscid hyperbolic conservation laws. SPIKE resolves a fundamental paradox: how strong‑form residual minimization can capture weak solutions containing discontinuities. SPIKE employs reproducing kernel representations with regularized parameter evolution, where Tikhonov regularization provides a smooth transition mechanism through shock formation, allowing the dynamics to traverse shock singularities. This approach automatically maintains conservation, tracks characteristics, and captures shocks satisfying Rankine‑Hugoniot conditions within a unified framework requiring no explicit shock detection or artificial viscosity. Numerical validation across scalar and vector‑valued conservation laws confirms the method's effectiveness.

Abstract:
The objective of designing a control system is to steer a dynamical system with a control signal, guiding it to exhibit the desired behavior. The Hamilton‑Jacobi‑Bellman (HJB) partial differential equation offers a framework for optimal control system design. However, numerical solutions to this equation are computationally intensive, and analytical solutions are frequently unavailable. Knowledge‑guided machine learning methodologies, such as physics‑informed neural networks (PINNs), offer new alternative approaches that can alleviate the difficulties of solving the HJB equation numerically. This work presents a multistage ensemble framework to learn the optimal cost‑to‑go, and subsequently the corresponding optimal control signal, through the HJB equation. Prior PINN‑based approaches rely on a stabilizing the HJB enforcement during training. Our framework does not use stabilizer terms and offers a means of controlling the nonlinear system, via either a singular learned control signal or an ensemble control signal policy. Success is demonstrated in closed‑loop control, using both ensemble‑ and singular‑control, of a steady‑state time‑invariant two‑state continuous nonlinear system with an infinite time horizon, accounting of noisy, perturbed system states and varying initial conditions.

Abstract:
Meshfree particle methods, such as Smoothed Particle Hydrodynamics (SPH) and the Moving Particle Semi‑Implicit (MPS) method, are widely used to simulate complex free‑surface and multiphase flows. A key challenge in these methods is the treatment of solid boundaries, where kernel truncation causes errors and instabilities. Traditional treatments, such as ghost particles and semi‑analytical wall corrections, restore kernel completeness but add significant computational cost and complexity, especially for irregular geometries. We propose a physics‑informed machine learning (ML) framework that directly predicts boundary correction terms for particle approximations, eliminating the need for ghost particles or analytical corrections. The framework is based on a hybrid convolutional neural network‑multilayer perceptron (CNN‑MLP) trained on physics‑informed features that capture local geometry, particle states, and kernel properties. Once trained, it provides consistent boundary contributions across all spatial differential operators, including gradients, divergences, and Laplacians. The approach is demonstrated with MPS but is readily extensible to other particle methods such as SPH. Tests with predefined fields, unsteady diffusion, and incompressible Navier‑Stokes flows demonstrate accuracy comparable to that of ghost‑particle methods while reducing computational overhead. The model generalizes well to unseen geometries, flow conditions, and particle distributions, including dynamically evolving domains. This work establishes a flexible, physics‑informed ML paradigm for boundary treatment in particle‑based PDE solvers, improving both accuracy and scalability across a broad class of meshfree methods.

Abstract:
We apply a physics‑informed neural network (PINN) to solve the two‑point boundary value problem (BVP) arising from the necessary conditions postulated by Pontryagin's Minimum Principle for optimal control. Such BVPs are known to be numerically difficult to solve by traditional shooting methods due to extremely high sensitivity to initial guesses. In the light of recent successes in applying PINNs for solving high‑dimensional differential equations, we develop a PINN to solve the problem of finding trajectories with minimum exposure to a spatiotemporal threat for a vehicle kinematic model. First, we implement PINNs that are trained to solve the BVP for a given pair of initial and final states for a given threat field. Next, we implement a PINN conditioned on the initial state for a given threat field, which eliminates the need for retraining for each initial state. We demonstrate that the PINN outputs satisfy the necessary conditions with low numerical error.

Abstract:
As an increasing amount of remote sensing data becomes available in the Arctic Ocean, data‑driven machine learning (ML) techniques are becoming widely used to predict sea ice velocity (SIV) and sea ice concentration (SIC). However, fully data‑driven ML models have limitations in generalizability and physical consistency due to their excessive reliance on the quantity and quality of training data. In particular, as Arctic sea ice entered a new phase with thinner ice and accelerated melting, there is a possibility that an ML model trained with historical sea ice data cannot fully represent the dynamically changing sea ice conditions in the future. In this study, we develop physics‑informed neural network (PINN) strategies to integrate physical knowledge of sea ice into the ML model. Based on the Hierarchical Information‑sharing U‑net (HIS‑Unet) architecture, we incorporate the physics loss function and the activation function to produce physically plausible SIV and SIC outputs. Our PINN model outperforms the fully data‑driven model in the daily predictions of SIV and SIC, even when trained with a small number of samples. The PINN approach particularly improves SIC predictions in melting and early freezing seasons and near fast‑moving ice regions.

Abstract:
Optimizing the energy management within a smart grids scenario presents significant challenges, primarily due to the complexity of real‑world systems and the intricate interactions among various components. Reinforcement Learning (RL) is gaining prominence as a solution for addressing the challenges of Optimal Power Flow in smart grids. However, RL needs to iterate compulsively throughout a given environment to obtain the optimal policy. This means obtaining samples from a, most likely, costly simulator, which can lead to a sample efficiency problem. In this work, we address this problem by substituting costly smart grid simulators with surrogate models built using Phisics‑informed Neural Networks (PINNs), optimizing the RL policy training process by arriving to convergent results in a fraction of the time employed by the original environment.

Abstract:
Grey‑box methods for system identification combine deep learning with physics‑informed constraints, capturing complex dependencies while improving out‑of‑distribution generalization. Despite the growing importance of floating‑base systems such as humanoids and quadrupeds, current grey‑box models ignore their specific physical constraints. For instance, the inertia matrix is not only positive definite but also exhibits branch‑induced sparsity and input independence. Moreover, the 6x6 composite spatial inertia of the floating base inherits properties of single‑rigid‑body inertia matrices. As we show, this includes the triangle inequality on the eigenvalues of the composite rotational inertia. To address the lack of physical consistency in deep learning models of floating‑base systems, we introduce a parameterization of inertia matrices that satisfies all these constraints. Inspired by Deep Lagrangian Networks (DeLaN), we train neural networks to predict physically plausible inertia matrices that minimize inverse dynamics error under Lagrangian mechanics. For evaluation, we collected and released a dataset on multiple quadrupeds and humanoids. In these experiments, our Floating‑Base Deep Lagrangian Networks (FeLaN) achieve better overall performance on both simulated and real robots, while providing greater physical interpretability.

Abstract:
Building on the first‑ever application of neural networks in TMD phenomenology: "Extraction of the Sivers function with deep neural networks", we now present a momentum space, physics‑informed deep learning framework for the direct extraction of unpolarized transverse momentum dependent parton distributions (TMDs) from fixed target Drell‑Yan data (E288, E605). Rather than transforming to impact‑parameter space, we remain in k and embed a normalized integrand s(x, k; Q) whose auto‑convolution produces the observed qT spectra. The extraction proceeds in two steps. Stage I learns the structure kernel S(qT , x1, x2; QM ) by regressing the cross‑section with known kinematic prefactors and charge‑weighted PDF combinations factored out; experimental and PDF uncertainties are propagated with Monte Carlo replicas. Stage II reconstructs s(x, k; Q) with an end‑to‑end differentiable k quadrature layer. Applied to Fermilab cross‑section data from experiments E288 and E605, the method reproduces the measured qT spectra across Q and yields x and Q dependent TMDs that broaden with Q, with uncertainty bands that consistently propagate experimental, PDF, algorithmic and methodological components. The approach is minimally biased (no factorized Ansatze and no bT transform) and provides a transferable template for polarized TMDs and related QCD inverse problems.

Abstract:
Heating, Ventilation, and Air‑Conditioning (HVAC) systems account for a substantial share of global building energy use, making reliable anomaly detection essential for improving efficiency and reducing emissions. Classical rule‑based approaches offer explainability but lack adaptability, while deep learning methods provide predictive power at the cost of transparency, efficiency, and physical plausibility. Recent attempts to use Large Language Models (LLMs) for anomaly detection improve interpretability but largely ignore the physical principles that govern HVAC operations. We present PILLM, a Physics‑Informed LLM framework that operates within an evolutionary loop to automatically generate, evaluate, and refine anomaly detection rules. Our approach introduces physics‑informed reflection and crossover operators that embed thermodynamic and control‑theoretic constraints, enabling rules that are both adaptive and physically grounded. Experiments on the public Building Fault Detection dataset show that PILLM achieves state‑of‑the‑art performance while producing diagnostic rules that are interpretable and actionable, advancing trustworthy and deployable AI for smart building systems.

Abstract:
Existing approaches for transporting and manipulating cable‑suspended loads using multiple UAVs along reference trajectories typically rely on either centralized control architectures or reliable inter‑agent communication. In this work, we propose a novel machine learning based method for decentralized kinodynamic planning that operates effectively under partial observability and without inter‑agent communication. Our method leverages imitation learning to train a decentralized student policy for each UAV by imitating a centralized kinodynamic motion planner with access to privileged global observations. The student policy generates smooth trajectories using physics‑informed neural networks that respect the derivative relationships in motion. During training, the student policies utilize the full trajectory generated by the teacher policy, leading to improved sample efficiency. Moreover, each student policy can be trained in under two hours on a standard laptop. We validate our method in both simulation and real‑world environments to follow an agile reference trajectory, demonstrating performance comparable to that of centralized approaches.

Abstract:
We propose an enhanced physics‑informed neural network (PINN), the Trace Regularity Physics‑Informed Neural Network (TRPINN), which enforces the boundary loss in the Sobolev‑Slobodeckij norm H^1/2(\partial Ω), the correct trace space associated with H^1(Ω). We reduce computational cost by computing only the theoretically essential portion of the semi‑norm and enhance convergence stability by avoiding denominator evaluations in the discretization. By incorporating the exact H^1/2(\partial Ω) norm, we show that the approximation converges to the true solution in the H^1(Ω) sense, and, through Neural Tangent Kernel (NTK) analysis, we demonstrate that TRPINN can converge faster than standard PINNs. Numerical experiments on the Laplace equation with highly oscillatory Dirichlet boundary conditions exhibit cases where TRPINN succeeds even when standard PINNs fail, and show performance improvements of one to three decimal digits.

Abstract:
Characterizing the temperature‑dependent thermal conductivity is challenging because the property varies strongly with temperature and reliable heat flow measurement, not just temperature sensing, is difficult under experimental conditions. Here, we present a physics informed deep learning framework that infers conductivity solely from sparse electric potential measurements. We first develop a deterministic physics‑informed neural network (PINN) that embeds coupled thermoelectric transport equations as soft constraints, enabling simultaneous recovery of spatial temperature, voltage, and conductivity profiles without temperature data. The deterministic PINN achieves accurate inference under noise‑free conditions, yet its predictions degrade when measurement noise is introduced. To address this, we extend the framework to a Bayesian PINN, which models network parameters probabilistically and employs Hamiltonian Monte Carlo (HMC) sampling for posterior inference. This extension produces robust thermal conductivity estimates and, importantly, provides credible intervals that quantify uncertainty from sparse and noisy data. Numerical experiments confirm that the Bayesian PINN not only preserves predictive accuracy under noise but also reveals inference bias and enables uncertainty aware interpretation of material properties. Together, the deterministic and Bayesian formulations establish a scalable and generalizable alternative to conventional methods for determining temperature‑dependent properties, offering physics‑ consistent and risk‑aware property inference for thermoelectric systems and other functional materials where direct temperature sensing is impractical

Abstract:
Solving the nonlinear AC optimal power flow (AC OPF) problem remains a major computational bottleneck for real‑time grid operations. In this paper, we propose a residual learning paradigm that uses fast DC optimal power flow (DC OPF) solutions as a baseline, and learns only the nonlinear corrections required to provide the full AC‑OPF solution. The method utilizes a topology‑aware Graph Neural Network with local attention and two‑level DC feature integration, trained using a physics‑informed loss that enforces AC power‑flow feasibility and operational limits. Evaluations on OPFData for 57‑, 118‑, and 2000‑bus systems show around 25% lower MSE, up to 3X reduction in feasibility error, and up to 13X runtime speedup compared to conventional AC OPF solvers. The model maintains accuracy under N‑1 contingencies and scales efficiently to large networks. These results demonstrate that residual learning is a practical and scalable bridge between linear approximations and AC‑feasible OPF, enabling near real‑time operational decision making.

Abstract:
Recent works have shown that natural gradient methods can significantly outperform standard optimizers when training physics‑informed neural networks (PINNs). In this paper, we analyze the training dynamics of PINNs optimized with ANaGRAM, a natural‑gradient‑inspired approach employing singular value decomposition with cutoff regularization. Building on this analysis, we propose a multi‑cutoff adaptation strategy that further enhances ANaGRAM's performance. Experiments on benchmark PDEs validate the effectiveness of our method, which allows to reach machine precision on some experiments. To provide theoretical grounding, we develop a framework based on spectral theory that explains the necessity of regularization and extend previous shown connections with Green's functions theory.

Abstract:
We propose a hybrid solver that fuses the dimensionality‑reduction strengths of the Method of Lines (MOL) with the flexibility of Physics‑Informed Neural Networks (PINNs). Instead of approximating spatial derivatives with fixed finite‑difference stencils ‑ whose truncation errors force extremely fine meshes ‑ our method trains a neural network to represent the initial spatial profile and then employs automatic differentiation to obtain spectrally accurate gradients at arbitrary nodes. These high‑fidelity derivatives define the right‑hand side of the MOL‑generated ordinary‑differential system, and time integration is replaced with a secondary temporal PINN while spatial accuracy is retained without mesh refinement. The resulting "boundary‑informed MOL‑PINN" matches or surpasses conventional MOL in accuracy using an order of magnitude fewer collocation points, thereby shrinking memory footprints, lessening dependence on large data sets, and increasing complexity robustness. Because it relies only on automatic differentiation and standard optimizers, the framework extends naturally to linear and nonlinear PDEs in any spatial dimension.

Abstract:
Clinical adoption of multi‑shot diffusion‑weighted magnetic resonance imaging (multi‑shot DWI) for body‑wide tumor diagnostics is limited by severe motion‑induced phase artifacts from respiration, peristalsis, and so on, compounded by multi‑organ, multi‑slice, multi‑direction and multi‑b‑value complexities. Here, we introduce a reconstruction framework, LoSP‑Prompt, that overcomes these challenges through physics‑informed modeling and synthetic‑data‑driven prompt learning. We model inter‑shot phase variations as a high‑order Locally Smooth Phase (LoSP), integrated into a low‑rank Hankel matrix reconstruction. Crucially, the algorithm's rank parameter is automatically set via prompt learning trained exclusively on synthetic abdominal DWI data emulating physiological motion. Validated across 10,000+ clinical images (43 subjects, 4 scanner models, 5 centers), LoSP‑Prompt: (1) Achieved twice the spatial resolution of clinical single‑shot DWI, enhancing liver lesion conspicuity; (2) Generalized to seven diverse anatomical regions (liver, kidney, sacroiliac, pelvis, knee, spinal cord, brain) with a single model; (3) Outperformed state‑of‑the‑art methods in image quality, artifact suppression, and noise reduction (11 radiologists' evaluations on a 5‑point scale, p<0.05), achieving 4‑5 points (excellent) on kidney DWI, 4 points (good to excellent) on liver, sacroiliac and spinal cord DWI, and 3‑4 points (good) on knee and tumor brain. The approach eliminates navigator signals and realistic data supervision, providing an interpretable, robust solution for high‑resolution multi‑organ multi‑shot DWI. Its scanner‑agnostic performance signifies transformative potential for precision oncology.

Abstract:
Two‑photon lithography (TPL) is a sophisticated additive manufacturing technology for creating three‑dimensional (3D) micro‑ and nano‑structures. Maintaining the health of TPL systems is critical for ensuring consistent fabrication quality. Current maintenance practices often rely on experience rather than informed monitoring of machine health, resulting in either untimely maintenance that causes machine downtime and poor‑quality fabrication, or unnecessary maintenance that leads to inefficiencies and avoidable downtime. To address this gap, this paper presents three methods for accurate and timely monitoring of TPL machine health. Through integrating physics‑informed data‑driven predictive models for structure dimensions with statistical approaches, the proposed methods are able to handle increasingly complex scenarios featuring different levels of generalizability. A comprehensive experimental dataset that encompasses six process parameter combinations and six structure dimensions under two machine health conditions was collected to evaluate the effectiveness of the proposed approaches. Across all test scenarios, the approaches are shown to achieve high accuracies, demonstrating excellent effectiveness, robustness, and generalizability. These results represent a significant step toward condition‑based maintenance for TPL systems.

Abstract:
Inverse problems involving partial differential equations (PDEs) with discontinuous coefficients are fundamental challenges in modeling complex spatiotemporal systems with heterogeneous structures and uncertain dynamics. Traditional numerical and machine learning approaches often face limitations in addressing these problems due to high dimensionality, inherent nonlinearity, and discontinuous parameter spaces. In this work, we propose a novel computational framework that synergistically integrates physics‑informed deep learning with Bayesian inference for accurate parameter identification in PDEs with jump discontinuities in coefficients. The core innovation of our framework lies in a dual‑network architecture employing a gradient‑adaptive weighting strategy: a main network approximates PDE solutions while a sub network samples its coefficients. To effectively identify mixture structures in parameter spaces, we employ Markovian dynamics methods to capture hidden state transitions of complex spatiotemporal systems. The framework has applications in reconstruction of solutions and identification of parameter‑varying regions. Comprehensive numerical experiments on various PDEs with jump‑varying coefficients demonstrate the framework's exceptional adaptability, accuracy, and robustness compared to existing methods. This study provides a generalizable computational approach of parameter identification for PDEs with discontinuous parameter structures, particularly in non‑stationary or heterogeneous systems.

Abstract:
The accurate representation of numerous physical, chemical, and biological processes relies heavily on differential equations (DEs), particularly nonlinear differential equations (NDEs). While understanding these complex systems necessitates obtaining solutions to their governing equations, the derivation of precise approximations for NDEs remains a formidable task in computational mathematics. Although established techniques such as the finite element method (FEM) have long been foundational, remarkable promise for approximating continuous functions with high efficacy has recently been demonstrated by advancements in physics‑informed deep‑learning feedforward neural networks. In this work, a novel application of PINNs is presented for the approximation of the challenging Electrohydrodynamic (EHD) problem. A specific L^2‑type total loss function is employed, notably without reliance on any prior knowledge of the exact solution. A comprehensive comparative study is conducted, juxtaposing the approximation capabilities of the proposed neural network with those of the conventional FEM. The PINN training regimen is composed of two critical steps: forward propagation for adjustments to gradient and curvature, and backpropagation for the refinement of hyperparameters. The critical challenge of identifying optimal neural network architectures and hyperparameter configurations for efficient optimization is meticulously investigated. Excellent performance is shown to be delivered by the neural network even with a limited training dataset. Simultaneously, it is demonstrated that the accuracy of the FEM can be substantially enhanced through the judicious selection of smaller mesh sizes.

Abstract:
The Keller‑Segel (KS) chemotaxis system is used to describe the overall behavior of a collection of cells under the influence of chemotaxis. However, solving the KS chemotaxis system and generating its aggregation patterns remain challenging due to the emergence of solutions exhibiting near‑singular behavior, such as finite‑time blow‑up or concentration phenomena. Building on a Lagrangian framework of the KS system, we develop DeepLagrangian, a self‑adaptive density estimation method that learns and generates aggregation patterns and near‑singular solutions of the KS system in two‑ and three‑dimensional (2D and 3D) space under different physical parameters. The main advantage of the Lagrangian framework is its inherent ability to adapt to near‑singular solutions. To develop this framework, we normalize the KS solution into a probability density function (PDF), derive the corresponding normalized KS system, and utilize the property of the continuity equation to rewrite the system into a Lagrangian framework. We then define a physics‑informed Lagrangian loss to enforce this framework and incorporate a flow‑based generative model, called the time‑dependent KRnet, to approximate the PDF by minimizing the loss. Furthermore, we integrate time‑marching strategies with the time‑dependent KRnet to enhance the accuracy of the PDF approximation. After obtaining the approximate PDF, we recover the original KS solution. We also prove that the Lagrangian loss effectively controls the Kullback‑Leibler (KL) divergence between the approximate PDF and the exact PDF. In the numerical experiments, we demonstrate the accuracy of our DeepLagrangian method for the 2D and 3D KS chemotaxis system with/without advection.

Abstract:
Shadow molecular dynamics provide an efficient and stable atomistic simulation framework for flexible charge models with long‑range electrostatic interactions. While previous implementations have been limited to atomic monopole charge distributions, we extend this approach to flexible multipole models. We derive detailed expressions for the shadow energy functions, potentials, and force terms, explicitly incorporating monopole‑monopole, dipole‑monopole, and dipole‑dipole interactions. In our formulation, both atomic monopoles and atomic dipoles are treated as extended dynamical variables alongside the propagation of the nuclear degrees of freedom. We demonstrate that introducing the additional dipole degrees of freedom preserves the stability and accuracy previously seen in monopole‑only shadow molecular dynamics simulations. Additionally, we present a shadow molecular dynamics scheme where the monopole charges are held fixed while the dipoles remain flexible. Our extended shadow dynamics provide a framework for stable, computationally efficient, and versatile molecular dynamics simulations involving long‑range interactions between flexible multipoles. This is of particular interest in combination with modern artificial intelligence and machine learning techniques, which are increasingly used to develop physics‑informed and data‑driven foundation models for atomistic simulations. These models aim to provide transferable, high‑accuracy representations of atomic interactions that are applicable across diverse sets of molecular systems, which requires accurate treatment of long‑range charge interactions.

Abstract:
Computational Fluid Dynamics (CFD) is central to science and engineering, but faces severe scalability challenges, especially in high‑dimensional, multiscale, and turbulent regimes. Traditional numerical methods often become prohibitively expensive under these conditions. Quantum computing and quantum‑inspired methods have been investigated as promising alternatives. This review surveys advances at the intersection of quantum computing, quantum algorithms, machine learning, and tensor network techniques for CFD. We discuss the use of Variational Quantum Algorithms as hybrid quantum‑classical solvers for PDEs, emphasizing their ability to incorporate nonlinearities through Quantum Nonlinear Processing Units. We further review Quantum Neural Networks and Quantum Physics‑Informed Neural Networks, which extend classical machine learning frameworks to quantum hardware and have shown advantages in parameter efficiency and solution accuracy for certain CFD benchmarks. Beyond quantum computing, we examine tensor network methods, originally developed for quantum many‑body systems and now adapted to CFD as efficient high‑dimensional compression and solver tools. Reported studies include several orders of magnitude reductions in memory and runtime while preserving accuracy. Together, these approaches highlight quantum and quantum‑inspired strategies that may enable more efficient CFD solvers. This review closes with perspectives: quantum CFD remains out of reach in the NISQ era, but quantum‑inspired tensor networks already show practical benefits, with hybrid approaches offering the most promising near‑term strategy.

Abstract:
DSC‑MRI perfusion is a medical imaging technique for diagnosing and prognosing brain tumors and strokes. Its analysis relies on mathematical deconvolution, but noise or motion artifacts in a clinical environment can disrupt this process, leading to incorrect estimate of perfusion parameters. Although deep learning approaches have shown promising results, their calibration typically rely on third‑party deconvolution algorithms to generate reference outputs and are bound to reproduce their limitations. To adress this problem, we propose a physics‑informed autoencoder that leverages an analytical model to decode the perfusion parameters and guide the learning of the encoding network. This autoencoder is trained in a self‑supervised fashion without any third‑party software and its performance is evaluated on a database with glioma patients. Our method shows reliable results for glioma grading in accordance with other well‑known deconvolution algorithms despite a lower computation time. It also achieved competitive performance even in the presence of high noise which is critical in a medical environment.

Abstract:
Physics‑Informed Neural Networks (PINNs) embed the partial differential equations (PDEs) governing the system under study directly into the training of Neural Networks, ensuring solutions that respect physical laws. While effective for single‑system problems, standard PINNs scale poorly to datasets containing many realizations of the same underlying physics with varying parameters. To address this limitation, we present a complementary approach by including auxiliary physically‑redundant information in loss (APRIL), i.e. augment the standard supervised output‑target loss with auxiliary terms which exploit exact physical redundancy relations among outputs. We mathematically demonstrate that these terms preserve the true physical minimum while reshaping the loss landscape, improving convergence toward physically consistent solutions. As a proof‑of‑concept, we benchmark APRIL on a fully‑connected neural network for gravitational wave (GW) parameter estimation (PE). We use simulated, noise‑free compact binary coalescence (CBC) signals, focusing on inspiral‑frequency waveforms to recover the chirp mass \mathcalM, the total mass M_\mathrmtot, and symmetric mass ratio η of the binary. In this controlled setting, we show that APRIL achieves up to an order‑of‑magnitude improvement in test accuracy, especially for parameters that are otherwise difficult to learn. This method provides physically consistent learning for large multi‑system datasets and is well suited for future GW analyses involving realistic noise and broader parameter ranges.

Abstract:
Accurate prediction of vehicle collision dynamics is crucial for advanced safety systems and post‑impact control applications, yet existing methods face inherent trade‑offs among computational efficiency, prediction accuracy, and data requirements. This paper proposes a dual Physics‑Informed Neural Network framework addressing these challenges through two complementary networks. The first network integrates Gaussian Mixture Models with PINN architecture to learn impact force distributions from finite element analysis data while enforcing momentum conservation and energy consistency constraints. The second network employs an adaptive PINN with dynamic constraint weighting to predict post‑collision vehicle dynamics, featuring an adaptive physics guard layer that prevents unrealistic predictions whil e preserving data‑driven learning capabilities. The framework incorporates uncertainty quantification through time‑varying parameters and enables rapid adaptation via fine‑tuning strategies. Validation demonstrates significant improvements: the impact force model achieves relative errors below 15.0% for force prediction on finite element analysis (FEA) datasets, while the vehicle dynamics model reduces average trajectory prediction error by 63.6% compared to traditional four‑degree‑of‑freedom models in scaled vehicle experiments. The integrated system maintains millisecond‑level computational efficiency suitable for real‑time applications while providing probabilistic confidence bounds essential for safety‑critical control. Comprehensive validation through FEA simulation, dynamic modeling, and scaled vehicle experiments confirms the framework's effectiveness for Precision Immobilization Technique scenarios and general collision dynamics prediction.

Abstract:
Physics‑informed neural networks (PINNs) are neural networks that embed the laws of dynamical systems modeled by differential equations into their loss function as constraints. In this work, we present a PINN framework applied to oncology. Here, we seek to learn time‑varying interactions due to a combination therapy in a tumor microenvironment. In oncology, experimental data are often sparse and composed of a few time points of tumor volume. By embedding inductive biases derived from prior information about a dynamical system, we extend the physics‑informed neural networks (PINN) and incorporate observed biological constraints as regularization agents. The modified PINN algorithm is able to steer itself to a reasonable solution and can generalize well with only a few training examples. We demonstrate the merit of our approach by learning the dynamics of treatment applied intermittently in an ordinary differential equation (ODE) model of a combination therapy. The algorithm yields a solution to the ODE and time‑varying forms of some of the ODE model parameters. We demonstrate a strong convergence using metrics such as the mean squared error (MSE), mean absolute error (MAE), and mean absolute percentage error (MAPE).

Abstract:
Discrete tensor train decomposition is widely employed to mitigate the curse of dimensionality in solving high‑dimensional PDEs through traditional methods. However, the direct application of the tensor train method typically requires uniform grids of regular domains, which limits its application on non‑uniform grids or irregular domains. To address the limitation, we develop a functional tensor train neural network (FTTNN) for solving high‑dimensional PDEs, which can represent PDE solutions on non‑uniform grids or irregular domains. An essential ingredient of our approach is to represent the PDE solutions by the functional tensor train format whose TT‑core functions are approximated by neural networks. To give the functional tensor train representation, we propose and study functional tensor train rank and employ it into a physics‑informed loss function for training. Because of tensor train representation, the resulting high‑dimensional integral in the loss function can be computed via one‑dimensional integrals by Gauss quadrature rules. Numerical examples including high‑dimensional PDEs on regular or irregular domains are presented to demonstrate that the performance of the proposed FTTNN is better than that of Physics Informed Neural Networks (PINN).

Abstract:
The global transition to hydrogen‑based energy infrastructures faces significant hurdles. Chief among these are the high costs and sustainability issues associated with acid‑based proton exchange membrane fuel cells. Anion exchange membrane (AEM) fuel cells offer promising cost‑effective alternatives, yet their widespread adoption is limited by rapid degradation in alkaline environments. Here, we develop a framework that integrates mechanistic insights with machine learning, enabling the identification of generalized degradation behavior across diverse polymeric AEM chemistries and operating conditions. Our model successfully predicts long‑term hydroxide conductivity degradation (up to 10,000 hours) from minimal early‑time experimental data. This capability significantly reduces experimental burdens and may expedite the design of high‑performance, durable AEM materials.

Abstract:
High‑harmonic generation (HHG) in solids provides a powerful platform to probe ultrafast electron dynamics and interband‑‑intraband coupling. However, disentangling the complex many‑body contributions in the HHG spectrum remains challenging. Here we introduce a machine‑learning approach based on a Transformer encoder to analyze and reconstruct HHG signals computed from a one‑dimensional Kronig‑‑Penney model. The self‑attention mechanism inherently highlights correlations between temporal dipole dynamics and high‑frequency spectral components, allowing us to identify signatures of nonadiabatic band coupling that are otherwise obscured in standard Fourier analysis. By combining attention maps with Gabor time‑‑frequency analysis, we extract and amplify weak coupling channels that contribute to even‑order harmonics and anomalous spectral features. Our results demonstrate that multi‑head self‑attention acts as a selective filter for strong‑coupling events in the time domain, enabling a physics‑informed interpretation of high‑dimensional quantum dynamics. This work establishes Transformer‑based attention as a versatile tool for solid‑state strong‑field physics, opening new possibilities for interpretable machine learning in attosecond spectroscopy and nonlinear photonics.

Abstract:
Electric Vehicles (EVs) offer substantial flexibility for grid services, yet large‑scale, uncoordinated charging can threaten voltage stability in distribution networks. Existing Reinforcement Learning (RL) approaches for smart charging often disregard physical grid constraints or have limited performance for complex large‑scale tasks, limiting their scalability and real‑world applicability. This paper introduces a physics‑informed (PI) RL algorithm that integrates a differentiable power flow model and voltage‑based reward design into the Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm, enabling EVs to deliver real‑time voltage support while meeting user demands. The resulting PI‑TD3 algorithm achieves faster convergence, improved sample efficiency, and reliable voltage magnitude regulation under uncertain and overloaded conditions. Benchmarks on the IEEE 34‑bus and 123‑bus networks show that the proposed PI‑TD3 outperforms both model‑free RL and optimization‑based baselines in grid constraint management, user satisfaction, and economic metrics, even as the system scales to hundreds of EVs. These advances enable robust, scalable, and practical EV charging strategies that enhance grid resilience and support distribution networks operation.

Abstract:
This paper presents a shape‑aware whole‑body control framework for tendon‑driven continuum robots with direct application to endoluminal surgical navigation. Endoluminal procedures, such as bronchoscopy, demand precise and safe navigation through tortuous, patient‑specific anatomy where conventional tip‑only control often leads to wall contact, tissue trauma, or failure to reach distal targets. To address these challenges, our approach combines a physics‑informed backbone model with residual learning through an Augmented Neural ODE, enabling accurate shape estimation and efficient Jacobian computation. A sampling‑based Model Predictive Path Integral (MPPI) controller leverages this representation to jointly optimize tip tracking, backbone conformance, and obstacle avoidance under actuation constraints. A task manager further enhances adaptability by allowing real‑time adjustment of objectives, such as wall clearance or direct advancement, during tele‑operation. Extensive simulation studies demonstrate millimeter‑level accuracy across diverse scenarios, including trajectory tracking, dynamic obstacle avoidance, and shape‑constrained reaching. Real‑robot experiments on a bronchoscopy phantom validate the framework, showing improved lumen‑following accuracy, reduced wall contacts, and enhanced adaptability compared to joystick‑only navigation and existing baselines. These results highlight the potential of the proposed framework to increase safety, reliability, and operator efficiency in minimally invasive endoluminal surgery, with broader applicability to other confined and safety‑critical environments.

Abstract:
Accurate rainfall forecasting, particularly for extreme events, remains a significant challenge in climatology and the Earth system. This paper presents novel physics‑informed Graph Neural Networks (GNNs) combined with extreme‑value analysis techniques to improve gauge‑station rainfall predictions across Thailand. The model leverages a graph‑structured representation of gauge stations to capture complex spatiotemporal patterns, and it offers explainability through teleconnections. We preprocess relevant climate indices that potentially influence regional rainfall. The proposed Graph Attention Network with Long Short‑Term Memory (Attention‑LSTM) applies the attention mechanism using initial edge features derived from simple orographic‑precipitation physics formulation. The embeddings are subsequently processed by LSTM layers. To address extremes, we perform Peak‑Over‑Threshold (POT) mapping using the novel Spatial Season‑aware Generalized Pareto Distribution (GPD) method, which overcomes limitations of traditional machine‑learning models. Experiments demonstrate that our method outperforms well‑established baselines across most regions, including areas prone to extremes, and remains strongly competitive with the state of the art. Compared with the operational forecasting system SEAS5, our real‑world application improves extreme‑event prediction and offers a practical enhancement to produce high‑resolution maps that support decision‑making in long‑term water management.

Abstract:
Conventional physics‑informed extreme learning machine (PIELM) often faces challenges in solving partial differential equations (PDEs) involving high‑frequency and variable‑frequency behaviors. To address these challenges, we propose a general Fourier feature physics‑informed extreme learning machine (GFF‑PIELM). We demonstrate that directly concatenating multiple Fourier feature mappings (FFMs) and an extreme learning machine (ELM) network makes it difficult to determine frequency‑related hyperparameters. Fortunately, we find an alternative to establish the GFF‑PIELM in three main steps. First, we integrate a variation of FFM into ELM as the Fourier‑based activation function, so there is still one hidden layer in the GFF‑PIELM framework. Second, we assign a set of frequency coefficients to the hidden neurons, which enables ELM network to capture diverse frequency components of target solutions. Finally, we develop an innovative, straightforward initialization method for these hyperparameters by monitoring the distribution of ELM output weights. GFF‑PIELM not only retains the high accuracy, efficiency, and simplicity of the PIELM framework but also inherits the ability of FFMs to effectively handle high‑frequency problems. We carry out five case studies with a total of ten numerical examples to highlight the feasibility and validity of the proposed GFF‑PIELM, involving high frequency, variable frequency, multi‑scale behaviour, irregular boundary and inverse problems. Compared to conventional PIELM, the GFF‑PIELM approach significantly improves predictive accuracy without additional cost in training time and architecture complexity. Our results confirm that that PIELM can be extended to solve high‑frequency and variable‑frequency PDEs with high accuracy, and our initialization strategy may further inspire advances in other physics‑informed machine learning (PIML) frameworks.

Abstract:
Batteries degrade with usage and continuous cycling. This aging is typically reflected through the resistance growth and the capacity fade of battery cells. Over the years, various charging methods have been presented in the literature that proposed current profiles in order to enable optimal, fast, and/or health‑conscious charging. However, very few works have attempted to make the ubiquitous Constant Current Constant Voltage (CCCV) charging protocol adaptive to the changing battery health as it cycles. This work aims to address this gap and proposes a framework that optimizes the constant current part of the CCCV protocol adapting to long‑term battery degradation. Specifically, a physics‑informed Reinforcement Learning (RL) approach has been used that not only estimates a key battery degradation mechanism, namely, Loss of Active Material (LAM), but also adjusts the current magnitude of CCCV as a result of this particular degradation. The proposed framework has been implemented by combining PyBamm, an open‑source battery modeling tool, and Stable‑baselines where the RL agent was trained using a Proximal Policy Optimization (PPO) network. Simulation results show the potential of the proposed framework for enhancing the widely used CCCV protocol by embedding physics information in RL algorithm. A comparative study of this proposed agent has also been discussed with 2 other charging protocols generated by a non‑physics‑based RL agent and a constant CCCV for all the cycles.

Abstract:
In the era of mega‑constellations, the need for accurate and publicly available information has become fundamental for satellite operators to guarantee the safety of spacecrafts and the Low Earth Orbit (LEO) space environment. This study critically evaluates the accuracy and reliability of publicly available ephemeris data for a LEO mega‑constellation ‑ Starlink. The goal of this work is twofold: (i) compare and analyze the quality of the data against high‑precision numerical propagation. (ii) Leverage Physics‑Informed Machine Learning to extract relevant satellite quantities, such as non‑conservative forces, during the decay process. By analyzing two months of real orbital data for approximately 1500 Starlink satellites, we identify discrepancies between high precision numerical algorithms and the published ephemerides, recognizing the use of simplified dynamics at fixed thresholds, planned maneuvers, and limitations in uncertainty propagations. Furthermore, we compare data obtained from multiple sources to track and analyze deorbiting satellites over the same period. Empirically, we extract the acceleration profile of satellites during deorbiting and provide insights relating to the effects of non‑conservative forces during reentry. For non‑deorbiting satellites, the position Root Mean Square Error (RMSE) was approximately 300 m, while for deorbiting satellites it increased to about 600 m. Through this in‑depth analysis, we highlight potential limitations in publicly available data for accurate and robust Space Situational Awareness (SSA), and importantly, we propose a data‑driven model of satellite decay in mega‑constellations.

Abstract:
Partial differential equations (PDEs) provide a mathematical foundation for simulating and understanding intricate behaviors in both physical sciences and engineering. With the growing capabilities of deep learning, data‑driven approaches like Physics‑Informed Neural Networks (PINNs) have been developed, offering a mesh‑free, analytic type framework for efficiently solving PDEs across a wide range of applications. However, traditional PINNs often struggle with challenges such as limited precision, slow training dynamics, lack of labeled data availability, and inadequate handling of multi‑physics interactions. To overcome these challenging issues of PINNs, we proposed a Gradient Enhanced Self‑Training PINN (gST‑PINN) method that specifically introduces a gradient based pseudo point self‑learning algorithm for solving PDEs. We tested the proposed method on three different types of PDE problems from various fields, each representing distinct scenarios. The effectiveness of the proposed method is evident, as the PINN approach for solving the Burgers' equation attains a mean square error (MSE) on the order of 10^‑3, while the diffusion‑sorption equation achieves an MSE on the order of 10^‑4 after 12,500 iterations, with no further improvement as the iterations increase. In contrast, the MSE for both PDEs in the gST‑PINN model continues to decrease, demonstrating better generalization and reaching an MSE on the order of 10^‑5 after 18,500 iterations. Furthermore, the results show that the proposed purely semi‑supervised gST‑PINN consistently outperforms the standard PINN method in all cases, even when solution of the PDEs are unavailable. It generalizes both PINN and Gradient‑enhanced PINN (gPINN), and can be effectively applied in scenarios prone to low accuracy and convergence issues, particularly in the absence of labeled data.

Abstract:
Lens flare occurs when shooting towards strong light sources, significantly degrading the visual quality of images. Due to the difficulty in capturing flare‑corrupted and flare‑free image pairs in the real world, existing datasets are typically synthesized in 2D by overlaying artificial flare templates onto background images. However, the lack of flare diversity in templates and the neglect of physical principles in the synthesis process hinder models trained on these datasets from generalizing well to real‑world scenarios. To address these challenges, we propose a new physics‑informed method for flare data generation, which consists of three stages: parameterized template creation, the laws of illumination‑aware 2D synthesis, and physical engine‑based 3D rendering, which finally gives us a mixed flare dataset that incorporates both 2D and 3D perspectives, namely FlareX. This dataset offers 9,500 2D templates derived from 95 flare patterns and 3,000 flare image pairs rendered from 60 3D scenes. Furthermore, we design a masking approach to obtain real‑world flare‑free images from their corrupted counterparts to measure the performance of the model on real‑world images. Extensive experiments demonstrate the effectiveness of our method and dataset.

Abstract:
Inferring the three‑dimensional (3D) solar atmospheric structures from observations is a critical task for advancing our understanding of the magnetic fields and electric currents that drive solar activity. In this work, we introduce a novel, Physics‑Informed Machine Learning method to reconstruct the 3D structure of the lower solar atmosphere based on the output of optical depth sampled spectropolarimetric inversions, wherein both the fully disambiguated vector magnetic fields and the geometric height associated with each optical depth are returned simultaneously. Traditional techniques typically resolve the 180‑degree azimuthal ambiguity assuming a single layer, either ignoring the intrinsic non‑planar physical geometry of constant optical‑depth surfaces (e.g., the Wilson depression in sunspots), or correcting the effect as a post‑processing step. In contrast, our approach simultaneously maps the optical depths to physical heights, and enforces the divergence‑free condition for magnetic fields fully in 3D. Tests on magnetohydrodynamic simulations of quiet Sun, plage, and a sunspot demonstrate that our method reliably recovers the horizontal magnetic field orientation in locations with appreciable magnetic field strength. By coupling the resolutions of the azimuthal ambiguity and the geometric heights problems, we achieve a self‑consistent reconstruction of the 3D vector magnetic fields and, by extension, the electric current density and Lorentz force. This physics‑constrained, label‑free training paradigm is a generalizable, physics‑anchored framework that extends across solar magnetic environments while improving the understanding of various solar puzzles.

Abstract:
We present a latent‑space formulation of adaptive temporal lifting for continuous‑time dynamical systems. The method introduces a smooth monotone mapping t \mapsto τ(t) that regularizes near‑singular behavior of the underlying flow while preserving its conservation laws. In the lifted coordinate, trajectories such as those of the incompressible Navier‑Stokes equations on the torus \mathbbT^3 become globally smooth. From the standpoint of machine‑learning dynamics, temporal lifting acts as a continuous‑time normalization operator that can stabilize physics‑informed neural networks and other latent‑flow architectures used in AI systems. The framework links analytic regularity theory with representation‑learning methods for stiff or turbulent processes.

Abstract:
Partial differential equations (PDEs) underpin models across science and engineering, yet analytical solutions are atypical and classical mesh‑based solvers can be costly in high dimensions. This dissertation presents a unified comparison of three mesh‑free neural PDE solvers, physics‑informed neural networks (PINNs), the deep Ritz method (DRM), and weak adversarial networks (WANs), on Poisson problems (up to 5D) and the time‑independent Schrödinger equation in 1D/2D (infinite well and harmonic oscillator), and extends the study to a laser‑driven case of Schrödinger's equation via the Kramers‑Henneberger (KH) transformation. Under a common protocol, all methods achieve low L_2 errors (10^‑6‑10^‑9) when paired with forced boundary conditions (FBCs), forced nodes (FNs), and orthogonality regularization (OG). Across tasks, PINNs are the most reliable for accuracy and recovery of excited spectra; DRM offers the best accuracy‑runtime trade‑off on stationary problems; WAN is more sensitive but competitive when weak‑form constraints and FN/OG are used effectively. Sensitivity analyses show that FBC removes boundary‑loss tuning, network width matters more than depth for single‑network solvers, and most gains occur within 5000‑10,000 epochs. The same toolkit solves the KH case, indicating transfer beyond canonical benchmarks. We provide practical guidelines for method selection and outline the following extensions: time‑dependent formulations for DRM and WAN, adaptive residual‑driven sampling, parallel multi‑state training, and neural domain decomposition. These results support physics‑guided neural solvers as credible, scalable tools for solving complex PDEs.

Abstract:
Modeling shock‑to‑detonation phenomena in energetic materials (EMs) requires capturing complex physical processes such as strong shocks, rapid changes in microstructural morphology, and nonlinear dynamics of chemical reaction fronts. These processes participate in energy localization at hotspots, which initiate chemical energy release leading to detonation. This study addresses the formation of hotspots in crystalline EMs subjected to weak‑to‑moderate shock loading, which, despite its critical relevance to the safe storage and handling of EMs, remains underexplored compared to the well‑studied strong shock conditions. To overcome the computational challenges associated with direct numerical simulations, we advance the Physics‑Aware Recurrent Convolutional Neural Network (PARCv2), which has been shown to be capable of predicting strong shock responses in EMs. We improved the architecture of PARCv2 to rapidly predict shear localizations and plastic heating, which play important roles in the weak‑to‑moderate shock regime. PARCv2 is benchmarked against two widely used physics‑informed models, namely, Fourier neural operator and neural ordinary differential equation; we demonstrate its superior performance in capturing the spatiotemporal dynamics of shear band formation. While all models exhibit certain failure modes, our findings underscore the importance of domain‑specific considerations in developing robust AI‑accelerated simulation tools for reactive materials.

Abstract:
To address the dual challenge of predicting multiphysics‑induced instability and optimizing drilling fluid parameters for open‑hole wellbores under long‑term exposure, a high‑fidelity system of coupled governing equations was developed. This system integrates seepage, hydration‑induced softening, thermal diffusion, and elasto‑plastic response to capture the nonlinear dynamics of wellbore stability evolution. A two‑dimensional numerical model in a polar coordinate system was established using COMSOL Multiphysics to simulate multi‑lithology and multi‑parameter perturbations. This process generated a high‑dimensional dataset characterizing the evolution of Von Mises stress, plastic strain, pore pressure, temperature, and water content, and its physical consistency was examined. Subsequently, the Seepage‑Thermal‑Water‑Mechanical Physics‑Informed Neural Network (STWM‑PINN) is proposed. This model embeds governing equation residuals and initial‑boundary constraints to achieve high‑precision, physically consistent predictions of the wellbore's spatio‑temporal evolution under the supervision of finite observational data, laying a foundation for parameter control. Building on this, a Double‑Noise Soft Actor‑Critic (DN‑SAC) algorithm is integrated. A reward function was designed to minimize the probability of instability while considering control smoothness and physical boundary constraints, enabling continuous‑space optimization of drilling fluid parameters. A case study demonstrates that the proposed method delays the onset of instability by an average of 32.33% and a maximum of 53.35%, significantly reducing instability risk. This study provides a decision‑support framework with engineering application potential for intelligent wellbore instability prediction and drilling fluid control.

Abstract:
We study the robustness and physical implications of a set of characteristic redshifts that capture key features of the late‑time Universe. Using both model‑independent reconstructions as well as different dark energy (DE) parameterizations, we show that these redshifts remain stable across cosmological models and reconstruction algorithm, making them reliable geometric anchors of the expansion history. Moreover, the Alcock‑Paczyński corrections at these redshift anchors are found to be unity with high statistical significance, making them natural isotropy points in the comoving distance‑redshift relation. We also find that certain redshifts anchors (z < 1) coincide with epochs where strong deviations from the Planck ΛCDM baseline are apparent irrespective of DE parametrisation like CPL or reconstruction algorithm, indicating their potential as probes of new physics in cosmological evolution. Finally, we demonstrate, for the first time, that a Raychaudhuri Equation Informed Reconstruction Algorithm, substantially enhances the precision of the inferred distance measures and the Hubble expansion rate as well as results tighter constraints in the DE parameter space. These results demonstrate that combining geometric reconstruction with physics‑informed kinematic information offers a powerful and consistent algorithm to probe new physics in the late‑time dynamics of our Universe.

Abstract:
Small‑Angle Neutron Scattering (SANS) data analysis often relies on fixed‑width binning schemes that overlook variations in signal strength and structural complexity. We introduce a statistically grounded approach based on the Freedman‑Diaconis (FD) rule, which minimizes the mean integrated squared error between the histogram estimate and the true intensity distribution. By deriving the competing scaling relations for counting noise (\propto h^‑1) and binning distortion (\propto h^2), we establish an optimal bin width that balances statistical precision and structural resolution. Application to synthetic data from the Debye scattering function of a Gaussian polymer chain demonstrates that the FD criterion quantitatively determines the most efficient binning, faithfully reproducing the curvature of I(Q) while minimizing random error. The optimal width follows the expected scaling h_\mathrmopt \propto N_\mathrmtotal^‑1/3, delineating the transition between noise‑ and resolution‑limited regimes. This framework provides a unified, physics‑informed basis for adaptive, statistically efficient binning in neutron scattering experiments.

Abstract:
Animals achieve energy‑efficient locomotion by their implicit passive dynamics, a marvel that has captivated roboticists for decades.Recently, methods incorporated Adversarial Motion Prior (AMP) and Reinforcement learning (RL) shows promising progress to replicate Animals' naturalistic motion. However, such imitation learning approaches predominantly capture explicit kinematic patterns, so‑called gaits, while overlooking the implicit passive dynamics. This work bridges this gap by incorporating a reward term guided by Impact Mitigation Factor (IMF), a physics‑informed metric that quantifies a robot's ability to passively mitigate impacts. By integrating IMF with AMP, our approach enables RL policies to learn both explicit motion trajectories from animal reference motion and the implicit passive dynamic. We demonstrate energy efficiency improvements of up to 32%, as measured by the Cost of Transport (CoT), across both AMP and handcrafted reward structure.

Abstract:
Thermal field reconstruction in post‑exposure bake (PEB) is critical for advanced lithography, yet current physics‑informed neural networks (PINNs) suffer from inconsistent accuracy due to a misalignment between geometric coordinates, physical fields, and differential operators. To resolve this, we introduce a novel architecture that unifies these elements on a single computation graph by integrating LSTM‑gated mechanisms within a Liquid Neural Network (LNN) backbone. This specific combination of gated liquid layers is necessary to dynamically regulate the network's spectral behavior and enforce operator‑level consistency, which ensures stable training and high‑fidelity predictions. Applied to a 2D PEB scenario with internal heat generation and convective boundaries, our model formulates residuals via differential forms and a composite loss functional. The results demonstrate rapid convergence, uniformly low errors, strong agreement with FEM benchmarks, and stable training without late‑stage oscillations, outperforming existing baselines in accuracy and robustness. Our framework thus establishes a reliable foundation for high‑fidelity thermal modeling and offers a transferable strategy for operator‑consistent neural surrogates in other physical domains.

Abstract:
In this work, we propose a data augmentation strategy aimed at improving the training phase of neural networks and, consequently, the accuracy of their predictions. Our approach relies on generating synthetic data through a suitable compartmental model combined with the incorporation of uncertainty. The available data are then used to calibrate the model, which is further integrated with deep learning techniques to produce additional synthetic data for training. The results show that neural networks trained on these augmented datasets exhibit significantly improved predictive performance. We focus in particular on two different neural network architectures: Physics‑Informed Neural Networks (PINNs) and Nonlinear Autoregressive (NAR) models. The NAR approach proves especially effective for short‑term forecasting, providing accurate quantitative estimates by directly learning the dynamics from data and avoiding the additional computational cost of embedding physical constraints into the training. In contrast, PINNs yield less accurate quantitative predictions but capture the qualitative long‑term behavior of the system, making them more suitable for exploring broader dynamical trends. Numerical simulations of the second phase of the COVID‑19 pandemic in the Lombardy region (Italy) validate the effectiveness of the proposed approach.

Abstract:
Learning complex network dynamics is fundamental to understanding, modelling and controlling real‑world complex systems. There are two main problems in the task of predicting the dynamic evolution of complex networks: on the one hand, existing methods usually use simple graphs to describe the relationships in complex networks; however, this approach can only capture pairwise relationships, while there may be rich non‑pairwise structured relationships in the network. First‑order GNNs have difficulty in capturing dynamic non‑pairwise relationships. On the other hand, theoretical prediction models lack accuracy and data‑driven prediction models lack interpretability. To address the above problems, this paper proposes a higher‑order network dynamics identification method for long‑term dynamic prediction of complex networks. Firstly, to address the problem that traditional graph machine learning can only deal with pairwise relations, dynamic hypergraph learning is introduced to capture the higher‑order non‑pairwise relations among complex networks and improve the accuracy of complex network modelling. Then, a dual‑driven dynamic prediction module for physical data is proposed. The Koopman operator theory is introduced to transform the nonlinear dynamical differential equations for the dynamic evolution of complex networks into linear systems for solving. Meanwhile, the physical information neural differential equation method is utilised to ensure that the dynamic evolution conforms to the physical laws. The dual‑drive dynamic prediction module ensures both accuracy and interpretability of the prediction. Validated on public datasets and self‑built industrial chain network datasets, the experimental results show that the method in this paper has good prediction accuracy and long‑term prediction performance.

Abstract:
We introduce adaptive‑basis physics‑informed neural networks (AB‑PINNs), a novel approach to domain decomposition for training PINNs in which existing subdomains dynamically adapt to the intrinsic features of the unknown solution. Drawing inspiration from classical mesh refinement techniques, we also modify the domain decomposition on‑the‑fly throughout training by introducing new subdomains in regions of high residual loss, thereby providing additional expressive power where the solution of the differential equation is challenging to represent. Our flexible approach to domain decomposition is well‑suited for multiscale problems, as different subdomains can learn to capture different scales of the underlying solution. Moreover, the ability to introduce new subdomains during training helps prevent convergence to unwanted local minima and can reduce the need for extensive hyperparameter tuning compared to static domain decomposition approaches. Throughout, we present comprehensive numerical results which demonstrate the effectiveness of AB‑PINNs at solving a variety of complex multiscale partial differential equations.

Abstract:
We propose PO‑CKAN, a physics‑informed deep operator framework based on Chunkwise Rational Kolmogorov‑‑Arnold Networks (KANs), for approximating the solution operators of partial differential equations. This framework leverages a Deep Operator Network (DeepONet) architecture that incorporates Chunkwise Rational Kolmogorov‑Arnold Network (CKAN) sub‑networks for enhanced function approximation. The principles of Physics‑Informed Neural Networks (PINNs) are integrated into the operator learning framework to enforce physical consistency. This design enables the efficient learning of physically consistent spatio‑temporal solution operators and allows for rapid prediction for parametric time‑dependent PDEs with varying inputs (e.g., parameters, initial/boundary conditions) after training. Validated on challenging benchmark problems, PO‑CKAN demonstrates accurate operator learning with results closely matching high‑fidelity solutions. PO‑CKAN adopts a DeepONet‑style branch‑‑trunk architecture with its sub‑networks instantiated as rational KAN modules, and enforces physical consistency via a PDE residual (PINN‑style) loss. On Burgers' equation with ν=0.01, PO‑CKAN reduces the mean relative L^2 error by approximately 48% compared to PI‑DeepONet, and achieves competitive accuracy on the Eikonal and diffusion‑‑reaction benchmarks.

Abstract:
Conventional time‑series generation often ignores domain‑specific physical constraints, limiting statistical and physical consistency. We propose a hierarchical framework that embeds the inherent hierarchy of physical laws‑conservation, dynamics, boundary, and empirical relations‑directly into deep generative models, introducing a new paradigm of physics‑informed inductive bias. Our method combines Fourier Neural Operators (FNOs) for learning physical operators with Conditional Flow Matching (CFM) for probabilistic generation, integrated via time‑dependent hierarchical constraints and FNO‑guided corrections. Experiments on harmonic oscillators, human activity recognition, and lithium‑ion battery degradation show 16.3% higher generation quality, 46% fewer physics violations, and 18.5% improved predictive accuracy over baselines.

Abstract:
Increase in the number of space exploration missions has led to the accumulation of space debris, posing risk of collision with the operational satellites. Addressing this challenge is crucial for the sustainability of space operations. To plan a safe trajectory in the presence of moving space debris, an integrated approach of artificial potential field and sliding mode controller is proposed and implemented in this paper. The relative 6‑DOF kinematics and dynamics of the spacecraft is modelled in the framework of geometric mechanics with the relative configuration expressed through exponential coordinates. Various collision avoidance guidance algorithms have been proposed in the literature but the Artificial Potential Field guidance algorithm is computationally efficient and enables real‑time path adjustments to avoid collision with obstacles. However, it is prone to issues such as local minima. In literature, local minima issue is typically avoided by either redefining the potential function such as adding vorticity or by employing search techniques which are computationally expensive. To address these challenges, a physics‑informed APF is proposed in this paper where Hamiltonian mechanics is used instead of the traditional Newtonian mechanics‑based approach. In this approach, instead of relying on attractive and repulsive forces for path planning, the Hamiltonian approach uses the potential field to define a path of minimum potential. Additionally, to track the desired trajectory planned by the guidance algorithm within a fixed‑time frame, a non‑singular fixed‑time sliding mode controller (FTSMC) is used. The proposed fixed‑time sliding surface not only ensures fixed‑time convergence of system states but also guarantees the global stability of the closed‑loop system without singularity. The simulation results presented support the claims made.

Abstract:
Persistent systematic errors in Earth system models (ESMs) arise from difficulties in representing the full diversity of subgrid, multiscale atmospheric convection and turbulence. Machine learning (ML) parameterizations trained on short high‑resolution simulations show strong potential to reduce these errors. However, stable long‑term atmospheric simulations with hybrid (physics + ML) ESMs remain difficult, as neural networks (NNs) trained offline often destabilize online runs. Training convection parameterizations directly on coarse‑grained data is challenging, notably because scales cannot be cleanly separated. This issue is mitigated using data from superparameterized simulations, which provide clearer scale separation. Yet, transferring a parameterization from one ESM to another remains difficult due to distribution shifts that induce large inference errors. Here, we present a proof‑of‑concept where a ClimSim‑trained, physics‑informed NN convection parameterization is successfully transferred to ICON‑A. The scheme is (a) trained on adjusted ClimSim data with subtracted radiative tendencies, and (b) integrated into ICON‑A. The NN parameterization predicts its own error, enabling mixing with a conventional convection scheme when confidence is low, thus making the hybrid AI‑physics model tunable with respect to observations and reanalysis through mixing parameters. This improves process understanding by constraining convective tendencies across column water vapor, lower‑tropospheric stability, and geographical conditions, yielding interpretable regime behavior. In AMIP‑style setups, several hybrid configurations outperform the default convection scheme (e.g., improved precipitation statistics). With additive input noise during training, both hybrid and pure‑ML schemes lead to stable simulations and remain physically consistent for at least 20 years.

Abstract:
Fixed‑wing unmanned aerial vehicles (UAVs) offer endurance and efficiency but lack low‑speed agility due to highly coupled dynamics. We present an end‑to‑end sensing‑to‑control pipeline that combines bio‑inspired hardware, physics‑informed dynamics learning, and convex control allocation. Measuring airflow on a small airframe is difficult because near‑body aerodynamics, propeller slipstream, control‑surface actuation, and ambient gusts distort pressure signals. Inspired by the narwhal's protruding tusk, we mount in‑house multi‑hole probes far upstream and complement them with sparse, carefully placed wing pressure sensors for local flow measurement. A data‑driven calibration maps probe pressures to airspeed and flow angles. We then learn a control‑affine dynamics model using the estimated airspeed/angles and sparse sensors. A soft left/right symmetry regularizer improves identifiability under partial observability and limits confounding between wing pressures and flaperon inputs. Desired wrenches (forces and moments) are realized by a regularized least‑squares allocator that yields smooth, trimmed actuation. Wind‑tunnel studies across a wide operating range show that adding wing pressures reduces force‑estimation error by 25‑30%, the proposed model degrades less under distribution shift (about 12% versus 44% for an unstructured baseline), and force tracking improves with smoother inputs, including a 27% reduction in normal‑force RMSE versus a plain affine model and 34% versus an unstructured baseline.

Abstract:
Scalable quantum information processing in spin‑based architectures necessitates the a bility to reliably shuttle quantum states across extended device regions with minimal decoherence. In this work, we present a physics‑informed algorithm for optimizing electrostatic bias equences that enable conveyor‑mode electron transport in silicon‑based quantum dot devices. Our approach combines self‑consistent Poisson and Schrodinger solvers to maintain a constant ground state energy and enable near‑constant velocity shuttling, with potential applicability to both single‑electron and hole transport. We validate the algorithm across three representative technologies: Fully‑Depleted Silicon on Insulator (FD‑SOI), Silicon Metal‑Oxide‑Seminconductor (SiMOS) and Silicon‑Germanium Heterostracture (Si/SiGe), highlighting key limitations and material‑specific effects that influence transport fidelity. Our findings underscore the impact of gate geometry, dielectric interfaces, and quantum dot size on the stability of shuttling operations, and offer pathways toward improving coherence preservation in large‑scale quantum systems.

Abstract:
AC Optimal Power Flow (ACOPF) is computationally intensive for large‑scale grids, often requiring prohibitive solution times with conventional solvers. Machine learning offers significant speedups, but existing models struggle with scalability and topology flexibility. To address these challenges, we propose a Hybrid Heterogeneous Message Passing Neural Network (HH‑MPNN) that integrates a heterogeneous graph neural network (GNN) with a scalable transformer and physics‑informed positional encodings. Our architecture explicitly models distinct power system components to capture local features while using global attention for long‑range dependencies. Evaluated on diverse benchmarks, including PGLearn and GridFM‑DataKit datasets, HH‑MPNN achieves less than 1% optimality gap on default topologies across grid sizes from 14 to 2,000 buses. For N‑1 contingencies, our approach demonstrates zero‑shot N‑1 generalization with less than 3% optimality gap on several test cases despite training only on default topologies. We further develop an approach that ensures robust N‑1 generalization to high‑impact contingencies through targeted augmentation of the training data, showing that exhaustive simulation is unnecessary for topologically flexible models. Finally, size generalization experiments demonstrate that pre‑training on small grids significantly improves performance on large‑scale systems. Achieving computational speedups of up to 5,000 times compared to interior point solvers, these results advance practical, generalizable machine learning for real‑time power system operations.

Abstract:
The COVID‑19 pandemic has highlighted the need for quantitative modeling and analysis to understand real‑world disease dynamics. In particular, post hoc analyses using compartmental models offer valuable insights into the effectiveness of public health interventions, such as vaccination strategies and containment policies. However, such compartmental models like SIR (Susceptible‑Infectious‑Recovered) often face limitations in directly incorporating noisy observational data. In this work, we employ Physics‑Informed Neural Networks (PINNs) to solve the inverse problem of the SIR model using infection data from the Robert Koch Institute (RKI). Our main contribution is a fine‑grained, spatio‑temporal analysis of COVID‑19 dynamics across all German federal states over a three‑year period. We estimate state‑specific transmission and recovery parameters and time‑varying reproduction number (R_t) to track the pandemic progression. The results highlight strong variations in transmission behavior across regions, revealing correlations with vaccination uptake and temporal patterns associated with major pandemic phases. Our findings demonstrate the utility of PINNs in localized, long‑term epidemiological modeling.

Abstract:
Physics‑Informed Neural Networks (PINNs) provide a powerful and general framework for solving Partial Differential Equations (PDEs) by embedding physical laws into loss functions. However, training PINNs is notoriously difficult due to the need to balance multiple loss terms, such as PDE residuals and boundary conditions, which often have conflicting objectives and vastly different curvatures. Existing methods address this issue by manipulating gradients before optimization (a "pre‑combine" strategy). We argue that this approach is fundamentally limited, as forcing a single optimizer to process gradients from spectrally heterogeneous loss landscapes disrupts its internal preconditioning. In this work, we introduce AutoBalance, a novel "post‑combine" training paradigm. AutoBalance assigns an independent adaptive optimizer to each loss component and aggregates the resulting preconditioned updates afterwards. Extensive experiments on challenging PDE benchmarks show that AutoBalance consistently outperforms existing frameworks, achieving significant reductions in solution error, as measured by both the MSE and L^\infty norms. Moreover, AutoBalance is orthogonal to and complementary with other popular PINN methodologies, amplifying their effectiveness on demanding benchmarks.

Abstract:
Symbolic regression aims to find interpretable analytical expressions by searching over mathematical formula spaces to capture underlying system behavior, particularly in scientific modeling governed by physical laws. However, traditional methods lack mechanisms for extracting structured physical priors from time series observations, making it difficult to capture symbolic expressions that reflect the system's global behavior. In this work, we propose a structure‑aware symbolic regression framework, called StruSR, that leverages trained Physics‑Informed Neural Networks (PINNs) to extract locally structured physical priors from time series data. By performing local Taylor expansions on the outputs of the trained PINN, we obtain derivative‑based structural information to guide symbolic expression evolution. To assess the importance of expression components, we introduce a masking‑based attribution mechanism that quantifies each subtree's contribution to structural alignment and physical residual reduction. These sensitivity scores steer mutation and crossover operations within genetic programming, preserving substructures with high physical or structural significance while selectively modifying less informative components. A hybrid fitness function jointly minimizes physics residuals and Taylor coefficient mismatch, ensuring consistency with both the governing equations and the local analytical behavior encoded by the PINN. Experiments on benchmark PDE systems demonstrate that StruSR improves convergence speed, structural fidelity, and expression interpretability compared to conventional baselines, offering a principled paradigm for physics‑grounded symbolic discovery.

Abstract:
Unmanned aerial vehicle (UAV) communications demand accurate yet interpretable air‑to‑ground (A2G) channel models that can adapt to nonstationary propagation environments. While deterministic models offer interpretability and deep learning (DL) models provide accuracy, both approaches suffer from either rigidity or a lack of explainability. To bridge this gap, we propose the Physics‑Inspired Kolmogorov‑Arnold Network (PIKAN) that embeds physical principles (e.g., free‑space path loss, two‑ray reflections) into the learning process. Unlike physics‑informed neural networks (PINNs), PIKAN is more flexible for applying physical information because it introduces them as flexible inductive biases. Thus, it enables a more flexible training process. Experiments on UAV A2G measurement data show that PIKAN achieves comparable accuracy to DL models while providing symbolic and explainable expressions aligned with propagation laws. Remarkably, PIKAN achieves this performance with only 232 parameters, making it up to 37 times lighter than multilayer perceptron (MLP) baselines with thousands of parameters, without sacrificing correlation with measurements and also providing symbolic expressions. These results highlight PIKAN as an efficient, interpretable, and scalable solution for UAV channel modelling in beyond‑5G and 6G networks.

Abstract:
To reliably project future sea level rise, ice sheet models require inputs that respect physics. Embedding physical principles like mass conservation into models that interpolate Antarctic ice flow vector fields from sparse & noisy measurements not only promotes physical adherence but can also improve accuracy and robustness. While physics‑informed neural networks (PINNs) impose physics as soft penalties, offering flexibility but no physical guarantees, we instead propose divergence‑free neural networks (dfNNs), which enforce local mass conservation exactly via a vector calculus trick. Our comparison of dfNNs, PINNs, and unconstrained NNs on ice flux interpolation over Byrd Glacier suggests that "mass conservation on rails" yields more reliable estimates, and that directional guidance, a learning strategy leveraging continent‑wide satellite velocity data, boosts performance across models.

Abstract:
Transferring the recent advancements in deep learning into scientific disciplines is hindered by the lack of the required large‑scale datasets for training. We argue that in these knowledge‑rich domains, the established body of scientific theory provides reliable inductive biases in the form of governing physical laws. We address the ill‑posed inverse problem of recovering Raman spectra from noisy Coherent Anti‑Stokes Raman Scattering (CARS) measurements, as the true Raman signal here is suppressed by a dominating non‑resonant background. We propose RamPINN, a model that learns to recover Raman spectra from given CARS spectra. Our core methodological contribution is a physics‑informed neural network that utilizes a dual‑decoder architecture to disentangle resonant and non‑resonant signals. This is done by enforcing the Kramers‑Kronig causality relations via a differentiable Hilbert transform loss on the resonant and a smoothness prior on the non‑resonant part of the signal. Trained entirely on synthetic data, RamPINN demonstrates strong zero‑shot generalization to real‑world experimental data, explicitly closing this gap and significantly outperforming existing baselines. Furthermore, we show that training with these physics‑based losses alone, without access to any ground‑truth Raman spectra, still yields competitive results. This work highlights a broader concept: formal scientific rules can act as a potent inductive bias, enabling robust, self‑supervised learning in data‑limited scientific domains.

Abstract:
Physics‑informed machine learning (PIML) is emerging as a potentially transformative paradigm for modeling complex biomedical systems by integrating parameterized physical laws with data‑driven methods. Here, we review three main classes of PIML frameworks: physics‑informed neural networks (PINNs), neural ordinary differential equations (NODEs), and neural operators (NOs), highlighting their growing role in biomedical science and engineering. We begin with PINNs, which embed governing equations into deep learning models and have been successfully applied to biosolid and biofluid mechanics, mechanobiology, and medical imaging among other areas. We then review NODEs, which offer continuous‑time modeling, especially suited to dynamic physiological systems, pharmacokinetics, and cell signaling. Finally, we discuss deep NOs as powerful tools for learning mappings between function spaces, enabling efficient simulations across multiscale and spatially heterogeneous biological domains. Throughout, we emphasize applications where physical interpretability, data scarcity, or system complexity make conventional black‑box learning insufficient. We conclude by identifying open challenges and future directions for advancing PIML in biomedical science and engineering, including issues of uncertainty quantification, generalization, and integration of PIML and large language models.

Abstract:
Physics‑Informed Neural Networks (PINNs) are a useful framework for approximating partial differential equation solutions using deep learning methods. In this paper, we propose a principled redesign of the PINNsformer, a Transformer‑based PINN architecture. We present the Spectral PINNSformer (S‑Pformer), a refinement of encoder‑decoder PINNSformers that addresses two key issues; 1. the redundancy (i.e. increased parameter count) of the encoder, and 2. the mitigation of spectral bias. We find that the encoder is unnecessary for capturing spatiotemporal correlations when relying solely on self‑attention, thereby reducing parameter count. Further, we integrate Fourier feature embeddings to explicitly mitigate spectral bias, enabling adaptive encoding of multiscale behaviors in the frequency domain. Our model outperforms encoder‑decoder PINNSformer architectures across all benchmarks, achieving or outperforming MLP performance while reducing parameter count significantly.

Abstract:
Arterial aneurysm (Fig.1) is a bulb‑shape local expansion of human arteries, the rupture of which is a leading cause of morbidity and mortality in US. Therefore, the prediction of arterial aneurysm rupture is of great significance for aneurysm management and treatment selection. The prediction of aneurysm rupture depends on the analysis of the time series of aneurysm growth history. However, due to the long time scale of aneurysm growth, the time series of aneurysm growth is not always accessible. We here proposed a method to reconstruct the aneurysm growth time series directly from patient parameters. The prediction is based on data pairs of [patient parameters, patient aneurysm growth time history]. To obtain the mapping from patient parameters to patient aneurysm growth time history, we first apply autoencoder to obtain a compact representation of the time series for each patient. Then a mapping is learned from patient parameters to the corresponding compact representation of time series via a five‑layer neural network. Moving average and convolutional output layer are implemented to explicitly taking account the time dependency of the time series. Apart from that, we also propose to use prior knowledge about the mechanism of aneurysm growth to improve the time series reconstruction results. The prior physics‑based knowledge is incorporated as constraints for the optimization problem associated with autoencoder. The model can handle both algebraic and differential constraints. Our results show that including physical model information about the data will not significantly improve the time series reconstruction results if the training data is error‑free. However, in the case of training data with noise and bias error, incorporating physical model constraints can significantly improve the predicted time series.

Abstract:
Physics‑informed neural networks (PINNs) provide a powerful approach for solving partial differential equations (PDEs), but constructing a usable PINN remains labor‑intensive and error‑prone. Scientists must interpret problems as PDE formulations, design architectures and loss functions, and implement stable training pipelines. Existing large language model (LLM) based approaches address isolated steps such as code generation or architecture suggestion, but typically assume a formal PDE is already specified and therefore lack an end‑to‑end perspective. We present Lang‑PINN, an LLM‑driven multi‑agent system that builds trainable PINNs directly from natural language task descriptions. Lang‑PINN coordinates four complementary agents: a PDE Agent that parses task descriptions into symbolic PDEs, a PINN Agent that selects architectures, a Code Agent that generates modular implementations, and a Feedback Agent that executes and diagnoses errors for iterative refinement. This design transforms informal task statements into executable and verifiable PINN code. Experiments show that Lang‑PINN achieves substantially lower errors and greater robustness than competitive baselines: mean squared error (MSE) is reduced by up to 3‑‑5 orders of magnitude, end‑to‑end execution success improves by more than 50%, and reduces time overhead by up to 74%.

Abstract:
Terahertz time‑domain spectroscopy (THz‑TDS) provides a non‑invasive and label‑free method for probing the internal structure and electromagnetic response of materials. Numerical simulation of THz‑TDS can help understanding wave‑matter interactions, guiding experimental design, and interpreting complex measurement data. However, existing simulation techniques face challenges in accurately modeling THz wave propagation with low computational cost. Additionally, conventional simulation solvers often require dense spatial‑temporal discretization, which limits their applicability to large‑scale and real‑time scenarios. Simplified analytical models may neglect dispersion, multiple scattering, and boundary effects. To address these limitations, we establish a novel computational framework that integrates frequency‑domain physics‑informed neural networks (FD‑PINNs) with less data‑driven. To validate our proposed FD‑PINNs, simulation results from finite‑difference time‑domain (FDTD) and time‑domain (TD)‑PINNs were used to compare with FD‑PINNs. Finally, experimental results from THz‑TDS systems were employed to further exhibit accurate reconstruction ability of FD‑PINNs.

Abstract:
This article proposes a data‑driven PID controller design based on the principle of adaptive gain optimization, leveraging Physics‑Informed Neural Networks (PINNs) generated for predictive modeling purposes. The proposed control design method utilizes gradients of the PID gain optimization, achieved through the automatic differentiation of PINNs, to apply model predictive control using a cost function based on tracking error and control inputs. By optimizing PINNs‑based PID gains, the method achieves adaptive gain tuning that ensures stability while accounting for system nonlinearities. The proposed method features a systematic framework for integrating PINNs‑based models of dynamical control systems into closed‑loop control systems, enabling direct application to PID control design. A series of numerical experiments is conducted to demonstrate the effectiveness of the proposed method from the control perspectives based on both time and frequency domains.

Abstract:
Partial differential equation (PDE) solvers are fundamental to engineering simulation. Classical mesh‑based approaches (finite difference/volume/element) are fast and accurate on high‑quality meshes but struggle with higher‑order operators and complex, hard‑to‑mesh geometries. Recently developed physics‑informed neural networks (PINNs) and their variants are mesh‑free and flexible, yet compute‑intensive and often less accurate. This paper systematically benchmarks RBF‑PIELM, a rapid PINN variant‑an extreme learning machine with radial‑basis activations‑for higher‑order PDEs. RBF‑PIELM replaces PINNs' time‑consuming gradient descent with a single‑shot least‑squares solve. We test RBF‑PIELM on the fourth‑order biharmonic equation using two benchmarks: lid‑driven cavity flow (streamfunction formulation) and a manufactured oscillatory solution. Our results show up to (350×) faster training than PINNs and over (10×) fewer parameters for comparable solution accuracy. Despite surpassing PINNs, RBF‑PIELM still lags mature mesh‑based solvers and its accuracy degrades on highly oscillatory solutions, highlighting remaining challenges for practical deployment.

Abstract:
Sound field reconstruction involves estimating sound fields from a limited number of spatially distributed observations. This work introduces a differentiable physics approach for sound field reconstruction, where the initial conditions of the wave equation are approximated with a neural network, and the differential operator is computed with a differentiable numerical solver. The use of a numerical solver enables a stable network training while enforcing the physics as a strong constraint, in contrast to conventional physics‑informed neural networks, which include the physics as a constraint in the loss function. We introduce an additional sparsity‑promoting constraint to achieve meaningful solutions even under severe undersampling conditions. Experiments demonstrate that the proposed approach can reconstruct sound fields under extreme data scarcity, achieving higher accuracy and better convergence compared to physics‑informed neural networks.

Abstract:
Partial differential equation (PDE) solvers underpin modern quantitative finance, governing option pricing and risk evaluation. Physics‑Informed Neural Networks (PINNs) have emerged as a promising approach for solving the forward and inverse problems of partial differential equations (PDEs) using deep learning. However they remain computationally expensive due to their iterative gradient descent based optimization and scale poorly with increasing model size. This paper introduces Physics‑Informed Extreme Learning Machines (PIELMs) as fast alternative to PINNs for solving both forward and inverse problems in financial PDEs. PIELMs replace iterative optimization with a single least‑squares solve, enabling deterministic and efficient training. We benchmark PIELM on the Black‑Scholes and Heston‑Hull‑White models for forward pricing and demonstrate its capability in inverse model calibration to recover volatility and interest rate parameters from noisy data. From experiments we observe that PIELM achieve accuracy comparable to PINNs while being up to 30× faster, highlighting their potential for real‑time financial modeling.

Abstract:
The Alternating Current Power Flow (ACPF) problem forces a trade‑off between the speed of data‑driven models and the reliability of analytical solvers. This paper introduces a hybrid framework that synergizes a Graph Neural Network (GNN) with the Implicit Z‑Bus Recursive (IZR) method, a robust, non‑iterative solver for radial distribution networks. The framework employs a physics‑informed GNN for rapid initial predictions and invokes the IZR solver as a failsafe for stressed cases identified by a two‑stage trigger. A failure is defined as any solution with a maximum power mismatch exceeding 0.1 p.u., a significant operational deviation. On a challenging test set of 7,500 stressed scenarios for the IEEE 33‑bus system, the GNN‑only model failed on 13.11 % of cases. In contrast, the hybrid framework identified all potential failures, delegating them to the IZR solver to achieve a 0.00 % failure rate, empirically matching the 100 % success rate of the analytical solver on this specific test set. An expanded ablation study confirms that both physics‑informed training and Z‑bus sensitivity features are critical, collaboratively reducing the GNN's failure rate from 98.72 % (data‑only) to 13.11 %. The hybrid approach demonstrates a pragmatic path to achieving the empirical reliability of an analytical solver while leveraging GNN speed, enabling a significant increase in the number of scenarios analyzable in near real‑time.

Abstract:
A major problem of kernel‑based methods (e.g., least squares support vector machines, LS‑SVMs) for solving linear/nonlinear ordinary differential equations (ODEs) is the prohibitive O(an^3) (a=1 for linear ODEs and 27 for nonlinear ODEs) part of their computational complexity with increasing temporal discretization points n. We propose a novel Nyström‑accelerated LS‑SVMs framework that breaks this bottleneck by reformulating ODEs as primal‑space constraints. Specifically, we derive for the first time an explicit Nyström‑based mapping and its derivatives from one‑dimensional temporal discretization points to a higher m‑dimensional feature space (1< m\le n), enabling the learning process to solve linear/nonlinear equation systems with m‑dependent complexity. Numerical experiments on sixteen benchmark ODEs demonstrate: 1) 10‑6000 times faster computation than classical LS‑SVMs and physics‑informed neural networks (PINNs), 2) comparable accuracy to LS‑SVMs (<0.13% relative MAE, RMSE, and \left \| y‑\haty \right \| _\infty difference) while maximum surpassing PINNs by 72% in RMSE, and 3) scalability to n=10^4 time steps with m=50 features. This work establishes a new paradigm for efficient kernel‑based ODEs learning without significantly sacrificing the accuracy of the solution.

Abstract:
A successful deep learning network is highly dependent not only on the training dataset, but the training algorithm used to condition the network for a given task. The loss function, dataset, and tuning of hyperparameters all play an essential role in training a network, yet there is not much discussion on the reliability or reproducibility of a training algorithm. With the rise in popularity of physics‑informed loss functions, this raises the question of how reliable one's loss function is in conditioning a network to enforce a particular boundary condition. Reporting the model variation is needed to assess a loss function's ability to consistently train a network to obey a given boundary condition, and provides a fairer comparison among different methods. In this work, a Pix2Pix network predicting the stress fields of high elastic contrast composites is used as a case study. Several different loss functions enforcing stress equilibrium are implemented, with each displaying different levels of variation in convergence, accuracy, and enforcing stress equilibrium across many training sessions. Suggested practices in reporting model variation are also shared.

Abstract:
Assessing turbulence control effects for wall friction numerically is a significant challenge since it requires expensive simulations of turbulent fluid dynamics. We instead propose an efficient deep reinforcement learning (RL) framework for modeling and control of turbulent flows. It is model‑based RL for predictive control (PC), where both the policy and the observer models for turbulence control are learned jointly using Physics Informed Neural Operators (PINO), which are discretization invariant and can capture fine scales in turbulent flows accurately. Our PINO‑PC outperforms prior model‑free reinforcement learning methods in various challenging scenarios where the flows are of high Reynolds numbers and unseen, i.e., not provided during model training. We find that PINO‑PC achieves a drag reduction of 39.0% under a bulk‑velocity Reynolds number of 15,000, outperforming previous fluid control methods by more than 32%.

Abstract:
Machine learning has been increasingly applied in climate modeling on system emulation acceleration, data‑driven parameter inference, forecasting, and knowledge discovery, addressing challenges such as physical consistency, multi‑scale coupling, data sparsity, robust generalization, and integration with scientific workflows. This paper analyzes a series of case studies from applied machine learning research in climate modeling, with a focus on design choices and workflow structure. Rather than reviewing technical details, we aim to synthesize workflow design patterns across diverse projects in ML‑enabled climate modeling: from surrogate modeling, ML parameterization, probabilistic programming, to simulation‑based inference, and physics‑informed transfer learning. We unpack how these workflows are grounded in physical knowledge, informed by simulation data, and designed to integrate observations. We aim to offer a framework for ensuring rigor in scientific machine learning through more transparent model development, critical evaluation, informed adaptation, and reproducibility, and to contribute to lowering the barrier for interdisciplinary collaboration at the interface of data science and climate modeling.

Abstract:
Bayesian physics‑informed neural networks (B‑PINNs) merge data with governing equations to solve differential equations under uncertainty. However, interpreting uncertainty and overconfidence in B‑PINNs requires care due to the poorly understood effects the physical constraints have on the network; overconfidence could reflect warranted precision, enforced by the constraints, rather than miscalibration. Motivated by the need to further clarify how individual physical constraints shape these networks, we introduce a scalable, matrix‑free Laplace framework that decomposes the posterior Hessian into contributions from each constraint and provides metrics to quantify their relative influence on the loss landscape. Applied to the Van der Pol equation, our method tracks how constraints sculpt the network's geometry and shows, directly through the Hessian, how changing a single loss weight non‑trivially redistributes curvature and effective dominance across the others.

Abstract:
Deep learning (DL) has demonstrated promise for accelerating and enhancing the accuracy of flow physics simulations, but progress is constrained by the scarcity of high‑fidelity training data, which is costly to generate and inherently limited to a small set of flow conditions. Consequently, closures trained in the conventional offline paradigm tend to overfit and fail to generalise to new regimes. We introduce an online optimisation framework for DL‑based Reynolds‑averaged Navier‑‑Stokes (RANS) closures which seeks to address the challenge of limited high‑fidelity datasets. Training data is dynamically generated by embedding a direct numerical simulation (DNS) within a subdomain of the RANS domain. The RANS solution supplies boundary conditions to the DNS, while the DNS provides mean velocity and turbulence statistics that are used to update a DL closure model during the simulation. This feedback loop enables the closure to adapt to the embedded DNS target flow, avoiding reliance on precomputed datasets and improving out‑of‑distribution performance. The approach is demonstrated for the stochastically forced Burgers equation and for turbulent channel flow at Re_τ=180, 270, 395 and 590 with varying embedded domain lengths 1\leq L_0/L\leq 8. Online‑optimised RANS models significantly outperform both offline‑trained and literature‑calibrated closures, with accurate training achieved using modest DNS subdomains. Performance degrades primarily when boundary‑condition contamination dominates or when domains are too short to capture low‑wavenumber modes. This framework provides a scalable route to physics‑informed machine learning closures, enabling data‑adaptive reduced‑order models that generalise across flow regimes without requiring large precomputed training datasets.

Abstract:
Physics‑informed neural networks (PINNs) offer a novel AI‑driven framework for integrating physical laws directly into neural network models, facilitating the solution of complex multiphysics problems in materials engineering. This study systematically explores the application of PINNs to simulate oxide film layer growth in halide‑free solutions using the point defect model (PDM). We identify and analyze four key failure modes in this context: imbalanced loss components across different physical processes, numerical instabilities due to variable scale disparities, challenges in enforcing boundary conditions within multiphysics systems, and convergence to mathematically valid but physically meaningless solutions. To overcome these challenges, we implement and validate established techniques including nondimensionalization for training stabilization, Neural Tangent Kernel‑based adaptive loss balancing, robust enforcement of boundary conditions and hybrid training with sparse data. Our results demonstrate the effectiveness of these strategies in enhancing the reliability and physical fidelity of PINNs, achieving sub 1% relative error as compared to Finite Element Benchmarks with the hybrid model. Thereby showing that PINNs can be used for high fidelity electrochemical simulations with minimal data requirements and highlight necesary factors for fully autonomous PINN simulations.

Abstract:
Weakly‑modelled searches for gravitational waves are essential for ensuring that all potential sources are accounted for in detection efforts, as they make minimal assumptions regarding source morphology. While these searches primarily target generic transient sources, they are also highly effective at identifying a broad range of compact binary coalescences, demonstrated by the weakly‑modelled search algorithm coherent WaveBurst being the first to detect GW150914. Despite their ability to detect compact binaries with diverse properties, the accurate estimation of source parameters from their output remains to be a challenging task. To overcome this, we leverage physics‑informed neural networks, which serve as a powerful tool for parameter estimation by applying physical constraints through the universal differential equation governing a compact binary system. With this approach, we rapidly infer the mass parameters of binary black hole merger systems to within 7% from only the time‑frequency representation of the gravitational wave signal.

Abstract:
This paper presents a novel approach to finding analytical approximations for bright‑soliton solutions in strongly magnetized plasmas. We leverage Physics‑Informed Symbolic Regression (PISR) to discover closed‑form expressions for the vector potential and number density profiles, governed by a reduced‑order model derived from Maxwell‑fluid equations. The PISR framework combines symbolic regression with physics‑based constraints, boundary conditions, and available simulation data to guide the search for solutions. We demonstrate the effectiveness of the approach by rediscovering approximate solutions consistent with previously published numerical results, showcasing the potential of PISR for reducing simulation costs of reduced‑order models in plasma physics.

Abstract:
Cyberattacks on pipeline operational technology systems pose growing risks to energy infrastructure. This study develops a physics‑informed simulation and optimization framework for analyzing cyber‑physical threats in petroleum pipeline networks. The model integrates networked hydraulic dynamics, SCADA‑based state estimation, model predictive control (MPC), and a bi‑level formulation for stealthy false‑data injection (FDI) attacks. Pipeline flow and pressure dynamics are modeled on a directed graph using nodal pressure evolution and edge‑based Weymouth‑type relations, including control‑aware equipment such as valves and compressors. An extended Kalman filter estimates the full network state from partial SCADA telemetry. The controller computes pressure‑safe control inputs via MPC under actuator constraints and forecasted demands. Adversarial manipulation is formalized as a bi‑level optimization problem where an attacker perturbs sensor data to degrade throughput while remaining undetected by bad‑data detectors. This attack‑control interaction is solved via Karush‑Kuhn‑Tucker (KKT) reformulation, which results in a tractable mixed‑integer quadratic program. Test gas pipeline case studies demonstrate the covert reduction of service delivery under attack. Results show that undetectable attacks can cause sustained throughput loss with minimal instantaneous deviation. This reveals the need for integrated detection and control strategies in cyber‑physical infrastructure.

Abstract:
Reconstructing high‑resolution rainfall fields is essential for flood forecasting, hydrological modeling, and climate analysis. However, existing spatial interpolation methods‑whether based on automatic weather station (AWS) measurements or enhanced with satellite/radar observations often over‑smooth critical structures, failing to capture sharp transitions and localized extremes. We introduce RainSeer, a structure‑aware reconstruction framework that reinterprets radar reflectivity as a physically grounded structural prior‑capturing when, where, and how rain develops. This shift, however, introduces two fundamental challenges: (i) translating high‑resolution volumetric radar fields into sparse point‑wise rainfall observations, and (ii) bridging the physical disconnect between aloft hydro‑meteors and ground‑level precipitation. RainSeer addresses these through a physics‑informed two‑stage architecture: a Structure‑to‑Point Mapper performs spatial alignment by projecting mesoscale radar structures into localized ground‑level rainfall, through a bidirectional mapping, and a Geo‑Aware Rain Decoder captures the semantic transformation of hydro‑meteors through descent, melting, and evaporation via a causal spatiotemporal attention mechanism. We evaluate RainSeer on two public datasets‑RAIN‑F (Korea, 2017‑2019) and MeteoNet (France, 2016‑2018)‑and observe consistent improvements over state‑of‑the‑art baselines, reducing MAE by over 13.31% and significantly enhancing structural fidelity in reconstructed rainfall fields.

Abstract:
Robot navigation in large, complex, and unknown indoor environments is a challenging problem. The existing approaches, such as traditional sampling‑based methods, struggle with resolution control and scalability, while imitation learning‑based methods require a large amount of demonstration data. Active Neural Time Fields (ANTFields) have recently emerged as a promising solution by using local observations to learn cost‑to‑go functions without relying on demonstrations. Despite their potential, these methods are hampered by challenges such as spectral bias and catastrophic forgetting, which diminish their effectiveness in complex scenarios. To address these issues, our approach decomposes the planning problem into a hierarchical structure. At the high level, a sparse graph captures the environment's global connectivity, while at the low level, a planner based on neural fields navigates local obstacles by solving the Eikonal PDE. This physics‑informed strategy overcomes common pitfalls like spectral bias and neural field fitting difficulties, resulting in a smooth and precise representation of the cost landscape. We validate our framework in large‑scale environments, demonstrating its enhanced adaptability and precision compared to previous methods, and highlighting its potential for online exploration, mapping, and real‑world navigation.

Abstract:
Efficient molecular dynamics (MD) simulation is vital for understanding atomic‑scale processes in materials science and biophysics. Traditional density functional theory (DFT) methods are computationally expensive, which limits the feasibility of long‑term simulations. We propose a novel approach that formulates MD simulation as a time‑series forecasting problem, enabling advanced forecasting models to predict atomic trajectories via displacements rather than absolute positions. We incorporate a physics‑informed loss and inference mechanism based on DFT‑parametrised pair‑wise Morse potential functions that penalize unphysical atomic proximity to enforce physical plausibility. Our method consistently surpasses standard baselines in simulation accuracy across diverse materials. The results highlight the importance of incorporating physics knowledge to enhance the reliability and precision of atomic trajectory forecasting. Remarkably, it enables stable modeling of thousands of MD steps in minutes, offering a scalable alternative to costly DFT simulations.

Abstract:
We present a data‑driven framework for turbulence modeling, applied to flow prediction in the FDA nozzle. In this study, the standard RANS equations have been modified using an implicit‑explicit hybrid approach. New variables were introduced, and a solver was developed within the OpenFOAM framework, integrating a machine learning module to estimate these variables. The invariant input features were derived based on Hilbert's basis theorem, and the outputs of the machine learning model were obtained through eigenvalue‑vector decomposition of the Reynolds stress tensor. Validation was performed using DNS data for turbulent flow in a square channel at various Reynolds numbers. A baseline MLP was first trained at Re=2900 and tested at Re=3500 to assess its ability to reproduce turbulence anisotropy and secondary flows. To further enhance generalization, three benchmark DNS datasets were transformed into images via the Deep‑Insight method, enabling the use of convolutional neural networks. The trained Deep‑Insight network demonstrated improved prediction of turbulence structures in the FDA blood nozzle, highlighting the promise of data‑driven augmentation in turbulence modeling.

Abstract:
Physics‑informed neural networks (PINNs) and related methods struggle to resolve sharp gradients in singularly perturbed boundary value problems without resorting to some form of domain decomposition, which often introduce complex interface penalties. While the Extreme Theory of Functional Connections (X‑TFC) avoids multi‑objective optimization by employing exact boundary condition enforcement, it remains computationally inefficient for boundary layers and incompatible with decomposition. We propose Gated X‑TFC, a novel framework for both forward and inverse problems, that overcomes these limitations through a soft, learned domain decomposition. Our method replaces hard interfaces with a differentiable logistic gate that dynamically adapts radial basis function (RBF) kernel widths across the domain, eliminating the need for interface penalties. This approach yields not only superior accuracy but also dramatic improvements in computational efficiency: on a benchmark one dimensional (1D) convection‑diffusion, Gated X‑TFC achieves an order‑of‑magnitude lower error than standard X‑TFC while using 80 percent fewer collocation points and reducing training time by 66 percent. In addition, we introduce an operator‑conditioned meta‑learning layer that learns a probabilistic mapping from PDE parameters to optimal gate configurations, enabling fast, uncertainty‑aware warm‑starting for new problem instances. We further demonstrate scalability to multiple subdomains and higher dimensions by solving a twin boundary‑layer equation and a 2D Poisson problem with a sharp Gaussian source. Overall, Gated X‑TFC delivers a simple alternative alternative to PINNs that is both accurate and computationally efficient for challenging boundar‑layer regimes. Future work will focus on nonlinear problems.

Abstract:
Physics‑informed machine learning has been a promising data‑driven and physics‑informed approach in geotechnical engineering. This study proposes a physics‑informed extreme learning machine (PIELM) framework for analyzing tunneling‑induced soil‑pile interactions. The pile foundation is modeled as an Euler‑Bernoulli beam, and the surrounding soil is modeled as a Pasternak foundation. The soil‑pile interaction is formulated into a fourth‑order ordinary differential equation (ODE) that constitutes the physics‑informed component, while measured data are incorporated into PIELM as the data‑driven component. Combining physics and data yields a loss vector of the extreme learning machine (ELM) network, which is trained within 1 second by the least squares method. After validating the PIELM approach by the boundary element method (BEM) and finite difference method (FDM), parametric studies are carried out to examine the effects of ELM network architecture, data monitoring locations and numbers on the performance of PIELM. The results indicate that monitored data should be placed at positions where the gradients of pile deflections are significant, such as at the pile tip/top and near tunneling zones. Two application examples highlight the critical role of physics‑informed and data‑driven approach for tunnelling‑induced soil‑pile interactions. The proposed approach shows great potential for real‑time monitoring and safety assessment of pile foundations, and benefits for intelligent early‑warning systems in geotechnical engineering.

Abstract:
We numerically investigate the feasibility and limits of jointly estimating flow fields and unknown particle properties (e.g., position, size, and density) from Lagrangian particle tracking (LPT) data. LPT offers time‑resolved, volumetric measurements of particle trajectories, which are markers of the carrier fluid motion. However, experimental tracks are spatially sparse and potentially noisy, and the problem of reconstructing flow fields may be further complicated by inertial particle transport, such that particle slip velocities must be determined to access the velocity field of the carrier fluid. To address this problem, we develop a data assimilation framework that couples an Eulerian representation of the flow with Lagrangian particle models, enabling the simultaneous inference of carrier fields and particle properties under the governing equations of disperse multiphase flow. We show that flow fields and particle properties can be jointly estimated in three representative regimes: (1) In a turbulent boundary layer with noisy tracer tracks (St to 0), flow fields and true particle positions are jointly estimated, which amounts to a physics‑informed particle tracking problem; (2) in homogeneous isotropic turbulence seeded with inertial particles (St ~ 1‑5), we demonstrate simultaneous recovery of flow states and particle diameters, showing the feasibility of implicit particle characterization; and (3) in a compressible, shock‑dominated flow, we report the first joint reconstructions of velocity, pressure, density, and inertial particle properties (diameter and density), highlighting both the potential and certain limits of joint estimation in supersonic regimes. A systematic sensitivity study reveals how the seeding density, noise level, and Stokes number govern reconstruction accuracy for our method.

Abstract:
With rapid urbanization, predicting urban microclimates has become critical, as it affects building energy demand and public health risks. However, existing generative and homogeneous graph approaches fall short in capturing physical consistency, spatial dependencies, and temporal variability. To address this, we introduce UrbanGraph, a framework founded on a novel structure‑based inductive bias. Unlike implicit graph learning, UrbanGraph transforms physical first principles into a dynamic causal topology, explicitly encoding time‑varying causalities (e.g., shading and convection) directly into the graph structure to ensure physical consistency and data efficiency. Results show that UrbanGraph achieves state‑of‑the‑art performance across all baselines. Specifically, the use of explicit causal pruning significantly reduces the model's floating‑point operations (FLOPs) by 73.8% and increases training speed by 21% compared to implicit graphs. Our contribution includes the first high‑resolution benchmark for spatio‑temporal microclimate modeling, and a generalizable explicit topological encoding paradigm applicable to urban spatio‑temporal dynamics governed by known physical equations.

Abstract:
Neural network training relies on gradient computation through backpropagation, yet memory requirements for storing layer activations present significant scalability challenges. We present the first adaptation of control‑theoretic matrix sketching to neural network layer activations, enabling memory‑efficient gradient reconstruction in backpropagation. This work builds on recent matrix sketching frameworks for dynamic optimization problems, where similar state trajectory storage challenges motivate sketching techniques. Our approach sketches layer activations using three complementary sketch matrices maintained through exponential moving averages (EMA) with adaptive rank adjustment, automatically balancing memory efficiency against approximation quality. Empirical evaluation on MNIST, CIFAR‑10, and physics‑informed neural networks demonstrates a controllable accuracy‑memory tradeoff. We demonstrate a gradient monitoring application on MNIST showing how sketched activations enable real‑time gradient norm tracking with minimal memory overhead. These results establish that sketched activation storage provides a viable path toward memory‑efficient neural network training and analysis.

Abstract:
Long‑horizon motion forecasting for multiple autonomous robots is challenging due to non‑linear agent interactions, compounding prediction errors, and continuous‑time evolution of dynamics. Learned dynamics of such a system can be useful in various applications such as travel time prediction, prediction‑guided planning and generative simulation. In this work, we aim to develop an efficient trajectory forecasting model conditioned on multi‑agent goals. Motivated by the recent success of physics‑guided deep learning for partially known dynamical systems, we develop a model based on neural Controlled Differential Equations (CDEs) for long‑horizon motion forecasting. Unlike discrete‑time methods such as RNNs and transformers, neural CDEs operate in continuous time, allowing us to combine physics‑informed constraints and biases to jointly model multi‑robot dynamics. Our approach, named PINCoDE (Physics‑Informed Neural Controlled Differential Equations), learns differential equation parameters that can be used to predict the trajectories of a multi‑agent system starting from an initial condition. PINCoDE is conditioned on future goals and enforces physics constraints for robot motion over extended periods of time. We adopt a strategy that scales our model from 10 robots to 100 robots without the need for additional model parameters, while producing predictions with an average ADE below 0.5 m for a 1‑minute horizon. Furthermore, progressive training with curriculum learning for our PINCoDE model results in a 2.7X reduction of forecasted pose error over 4 minute horizons compared to analytical models.

Abstract:
Solving nonlinear algebraic equations is a fundamental but challenging problem in scientific computations and also has many applications in system engineering. Though traditional iterative methods and modern optimization algorithms have exerted effective roles in addressing certain specific problems, there still exist certain weaknesses such as the initial value sensitivity, limited accuracy and slow convergence rate, particulary without flexible input for the neural network methods. In this paper, we propose a homotopy auxiliary neural network (HANN) for solving nonlinear algebraic equations which integrates the classical homotopy continuation method and popular physics‑informed neural network. Consequently, the HANN‑1 has strong learning ability and can rapidly give an acceptable solution for the problem which outperforms some known methods, while the HANN‑2 can further improve its accuracy. Numerical results on the benchmark problems confirm that the HANN method can effectively solve the problems of determining the total number of solutions of a single equation, finding solutions of transcendental systems involving the absolute value function or trigonometric function, ill‑conditioned and normal high‑dimensional nonlinear systems and time‑varying nonlinear problems, for which the Python's built‑in Fsolve function exhibits significant limitations, even fails to work.

Abstract:
This paper presents intersection of Physics informed neural networks (PINNs) and Lie symmetry group to enhance the accuracy and efficiency of solving partial differential equation (PDEs). Various methods have been developed to solve these equations. A Lie group is an efficient method that can lead to exact solutions for the PDEs that possessing Lie Symmetry. Leveraging the concept of infinitesimal generators from Lie symmetry group in a novel manner within PINN leads to significant improvements in solution of PDEs. In this study three distinct cases are discussed, each showing progressive improvements achieved through Lie symmetry modifications and adaptive techniques. State‑of‑the‑art numerical methods are adopted for comparing the progressive PINN models. Numerical experiments demonstrate the key role of Lie symmetry in enhancing PINNs performance, emphasizing the importance of integrating abstract mathematical concepts into deep learning for addressing complex scientific problems adequately.

Abstract:
We introduce a formal active learning methodology for guiding the placement of Lagrangian observers to infer time‑dependent vector fields ‑‑ a key task in oceanography, marine science, and ocean engineering ‑‑ using a physics‑informed spatio‑temporal Gaussian process surrogate model. The majority of existing placement campaigns either follow standard `space‑filling' designs or relatively ad‑hoc expert opinions. A key challenge to applying principled active learning in this setting is that Lagrangian observers are continuously advected through the vector field, so they make measurements at different locations and times. It is, therefore, important to consider the likely future trajectories of placed observers to account for the utility of candidate placement locations. To this end, we present BALLAST: Bayesian Active Learning with Look‑ahead Amendment for Sea‑drifter Trajectories. We observe noticeable benefits of BALLAST‑aided sequential observer placement strategies on both synthetic and high‑fidelity ocean current models. In addition, we developed a novel GP inference method ‑‑ the Vanilla SPDE Exchange (VaSE) ‑‑ to boost the GP posterior sampling efficiency, which is also of independent interest.

Abstract:
Ship traffic is an increasing source of underwater radiated noise in coastal waters, motivating real‑time digital twins of ocean acoustics for operational noise mitigation. We present a physics‑guided probabilistic framework to predict three‑dimensional transmission loss in realistic ocean environments. As a case study, we consider the Salish Sea along shipping routes from the Pacific Ocean to the Port of Vancouver. A dataset of over 30 million source‑receiver pairs was generated with a Gaussian beam solver across seasonal sound speed profiles and one‑third‑octave frequency bands spanning 12.5 Hz to 8 kHz. We first assess sparse variational Gaussian processes (SVGP) and then incorporate physics‑based mean functions combining spherical spreading with frequency‑dependent absorption. To capture nonlinear effects, we examine deep sigma‑point processes and stochastic variational deep kernel learning. The final framework integrates four components: (i) a learnable physics‑informed mean that represents dominant propagation trends, (ii) a convolutional encoder for bathymetry along the source‑receiver track, (iii) a neural encoder for source, receiver, and frequency coordinates, and (iv) a residual SVGP layer that provides calibrated predictive uncertainty. This probabilistic digital twin facilitates the construction of sound‑exposure bounds and worst‑case scenarios for received levels. We further demonstrate the application of the framework to ship speed optimization, where predicted transmission loss combined with near‑field source models provides sound exposure level estimates for minimizing acoustic impacts on marine mammals. The proposed framework advances uncertainty‑aware digital twins for ocean acoustics and illustrates how physics‑guided machine learning can support sustainable maritime operations.

Abstract:
Accurate and physically feasible human motion prediction is crucial for safe and seamless human‑robot collaboration. While recent advancements in human motion capture enable real‑time pose estimation, the practical value of many existing approaches is limited by the lack of future predictions and consideration of physical constraints. Conventional motion prediction schemes rely heavily on past poses, which are not always available in real‑world scenarios. To address these limitations, we present a physics‑informed learning framework that integrates domain knowledge into both training and inference to predict human motion using inertial measurements from only 5 IMUs. We propose a network that accounts for the spatial characteristics of human movements. During training, we incorporate forward and differential kinematics functions as additional loss components to regularize the learned joint predictions. At the inference stage, we refine the prediction from the previous iteration to update a joint state buffer, which is used as extra inputs to the network. Experimental results demonstrate that our approach achieves high accuracy, smooth transitions between motions, and generalizes well to unseen subjects

Abstract:
This work develops a computational framework that combines physics‑informed neural networks with multi‑patch isogeometric analysis to solve partial differential equations on complex computer‑aided design geometries. The method utilizes patch‑local neural networks that operate on the reference domain of isogeometric analysis. A custom output layer enables the strong imposition of Dirichlet boundary conditions. Solution conformity across interfaces between non‑uniform rational B‑spline patches is enforced using dedicated interface neural networks. Training is performed using the variational framework by minimizing the energy functional derived after the weak form of the partial differential equation. The effectiveness of the suggested method is demonstrated on two highly non‑trivial and practically relevant use‑cases, namely, a 2D magnetostatics model of a quadrupole magnet and a 3D nonlinear solid and contact mechanics model of a mechanical holder. The results show excellent agreement to reference solutions obtained with high‑fidelity finite element solvers, thus highlighting the potential of the suggested neural solver to tackle complex engineering problems given the corresponding computer‑aided design models.

Abstract:
We implement physics‑informed‑neural‑networks (PINNs) to compute holographic entanglement entropy and entanglement wedge cross section. This technique allows us to compute these quantities for arbitrary shapes of the subregions in any asymptotically AdS metric. We test our computations against some known results and further demonstrate the utility of PINNs in examples, where it is not straightforward to perform such computations.

Abstract:
This paper presents adaptive weighted Euler‑Lagrange theorem combined physics‑informed neural networks (AW‑EL‑PINNs) for solving Euler‑Lagrange systems in optimal control problems. The framework systematically converts optimal control frameworks into two‑point boundary value problems (TPBVPs) while establishing a multi‑task learning paradigm through innovative integration of the Euler‑Lagrange theorem with deep learning architecture. An adaptive loss weighting mechanism dynamically balances loss function components during training, decreasing tedious manual tuning of weighting the loss functions compared to the conventional physics‑informed neural networks (PINNs). Based on six numerical examples, it's clear that AW‑EL‑PINNs achieve enhanced solution accuracy compared to baseline methods while maintaining stability throughout the optimization process. These results highlight the framework's capability to improve precision and ensure stability in solving Euler‑Lagrange systems in optimal control problems, offering potential strategies for problems under physical applications.

Abstract:
We propose a physics‑informed machine‑learned framework for sensor‑based flow estimation for drone trajectories in complex urban terrain. The input is a rich set of flow simulations at many wind conditions. The outputs are velocity and uncertainty estimates for a target domain and subsequent sensor optimization for minimal uncertainty. The framework has three innovations compared to traditional flow estimators. First, the algorithm scales proportionally to the domain complexity, making it suitable for flows that are too complex for any monolithic reduced‑order representation. Second, the framework extrapolates beyond the training data, e.g., smaller and larger wind velocities. Last, and perhaps most importantly, the sensor location is a free input, significantly extending the vast majority of the literature. The key enablers are (1) a Reynolds number‑based scaling of the flow variables, (2) a physics‑based domain decomposition, (3) a cluster‑based flow representation for each subdomain, (4) an information entropy correlating the subdomains, and (5) a multi‑variate probability function relating sensor input and targeted velocity estimates. This framework is demonstrated using drone flight paths through a three‑building cluster as a simple example. We anticipate adaptations and applications for estimating complete cities and incorporating weather input.

Abstract:
Voltage prediction in distribution grids is a critical yet difficult task for maintaining power system stability. Machine learning approaches, particularly Graph Neural Networks (GNNs), offer significant speedups but suffer from poor generalization when trained on limited or incomplete data. In this work, we systematically investigate the role of inductive biases in improving a model's ability to reliably learn power flow. Specifically, we evaluate three physics‑informed strategies: (i) power‑flow‑constrained loss functions, (ii) complex‑valued neural networks, and (iii) residual‑based task reformulation. Using the ENGAGE dataset, which spans multiple low‑ and medium‑voltage grid configurations, we conduct controlled experiments to isolate the effect of each inductive bias and assess both standard predictive performance and out‑of‑distribution generalization. Our study provides practical insights into which model assumptions most effectively guide learning for reliable and efficient voltage prediction in modern distribution networks.

Abstract:
X‑ray ptychography is a data‑intensive imaging technique expected to become ubiquitous at next‑generation light sources delivering many‑fold increases in coherent flux. The need for real‑time feedback under accelerated acquisition rates motivates surrogate reconstruction models like deep neural networks, which offer orders‑of‑magnitude speedup over conventional methods. However, existing deep learning approaches lack robustness across diverse experimental conditions. We propose an unsupervised training workflow emphasizing probe learning by combining experimentally‑measured probes with synthetic, procedurally generated objects. This probe‑centric approach enables a single physics‑informed neural network to reconstruct unseen experiments across multiple beamlines; among the first demonstrations of multi‑probe generalization. We find probe learning is equally important as in‑distribution learning; models trained using this synthetic workflow achieve reconstruction fidelity comparable to those trained exclusively on experimental data, even when changing the type of synthetic training object. The proposed approach enables training of experiment‑steering models that provide real‑time feedback under dynamic experimental conditions.

Abstract:
Learning control policies for multi‑robot systems (MRS) remains a major challenge due to long‑term coordination and the difficulty of obtaining realistic training data. In this work, we address both limitations within an imitation learning framework. First, we shift the typical role of Curriculum Learning in MRS, from scalability with the number of robots, to focus on improving long‑term coordination. We propose a curriculum strategy that gradually increases the length of expert trajectories during training, stabilizing learning and enhancing the accuracy of long‑term behaviors. Second, we introduce a method to approximate the egocentric perception of each robot using only third‑person global state demonstrations. Our approach transforms idealized trajectories into locally available observations by filtering neighbors, converting reference frames, and simulating onboard sensor variability. Both contributions are integrated into a physics‑informed technique to produce scalable, distributed policies from observations. We conduct experiments across two tasks with varying team sizes and noise levels. Results show that our curriculum improves long‑term accuracy, while our perceptual estimation method yields policies that are robust to realistic uncertainty. Together, these strategies enable the learning of robust, distributed controllers from global demonstrations, even in the absence of expert actions or onboard measurements.

Abstract:
A major challenge in physics‑informed machine learning is to understand how the incorporation of prior domain knowledge affects learning rates when data are dependent. Focusing on empirical risk minimization with physics‑informed regularization, we derive complexity‑dependent bounds on the excess risk in probability and in expectation. We prove that, when the physical prior information is aligned, the learning rate improves from the (slow) Sobolev minimax rate to the (fast) optimal i.i.d. one without any sample‑size deflation due to data dependence.

Abstract:
Recent trends in humanoid robot control have successfully employed imitation learning to enable the learned generation of smooth, human‑like trajectories from human data. While these approaches make more realistic motions possible, they are limited by the amount of available motion data, and do not incorporate prior knowledge about the physical laws governing the system and its interactions with the environment. Thus they may violate such laws, leading to divergent trajectories and sliding contacts which limit real‑world stability. We address such limitations via a two‑pronged learning strategy which leverages the known physics of the system and fundamental control principles. First, we encode physics priors during supervised imitation learning to promote trajectory feasibility. Second, we minimize drift at inference time by applying a proportional‑integral controller directly to the generated output state. We validate our method on various locomotion behaviors for the ergoCub humanoid robot, where a physics‑informed loss encourages zero contact foot velocity. Our experiments demonstrate that the proposed approach is compatible with multiple controllers on a real robot and significantly improves the accuracy and physical constraint conformity of generated trajectories.

Abstract:
The current work aims to incorporate physics‑based loss in Physics Informed Neural Network (PINN) directly using the numerical residual obtained from the governing equation in any dicretized forward solver. PINN's major difficulties in coupling with external forward solvers arise from the inability to access the discretized form (Finite difference, finite volume, finite element, etc.) of the governing equation directly through the network and to include them in its computational graph. This poses a significant challenge to conventional automatic‑differentiation‑based derivative computation of physics‑based loss terms concerning the neural network hyperparameters if gradient‑based optimization techniques are adopted. Therefore, we propose modifying the physics‑based loss term to account for the residual arising from the external solver and to compute the derivative required for the optimization machinery. The proposed methodologies are demonstrated on benchmark full‑order and reduced‑order systems.

Abstract:
We examine the use of a novel variant of Physics‑Informed Neural Networks to predict cosmological parameters from recent supernovae and baryon acoustic oscillations (BAO) datasets. Our machine learning framework generates uncertainty estimates for target variables and the inferred unknown parameters of the underlying PDE descriptions. Built upon a hybrid of the principles of Evidential Deep Learning, Physics‑Informed Neural Networks, Bayesian Neural Networks and Gaussian Processes, our model enables learning of the posterior distribution of the unknown PDE parameters through standard gradient‑descent based training. We apply our model to an up‑to‑date BAO dataset (Bousis et al. 2024) calibrated with the CMB‑inferred sound horizon, and the Pantheon+ Sne Ia distances (Scolnic et al. 2018), examining the relative effectiveness and mutual consistency among the standard ΛCDM, wCDM and Λ_sCDM models. Unlike previous results arising from the standard approach of minimizing an appropriate χ^2 function, the posterior distributions for parameters in various models trained purely on Pantheon+ data were found to be largely contained within the 2σ contours of their counterparts trained on BAO data. Their posterior medians for h_0 were within about 2σ of one another, indicating that our machine learning‑guided approach provides a different measure of the Hubble tension.

Abstract:
Co‑optimizing safety and performance in large‑scale multi‑agent systems remains a fundamental challenge. Existing approaches based on multi‑agent reinforcement learning (MARL), safety filtering, or Model Predictive Control (MPC) either lack strict safety guarantees, suffer from conservatism, or fail to scale effectively. We propose MAD‑PINN, a decentralized physics‑informed machine learning framework for solving the multi‑agent state‑constrained optimal control problem (MASC‑OCP). Our method leverages an epigraph‑based reformulation of SC‑OCP to simultaneously capture performance and safety, and approximates its solution via a physics‑informed neural network. Scalability is achieved by training the SC‑OCP value function on reduced‑agent systems and deploying them in a decentralized fashion, where each agent relies only on local observations of its neighbours for decision‑making. To further enhance safety and efficiency, we introduce an Hamilton‑Jacobi (HJ) reachability‑based neighbour selection strategy to prioritize safety‑critical interactions, and a receding‑horizon policy execution scheme that adapts to dynamic interactions while reducing computational burden. Experiments on multi‑agent navigation tasks demonstrate that MAD‑PINN achieves superior safety‑performance trade‑offs, maintains scalability as the number of agents grows, and consistently outperforms state‑of‑the‑art baselines.

Abstract:
Solving differential equations from limited or noisy data remains a key challenge for physics‑informed neural networks (PINNs), which are typically applied to already known and smooth solutions. In this work, we explore Bayesian PINNs and extended PINNs, (B‑(X)PINNs), to solve non‑linear second order differential equation typical for high energy theory, where data is only available from the boundary domain, to benchmark suitable approaches to PINNs in this category. In particular, we consider an entangling surface; a differential equation typical in holography. We perform asymptotic analysis to generate analytical training data from the boundary domain. We also explore the meaning of overconfidence in models that are constrained by physical priors and argue that standard overconfidence metrics are not suitable to consider when dealing with B‑PINNs. Overconfidence can be a natural feature and not a bug in systems with soft or hard constraints on the loss function; one have to look at when the overconfidence is an artifact of the model adhering to the physics constraints. To diagnose this effect, we introduce an information density quantity, and a local physics‑constraint coupling (PCC) metric, to capture locally to what extent the enforced physics collapses the posterior distribution. We also consider these quantities for a Liouville‑type equation and the Van der Pol equation to probe apparent overconfidence further.

Abstract:
Accurate aircraft trajectory prediction is critical for air traffic management, airline operations, and environmental assessment. This paper introduces NODE‑FDM, a Neural Ordinary Differential Equations‑based Flight Dynamics Model trained on Quick Access Recorder (QAR) data. By combining analytical kinematic relations with data‑driven components, NODE‑FDM achieves a more accurate reproduction of recorded trajectories than state‑of‑the‑art models such as a BADA‑based trajectory generation methodology (BADA4 performance model combined with trajectory control routines), particularly in the descent phase of the flight. The analysis demonstrates marked improvements across altitude, speed, and mass dynamics. Despite current limitations, including limited physical constraints and the limited availability of QAR data, the results demonstrate the potential of physics‑informed neural ordinary differential equations as a high‑fidelity, data‑driven approach to aircraft performance modelling. Future work will extend the framework to incorporate a full modelling of the lateral dynamics of the aircraft.

Abstract:
Activation functions critically influence trainability and expressivity, and recent work has therefore explored a broad range of nonlinearities. However, widely used Gaussian i.i.d. initializations are designed to preserve activation variance under wide or infinite width assumptions. In deep and relatively narrow networks with sigmoidal nonlinearities, these schemes often drive preactivations into saturation, and collapse gradients. To address this, we introduce an odd‑sigmoid activations and propose an activation aware initialization tailored to any function in this class. Our method remains robust over a wide band of variance scales, preserving both forward signal variance and backpropagated gradient norms even in very deep and narrow networks. Empirically, across standard image benchmarks we find that the proposed initialization is substantially less sensitive to depth, width, and activation scale than Gaussian initializations. In physics informed neural networks (PINNs), scaled odd‑sigmoid activations combined with our initialization achieve lower losses than Gaussian based setups, suggesting that diagonal‑plus‑noise weights provide a practical alternative when Gaussian initialization breaks down.

Abstract:
We develop a physics‑informed neural network (PINN) framework for parameter estimation in fractional‑order SEIRD epidemic models. By embedding the Caputo fractional derivative into the network residuals via the L1 discretization scheme, our method simultaneously reconstructs epidemic trajectories and infers both epidemiological parameters and the fractional memory order α. The fractional formulation extends classical integer‑order models by capturing long‑range memory effects in disease progression, incubation, and recovery. Our framework learns the fractional memory order α as a trainable parameter while simultaneously estimating the epidemiological rates (β, σ, γ, μ). A composite loss combining data misfit, physics residuals, and initial conditions, with constraints on positivity and population conservation, ensures both accuracy and biological consistency. Tests on synthetic Mpox data confirm reliable recovery of α and parameters under noise, while applications to COVID‑19 show that optimal α\in (0, 1] captures memory effects and improves predictive performance over the classical SEIRD model. This work establishes PINNs as a robust tool for learning memory effects in epidemic dynamics, with implications for forecasting, control strategies, and the analysis of non‑Markovian epidemic processes.

Abstract:
The lattice Boltzmann equation (LBE), rooted in kinetic theory, provides a powerful framework for capturing complex flow behaviour by describing the evolution of single‑particle distribution functions (PDFs). Despite its success, solving the LBE numerically remains computationally intensive due to strict time‑step restrictions imposed by collision kernels. Here, we introduce a physics‑informed neural operator framework for the LBE that enables prediction over large time horizons without step‑by‑step integration, effectively bypassing the need to explicitly solve the collision kernel. We incorporate intrinsic moment‑matching constraints of the LBE, along with global equivariance of the full distribution field, enabling the model to capture the complex dynamics of the underlying kinetic system. Our framework is discretization‑invariant, enabling models trained on coarse lattices to generalise to finer ones (kinetic super‑resolution). In addition, it is agnostic to the specific form of the underlying collision model, which makes it naturally applicable across different kinetic datasets regardless of the governing dynamics. Our results demonstrate robustness across complex flow scenarios, including von Karman vortex shedding, ligament breakup, and bubble adhesion. This establishes a new data‑driven pathway for modelling kinetic systems.

Abstract:
Four‑dimensional variational data assimilation (4DVAR) is a cornerstone of numerical weather prediction, but its cost function is difficult to optimize and computationally intensive. We propose a neural field‑based reformulation in which the full spatiotemporal state is represented as a continuous function parameterized by a neural network. This reparameterization removes the time‑sequential dependency of classical 4DVAR, enabling parallel‑in‑time optimization in parameter space. Physical constraints are incorporated directly through a physics‑informed loss, simplifying implementation and reducing computational cost. We evaluate the method on the two‑dimensional incompressible Navier‑‑Stokes equations with Kolmogorov forcing. Compared to a baseline 4DVAR implementation, the neural reparameterized variants produce more stable initial condition estimates without spurious oscillations. Notably, unlike most machine learning‑based approaches, our framework does not require access to ground‑truth states or reanalysis data, broadening its applicability to settings with limited reference information.

Abstract:
Hair simulation and rendering are challenging due to complex strand dynamics, diverse material properties, and intricate light‑hair interactions. Recent video diffusion models can generate high‑quality videos, but they lack fine‑grained control over hair dynamics. We present ControlHair, a hybrid framework that integrates a physics simulator with conditional video diffusion to enable controllable dynamic hair rendering. ControlHair adopts a three‑stage pipeline: it first encodes physics parameters (e.g., hair stiffness, wind) into per‑frame geometry using a simulator, then extracts per‑frame control signals, and finally feeds control signals into a video diffusion model to generate videos with desired hair dynamics. This cascaded design decouples physics reasoning from video generation, supports diverse physics, and makes training the video diffusion model easy. Trained on a curated 10K video dataset, ControlHair outperforms text‑ and pose‑conditioned baselines, delivering precisely controlled hair dynamics. We further demonstrate three use cases of ControlHair: dynamic hairstyle try‑on, bullet‑time effects, and cinemagraphic. ControlHair introduces the first physics‑informed video diffusion framework for controllable dynamics. We provide a teaser video and experimental results on our website.

Abstract:
This work addresses object identification under known dynamics in unmanned aerial vehicle applications, where learning and classification are combined through a physics‑informed residual neural network. The proposed framework leverages physics‑informed learning for state mapping and state‑derivative prediction, while a softmax layer enables multi‑class confidence estimation. Quadcopter, fixed‑wing, and helicopter aerial vehicles are considered as case studies. The results demonstrate high classification accuracy with reduced training time, offering a promising solution for system identification problems in domains where the underlying dynamics are well understood.

Abstract:
Physics Informed Neural Networks offer a mesh free framework for solving PDEs but are highly sensitive to loss weight selection. We propose two dimensional analysis based weighting schemes, one based on quantifiable terms, and another also incorporating unquantifiable terms for more balanced training. Benchmarks on heat conduction, convection diffusion, and lid driven cavity flows show that the second scheme consistently improves stability and accuracy over equal weighting. Notably, in high Peclet number convection diffusion, where traditional solvers fail, PINNs with our scheme achieve stable, accurate predictions, highlighting their robustness and generalizability in CFD problems.

Abstract:
Prognostics and Health Management ensures the reliability, safety, and efficiency of complex engineered systems by enabling fault detection, anticipating equipment failures, and optimizing maintenance activities throughout an asset lifecycle. However, real‑world PHM presents persistent challenges: sensor data is often noisy or incomplete, available labels are limited, and degradation behaviors and system interdependencies can be highly complex and nonlinear. Physics‑informed machine learning has emerged as a promising approach to address these limitations by embedding physical knowledge into data‑driven models. This review examines how incorporating learning and observational biases through physics‑informed modeling and data strategies can guide models toward physically consistent and reliable predictions. Learning biases embed physical constraints into model training through physics‑informed loss functions and governing equations, or by incorporating properties like monotonicity. Observational biases influence data selection and synthesis to ensure models capture realistic system behavior through virtual sensing for estimating unmeasured states, physics‑based simulation for data augmentation, and multi‑sensor fusion strategies. The review then examines how these approaches enable the transition from passive prediction to active decision‑making through reinforcement learning, which allows agents to learn maintenance policies that respect physical constraints while optimizing operational objectives. This closes the loop between model‑based predictions, simulation, and actual system operation, empowering adaptive decision‑making. Finally, the review addresses the critical challenge of scaling PHM solutions from individual assets to fleet‑wide deployment. Fast adaptation methods including meta‑learning and few‑shot learning are reviewed alongside domain generalization techniques ...

Abstract:
Future high‑luminosity hadron colliders demand tracking detectors with extreme radiation tolerance, high spatial precision, and sub‑nanosecond timing. 3D diamond pixel sensors offer these capabilities due to diamond's radiation hardness and high carrier mobility. Conductive electrodes, produced via femtosecond IR laser pulses, exhibit high resistivity that delays signal propagation. This effect necessitates extending the classical Ramo‑Shockley weighting potential formalism. We model the phenomenon through a 3rd‑order, 3+1D PDE derived as a quasi‑stationary approximation of Maxwell's equations. The PDE is solved numerically and coupled with charge transport simulations for realistic 3D sensor geometries. A Mixture‑of‑Experts Physics‑Informed Neural Network, trained on Spectral Method data, provides a meshless solver to assess timing degradation from electrode resistance.

Abstract:
Variational quantum algorithms (VQAs) are leading strategies to reach practical utilities of near‑term quantum devices. However, the no‑cloning theorem in quantum mechanics precludes standard backpropagation, leading to prohibitive quantum resource costs when applying VQAs to large‑scale tasks. To address this challenge, we reformulate the training dynamics of VQAs as a nonlinear partial differential equation and propose a novel protocol that leverages physics‑informed neural networks (PINNs) to model this dynamical system efficiently. Given a small amount of training trajectory data collected from quantum devices, our protocol predicts the parameter updates of VQAs over multiple iterations on the classical side, dramatically reducing quantum resource costs. Through systematic numerical experiments, we demonstrate that our method achieves up to a 30x speedup compared to conventional methods and reduces quantum resource costs by as much as 90% for tasks involving up to 40 qubits, including ground state preparation of different quantum systems, while maintaining competitive accuracy. Our approach complements existing techniques aimed at improving the efficiency of VQAs and further strengthens their potential for practical applications.

Abstract:
Diffusion models have demonstrated strong generative capabilities across scientific domains, but often produce outputs that violate physical laws. We propose a new perspective by framing physics‑informed generation as a sparse reward optimization problem, where adherence to physical constraints is treated as a reward signal. This formulation unifies prior approaches under a reward‑based paradigm and reveals a shared bottleneck: reliance on diffusion posterior sampling (DPS)‑style value function approximations, which introduce non‑negligible errors and lead to training instability and inference inefficiency. To overcome this, we introduce Physics‑Informed Reward Fine‑tuning (PIRF), a method that bypasses value approximation by computing trajectory‑level rewards and backpropagating their gradients directly. However, a naive implementation suffers from low sample efficiency and compromised data fidelity. PIRF mitigates these issues through two key strategies: (1) a layer‑wise truncated backpropagation method that leverages the spatiotemporally localized nature of physics‑based rewards, and (2) a weight‑based regularization scheme that improves efficiency over traditional distillation‑based methods. Across five PDE benchmarks, PIRF consistently achieves superior physical enforcement under efficient sampling regimes, highlighting the potential of reward fine‑tuning for advancing scientific generative modeling.

Abstract:
Accretion disks are ubiquitous in astrophysics, appearing in diverse environments from planet‑forming systems to X‑ray binaries and active galactic nuclei. Traditionally, modeling their dynamics requires computationally intensive (magneto)hydrodynamic simulations. Recently, Physics‑Informed Neural Networks (PINNs) have emerged as a promising alternative. This approach trains neural networks directly on physical laws without requiring data. We for the first time demonstrate PINNs for solving the two‑dimensional, time‑dependent hydrodynamics of non‑self‑gravitating accretion disks. Our models provide solutions at arbitrary times and locations within the training domain, and successfully reproduce key physical phenomena, including the excitation and propagation of spiral density waves and gap formation from disk‑companion interactions. Notably, the boundary‑free approach enabled by PINNs naturally eliminates the spurious wave reflections at disk edges, which are challenging to suppress in numerical simulations. These results highlight how advanced machine learning techniques can enable physics‑driven, data‑free modeling of complex astrophysical systems, potentially offering an alternative to traditional numerical simulations in the future.

Abstract:
Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics‑informed neural networks (PINNs) are a recent machine learning‑based approach, for which many properties and limitations remain unknown. PINNs are widely accepted as inferior to traditional methods for solving PDEs, such as the finite element method, both with regard to computation time and accuracy. However, PINNs are commonly claimed to show promise in solving inverse problems and handling noisy or incomplete data. We compare the performance of PINNs in solving inverse problems with that of a traditional approach using the finite element method combined with a numerical optimizer. The models are tested on a series of increasingly difficult fluid mechanics problems, with and without noise. We find that while PINNs may require less human effort and specialized knowledge, they are outperformed by the traditional approach. However, the difference appears to decrease with higher dimensions and more data. We identify common failures during training to be addressed if the performance of PINNs on noisy inverse problems is to become more competitive.

Abstract:
Most refrigerants currently used in air‑conditioning systems, such as hydrofluorocarbons, are potent greenhouse gases and are being phased down. Large‑scale molecular screening has been applied to the search for alternatives, but in practice only about 300 refrigerants are known, and only a few additional candidates have been suggested without experimental validation. This scarcity of reliable data limits the effectiveness of purely data‑driven methods. We present Refgen, a generative pipeline that integrates machine learning with physics‑grounded inductive biases. Alongside fine‑tuning for valid molecular generation, Refgen incorporates predictive models for critical properties, equations of state, thermochemical polynomials, and full vapor compression cycle simulations. These models enable reinforcement learning fine‑tuning under thermodynamic constraints, enforcing consistency and guiding discovery toward molecules that balance efficiency, safety, and environmental impact. By embedding physics into the learning process, Refgen leverages scarce data effectively and enables de novo refrigerant discovery beyond the known set of compounds.

Abstract:
Physics‑Informed Neural Networks (PINNs) are a class of deep learning models aiming to approximate solutions of PDEs by training neural networks to minimize the residual of the equation. Focusing on non‑equilibrium fluctuating systems, we propose a physically informed choice of penalization that is consistent with the underlying fluctuation structure, as characterized by a large deviations principle. This approach yields a novel formulation of PINNs in which the penalty term is chosen to penalize improbable deviations, rather than being selected heuristically. The resulting thermodynamically consistent extension of PINNs, termed THINNs, is subsequently analyzed by establishing analytical a posteriori estimates, and providing empirical comparisons to established penalization strategies.

Abstract:
In the context of the energy transition, with increasing integration of renewable sources and cross‑border electricity exchanges, power grids are encountering greater uncertainty and operational risk. Maintaining grid stability under varying conditions is a complex task, and power flow simulators are commonly used to support operators by evaluating potential actions before implementation. However, traditional physical solvers, while accurate, are often too slow for near real‑time use. Machine learning models have emerged as fast surrogates, and to improve their adherence to physical laws (e.g., Kirchhoff's laws), they are often trained with embedded constraints which are also known as physics‑informed or hybrid models. This paper presents an ablation study to demystify hybridization strategies, ranging from incorporating physical constraints as regularization terms or unsupervised losses, and exploring model architectures from simple multilayer perceptrons to advanced graph‑based networks enabling the direct optimization of physics equations. Using our custom benchmarking pipeline for hybrid models called LIPS, we evaluate these models across four dimensions: accuracy, physical compliance, industrial readiness, and out‑of‑distribution generalization. The results highlight how integrating physical knowledge impacts performance across these criteria. All the implementations are reproducible and provided in the corresponding Github page.

Abstract:
Data‑driven methods keep increasing their popularity in engineering applications, given the developments in data analysis techniques. Some of these approaches, such as Field Inversion Machine Learning (FIML), suggest correcting low‑fidelity models by leveraging available observations of the problem. However, the solely data‑driven field inversion stage of the method generally requires dense observations that limit the usage of sparse data. In this study, we propose a physical loss term addition to the field inversion stage of the FIML technique similar to the physics‑informed machine learning applications. This addition embeds the complex physics of the problem into the low‑fidelity model, which allows for obtaining dense gradient information for every correction parameter and acts as an adaptive regularization term improving inversion accuracy. The proposed Physics‑Informed Field Inversion approach is tested using three different examples and highlights that incorporating physical loss can enhance the reconstruction performance for limited data cases, such as sparse, truncated, and noisy observations. Additionally, this modification enables us to obtain accurate posterior correction parameter distribution with limited realizations, making it data‑efficient. The increase in the computational cost caused by the physical loss calculation is at an acceptable level given the relaxed grid and numerical scheme requirements.

Abstract:
We introduce a novel convolutional neural network architecture, termed the \emphperiodic CNN, which incorporates periodic boundary conditions into the convolutional layers. Our main theoretical contribution is a rigorous approximation theorem: periodic CNNs can approximate ridge functions depending on d‑1 linear variables in a d‑dimensional input space, while such approximation is impossible in lower‑dimensional ridge settings (d‑2 or fewer variables). This result establishes a sharp characterization of the expressive power of periodic CNNs. Beyond the theory, our findings suggest that periodic CNNs are particularly well‑suited for problems where data naturally admits a ridge‑like structure of high intrinsic dimension, such as image analysis on wrapped domains, physics‑informed learning, and materials science. The work thus both expands the mathematical foundation of CNN approximation theory and highlights a class of architectures with surprising and practically relevant approximation capabilities.

Abstract:
The prediction of quantum dynamical responses lies at the heart of modern physics. Yet, modeling these time‑dependent behaviors remains a formidable challenge because quantum systems evolve in high‑dimensional Hilbert spaces, often rendering traditional numerical methods computationally prohibitive. While large language models have achieved remarkable success in sequential prediction, quantum dynamics presents a fundamentally different challenge: forecasting the entire temporal evolution of quantum systems rather than merely the next element in a sequence. Existing neural architectures such as recurrent and convolutional networks often require vast training datasets and suffer from spurious oscillations that compromise physical interpretability. In this work, we introduce a fundamentally new approach: Kolmogorov Arnold Networks (KANs) augmented with physics‑informed loss functions that enforce the Ehrenfest theorems. Our method achieves superior accuracy with significantly less training data: it requires only 5.4 percent of the samples (200) compared to Temporal Convolution Networks (3,700). We further introduce the Chain of KANs, a novel architecture that embeds temporal causality directly into the model design, making it particularly well‑suited for time series modeling. Our results demonstrate that physics‑informed KANs offer a compelling advantage over conventional black‑box models, maintaining both mathematical rigor and physical consistency while dramatically reducing data requirements.

Abstract:
Physics‑informed neural networks (PINNs) often exhibit weight matrices that appear statistically random after training, yet their implications for signal propagation and stability remain unsatisfactorily understood, let alone the interpretability. In this work, we analyze the spectral and statistical properties of trained PINN weights using viscous and inviscid variants of the one‑dimensional Burgers' equation, and show that the learned weights reside in a high‑entropy regime consistent with predictions from random matrix theory. To investigate the dynamical consequences of such weight structures, we study the evolution of signal features inside a network through the lens of neural partial differential equations (neural PDEs). We show that random and structured weight matrices can be associated with specific discretizations of neural PDEs, and that the numerical stability of these discretizations governs the stability of signal propagation through the network. In particular, explicit unstable schemes lead to degraded signal evolution, whereas stable implicit and higher‑order schemes yield well‑behaved dynamics for the same underlying neural PDE. Our results offer an explicit example of how numerical stability and network architecture shape signal propagation in deep networks, in relation to random matrix and neural PDE descriptions in PINNs.

Abstract:
We study learning of continuous‑time inventory dynamics under stochastic demand and quantify when structure helps or hurts forecasting of the bullwhip effect. BULL‑ODE compares a fully learned Neural ODE (NODE) that models the entire right‑hand side against a physics‑informed Universal Differential Equation (UDE) that preserves conservation and order‑up‑to structure while learning a small residual policy term. Classical supply chain models explain the bullwhip through control/forecasting choices and information sharing, while recent physics‑informed and neural differential equation methods blend domain constraints with learned components. It is unclear whether structural bias helps or hinders forecasting under different demand regimes. We address this by using a single‑echelon testbed with three demand regimes ‑ AR(1) (autocorrelated), i.i.d. Gaussian, and heavy‑tailed lognormal. Training is done on varying fractions of each trajectory, followed by evaluation of multi‑step forecasts for inventory I, order rate O, and demand D. Across the structured regimes, UDE consistently generalizes better: with 90% of the training horizon, inventory RMSE drops from 4.92 (NODE) to 0.26 (UDE) under AR(1) and from 5.96 to 0.95 under Gaussian demand. Under heavy‑tailed lognormal shocks, the flexibility of NODE is better. These trends persist as train18 ing data shrinks, with NODE exhibiting phase drift in extrapolation while UDE remains stable but underreacts to rare spikes. Our results provide concrete guidance: enforce structure when noise is light‑tailed or temporally correlated; relax structure when extreme events dominate. Beyond inventory control, the results offer guidance for hybrid modeling in scientific and engineering systems: enforce known structure when conservation laws and modest noise dominate, and relax structure to capture extremes in settings where rare events drive dynamics.

Abstract:
In experimental nuclear and particle physics, the extraction of high‑purity samples of rare events critically depends on the efficiency and accuracy of particle identification (PID). In this work, we present a PID method applied to HADES data at the level of fully reconstructed particle track candidates. The results demonstrate a significant improvement in PID performance compared to conventional techniques, highlighting the potential of physics‑informed neural networks as a powerful tool for future data analyses.

Abstract:
Accurate battery modeling is essential for reliable state estimation in modern applications, such as predicting the remaining discharge time and remaining discharge energy in battery management systems. Existing approaches face several limitations: model‑based methods require a large number of parameters; data‑driven methods rely heavily on labeled datasets; and current physics‑informed neural networks (PINNs) often lack aging adaptation, or still depend on many parameters, or continuously regenerate states. In this work, we propose SeqBattNet, a discrete‑state PINN with built‑in aging adaptation for battery modeling, to predict terminal voltage during the discharge process. SeqBattNet consists of two components: (i) an encoder, implemented as the proposed HRM‑GRU deep learning module, which generates cycle‑specific aging adaptation parameters; and (ii) a decoder, based on the equivalent circuit model (ECM) combined with deep learning, which uses these parameters together with the input current to predict voltage. The model requires only three basic battery parameters and, when trained on data from a single cell, still achieves robust performance. Extensive evaluations across three benchmark datasets (TRI, RT‑Batt, and NASA) demonstrate that SeqBattNet significantly outperforms classical sequence models and PINN baselines, achieving consistently lower RMSE while maintaining computational efficiency.

Abstract:
Lane‑change maneuvers are a leading cause of highway accidents, underscoring the need for accurate intention prediction to improve the safety and decision‑making of autonomous driving systems. While prior studies using machine learning and deep learning methods (e.g., SVM, CNN, LSTM, Transformers) have shown promise, most approaches remain limited by binary classification, lack of scenario diversity, and degraded performance under longer prediction horizons. In this study, we propose a physics‑informed AI framework that explicitly integrates vehicle kinematics, interaction feasibility, and traffic‑safety metrics (e.g., distance headway, time headway, time‑to‑collision, closing gap time) into the learning process. lane‑change prediction is formulated as a three‑class problem that distinguishes left change, right change, and no change, and is evaluated across both straight highway segments (highD) and complex ramp scenarios (exiD). By integrating vehicle kinematics with interaction features, our machine learning models, particularly LightGBM, achieve state‑of‑the‑art accuracy and strong generalization. Results show up to 99.8% accuracy and 93.6% macro F1 on highD, and 96.1% accuracy and 88.7% macro F1 on exiD at a 1‑second horizon, outperforming a two‑layer stacked LSTM baseline. These findings demonstrate the practical advantages of a physics‑informed and feature‑rich machine learning framework for real‑time lane‑change intention prediction in autonomous driving systems.

Abstract:
Accurate modeling of personalized cardiovascular dynamics is crucial for non‑invasive monitoring and therapy planning. State‑of‑the‑art physics‑informed neural network (PINN) approaches employ deep, multi‑branch architectures with adversarial or contrastive objectives to enforce partial differential equation constraints. While effective, these enhancements introduce significant training and implementation complexity, limiting scalability and practical deployment. We investigate physics‑informed neural operator learning models as efficient supervisory signals for training simplified architectures through knowledge distillation. Our approach pre‑trains a physics‑informed DeepONet (PI‑DeepONet) on high‑fidelity cuffless blood pressure recordings to learn operator mappings from raw wearable waveforms to beat‑to‑beat pressure signals under embedded physics constraints. This pre‑trained operator serves as a frozen supervisor in a lightweight knowledge‑distillation pipeline, guiding streamlined base models that eliminate complex adversarial and contrastive learning components while maintaining performance. We characterize the role of physics‑informed regularization in operator learning and demonstrate its effectiveness for supervisory guidance. Through extensive experiments, our operator‑supervised approach achieves performance parity with complex baselines (correlation: 0.766 vs. 0.770, RMSE: 4.452 vs. 4.501), while dramatically reducing architectural complexity from eight critical hyperparameters to a single regularization coefficient and decreasing training overhead by 4%. Our results demonstrate that operator‑based supervision effectively replaces intricate multi‑component training strategies, offering a more scalable and interpretable approach to physiological modeling with reduced implementation burden.

Abstract:
We studied the reconstruction of turbulent flow fields from trajectory data recorded by actively migrating Lagrangian agents. We propose a deep‑learning model, track‑to‑flow (T2F), which employs a vision transformer as the encoder to capture the spatiotemporal features of a single agent trajectory, and a convolutional neural network as the decoder to reconstruct the flow field. To enhance the physical consistency of the T2F model, we further incorporate a physics‑informed loss function inspired by the framework of physics‑informed neural network (PINN), yielding a variant model referred to as T2F+PINN. We first evaluate both models in a laminar cylinder wake flow at a Reynolds number of Re = 800 as a proof of concept. The results show that the T2F model achieves velocity reconstruction accuracy comparable to that of existing flow reconstruction methods, while the T2F+PINN model reduces the normalised error in vorticity reconstruction relative to the T2F model. We then apply the models in a turbulent Rayleigh‑Bénard convection at a Rayleigh number of Ra = 10^8 and a Prandtl number of Pr = 0.71. The results show that the T2F model accurately reconstructs both the velocity and temperature fields, whereas the T2F+PINN model further improves the reconstruction accuracy of gradient‑related physical quantities, such as temperature gradients, vorticity and the Q value, with a maximum improvement of approximately 60 % compared to the T2F model. Overall, the T2F model is better suited for reconstructing primitive flow variables, while the T2F+PINN model provides advantages in reconstructing gradient‑related quantities. Our models open a promising avenue for accurate flow reconstruction from a single Lagrangian trajectory.

Abstract:
Refractive index matching (RIM) is a powerful tool for multiphase flow studies as it eliminates optical distortions and enables high‑fidelity tomographic measurements near solid‑fluid interfaces of freely moving solids in the flow. However, by improving the RIM and optical quality, the solids become effectively invisible, preventing direct identification of their location. To address this limitation, we develop a physics‑informed detection framework that locates transparent spheres within time‑resolved tomographic Particle Tracking Velocimetry by combining tracer voids, vertical velocity signatures, and vortex structures into a unified optimization problem. Integrated with volumetric reconstructions, the method provides simultaneous analysis of velocity, pressure, and force on the sphere. Applied to an example case of an 11.11 mm acrylic sphere rising in a RIM sodium iodide solution, the technique reveals a clear phase‑locked relation between double‑thread wake structures, surface‑pressure distributions, and unsteady hydrodynamic forces over half a cycle of the sphere motion in the 4R vortex shedding regime. For the first time, this enables direct calculation of drag and lift histories on a freely moving sphere. The framework can be extended to dynamic masking for improved tomographic reconstruction and pressure‑field calculations, to non‑spherical bodies with more complex motions, and to multi‑body interactions, advancing RIM from a flow‑only diagnostic to a tool for fully coupled body‑wake measurements.

Abstract:
I present a simple hybrid framework that combines physics informed neural networks (PINNs) with features generated from small quantum circuits. As a proof of concept, a first‑order equation is solved by feeding quantum measurement probabilities into the neural model. The architecture enforces the initial condition exactly, and training is guided by the ODE residual loss. Numerical results show that the hybrid model reproduces the analytical solution, illustrating the potential of quantum‑enhanced PINNs for differential equation solving.

Abstract:
Laser Powder Bed Fusion (L‑PBF) is a widely adopted additive manufacturing process for fabricating complex metallic parts layer by layer. Effective thermal management is essential to ensure part quality and structural integrity, as thermal gradients and residual stresses can lead to defects such as warping and cracking. However, existing experimental or computational techniques lack the ability to forecast future temperature distributions in real time, an essential capability for proactive process control. This paper presents a real‑time thermal state forecasting framework for L‑PBF, based on a physics‑informed reduced‑order thermal model integrated with a Kalman filtering scheme. The proposed approach efficiently captures inter‑layer heat transfer dynamics and enables accurate tracking and forecasting of spatial and temporal temperature evolution. Validation across multiple part geometries using measured data demonstrates that the method reliably estimates and forecasts peak temperatures and cooling trends. By enabling predictive thermal control, this framework offers a practical and computationally efficient solution for thermal management in L‑PBF, paving the way toward closed‑loop control in L‑PBF.

Abstract:
In the past, we have presented a systematic computational framework for analyzing self‑similar and traveling wave dynamics in nonlinear partial differential equations (PDEs) by dynamically factoring out continuous symmetries such as translation and scaling. This is achieved through the use of time‑dependent transformations ‑‑ what can be viewed as dynamic pinning conditions ‑‑ that render the symmetry‑invariant solution stationary or slowly varying in rescaled coordinates. The transformation process yields a modified evolution equation coupled with algebraic constraints on the symmetry parameters, resulting in index‑2 differential‑algebraic equation (DAE) systems. The framework accommodates both first‑kind and second‑kind self‑similarity, and directly recovers the self‑similarity exponents or wave speeds as part of the solution, upon considering steady‑state solutions in the rescaled coordinate frame. To solve the resulting high‑index DAE systems, we employ Physics‑Informed Neural Networks (PINNs), which naturally integrate PDE residuals and algebraic constraints into a unified loss function. This allows simultaneous inference of both the invariant solution and the transformation properties (such as the speed or the scaling rate without the need for large computational domains, mesh adaptivity, or front tracking. We demonstrate the effectiveness of the method on four canonical problems: (i) the Nagumo equation exhibiting traveling waves, (ii) the diffusion equation (1D and 2D) with first‑kind self‑similarity, (iii) the 2D axisymmetric porous medium equation showcasing second‑kind self‑similarity, and (iv) the Burgers equation, which involves both translational and scaling invariance. The results demonstrate the capability of PINNs to effectively solve these complex PDE‑DAE systems, providing a promising tool for studying nonlinear wave and scaling phenomena.

Abstract:
Scientific Machine Learning (SciML) integrates physics and data into the learning process, offering improved generalization compared with purely data‑driven models. Despite its potential, applications of SciML in prognostics remain limited, partly due to the complexity of incorporating partial differential equations (PDEs) for ageing physics and the scarcity of robust uncertainty quantification methods. This work introduces a Bayesian Physics‑Informed Neural Network (B‑PINN) framework for probabilistic prognostics estimation. By embedding Bayesian Neural Networks into the PINN architecture, the proposed approach produces principled, uncertainty‑aware predictions. The method is applied to a transformer ageing case study, where insulation degradation is primarily driven by thermal stress. The heat diffusion PDE is used as the physical residual, and different prior distributions are investigated to examine their impact on predictive posterior distributions and their ability to encode a priori physical knowledge. The framework is validated against a finite element model developed and tested with real measurements from a solar power plant. Results, benchmarked against a dropout‑PINN baseline, show that the proposed B‑PINN delivers more reliable prognostic predictions by accurately quantifying predictive uncertainty. This capability is crucial for supporting robust and informed maintenance decision‑making in critical power assets.

Abstract:
This paper presents a unified framework that integrates modeling, optimization, and sensorless control of an all‑electric heavy‑duty robotic manipulator (HDRM) driven by electromechanical linear actuators (EMLAs). An EMLA model is formulated to capture motor electromechanics and direction‑dependent transmission efficiencies, while a mathematical model of the HDRM, incorporating both kinematics and dynamics, is established to generate joint‑space motion profiles for prescribed TCP trajectories. A safety‑ensured trajectory generator, tailored to this model, maps Cartesian goals to joint space while enforcing joint‑limit and velocity margins. Based on the resulting force and velocity demands, a multi‑objective Non‑dominated Sorting Genetic Algorithm II (NSGA‑II) is employed to select the optimal EMLA configuration. To accelerate this optimization, a deep neural network, trained with EMLA parameters, is embedded in the optimization process to predict steady‑state actuator efficiency from trajectory profiles. For the chosen EMLA design, a physics‑informed Kriging surrogate, anchored to the analytic model and refined with experimental data, learns residuals of EMLA outputs to support force and velocity sensorless control. The actuator model is further embedded in a hierarchical virtual decomposition control (VDC) framework that outputs voltage commands. Experimental validation on a one‑degree‑of‑freedom EMLA testbed confirms accurate trajectory tracking and effective sensorless control under varying loads.

Abstract:
Modelling the underlying mechanisms of neurodegenerative diseases demands methods that capture heterogeneous and spatially varying dynamics from sparse, high‑dimensional neuroimaging data. Integrating partial differential equation (PDE) based physics knowledge with machine learning provides enhanced interpretability and utility over classic numerical methods. However, current physics‑integrated machine learning methods are limited to considering a single PDE, severely limiting their application to diseases where multiple mechanisms are responsible for different groups (i.e., subtypes) and aggravating problems with model misspecification and degeneracy. Here, we present a deep generative model for learning mixtures of latent dynamic models governed by physics‑based PDEs, going beyond traditional approaches that assume a single PDE structure. Our method integrates reaction‑diffusion PDEs within a variational autoencoder (VAE) mixture model framework, supporting inference of subtypes of interpretable latent variables (e.g. diffusivity and reaction rates) from neuroimaging data. We evaluate our method on synthetic benchmarks and demonstrate its potential for uncovering mechanistic subtypes of Alzheimer's disease progression from positron emission tomography (PET) data.

Abstract:
This paper presents a physics‑informed framework that integrates graph convolutional networks (GCN) with long short‑term memory (LSTM) architecture to forecast microstructure evolution over long time horizons in both 2D and 3D with remarkable performance across varied metrics. The proposed framework is composition‑aware, trained jointly on datasets with different compositions, and operates in latent graph space, which enables the model to capture compositions and morphological dynamics while remaining computationally efficient. Compressing and encoding phase‑field simulation data with convolutional autoencoders and operating in Latent graph space facilitates efficient modeling of microstructural evolution across composition, dimensions, and long‑term horizons. The framework captures the spatial and temporal patterns of evolving microstructures while enabling long‑range forecasting at reduced computational cost after training.

Abstract:
Physics‑informed neural networks (PINNs) have been widely utilized for solving a range of partial differential equations (PDEs) in various scientific and engineering disciplines. This paper presents a Fourier heuristic‑enhanced PINN (termed FCPINN) designed to address a specific class of biharmonic equations with Dirichlet and Navier boundary conditions. The method achieves this by decomposing the high‑order equations into two Poisson equations. FCPINN integrates Fourier spectral theory with a reduced‑order formulation for high‑order PDEs, significantly improving approximation accuracy and reducing computational complexity. This approach is especially beneficial for problems with intricate boundary constraints and high‑dimensional inputs. To assess the effectiveness and robustness of the FCPINN algorithm, we conducted several numerical experiments on both linear and nonlinear biharmonic problems across different Euclidean spaces. The results show that FCPINN provides an optimal trade‑off between speed and accuracy for high‑order PDEs, surpassing the performance of conventional PINN and deep mixed residual method (MIM) approaches, while also maintaining stability and robustness with varying numbers of hidden layer nodes.

Abstract:
In this paper, we propose a novel physics‑informed deep learning architecture for predicting radio maps over lunar terrain. Our approach integrates a physics‑based lunar terrain generator, which produces realistic topography informed by publicly available NASA data, with a ray‑tracing engine to create a high‑fidelity dataset of radio propagation scenarios. Building on this dataset, we introduce a triplet‑UNet architecture, consisting of two standard UNets and a diffusion network, to model complex propagation effects. Experimental results demonstrate that our method outperforms existing deep learning approaches on our terrain dataset across various metrics.

Abstract:
Granular material has significant implications for industrial and geophysical processes. A long‑lasting challenge, however, is seeking a unified rheology for its solid‑ and liquid‑like behaviors under quasi‑static, inertial, and even unsteady shear conditions. Here, we present a data‑driven framework to discover the hidden governing equation of sheared granular materials. The framework, PINNSR‑DA, addresses noisy discrete particle data via physics‑informed neural networks with sparse regression (PINNSR) and ensures dimensional consistency via machine learning‑based dimensional analysis (DA). Applying PINNSR‑DA to our discrete element method simulations of oscillatory shear flow, a general differential equation is found to govern the effective friction across steady and transient states. The equation consists of three interpretable terms, accounting respectively for linear response, nonlinear response and energy dissipation of the granular system, and the coefficients depends primarily on a dimensionless relaxation time, which is shorter for stiffer particles and thicker flow layers. This work pioneers a pathway for discovering physically interpretable governing laws in granular systems and can be readily extended to more complex scenarios involving jamming, segregation, and fluid‑particle interactions.

Abstract:
We develop a framework for non‑invasive volumetric indoor airflow estimation from a single viewpoint using background‑oriented schlieren (BOS) measurements and physics‑informed reconstruction. Our framework utilizes a light projector that projects a pattern onto a target back‑wall and a camera that observes small distortions in the light pattern. While the single‑view BOS tomography problem is severely ill‑posed, our proposed framework addresses this using: (1) improved ray tracing, (2) a physics‑based light rendering approach and loss formulation, and (3) a physics‑based regularization using a physics‑informed neural network (PINN) to ensure that the reconstructed airflow is consistent with the governing equations for buoyancy‑driven flows.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a promising machine learning approach for solving partial differential equations (PDEs). However, PINNs face significant challenges in balancing multi‑objective losses, as multiple competing loss terms such as physics residuals, boundary conditions, and initial conditions must be appropriately weighted. While various loss balancing schemes have been proposed, they have been implemented within neural network architectures with fixed activation functions, and their effectiveness has been assessed using simpler PDEs. We hypothesize that the effectiveness of loss balancing schemes depends not only on the balancing strategy itself, but also on the loss function design and the neural network's inherent function approximation capabilities, which are influenced by the choice of activation function. In this paper, we extend existing solutions by incorporating trainable activation functions within the neural network architecture and evaluate the proposed approach on complex fluid flow applications modeled by the Navier‑Stokes equations. Our evaluation across diverse Navier‑Stokes problems demonstrates that this proposed solution achieves root mean square error (RMSE) improvements ranging from 7.4% to 95.2% across different scenarios. These findings highlight the importance of carefully designing the loss function and selecting activation functions for effective loss balancing.

Abstract:
This paper presents eXtended Physics‑Informed Neural Network (X‑PINN), a novel and robust framework for addressing fracture mechanics problems involving multiple cracks in fractured media. To address this, an energy‑based loss function, customized integration schemes, and domain decomposition procedures are proposed. Inspired by the Extended Finite Element Method (XFEM), the neural network solution space is enriched with specialized functions that allow crack body discontinuities and singularities at crack tips to be explicitly captured. Furthermore, a structured framework is introduced in which standard and enriched solution components are modeled using distinct neural networks, enabling flexible and effective simulations of complex multiple‑crack problems in 1D and 2D domains, with convenient extensibility to 3D problems. Numerical experiments are conducted to validate the effectiveness and robustness of the proposed method.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful framework for solving PDEs, yet existing uncertainty quantification (UQ) approaches for PINNs generally lack rigorous statistical guarantees. In this work, we bridge this gap by introducing a distribution‑free conformal prediction (CP) framework for UQ in PINNs. This framework calibrates prediction intervals by constructing nonconformity scores on a calibration set, thereby yielding distribution‑free uncertainty estimates with rigorous finite‑sample coverage guarantees for PINNs. To handle spatial heteroskedasticity, we further introduce local conformal quantile estimation, enabling spatially adaptive uncertainty bands while preserving theoretical guarantee. Through systematic evaluations on typical PDEs (damped harmonic oscillator, Poisson, Allen‑Cahn, and Helmholtz equations) and comprehensive testing across multiple uncertainty metrics, our results demonstrate that the proposed framework achieves reliable calibration and locally adaptive uncertainty intervals, consistently outperforming heuristic UQ approaches. By bridging PINNs with distribution‑free UQ, this work introduces a general framework that not only enhances calibration and reliability, but also opens new avenues for uncertainty‑aware modeling of complex PDE systems.

Abstract:
Accurate localized wireless channel modeling is a cornerstone of cellular network optimization, enabling reliable prediction of network performance during parameter tuning. Localized statistical channel modeling (LSCM) is the state‑of‑the‑art channel modeling framework tailored for cellular network optimization. However, traditional LSCM methods, which infer the channel's Angular Power Spectrum (APS) from Reference Signal Received Power (RSRP) measurements, suffer from critical limitations: they are typically confined to single‑cell, single‑grid and single‑carrier frequency analysis and fail to capture complex cross‑domain interactions. To overcome these challenges, we propose RF‑LSCM, a novel framework that models the channel APS by jointly representing large‑scale signal attenuation and multipath components within a radiance field. RF‑LSCM introduces a multi‑domain LSCM formulation with a physics‑informed frequency‑dependent Attenuation Model (FDAM) to facilitate the cross frequency generalization as well as a point‑cloud‑aided environment enhanced method to enable multi‑cell and multi‑grid channel modeling. Furthermore, to address the computational inefficiency of typical neural radiance fields, RF‑LSCM leverages a low‑rank tensor representation, complemented by a novel Hierarchical Tensor Angular Modeling (HiTAM) algorithm. This efficient design significantly reduces GPU memory requirements and training time while preserving fine‑grained accuracy. Extensive experiments on real‑world multi‑cell datasets demonstrate that RF‑LSCM significantly outperforms state‑of‑the‑art methods, achieving up to a 30% reduction in mean absolute error (MAE) for coverage prediction and a 22% MAE improvement by effectively fusing multi‑frequency data.

Abstract:
Neural operators have emerged as cost‑effective surrogates for expensive fluid‑flow simulators, particularly in computationally intensive tasks such as permeability inversion from time‑lapse seismic data, and uncertainty quantification. In these applications, the fidelity of the surrogate's gradients with respect to system parameters is crucial, as the accuracy of downstream tasks, such as optimization and Bayesian inference, relies directly on the quality of the derivative information. Recent advances in physics‑informed methods have leveraged derivative information to improve surrogate accuracy. However, incorporating explicit Jacobians can become computationally prohibitive, as the complexity typically scales quadratically with the number of input parameters. To address this limitation, we propose DeFINO (Derivative‑based Fisher‑score Informed Neural Operator), a reduced‑order, derivative‑informed training framework. DeFINO integrates Fourier neural operators (FNOs) with a novel derivative‑based training strategy guided by the Fisher Information Matrix (FIM). By projecting Jacobians onto dominant eigen‑directions identified by the FIM, DeFINO captures critical sensitivity information directly informed by observational data, significantly reducing computational expense. We validate DeFINO through synthetic experiments in the context of subsurface multi‑phase fluid‑flow, demonstrating improvements in gradient accuracy while maintaining robust forward predictions of underlying fluid dynamics. These results highlight DeFINO's potential to offer practical, scalable solutions for inversion problems in complex real‑world scenarios, all at substantially reduced computational cost.

Abstract:
Coronagraphic observations enable direct monitoring of coronal mass ejections (CMEs) through scattered light from free electrons, but determining the 3D plasma distribution from 2D imaging data is challenging due to the optically‑thin plasma and the complex image formation processes. We introduce SuNeRF‑CME, a framework for 3D tomographic reconstructions of the heliosphere using multi‑viewpoint coronagraphic observations. The method leverages Neural Radiance Fields (NeRFs) to estimate the electron density in the heliosphere through a ray‑tracing approach, while accounting for the underlying Thomson scattering of image formation. The model is optimized by iteratively fitting the time‑dependent observational data. In addition, we apply physical constraints in terms of continuity, propagation direction, and speed of the heliospheric plasma to overcome limitations imposed by the sparse number of viewpoints. We utilize synthetic observations of a CME simulation to fully quantify the model's performance for different viewpoint configurations. The results demonstrate that our method can reliably estimate the CME parameters from only two viewpoints, with a mean velocity error of 3.01\pm1.94% and propagation direction errors of 3.39\pm1.94^\circ in latitude and 1.76\pm0.79^\circ in longitude. We further show that our approach can achieve a full 3D reconstruction of the simulated CME from two viewpoints, where we correctly model the three‑part structure, deformed CME front, and internal plasma variations. Additional viewpoints can be seamlessly integrated, directly enhancing the reconstruction of the plasma distribution in the heliosphere. This study underscores the value of physics‑informed methods for reconstructing the heliospheric plasma distribution, paving the way for unraveling the dynamic 3D structure of CMEs and enabling advanced space weather monitoring.

Abstract:
Ecological systems exhibit complex multi‑scale dynamics that challenge traditional modeling. New methods must capture temporal oscillations and emergent spatiotemporal patterns while adhering to conservation principles. We present the Unified Spatiotemporal Physics‑Informed Learning (USPIL) framework, a deep learning architecture integrating physics‑informed neural networks (PINNs) and conservation laws to model predator‑prey dynamics across dimensional scales. The framework provides a unified solution for both ordinary (ODE) and partial (PDE) differential equation systems, describing temporal cycles and reaction‑diffusion patterns within a single neural network architecture. Our methodology uses automatic differentiation to enforce physics constraints and adaptive loss weighting to balance data fidelity with physical consistency. Applied to the Lotka‑Volterra system, USPIL achieves 98.9% correlation for 1D temporal dynamics (loss: 0.0219, MAE: 0.0184) and captures complex spiral waves in 2D systems (loss: 4.7656, pattern correlation: 0.94). Validation confirms conservation law adherence within 0.5% and shows a 10‑50x computational speedup for inference compared to numerical solvers. USPIL also enables mechanistic understanding through interpretable physics constraints, facilitating parameter discovery and sensitivity analysis not possible with purely data‑driven methods. Its ability to transition between dimensional formulations opens new avenues for multi‑scale ecological modeling. These capabilities make USPIL a transformative tool for ecological forecasting, conservation planning, and understanding ecosystem resilience, establishing physics‑informed deep learning as a powerful and scientifically rigorous paradigm.

Abstract:
We present VEGA, a vehicle‑adaptive energy‑aware routing system for electric vehicles (EVs) that integrates physics‑informed parameter estimation with RL‑based charge‑aware path planning. VEGA consists of two copupled modules: (1) a physics‑informed neural operator (PINO) that estimates vehicle‑specific physical parameters‑drag, rolling resistance, mass, motor and regenerative‑braking efficiencies, and auxiliary load‑from short windows of onboard speed and acceleration data; (2) a Proximal Policy Optimization (PPO) agent that navigates a charger‑annotated road graph, jointly selecting routes and charging stops under state‑of‑charge constraints. The agent is initialized via behavior cloning from an A teacher and fine‑tuned with cirriculum‑guided PPO on the full U.S. highway network with Tesla Supercharger locations. On a cross‑country San Francisco‑to‑New York route (~4,860km), VEGA produces a feasible 20‑stop plan with 56.12h total trip time and minimum SoC 11.41%. Against the controlled Energy‑aware A baseline, the distance and driving‑time gaps are small (‑8.49km and +0.37h), while inference is >20x faster. The learned policy generalizes without retraining to road networks in France and Japan.

Abstract:
The relaxation of moiré superlattices in twisted bilayers of transition metal dichalcogenides (TMDs) has been modeled using a set of neural‑network‑based approaches. We implemented and compared several architectures, including (i) an interpolator combined with an autoencoder, (ii) an interpolator combined with a decoder, (iii) a direct generator mapping input parameters to displacement fields, and (iv) a physics‑informed neural network (PINN). Among these, the direct generator architecture demonstrated the best performance, achieving machine‑level precision with minimal training data. Remarkably, once trained, this simple fully connected network is able to predict the full displacement field of a moiré bilayer within a fraction of a second, whereas conventional continuum simulations require hours or even days. This finding highlights the low‑dimensional nature of the relaxation process and establishes neural networks as a practical and efficient alternative to ab initio approaches for rapid modeling and high‑throughput screening of 2D twisted heterostructures.

Abstract:
We address a wide spectrum of quantum control strategies, including various open‑loop protocols and advanced adaptive methods. These methodologies apply to few‑qubit scenarios and naturally scale to larger N‑qubit systems. We benchmark them across fundamental quantum tasks: entanglement preservation/generation, and directed quantum transport in a disordered quantum walk. All simulations are performed in a challenging environment featuring non‑Markov colored noise, imperfections, and the Markov Lindblad equation. With a complex task‑dependent performance hierarchy, our deterministic protocols proved highly effective for entanglement generation/preservation, and in specific pulse configurations, they even outperformed the RL‑optimization. In contrast, more advanced methods demonstrate a marked specialization. For entanglement preservation, a physics‑informed hybrid Quantum Error Correction and Dynamical Decoupling (QEC‑DD) protocol provides the most stable and effective solution, outperforming all other approaches. Conversely, for dynamic tasks requiring the discovery of non‑trivial control sequences, such as DD, Floquet engineering, and rapid entanglement generation or coherent transport, the model‑free Reinforcement Learning (RL) agents consistently find superior solutions. We further demonstrate that the control pulse envelope is a non‑trivial factor that actively shapes the control landscape, which determines the difficulty for all protocols and highlights the adaptability of the RL agent. We conclude that no single strategy is universally dominant. A clear picture emerges: the future of high‑fidelity quantum control lies in a synthesis of physics‑informed design, as exemplified by robust hybrid methods, and the specialized, high‑performance optimization power of adaptive machine learning.

Abstract:
Physics‑Informed Neural Networks (PINNs) integrate machine learning with differential equations to solve forward and inverse problems while ensuring that predictions adhere to physical laws. Physiologically based pharmacokinetic (PBPK) modeling advances beyond classical compartmental approaches by employing a mechanistic, physiology‑focused framework. Such models involve many unknown parameters that are difficult to measure directly in humans due to ethical and practical constraints. PBPK models are constructed as systems of ordinary differential equations (ODEs) and these parametric ODEs are often stiff, and traditional numerical and statistical methods frequently fail to converge. In this study, we consider a permeability‑limited, four‑compartment PBPK brain model that mimics human brain functionality in drug delivery. We introduce PBPK‑iPINN, a method for estimating drug‑specific or patient‑specific parameters and drug concentration profiles using inverse PINNs. We also conducted parameter identifiability analysis to determines whether the parameters can be uniquely and reliably estimated from the available data. We demonstrate that, for the inverse problem to converge to the correct solution, the components of the loss function (data loss, initial condition loss, and residual loss) must be appropriately weighted, and the hyperparameters including the number of layers and neurons, activation functions, learning rate, optimizer, and collocation points must be carefully tuned. The performance of the PBPK‑iPINN approach is then compared with established numerical and statistical methods. Accurate parameter estimation yields precise drug concentration‑time profiles, which in turn enable the calculation of pharmacokinetic metrics. These metrics support drug developers and clinicians in designing and optimizing therapies for brain cancer.

Abstract:
Various neural network architectures are used in many of the state‑of‑the‑art approaches for real‑time nonlinear state estimation in dynamical systems. With the ever‑increasing incorporation of these data‑driven models into the estimation domain, models with reliable margins of error are required ‑‑ especially for safety‑critical applications. This paper discusses a novel hybrid, data‑driven state estimation approach based on the physics‑informed attentive neural process (PI‑AttNP), a model‑informed extension of the attentive neural process (AttNP). We augment this estimation approach with the regression‑based split conformal prediction (CP) framework to obtain quantified model uncertainty with probabilistic guarantees. After presenting the algorithm in a generic form, we validate its performance in the task of grey‑box state estimation of a simulated under‑actuated six‑degree‑of‑freedom quadrotor with multimodal Gaussian sensor noise and several external perturbations typical to quadrotors. Further, we compare outcomes with state‑of‑the‑art data‑driven methods, which provide significant evidence of the physics‑informed neural process as a viable novel approach for model‑driven estimation.

Abstract:
This study compares the Boundary Element Method (BEM) and Physics‑Informed Neural Networks (PINNs) for solving the two‑dimensional Helmholtz equation in wave scattering problems. The objective is to evaluate the performance of both methods under the same conditions. We solve the Helmholtz equation using BEM and PINNs for the same scattering problem. PINNs are trained by minimizing the residual of the governing equations and boundary conditions with their configuration determined through hyperparameter optimization, while BEM is applied using boundary discretization. Both methods are evaluated in terms of solution accuracy and computation time. We conducted numerical experiments by varying the number of boundary integration points for the BEM and the number of hidden layers and neurons per layer for the PINNs. We performed a hyperparameter tuning to identify an adequate PINN configuration for this problem as a network with 3 hidden layers and 25 neurons per layer, using a learning rate of 10^‑2 and a sine activation function. At comparable levels of accuracy, the assembly and solution of the BEM system required a computational time on the order of 10^‑2~s, whereas the training time of the PINN was on the order of 10^2~s, corresponding to a difference of approximately four orders of magnitude. However, once trained, the PINN achieved evaluation times on the order of 10^‑2~s, which is about two orders of magnitude faster than the evaluation of the BEM solution at interior points. This work establishes a procedure for comparing BEM and PINNs. It also presents a direct comparison between the two methods for the scattering problem. The analysis provides quantitative data on their performance, supporting their use in future research on wave propagation problems and outlining challenges and directions for further investigation.

Abstract:
A major challenge in deploying world models is the trade‑off between size and performance. Large world models can capture rich physical dynamics but require massive computing resources, making them impractical for edge devices. Small world models are easier to deploy but often struggle to learn accurate physics, leading to poor predictions. We propose the Physics‑Informed BEV World Model (PIWM), a compact model designed to efficiently capture physical interactions in bird's‑eye‑view (BEV) representations. PIWM uses Soft Mask during training to improve dynamic object modeling and future prediction. We also introduce a simple yet effective technique, Warm Start, for inference to enhance prediction quality with a zero‑shot model. Experiments show that at the same parameter scale (400M), PIWM surpasses the baseline by 60.6% in weighted overall score. Moreover, even when compared with the largest baseline model (400M), the smallest PIWM (130M Soft Mask) achieves a 7.4% higher weighted overall score with a 28% faster inference speed.

Abstract:
This work proposes a Variational Physics‑Informed Neural Network (VPINN) framework that integrates the Petrov‑Galerkin formulation with deep neural networks (DNNs) for solving one‑dimensional singularly perturbed boundary value problems (BVPs) and parabolic partial differential equations (PDEs) involving one or two small parameters. The method adopts a nonlinear approximation in which the trial space is defined by neural network functions, while the test space is constructed from hat functions. The weak formulation is constructed using localized test functions, with interface penalty terms introduced to enhance numerical stability and accurately capture boundary layers. Dirichlet boundary conditions are imposed via hard constraints, and source terms are computed using automatic differentiation. Numerical experiments on benchmark problems demonstrate the effectiveness of the proposed method, showing significantly improved accuracy in both the L_2 and maximum norms compared to the standard VPINN approach for one‑dimensional singularly perturbed differential equations (SPDEs).

Abstract:
Non‑invasive glucose monitoring outside controlled settings is dominated by low signal‑to‑noise ratio (SNR): hardware drift, environmental variation, and physiology suppress the glucose signature in NIR signals. We present a noise‑stressed NIR simulator that injects 12‑bit ADC quantisation, LED drift, photodiode dark noise, temperature/humidity variation, contact‑pressure noise, Fitzpatrick I‑VI melanin, and glucose variability to create a low‑correlation regime (rho_glucose‑NIR = 0.21). Using this platform, we benchmark six methods: Enhanced Beer‑Lambert (physics‑engineered ridge regression), Original PINN, Optimised PINN, RTE‑inspired PINN, Selective RTE PINN, and a shallow DNN. The physics‑engineered Beer Lambert model achieves the lowest error (13.6 mg/dL RMSE) with only 56 parameters and 0.01 ms inference, outperforming deeper PINNs and the SDNN baseline under low‑SNR conditions. The study reframes the task as noise suppression under weak signal and shows that carefully engineered physics features can outperform higher‑capacity models in this regime.

Abstract:
This work introduces a physics‑informed neural networks (PINNs)‑based model predictive control (MPC) framework for susceptible‑infected‑recovered (SIR) spreading models. Existing studies in MPC design for epidemic control often assume either 1) measurable states of the dynamics, where the parameters are learned, or 2) known parameters of the model, where the states are learned. In this work, we address the joint real‑time estimation of states and parameters within the MPC framework using only noisy infected states, under the assumption that 1) only the recovery rate is known, or 2) only the basic reproduction number is known. Under the first assumption, we propose MPC‑PINNs and two novel PINNs algorithms, all of which are integrated into the MPC framework. First, we introduce MPC‑PINNs, which are designed for SIR models with control. We then propose log‑scaled PINNs (MPC‑LS‑PINNs), which incorporate a log‑scaled loss function to improve robustness against noise. Next, we present split‑integral PINNs (MPC‑SI‑PINNs), which leverage integral operators and state coupling in the neural network training process to effectively reconstruct the complete epidemic state information. Building upon these methods, we further extend our framework for the second assumption. We establish the necessary conditions and extend our PINNs algorithms, where MPC‑SI‑PINNs are simplified as split‑PINNs (MPC‑S‑PINNs). By incorporating these algorithms into the MPC framework, we simultaneously estimate the epidemic states and parameters while generating optimal control strategies. Experiment results demonstrate the effectiveness of the proposed methods under different settings.

Abstract:
Characterizing the environmental interactions of quantum systems is a critical bottleneck in the development of robust quantum technologies. Traditional tomographic methods are often data‑intensive and struggle with scalability. In this work, we introduce a novel framework for performing Lindblad tomography using Physics‑Informed Neural Networks (PINNs). By embedding the Lindblad master equation directly into the neural network's loss function, our approach simultaneously learns the quantum state's evolution and infers the underlying dissipation parameters from sparse, time‑series measurement data. Our results show that PINNs can reconstruct both the system dynamics and the functional form of unknown noise parameters, presenting a sample‑efficient and scalable solution for quantum device characterization. Ultimately, our method produces a fully‑differentiable digital twin of a noisy quantum system by learning its governing master equation.

Abstract:
We present a new extraction of unpolarized Dihadron Fragmentation Functions, which describe the probability density for an unpolarized parton to fragment into a π^+ π^‑ pair. Our analysis is based on data from the BELLE collaboration. We improve on previous determinations in several key aspects: we employ state‑of‑the‑art perturbative QCD calculations up to next‑to‑next‑to‑leading order (NNLO); we limit the use of Monte Carlo event generators to estimating the relative contributions of different flavors, a necessary input due to the limited flavor sensitivity of the available data; and, in addition to a traditional fit based on a physics‑informed functional form, we explore a Neural Network parametrization. This latter approach paves the way for more robust and flexible determinations of Dihadron Fragmentation Functions using machine learning techniques.

Abstract:
Magnetic resonance imaging (MRI) is the gold standard imaging modality for numerous diagnostic tasks, yet its usefulness is tempered due to its high cost and infrastructural requirements. Low‑cost very‑low‑field portable scanners offer new opportunities, while enabling imaging outside conventional MRI suites. However, achieving diagnostic‑quality images in clinically acceptable scan times remains challenging with these systems. Therefore methods for improving the image quality while reducing the scan duration are highly desirable. Here, we investigate a physics‑informed 3D deep unrolled network for the reconstruction of portable MR acquisitions. Our approach includes a novel network architecture that utilizes momentum‑based acceleration and leverages complex conjugate symmetry of k‑space for improved reconstruction performance. Comprehensive evaluations on emulated datasets as well as 47mT portable MRI acquisitions demonstrate the improved reconstruction quality of the proposed method compared to existing methods.

Abstract:
It was recently shown that the loss function used for training physics‑informed neural networks (PINNs) exhibits local minima at solutions corresponding to fixed points of dynamical systems. In the forward setting, where the PINN is trained to solve initial value problems, these local minima can interfere with training and potentially leading to physically incorrect solutions. Building on stability theory, this paper proposes a regularization scheme that penalizes solutions corresponding to unstable fixed points. Experimental results on four dynamical systems, including the Lotka‑Volterra model and the van der Pol oscillator, show that our scheme helps avoiding physically incorrect solutions and substantially improves the training success rate of PINNs.

Abstract:
Developing physically consistent closure models is a longstanding challenge in simulating plasma turbulence, even in minimal systems such as the two‑field Hasegawa‑Wakatani (HW) model, which captures essential features of drift‑wave turbulence with a reduced set of variables. In this work, we leverage theoretical insights from Direct Interaction Approximation (DIA) to construct a six‑term closure structure that captures the dominant turbulent transport processes, including both diffusion and hyper‑diffusion. While the mathematical form of the closure is fully prescribed by DIA, the corresponding transport coefficients are learned from data using physics‑informed neural networks (PINNs). The resulting Extended HW model with Closure (EHW‑C) model reveals several nontrivial features of plasma turbulence: notably, some inferred coefficients become negative in certain regimes, indicating inverse transport, a phenomenon absent in conventional closure models. Moreover, the EHW‑C model accurately reproduces the spectral and flux characteristics of high‑resolution Direct Numerical Simulations (DNS), while requiring only one‑eighth the spatial resolution per direction, yielding a tenfold speed‑up. This work demonstrates how theory‑guided machine learning can both enhance computational efficiency and uncover emergent transport mechanisms in strongly nonlinear plasma systems.

Abstract:
We introduce PINGS (Physics‑Informed Neural Network for Fast Generative Sampling), a framework that amortizes diffusion sampling by training a physics‑informed network to approximate reverse‑time probability‑flow dynamics, reducing sampling to a single forward pass (NFE = 1). As a proof of concept, we learn a direct map from a 3D standard normal to a non‑Gaussian Gaussian Mixture Model (GMM). PINGS preserves the target's distributional structure (multi‑bandwidth kernel MMD^2 = 1.88 × 10^‑2 with small errors in mean, covariance, skewness, and excess kurtosis) and achieves constant‑time generation: 10^4 samples in 16.54 \pm 0.56 millisecond on an RTX 3090, versus 468‑843 millisecond for DPM‑Solver (10/20) and 960 millisecond for DDIM (50) under matched conditions. We also sanity‑check the PINN/automatic‑differentiation pipeline on a damped harmonic oscillator, obtaining MSEs down to \mathcalO(10^‑5). Compared to fast but iterative ODE solvers and direct‑map families (Flow, Rectified‑Flow, Consistency), PINGS frames generative sampling as a PINN‑style residual problem with endpoint anchoring, yielding a white‑box, differentiable map with NFE = 1. These proof‑of‑concept results position PINGS as a promising route to fast, function‑based generative sampling with potential extensions to scientific simulation (e.g., fast calorimetry).

Abstract:
In this article, we employ Chien‑Physics Informed Neural Networks (C‑PINNs) to obtain solutions for singularly perturbed convection‑diffusion equations, reaction‑diffusion equations, and their coupled forms in both one and two‑dimensional settings. While PINNs have emerged as a powerful tool for solving various types of differential equations, their application to singular perturbation problems (SPPs) presents significant challenges. These challenges arise because a small perturbation parameter multiplies the highest‑order derivatives, leading to sharp gradient changes near the boundary layer. To overcome these difficulties, we apply C‑PINNs, a modified version of the standard PINNs framework, which is specifically designed to address singular perturbation problems. Our study shows that C‑PINNs provide a more accurate solution for SPPs, demonstrating better performance than conventional methods.

Abstract:
The thermal diffusivity measurement of impacted composites using pulsed methods presents an ill‑posed inverse problem influenced by multiple factors such as sample thickness, cooling duration, and excitation energy. In this study, a novel excitation method, evaporative cryocooling, was introduced for measuring the thermal diffusivity of tested samples. Compared to conventional excitation modalities, evaporative cryocooling excitation is compact, portable, and low cost. However, evaporative cryocooling cannot be considered a pulsed method due to its prolonged excitation duration. In general, it is difficult to measure thermal diffusivity based on non‑impulsive pulsed excitation at times commensurate with the pulse duration, often due to ill‑defined pulse shape and width and the subsequent potentially complicated thermal response which may be subject to diffusive broadening. To address this challenge, inverse physics‑informed neural networks (IPINNs) were introduced in this work and integrated with an evaporative cryocooling method. The Parker method combined with a photothermal method was employed as a reference. To improve the accuracy of both IPINNs and Parker methods, terahertz time‑domain spectroscopy (THz‑TDS) was employed for measuring the thickness of impacted composites. Simulations and experimental results demonstrated the feasibility and accuracy of the IPINN‑based approach.

Abstract:
We develop a flexible framework based on physics‑informed neural networks (PINNs) for solving boundary value problems involving minimal surfaces in curved spacetimes, with a particular emphasis on singularities and moving boundaries. By encoding the underlying physical laws into the loss function and designing network architectures that incorporate the singular behavior and dynamic boundaries, our approach enables robust and accurate solutions to both ordinary and partial differential equations with complex boundary conditions. We demonstrate the versatility of this framework through applications to minimal surface problems in anti‑de Sitter (AdS) spacetime, including examples relevant to the AdS/CFT correspondence (e.g. Wilson loops and gluon scattering amplitudes) popularly used in the context of string theory in theoretical physics. Our methods efficiently handle singularities at boundaries, and also support both "soft" (loss‑based) and "hard" (formulation‑based) imposition of boundary conditions, including cases where the position of a boundary is promoted to a trainable parameter. The techniques developed here are not limited to high‑energy theoretical physics but are broadly applicable to boundary value problems encountered in mathematics, engineering, and the natural sciences, wherever singularities and moving boundaries play a critical role.

Abstract:
Accurate and fast thermophysical models are needed to embed vapor‑liquid equilibrium (VLE) calculations in design, optimization, and control loops for cryogenic mixtures. This study asks whether a structure‑aware graph neural network (GNN; DimeNet++) trained on GERG‑2008/CoolProp data can act as a practical surrogate for an equation of state (EoS). We generate a ternary dataset over 90‑200 K and pressures to 100 bar, curate it with a 15% density filter (reducing 5,200 states to 1,516), and pair each state with a lightweight molecular‑dynamics snapshot to supply structural features. The model is trained in two stages; pretraining on residual Helmholtz energy followed by pressure fine‑tuning with a stability penalty; and evaluated via single‑phase interpolation tests, solver‑free derivative‑quality diagnostics, an audited VLE driver, and a latency benchmark. Within its regime, the GNN interpolates single‑phase properties reasonably well; however, the VLE driver accepts no GNN equilibria on tested binaries (all plotted VLE points are CoolProp fallback or the solver fails), and diagnostic probes reveal jagged P(V|T) paths and thermal‑stability flags concentrated in dense/cold regions, indicating insufficient derivative smoothness/consistency for robust equilibrium solving. An end‑to‑end timing comparison shows no single‑phase speed advantage relative to CoolProp (tens of milliseconds vs sub‑millisecond). We conclude that, as configured, the surrogate in this study is not solver‑ready for VLE and offers no runtime benefit; its value is methodological, delineating failure modes and pointing to remedies such as physics‑informed training signals and targeted coverage near phase boundaries.

Abstract:
We present a machine learning framework for adaptive source localization in which agents use a structure‑preserving digital twin of a coupled hydrodynamic‑transport system for real‑time trajectory planning and data assimilation. The twin is constructed with conditional neural Whitney forms (CNWF), coupling the numerical guarantees of finite element exterior calculus (FEEC) with transformer‑based operator learning. The resulting model preserves discrete conservation, and adapts in real time to streaming sensor data. It employs a conditional attention mechanism to identify: a reduced Whitney‑form basis; reduced integral balance equations; and a source field, each compatible with given sensor measurements. The induced reduced‑order environmental model retains the stability and consistency of standard finite‑element simulation, yielding a physically realizable, regular mapping from sensor data to the source field. We propose a staggered scheme that alternates between evaluating the digital twin and applying Lloyd's algorithm to guide sensor placement, with analysis providing conditions for monotone improvement of a coverage functional. Using the predicted source field as an importance function within an optimal‑recovery scheme, we demonstrate recovery of point sources under continuity assumptions, highlighting the role of regularity as a sufficient condition for localization. Experimental comparisons with physics‑agnostic transformer architectures show improved accuracy in complex geometries when physical constraints are enforced, indicating that structure preservation provides an effective inductive bias for source identification.

Abstract:
Recent work in scientific machine learning aims to tackle scientific tasks directly by predicting target values with neural networks (e.g., physics‑informed neural networks, neural ODEs, neural operators, etc.), but attaining high accuracy and robustness has been challenging. We explore an alternative view: use LLMs to write code that leverages decades of numerical algorithms. This shifts the burden from learning a solution function to making domain‑aware numerical choices. We ask whether LLMs can act as SciML agents that, given a natural‑language ODE description, generate runnable code that is scientifically appropriate, selecting suitable solvers (stiff vs. non‑stiff), and enforcing stability checks. There is currently no benchmark to measure this kind of capability for scientific computing tasks. As such, we first introduce two new datasets: a diagnostic dataset of adversarial "misleading" problems; and a large‑scale benchmark of 1,000 diverse ODE tasks. The diagnostic set contains problems whose superficial appearance suggests stiffness, and that require algebraic simplification to demonstrate non‑stiffness; and the large‑scale benchmark spans stiff and non‑stiff ODE regimes. We evaluate open‑ and closed‑source LLM models along two axes: (i) unguided versus guided prompting with domain‑specific knowledge; and (ii) off‑the‑shelf versus fine‑tuned variants. Our evaluation measures both executability and numerical validity against reference solutions. We find that with sufficient context and guided prompts, newer instruction‑following models achieve high accuracy on both criteria. In many cases, recent open‑source systems perform strongly without fine‑tuning, while older or smaller models still benefit from fine‑tuning. Overall, our preliminary results indicate that careful prompting and fine‑tuning can yield a specialized LLM agent capable of reliably solving simple ODE problems.

Abstract:
We propose a novel data‑lean operator learning algorithm, the Reduced Basis Neural Operator (ReBaNO), to solve a group of PDEs with multiple distinct inputs. Inspired by the Reduced Basis Method and the recently introduced Generative Pre‑Trained Physics‑Informed Neural Networks, ReBaNO relies on a mathematically rigorous greedy algorithm to build its network structure offline adaptively from the ground up. Knowledge distillation via task‑specific activation function allows ReBaNO to have a compact architecture requiring minimal computational cost online while embedding physics. In comparison to state‑of‑the‑art operator learning algorithms such as PCA‑Net, DeepONet, FNO, and CNO, numerical results demonstrate that ReBaNO significantly outperforms them in terms of eliminating/shrinking the generalization gap for both in‑ and out‑of‑distribution tests and being the only operator learning algorithm achieving strict discretization invariance.

Abstract:
A comprehensive experimental and physics informed neural network numerical inverse diffusion analysis is conducted in technologically important NiAlTi ternary and NiCoFeAlTi quinary solid solutions for estimating and extracting composition dependent diffusion coefficients. A systematic variation of tracer, intrinsic and interdiffusion coefficients with composition could be estimated in the ternary solid solution. Following, the possibility of producing Al Ti constant PB diffusion profiles keeping constant Ni, Co, Fe in the quinary system is demonstrated. The estimation of diffusion coefficients of all the elements at the Kirkendall marker plane of a single diffusion couple profile is elaborated. PINN optimisation parameters are established using self and impurity diffusion coefficients in Ni and tracer diffusion coefficients at the Kirkendall marker plane. The reliability of optimized parameters is validated by comparing with the interdiffusion coefficients estimated from binary NiTi, NiAl and PB diffusion profiles, indicating extendibility to even lower order systems.

Abstract:
Low‑light Object detection is crucial for many real‑world applications but remains challenging due to degraded image quality. While recent studies have shown that RAW images offer superior potential over RGB images, existing approaches either use RAW‑RGB images with information loss or employ complex frameworks. To address these, we propose a lightweight and self‑adaptive Image Signal Processing (ISP) plugin, Dark‑ISP, which directly processes Bayer RAW images in dark environments, enabling seamless end‑to‑end training for object detection. Our key innovations are: (1) We deconstruct conventional ISP pipelines into sequential linear (sensor calibration) and nonlinear (tone mapping) sub‑modules, recasting them as differentiable components optimized through task‑driven losses. Each module is equipped with content‑aware adaptability and physics‑informed priors, enabling automatic RAW‑to‑RGB conversion aligned with detection objectives. (2) By exploiting the ISP pipeline's intrinsic cascade structure, we devise a Self‑Boost mechanism that facilitates cooperation between sub‑modules. Through extensive experiments on three RAW image datasets, we demonstrate that our method outperforms state‑of‑the‑art RGB‑ and RAW‑based detection approaches, achieving superior results with minimal parameters in challenging low‑light environments.

Abstract:
Existing reinforcement learning (RL) methods struggle with complex dynamical systems that demand interactions at high frequencies or irregular time intervals. Continuous‑time RL (CTRL) has emerged as a promising alternative by replacing discrete‑time Bellman recursion with differential value functions defined as viscosity solutions of the Hamilton‑‑Jacobi‑‑Bellman (HJB) equation. While CTRL has shown promise, its applications have been largely limited to the single‑agent domain. This limitation stems from two key challenges: (i) conventional solution methods for HJB equations suffer from the curse of dimensionality (CoD), making them intractable in high‑dimensional systems; and (ii) even with HJB‑based learning approaches, accurately approximating centralized value functions in multi‑agent settings remains difficult, which in turn destabilizes policy training. In this paper, we propose a CT‑MARL framework that uses physics‑informed neural networks (PINNs) to approximate HJB‑based value functions at scale. To ensure the value is consistent with its differential structure, we align value learning with value‑gradient learning by introducing a Value Gradient Iteration (VGI) module that iteratively refines value gradients along trajectories. This improves gradient fidelity, in turn yielding more accurate values and stronger policy learning. We evaluate our method using continuous‑time variants of standard benchmarks, including multi‑agent particle environment (MPE) and multi‑agent MuJoCo. Our results demonstrate that our approach consistently outperforms existing continuous‑time RL baselines and scales to complex multi‑agent dynamics.

Abstract:
Full waveform inversion (FWI) is crucial for reconstructing high‑resolution subsurface models, but it is often hindered, considering the limited data, by its null space resulting in low‑resolution models, and more importantly, by its computational cost, especially if needed for real‑time applications. Recent attempts to accelerate FWI using learned wavefield neural operators have shown promise in efficiency and differentiability, but typically suffer from noisy and unstable inversion performance. To address these limitations, we introduce a novel physics‑informed FWI framework to enhance the inversion in accuracy while maintaining the efficiency of neural operator‑based FWI. Instead of relying only on the L2 norm objective function via automatic differentiation, resulting in noisy model reconstruction, we integrate a physics constraint term in the loss function of FWI, improving the quality of the inverted velocity models. Specifically, starting with an initial model to simulate wavefields and then evaluating the loss over how much the resulting wavefield obeys the physical laws (wave equation) and matches the recorded data, we achieve a reduction in noise and artifacts. Numerical experiments using the OpenFWI and Overthrust models demonstrate our method's superior performance, offering cleaner and more accurate subsurface velocity than vanilla approaches. Considering the efficiency of the approach compared to FWI, this advancement represents a significant step forward in the practical application of FWI for real‑time subsurface monitoring.

Abstract:
The contractile motion of the heart is strongly determined by the distribution of the fibers that constitute cardiac tissue. Strain analysis informed with the orientation of fibers allows to describe several pathologies that are typically associated with impaired mechanics of the myocardium, such as cardiovascular disease. Several methods have been developed to estimate strain‑derived metrics from traditional imaging techniques. However, the physical models underlying these methods do not include fiber mechanics, restricting their capacity to accurately explain cardiac function. In this work, we introduce WarpPINN‑fibers, a physics‑informed neural network framework to accurately obtain cardiac motion and strains enhanced by fiber information. We train our neural network to satisfy a hyper‑elastic model and promote fiber contraction with the goal to predict the deformation field of the heart from cine magnetic resonance images. For this purpose, we build a loss function composed of three terms: a data‑similarity loss between the reference and the warped template images, a regularizer enforcing near‑incompressibility of cardiac tissue and a fiber‑stretch penalization that controls strain in the direction of synthetically produced fibers. We show that our neural network improves the former WarpPINN model and effectively controls fiber stretch in a synthetic phantom experiment. Then, we demonstrate that WarpPINN‑fibers outperforms alternative methodologies in landmark‑tracking and strain curve prediction for a cine‑MRI benchmark with a cohort of 15 healthy volunteers. We expect that our method will enable a more precise quantification of cardiac strains through accurate deformation fields that are consistent with fiber physiology, without requiring imaging techniques more sophisticated than MRI.

Abstract:
Inverse microstructure design plays a central role in materials discovery, yet remains challenging due to the complexity of structure‑property linkages and the scarcity of labeled training data. We propose Design‑GenNO, a physics‑informed generative neural operator framework that unifies generative modeling with operator learning to address these challenges. In Design‑GenNO, microstructures are encoded into a low‑dimensional, well‑structured latent space, which serves as the generator for both reconstructing microstructures and predicting solution fields of governing PDEs. MultiONet‑based decoders enable functional mappings from latent variables to both microstructures and full PDE solution fields, allowing a multitude of design objectives to be addressed without retraining. A normalizing flow prior regularizes the latent space, facilitating efficient sampling and robust gradient‑based optimization. A distinctive feature of the framework is its physics‑informed training strategy: by embedding PDE residuals directly into the learning objective, Design‑GenNO significantly reduces reliance on labeled datasets and can even operate in a self‑supervised setting. We validate the method on a suite of inverse design tasks in two‑phase materials, including effective property matching, recovery of microstructures from sparse field measurements, and maximization of conductivity ratios. Across all tasks, Design‑GenNO achieves high accuracy, generates diverse and physically meaningful designs, and consistently outperforms the state‑of‑the‑art method. Moreover, it demonstrates strong extrapolation by producing microstructures with effective properties beyond the training distribution. These results establish Design‑GenNO as a robust and general framework for physics‑informed inverse design, offering a promising pathway toward accelerated materials discovery.

Abstract:
Particle‑induced wear is a critical concern in bulk material handling systems, where abrasive interactions accelerate equipment degradation, increase maintenance needs, and raise operational costs. The Discrete Element Method (DEM) and Archard's wear model are widely adopted for predicting particle‑surface wear processes. However, DEM is computationally prohibitive for real‑time design and predictive maintenance, often requiring hours to days for a single parametric analysis. We propose a DEM‑machine learning (ML) framework to address this limitation that combines physics‑based simulations with data‑driven efficiency. A dataset of 200 DEM simulations is generated by systematically varying particle size, material, and contacting plate geometric parameters. A few ML models ‑‑ linear regression, Lasso and Ridge regularization, decision trees, and a genetic algorithm‑optimized artificial neural network (GA‑ANN) ‑‑ were trained and evaluated. Feature selection revealed that Archard's wear constant, particle size, plate angle, and impingement velocity are the dominant predictors of wear. While linear models offered interpretability, their accuracy was limited. The GA‑ANN achieved the highest performance (R^2 = 0.91), effectively capturing nonlinear wear dynamics while reducing computational cost by orders of magnitude. This study demonstrates that physics‑informed ML provides a scalable pathway for accurate, real‑time wear prediction, enabling predictive maintenance and optimized design in bulk material handling industries.

Abstract:
The geodesy of irregularly shaped small bodies presents fundamental challenges for gravitational field modeling, particularly as deep space exploration missions increasingly target asteroids and comets. Traditional approaches suffer from critical limitations: spherical harmonics diverge within the Brillouin sphere where spacecraft typically operate, polyhedral models assume unrealistic homogeneous density distributions, and existing machine learning methods like GeodesyNets and Physics‑Informed Neural Networks (PINN‑GM) require extensive computational resources and training time. This work introduces MasconCubes, a novel self‑supervised learning approach that formulates gravity inversion as a direct optimization problem over a regular 3D grid of point masses (mascons). Unlike implicit neural representations, MasconCubes explicitly model mass distributions while leveraging known asteroid shape information to constrain the solution space. Comprehensive evaluation on diverse asteroid models including Bennu, Eros, Itokawa, and synthetic planetesimals demonstrates that MasconCubes achieve superior performance across multiple metrics. Most notably, MasconCubes demonstrate computational efficiency advantages with training times approximately 40 times faster than GeodesyNets while maintaining physical interpretability through explicit mass distributions. These results establish MasconCubes as a promising approach for mission‑critical gravitational modeling applications requiring high accuracy, computational efficiency, and physical insight into internal mass distributions of irregular celestial bodies.

Abstract:
Achieving clean combustion systems is crucial in terms of solving environmental impacts, decarbonization needs and sustainability matters. Traditional combustion modeling techniques via computational fluid dynamics with accurate chemical kinetics face obstacles in computational cost and accurate representation of turbulence‑chemistry interactions. Physically Informed Neural Networks (PINNs) as a new framework, merges physical laws with data‑driven learning and shows great potential as an alternative methodology. By directly integrating conservation equations into their training process, PINNs achieve accurate mesh‑free modeling of complex combustion phenomena despite having limited data sets. This review examines how this approach applies to clean combustion systems while focusing on their impact in aerospace applications including flame dynamics, turbulent combustion, emission prediction, and instability management in propulsion systems. Next‑generation aerospace engines rely on PINNs to reduce computational costs while increasing predictive performance and enabling real‑time control methods. This analysis concludes by exploring current barriers and future paths, while demonstrating how PINNs can revolutionize sustainable and efficient combustion technologies in aerospace propulsion systems.

Abstract:
The accurate modeling of semiconductor devices plays a critical role in the development of new technology nodes and next‑generation devices. Semiconductor device designers largely rely on advanced simulation software to solve the drift‑diffusion equations, a coupled system of nonlinear partial differential equations that describe carrier transport in semiconductor devices. While these tools perform well for forward modeling, they are not suitable to address inverse problems, for example, determining doping profiles, material, and geometrical parameters given a desired device performance. Meanwhile, physics‑informed neural networks (PINNs) have grown in popularity in recent years thanks to their ability to efficiently and accurately solve inverse problems at minimal computational cost compared to forward problems. In this study, we introduce the Drift‑Diffusion Network (DDNet), a unified physics‑informed deep learning solver for the forward and inverse mesh‑free solutions of the drift‑diffusion equations of semiconductor device modeling. Using prototypical device configurations in one‑ and two spatial dimensions, we show that DDNet achieves low absolute and relative error compared to traditional simulation software while additionally solving user‑defined inverse problems with minimal computational overhead. We expect that DDNet will benefit semiconductor device modeling by facilitating exploration and discovery of novel device structures across comprehensive parameter sets in a fully automated way.

Abstract:
We present the Physics‑Informed Low‑Rank Neural Operator (PILNO), a neural operator framework for efficiently approximating solution operators of partial differential equations (PDEs) on point cloud data. PILNO combines low‑rank kernel approximations with an encoder‑‑decoder architecture, enabling fast, continuous one‑shot predictions while remaining independent of specific discretizations. The model is trained using a physics‑informed penalty framework, ensuring that PDE constraints and boundary conditions are satisfied in both supervised and unsupervised settings. We demonstrate its effectiveness on diverse problems, including function fitting, the Poisson equation, the screened Poisson equation with variable coefficients, and parameterized Darcy flow. The low‑rank structure provides computational efficiency in high‑dimensional parameter spaces, establishing PILNO as a scalable and flexible surrogate modeling tool for PDEs.

Abstract:
This paper presents a kernel‑based framework for physics‑informed nonlinear system identification. The key contribution is a structured methodology that extends kernel‑based techniques to seamlessly embed partially known physics‑based models, improving parameter estimation and overall model accuracy. The proposed method enhances traditional modeling approaches by embedding a parametric model, which provides physical interpretability, with a kernel‑based function, which accounts for unmodeled dynamics. The two models' components are identified from the data simultaneously, thereby minimizing a suitable cost that balances the relative importance of the physical and the black‑box parts of the model. Additionally, nonlinear state smoothing is employed to address scenarios involving state‑space models with not fully measurable states. Numerical simulations on an experimental benchmark system demonstrate the effectiveness of the proposed approach, achieving up to 51% reduction in simulation root mean square error compared to physics‑only models and 31% performance improvement over state‑of‑the‑art identification techniques.

Abstract:
This paper presents an innovative Transformer‑based deep learning strategy for optimizing the placement of sensors aiming at structural health monitoring of semiconductor probe cards. Failures in probe cards, including substrate cracks and loosened screws, would critically affect semiconductor manufacturing yield and reliability. Some failure modes could be detected by equipping a probe card with adequate sensors. Frequency response functions from simulated failure scenarios are adopted within a finite element model of a probe card. A comprehensive dataset, enriched by physics‑informed scenario expansion and physics‑aware statistical data augmentation, is exploited to train a hybrid Convolutional Neural Network and Transformer model. The model achieves high accuracy (99.83%) in classifying the probe card health states (baseline, loose screw, crack) and an excellent crack detection recall (99.73%). Model robustness is confirmed through a rigorous framework of 3 repetitions of 10‑fold stratified cross‑validation. The attention mechanism also pinpoints critical sensor locations: an analysis of the attention weights offers actionable insights for designing efficient, cost‑effective monitoring systems by optimizing sensor configurations. This research highlights the capability of attention‑based deep learning to advance proactive maintenance, enhancing operational reliability and yield in semiconductor manufacturing.

Abstract:
Physics‑informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs) relevant to multiscale modeling, but they often fail when applied to materials with discontinuous coefficients, such as media with piecewise constant properties. This paper introduces a dual formulation for the PINN framework to improve the reliability of the homogenization of periodic thermo‑conductive composites, for both strong and variational (weak) formulations. The dual approach facilitates the derivation of guaranteed upper and lower error bounds, enabling more robust detection of PINN failure. We compare standard PINNs applied to smoothed material approximations with variational PINNs (VPINNs) using both spectral and neural network‑based test functions. Our results indicate that while strong‑form PINNs may outperform VPINNs in controlled settings, they are sensitive to material discontinuities and may fail without clear diagnostics. In contrast, VPINNs accommodate piecewise constant material parameters directly but require careful selection of test functions to avoid instability. Dual formulation serves as a reliable indicator of convergence quality, and its integration into PINN frameworks enhances their applicability to homogenization problems in micromechanics.

Abstract:
Solving inverse problems with Physics‑Informed Neural Networks (PINNs) is computationally expensive for multi‑query scenarios, as each new set of observed data requires a new, expensive training procedure. We present Inverse‑Parameter Basis PINNs (IP‑Basis PINNs), a meta‑learning framework that extends the foundational work of Desai et al. (2022) to enable rapid and efficient inference for inverse problems. Our method employs an offline‑online decomposition: a deep network is first trained offline to produce a rich set of basis functions that span the solution space of a parametric differential equation. For each new inverse problem online, this network is frozen, and solutions and parameters are inferred by training only a lightweight linear output layer against observed data. Key innovations that make our approach effective for inverse problems include: (1) a novel online loss formulation for simultaneous solution reconstruction and parameter identification, (2) a significant reduction in computational overhead via forward‑mode automatic differentiation for PDE loss evaluation, and (3) a non‑trivial validation and early‑stopping mechanism for robust offline training. We demonstrate the efficacy of IP‑Basis PINNs on three diverse benchmarks, including an extension to universal PINNs for unknown functional terms‑showing consistent performance across constant and functional parameter estimation, a significant speedup per query over standard PINNs, and robust operation with scarce and noisy data.

Abstract:
Terahertz time‑domain spectroscopy (THz‑TDS) is a powerful tool for extracting optical and electrical parameters, as well as probing internal structures and surface morphology of materials. However, conventional simulation techniques, such as the finite element method (FEM) and finite‑difference time‑domain (FDTD), face challenges in accurately modeling THz wave propagation. These difficulties arise from the broad operating bandwidth of THz‑TDS (~0.1‑10 THz), which demands extremely high temporal resolution to capture pulse dynamics and fine spatial resolution to resolve microstructures and interface effects. To address these limitations, we introduce physics‑informed neural networks (PINNs) into THz‑TDS modeling for the first time. Through a comprehensive analysis of forward problems in the time domain, we demonstrate the feasibility and potential of PINNs as a powerful framework for advancing THz wave simulation and analysis.

Abstract:
Offline Goal‑Conditioned Reinforcement Learning (GCRL) holds great promise for domains such as autonomous navigation and locomotion, where collecting interactive data is costly and unsafe. However, it remains challenging in practice due to the need to learn from datasets with limited coverage of the state‑action space and to generalize across long‑horizon tasks. To improve on these challenges, we propose a \emphPhysics‑informed (Pi) regularized loss for value learning, derived from the Eikonal Partial Differential Equation (PDE) and which induces a geometric inductive bias in the learned value function. Unlike generic gradient penalties that are primarily used to stabilize training, our formulation is grounded in continuous‑time optimal control and encourages value functions to align with cost‑to‑go structures. The proposed regularizer is broadly compatible with temporal‑difference‑based value learning and can be integrated into existing Offline GCRL algorithms. When combined with Hierarchical Implicit Q‑Learning (HIQL), the resulting method, Eikonal‑regularized HIQL (Eik‑HIQL), yields significant improvements in both performance and generalization, with pronounced gains in stitching regimes and large‑scale navigation tasks.

Abstract:
Capturing the dynamics of active particles, i.e., small self‑propelled agents that both deform and are deformed by a fluid in which they move is a formidable problem as it requires coupling fine scale hydrodynamics with large scale collective effects. So we present a multi‑scale framework that combines the three learning‑driven tools to learn in concert within one pipeline. We use high‑resolution Lattice Boltzmann snapshots of fluid velocity and particle stresses in a periodic box as input to the learning pipeline. the second step takes the morphology and positions orientations of particles to predict pairwise interaction forces between them with a E(2)‑equivariant graph neural network that necessarily respect flat symmetries. Then, a physics‑informed neural network further updates these local estimates by summing over them with a stress data using Fourier feature mappings and residual blocks that is additionally regularized with a topological term (introduced by persistent homology) to penalize unrealistically tangled or spurious connections. In concert, these stages deliver an holistic highly‑data driven full force network prediction empathizing on the physical underpinnings together with emerging multi‑scale structure typical for active matter.

Abstract:
Rapid and accurate body weight estimation is critical in emergency medical care, as it directly influences treatment decisions, such as drug dosing, defibrillation energy selection, and fluid resuscitation. Traditional methods such as stand‑on scales, length‑based tapes, or transfer‑based weighing scales are often impractical for immobilized patients, inaccurate, or labor‑intensive and time‑consuming. This paper introduces MelodyBedScale, a non‑intrusive and rapid on‑bed weight estimation system that leverages bed vibration induced by music. The core insight is that body weight affects the vibration transfer function of the bed‑body system, which is captured using vibration sensors placed on opposite sides of the bed. First, we identify weight‑sensitive frequency bands and compose clinically acceptable soft, natural music with high signal energy in these frequency bands. This music is then played through a speaker mounted on the bed to induce bed vibrations. Additionally, to efficiently capture the complex weight‑vibration relationship with limited data and enhance generalizability to unseen individuals and weights, we theoretically analyze the weight‑vibration relationship and integrate the results into the activation functions of the neural network for physics‑informed weight regression. We evaluated MelodyBedScale on both wooden and steel beds across 11 participants, achieving a mean absolute error of up to 1.55 kg.

Abstract:
Neural networks have been demonstrated to be able to accelerate the modeling and inverse design of optical and electromagnetic devices by serving as fast surrogates for electromagnetic solvers. Nevertheless, such neural networks can be unreliable and normally require extreme amounts of data to train. Here it is shown that these limitations can be alleviated by constraining neural‑network models using prior knowledge about the governing physics. We propose a universal physics‑informed neural network framework for electromagnetic scattering based on the quasinormal mode expansion of the scattering matrix. The neural networks learn the resonant structure underlying the scattering spectrum, are guaranteed to obey energy conservation and causality, and are shown to have significantly improved data efficiency for photonic‑crystal slabs and all‑dielectric free‑form metasurfaces. Furthermore, the framework allows additional problem‑specific constraints, such as losslessness, symmetries, and number of modes, to be imposed manually when they are available. The method can be applied to a wide range of optical and electromagnetic devices owing to the generality of the quasinormal mode formalism.

Abstract:
A surrogate model is developed to predict the convective heat transfer coefficient of liquid sodium (Na) flow within rectangular miniature heat sinks. Initially, kernel‑based machine learning techniques and shallow neural network are applied to a dataset with 87 Nusselt numbers for liquid sodium in rectangular miniature heat sinks. Subsequently, a self‑supervised physics‑informed neural network and transfer learning approach are used to increase the estimation performance. In the self‑supervised physics‑informed neural network, an additional layer determines the weight the of physics in the loss function to balance data and physics based on their uncertainty for a better estimation. For transfer learning, a shallow neural network trained on water is adapted for use with Na. Validation results show that the self‑supervised physics‑informed neural network successfully estimate the heat transfer rates of Na with an error margin of approximately +8%. Using only physics for regression, the error remains between 5% to 10%. Other machine learning methods specify the prediction mostly within +8%. High‑fidelity modeling of turbulent forced convection of liquid metals using computational fluid dynamics (CFD) is both time‑consuming and computationally expensive. Therefore, machine learning based models offer a powerful alternative tool for the design and optimization of liquid‑metal‑cooled miniature heat sinks.

Abstract:
In this paper, we integrate neural networks and Gaussian wave packets to numerically solve the Schrödinger equation with a smooth potential near the semi‑classical limit. Our focus is not only on accurately obtaining solutions when the non‑dimensional Planck's constant, \varepsilon, is small, but also on constructing an operator that maps initial values to solutions for the Schrödinger equation with multiscale properties. Using Gaussian wave packets framework, we first reformulate the Schrödinger equation as a system of ordinary differential equations. For a single initial condition, we solve the resulting system using PINNs or MscaleDNNs. Numerical simulations indicate that MscaleDNNs outperform PINNs, improving accuracy by one to two orders of magnitude. When dealing with a set of initial conditions, we adopt an operator‑learning approach, such as physics‑informed DeepONets. Numerical examples validate the effectiveness of physics‑informed DeepONets with Gaussian wave packets in accurately mapping initial conditions to solutions.

Abstract:
X‑ray ptychography provides exceptional nanoscale resolution and is widely applied in materials science, biology, and nanotechnology. However, its full potential is constrained by the critical challenge of accurately reconstructing images when the illuminating probe is unknown. Conventional iterative methods and deep learning approaches are often suboptimal, particularly under the low‑signal conditions inherent to low‑dose and high‑speed experiments. These limitations compromise reconstruction fidelity and restrict the broader adoption of the technique. In this work, we introduce the Ptychographic Implicit Neural Representation (PtyINR), a self‑supervised framework that simultaneously addresses the object and probe recovery problem. By parameterizing both as continuous neural representations, PtyINR performs end‑to‑end reconstruction directly from raw diffraction patterns without requiring any pre‑characterization of the probe. Extensive evaluations demonstrate that PtyINR achieves superior reconstruction quality on both simulated and experimental data, with remarkable robustness under challenging low‑signal conditions. Furthermore, PtyINR offers a generalizable, physics‑informed framework for addressing probe‑dependent inverse problems, making it applicable to a wide range of computational microscopy problems.

Abstract:
The present study investigates the dynamics of nonlocal beams by establishing a consistent stress‑driven integral elastic using the Physics‑Informed Neural Network (PINN) approach. Specifically, a PINN is developed to compute the first eigenfunction and eigenvalue arising from the underlying sixth‑order ordinary differential equation. The PINN is based on a feedforward neural network, with a loss function composed of terms from the differential equation, the normalization condition, and both boundary and constitutive boundary conditions. Relevant eigenvalues are treated as separate trainable variables. The results demonstrate that the proposed method is a powerful and robust tool for addressing the complexity of the problem. Once trained, the neural network is less computationally intensive than analytical methods. The obtained results are compared with benchmark analytical solutions and show strong agreement.

Abstract:
Heat propagation is governed by phonon interactions and mathematically described by partial differential equations (PDEs), which link thermal transport to the intrinsic properties of materials. Conventional experimental techniques infer thermal responses based on surface emissions, limiting their ability to fully resolve subsurface structures and internal heat distribution. Additionally, existing thermal tomographic techniques can only shoot one frame from each layer. Physics‑informed neural networks (PINNs) have recently emerged as powerful tools for solving inverse problems in heat transfer by integrating observational data with physical constraints. However, standard PINNs are primarily focused on fitting the given external temperature data, without explicit knowledge of the unknown internal temperature distribution. In this study, we introduce a Helmholtz‑informed neural network (HINN) to predict internal temperature distributions without requiring internal measurements. The time‑domain heat diffusion equation was converted to the frequency‑domain and becomes the pseudo‑Helmholtz equation. HINN embeds this pseudo‑Helmholtz equation into the learning framework, leveraging both real and imaginary components of the thermal field. Finally, an inverse Fourier transform brings real‑part and imagery‑part back to the time‑domain and can be used to map 3D thermal fields with interior defects. Furthermore, a truncated operation was conducted to improve computational efficiency, and the principle of conjugate symmetry was employed for repairing the discarded data. This approach significantly enhances predictive accuracy and computational efficiency. Our results demonstrate that HINN outperforms state‑of‑the‑art PINNs and inverse heat solvers, offering a novel solution for non‑invasive thermography in applications spanning materials science, biomedical diagnostics, and nondestructive evaluation.

Abstract:
As the number of satellites in orbit has increased exponentially in recent years, ensuring their correct functionality has started to require automated methods to decrease human workload. In this work, we present an algorithm that analyzes the on‑board data related to friction from the Reaction Wheel Assemblies (RWA) of a satellite and determines their operating status, distinguishing between nominal status and several possible anomalies that require preventive measures to be taken. The algorithm first uses a model based on hybrid systems theory to extract the information relevant to the problem. The extraction process combines techniques in changepoint detection, dynamic programming, and maximum likelihood in a structured way. A classifier then uses the extracted information to determine the status of the RWA. This last classifier has been previously trained with a labelled dataset produced by a high‑fidelity simulator, comprised for the most part of nominal data. The final algorithm combines model‑based and data‑based approaches to obtain satisfactory results with an accuracy around 95%.

Abstract:
Although speckle is a powerful tool for high‑precision metrology, large datasets and cumbersome training are always required to learn from the encoded speckle patterns, which is unfavorable for rapid deployment and multi‑dimensional metrology. To enable high accuracy and fast training, physics‑informed machine learning enforces physical laws to address high‑dimensional problems. Here, we harness the modal fields in a few‑mode fiber, which follow the law of beam propagation, to enable high‑accuracy and fast‑training parameter estimation. Anti‑noise fast mode decomposition is implemented to retrieve the modal fields from the speckles. The accuracy is enhanced since the modal fields enable parameter estimation at random points in the continuous space‑time domain. Artificial tactile perception and multi‑dimensional metrology are achieved with high accuracy because the modal fields respond diversely to different parameters. Meanwhile, the number of specklegrams for training is reduced by around 5 times. The training time of machine learning is significantly reduced by 800 times, from 9 hours and 45 minutes to 40 seconds. Therefore, harnessing the modal fields paves a new way for the speckle‑based metrology to develop efficient, low‑cost, multi‑dimensional sensors, making it suitable for intelligent wearable devices, industrial robots and healthcare applications.

Abstract:
We introduce the Neural Field Turing Machine (NFTM), a differentiable architecture that unifies symbolic computation, physical simulation, and perceptual inference within continuous spatial fields. NFTM combines a neural controller, continuous memory field, and movable read/write heads that perform local updates. At each timestep, the controller reads local patches, computes updates via learned rules, and writes them back while updating head positions. This design achieves linear O(N) scaling through fixed‑radius neighborhoods while maintaining Turing completeness under bounded error. We demonstrate three example instantiations of NFTM: cellular automata simulation (Rule 110), physics‑informed PDE solvers (2D heat equation), and iterative image refinement (CIFAR‑10 inpainting). These instantiations learn local update rules that compose into global dynamics, exhibit stable long‑horizon rollouts, and generalize beyond training horizons. NFTM provides a unified computational substrate bridging discrete algorithms and continuous field dynamics within a single differentiable framework.

Abstract:
Physics‑informed machine learning (PIML) represents an emerging paradigm that integrates various forms of physical knowledge into machine learning (ML) components, thereby enhancing the physical consistency of ML models compared to purely data‑driven paradigms. The field of combustion, characterized by a rich foundation of physical laws and abundant data, is undergoing a transformation due to PIML. This paper aims to provide a comprehensive overview of PIML for combustion, systematically outlining fundamental principles, significant contributions, key advancements, and available resources. The application of PIML in combustion is categorized into three domains: combustion chemical kinetics, combustion reacting flows, and other combustion‑related scenarios. Additionally, current challenges, potential solutions, and practical guidelines for researchers and engineers will be discussed. A primary focus of this review is to demonstrate how combustion laws can be integrated into ML, either through soft or hard constraints, via loss functions or representation models, and within coordinate‑to‑variable or field‑to‑field paradigms. This paper shows that PIML offers a unified framework linking physics, model, and data in combustion‑‑integrating physical knowledge in model‑to‑data simulation and reconstruction tasks, as well as data‑to‑model modeling tasks‑‑resulting in enhanced data, improved physical models, and more reliable ML models. PIML for combustion presents significant opportunities for both the combustion and ML communities, encouraging greater collaboration and cross‑disciplinary engagement.

Abstract:
Self‑supervised pretraining from static structures of drug‑like compounds and proteins enable powerful learned feature representations. Learned features demonstrate state of the art performance on a range of predictive tasks including molecular properties, structure generation, and protein‑ligand interactions. The majority of approaches are limited by their use of static structures and it remains an open question, how best to use atomistic molecular dynamics (MD) simulations to develop more generalized models to improve prediction accuracy for novel molecular structures. We present SURrogate mmGBSA (SurGBSA) as a new modeling approach for MD‑based representation learning, which learns a surrogate function of the Molecular Mechanics Generalized Born Surface Area (MMGBSA). We show for the first time the benefits of physics‑informed pre‑training to train a surrogate MMGBSA model on a collection of over 1.4 million 3D trajectories collected from MD simulations of the CASF‑2016 benchmark. SurGBSA demonstrates a dramatic 27,927x speedup versus a traditional physics‑based single‑point MMGBSA calculation while nearly matching single‑point MMGBSA accuracy on the challenging pose ranking problem for identification of the correct top pose (‑0.4% difference). Our work advances the development of molecular foundation models by showing model improvements when training on MD simulations. Models, code and training data are made publicly available.

Abstract:
Symbolic regression (SR) has emerged as a powerful tool for automated scientific discovery, enabling the derivation of governing equations from experimental data. A growing body of work illustrates the promise of integrating domain knowledge into the SR to improve the discovered equation's generality and usefulness. Physics‑informed SR (PiSR) addresses this by incorporating domain knowledge, but current methods often require specialized formulations and manual feature engineering, limiting their adaptability only to domain experts. In this study, we leverage pre‑trained Large Language Models (LLMs) to facilitate knowledge integration in PiSR. By harnessing the contextual understanding of LLMs trained on vast scientific literature, we aim to automate the incorporation of domain knowledge, reducing the need for manual intervention and making the process more accessible to a broader range of scientific problems. Namely, the LLM is integrated into the SR's loss function, adding a term of the LLM's evaluation of the SR's produced equation. We extensively evaluate our method using three SR algorithms (DEAP, gplearn, and PySR) and three pre‑trained LLMs (Falcon, Mistral, and LLama 2) across three physical dynamics (dropping ball, simple harmonic motion, and electromagnetic wave). The results demonstrate that LLM integration consistently improves the reconstruction of physical dynamics from data, enhancing the robustness of SR models to noise and complexity. We further explore the impact of prompt engineering, finding that more informative prompts significantly improve performance.

Abstract:
Kernel methods are powerful tools in statistical learning, but their cubic complexity in the sample size n limits their use on large‑scale datasets. In this work, we introduce a scalable framework for kernel regression with O(n log n) complexity, fully leveraging GPU acceleration. The approach is based on a Fourier representation of kernels combined with non‑uniform fast Fourier transforms (NUFFT), enabling exact, fast, and memory‑efficient computations. We instantiate our framework in three settings: Sobolev kernel regression, physics‑informed regression, and additive models. When known, the proposed estimators are shown to achieve minimax convergence rates, consistent with classical kernel theory. Empirical results demonstrate that our methods can process up to tens of billions of samples within minutes, providing both statistical accuracy and computational scalability. These contributions establish a flexible approach, paving the way for the routine application of kernel methods in large‑scale learning tasks.

Abstract:
Parametric partial differential equations (PDEs) serve as fundamental mathematical tools for modeling complex physical phenomena, yet repeated high‑fidelity numerical simulations across parameter spaces remain computationally prohibitive. In this work, we propose a physical law‑corrected prior Gaussian process (LC‑prior GP) for efficient surrogate modeling of parametric PDEs. The proposed method employs proper orthogonal decomposition (POD) to represent high‑dimensional discrete solutions in a low‑dimensional modal coefficient space, significantly reducing the computational cost of kernel optimization compared with standard GP approaches in full‑order spaces. The governing physical laws are further incorporated to construct a law‑corrected prior to overcome the limitation of existing physics‑informed GP methods that rely on linear operator invariance, which enables applications to nonlinear and multi‑coupled PDE systems without kernel redesign. Furthermore, the radial basis function‑finite difference (RBF‑FD) method is adopted for generating training data, allowing flexible handling of irregular spatial domains. The resulting differentiation matrices are independent of solution fields, enabling efficient optimization in the physical correction stage without repeated assembly. The proposed framework is validated through extensive numerical experiments, including nonlinear multi‑parameter systems and scenarios involving multi‑coupled physical variables defined on different two‑dimensional irregular domains to highlight the accuracy and efficiency compared with baseline approaches.

Abstract:
Radioembolization is a localized liver cancer treatment that delivers radioactive microspheres (30 micron) to tumors via a catheter inserted in the hepatic arterial tree. The goal is to maximize therapeutic efficacy while minimizing damage to healthy liver tissue. However, optimization is challenging due to complex hepatic artery anatomy, variable blood flow, and uncertainty in microsphere transport. The creation of dynamic, patient‑specific digital twins may provide a transformative solution to these challenges. This work outlines a framework for a liver radioembolization digital twin using high‑fidelity computational fluid dynamics (CFD) and/or recent physics‑informed machine learning approaches. The CFD approach involves microsphere transport calculations in the hepatic arterial tree with individual patient data, which enables personalized treatment planning. Although accurate, traditional CFD is computationally expensive and limits clinical applicability. To accelerate simulations, physics‑informed neural networks (PINNs) and their generative extensions play an increasingly important role. PINNs integrate governing equations, such as the Navier‑Stokes equations, directly into the neural network training process, enabling mesh‑free, data‑efficient approximation of blood flow and microsphere transport. Physics‑informed generative adversarial networks (PI‑GANs), diffusion models (PI‑DMs), and transformer‑based architectures further enable uncertainty‑aware, temporally resolved predictions with reduced computational cost. These AI surrogates not only maintain physical fidelity but also support rapid sampling of diverse flow scenarios, facilitating real‑time decision support. Together, CFD and physics‑informed AI methods form the foundation of dynamic, patient‑specific digital twin to optimize radioembolization planning and ultimately improve clinical outcomes.

Abstract:
We perform a physics‑informed Bayesian analyses of the equation of state of hybrid neutron stars that incorporates color‑flavor‑locked quark matter modeled by a three‑flavor non‑local Nambu‑Jona‑Lasinio framework with vector repulsion and diquark pairing. Contrary to the model‑agnostic Bayesian analyses our scheme allows for distinguishing between the scenarios of neutron stars with quark cores and without them. The used quark model realizes asymptotic conformality at high densities in accordance with perturbative QCD. The hadronic sector is described by the density‑dependent relativistic functional DD2Y‑T, which satisfies chiral effective field theory constraints and includes hyperonic degrees of freedom. We construct a large set of candidate hybrid EOSs by varying the vector and diquark couplings and apply a Maxwell construction for the quark‑hadron phase transition. Observational constraints from recent NICER pulsar mass‑radius measurements and tidal deformability from GW170817 are incorporated into the likelihood. Depending on whether the observational data from the black widow pulsar PSR J0952‑0607 and the HESS J1731‑347 object are included to the analysis or not, the posterior distribution favors vector and diquark couplings around (η_V,η_D)～eq (0.82,0.40) or (η_V,η_D)～eq (0.64,0.36), respectively. This corresponds to equations of state that support two‑solar‑mass neutron stars with superconformal speed of sound and relatively low onset densities for deconfinement. Our findings indicate that the most probable hybrid EOSs are statistically preferred over the purely hadronic baseline. The corresponding probabilities of agreeing with the observational data differ by one or two orders of magnitude depending on the data set used. This suggests that quark cores may exist in all observed neutron stars.

Abstract:
As digital twin technologies are increasingly incorporated into battery management systems to meet the growing need for transparent and lifecycle‑aware operation, existing battery digital twins still suffer from fragmented operational processes and lack an architectural perspective to coordinate modeling, inference, and decision‑making throughout the battery lifecycle. To this end, we develop a unified five‑tier battery digital twin framework that integrates key functionalities into a coherent pipeline and facilitates a clearer architectural understanding of digital twins. The five‑tier comprises geometric modeling, descriptive analytics, physics‑informed prediction, prescriptive optimization, and autonomous control. In quantitative evaluation, the resulting architecture achieves high‑fidelity multi‑physics calibration with 0.92% voltage and 0.18% temperature prediction error, and provides state‑of‑health estimation with 1.09% MAPE and calibrated uncertainty. As the first battery digital twin system empowered by the NVIDIA ecosystem with physics‑AI technologies, our proposed five‑tier framework shifts battery management from reactive protection to an interpretable, predictive, and autonomous paradigm, paving the path to develop next‑generation battery management and energy management systems.

Abstract:
Physics‑informed Neural Network (PINN) faces significant challenges when approximating solutions to conservation laws, particularly in ensuring conservation and accurately resolving discontinuities. To address these limitations, we propose Conservation Law‑informed Neural Network (CLINN), a novel framework that incorporates the boundedness constraint, implicit solution form, and Rankine‑Hugoniot condition of scalar conservation laws into the loss function, thereby enforcing exact conservation properties. Furthermore, we integrate a residual‑based adaptive refinement (RAR) strategy to dynamically prioritize training near discontinuities, substantially improving the network's ability to capture sharp gradients. Numerical experiments are conducted on benchmark problems, including the inviscid Burgers equation, the Lighthill‑Whitham‑Richards (LWR) traffic flow model, and the Buckley‑Leverett problem. Results demonstrate that CLINN achieves superior accuracy in resolving solution profiles and discontinuity locations while reducing numeral oscillations. Compared to conventional PINN, CLINN yields a maximum reduction of 99.2% in mean squared error (MSE).

Abstract:
In response to recent FIA regulations reducing Formula 1 team wind tunnel hours (from 320 hours for last‑place teams to 200 hours for championship leaders) and strict budget caps of 135 million USD per year, more efficient aerodynamic development tools are needed by teams. Conventional computational fluid dynamics (CFD) simulations, though offering high fidelity results, require large computational resources with typical simulation durations of 8‑24 hours per configuration analysis. This article proposes a Physics‑Informed Neural Network (PINN) for the fast prediction of Formula 1 front wing aerodynamic coefficients. The suggested methodology combines CFD simulation data from SimScale with first principles of fluid dynamics through a hybrid loss function that constrains both data fidelity and physical adherence based on Navier‑Stokes equations. Training on force and moment data from 12 aerodynamic features, the PINN model records coefficient of determination (R‑squared) values of 0.968 for drag coefficient and 0.981 for lift coefficient prediction while lowering computational time. The physics‑informed framework guarantees that predictions remain adherent to fundamental aerodynamic principles, offering F1 teams an efficient tool for the fast exploration of design space within regulatory constraints.

Abstract:
Operator learning has emerged as a promising tool for accelerating the solution of partial differential equations (PDEs). The Deep Operator Networks (DeepONets) represent a pioneering framework in this area: the "vanilla" DeepONet is valued for its simplicity and efficiency, while the modified DeepONet achieves higher accuracy at the cost of increased training time. In this work, we propose a series of Transformer‑inspired DeepONet variants that introduce bidirectional cross‑conditioning between the branch and trunk networks in DeepONet. Query‑point information is injected into the branch network and input‑function information into the trunk network, enabling dynamic dependencies while preserving the simplicity and efficiency of the "vanilla" DeepONet in a non‑intrusive manner. Experiments on four PDE benchmarks ‑‑ advection, diffusion‑reaction, Burgers', and Korteweg‑de Vries equations ‑‑ show that for each case, there exists a variant that matches or surpasses the accuracy of the modified DeepONet while offering improved training efficiency. Moreover, the best‑performing variant for each equation aligns naturally with the equation's underlying characteristics, suggesting that the effectiveness of cross‑conditioning depends on the characteristics of the equation and its underlying physics. To ensure robustness, we validate the effectiveness of our variants through a range of rigorous statistical analyses, among them the Wilcoxon Two One‑Sided Test, Glass's Delta, and Spearman's rank correlation.

Abstract:
Physics‑informed neural networks (PINNs) and neural operators, two leading scientific machine learning (SciML) paradigms, have emerged as powerful tools for solving partial differential equations (PDEs). Although increasing the training sample size generally enhances network performance, it also increases computational costs for physics‑informed or data‑driven training. To address this trade‑off, different sampling strategies have been developed to sample more points in regions with high PDE residuals. However, existing sampling methods are computationally demanding for high‑dimensional problems, such as high‑dimensional PDEs or operator learning tasks. Here, we propose a residual‑based adversarial‑gradient moving sample (RAMS) method, which moves samples according to the adversarial gradient direction to maximize the PDE residual via gradient‑based optimization. RAMS can be easily integrated into existing sampling methods. Extensive experiments, ranging from PINN applied to high‑dimensional PDEs to physics‑informed and data‑driven operator learning problems, have been conducted to demonstrate the effectiveness of RAMS. Notably, RAMS represents the first efficient adaptive sampling approach for operator learning, marking a significant advancement in the SciML field.

Abstract:
Diabetic retinopathy (DR) is a leading cause of blindness worldwide, necessitating early detection to prevent vision loss. Current automated DR detection systems often struggle with poor‑quality images, lack interpretability, and insufficient integration of domain‑specific knowledge. To address these challenges, we introduce a novel framework that integrates three innovative contributions: (1) Adaptive Retinal Image Enhancement Using Physics‑Informed Neural Networks (PINNs): this technique dynamically enhances retinal images by incorporating physical constraints, improving the visibility of critical features such as microaneurysms, hemorrhages, and exudates; (2) Hybrid Feature Fusion Network (HFFN): by combining deep learning embeddings with handcrafted features, HFFN leverages both learned representations and domain‑specific knowledge to enhance generalization and accuracy; (3) Multi‑Stage Classifier with Uncertainty Quantification: this method breaks down the classification process into logical stages, providing interpretable predictions and confidence scores, thereby improving clinical trust. The proposed framework achieves an accuracy of 92.7%, a precision of 92.5%, a recall of 92.6%, an F1‑score of 92.5%, an AUC of 97.8%, a mAP of 0.96, and an MCC of 0.85. Ophthalmologists rated the framework's predictions as highly clinically relevant (4.8/5), highlighting its alignment with real‑world diagnostic needs. Qualitative analyses, including Grad‑CAM visualizations and uncertainty heatmaps, further enhance the interpretability and trustworthiness of the system. The framework demonstrates robust performance across diverse conditions, including low‑quality images, noisy data, and unseen datasets. These features make the proposed framework a promising tool for clinical adoption, enabling more accurate and reliable DR detection in resource‑limited settings.

Abstract:
Cities today generate enormous streams of data from sensors, cameras, and connected infrastructure. While this information offers unprecedented opportunities to improve urban life, most existing systems struggle with scale, latency, and fragmented insights. This work introduces a framework that blends physics‑informed machine learning, multimodal data fusion, and knowledge graph representation with adaptive, rule‑based intelligence powered by large language models (LLMs). Physics‑informed methods ground learning in real‑world constraints, ensuring predictions remain meaningful and consistent with physical dynamics. Knowledge graphs act as the semantic backbone, integrating heterogeneous sensor data into a connected, queryable structure. At the edge, LLMs generate context‑aware rules that adapt filtering and decision‑making in real time, enabling efficient operation even under constrained resources. Together, these elements form a foundation for digital twin systems that go beyond passive monitoring to provide actionable insights. By uniting physics‑based reasoning, semantic data fusion, and adaptive rule generation, this approach opens new possibilities for creating responsive, trustworthy, and sustainable smart infrastructures.

Abstract:
Neural operators (NOs) provide a new paradigm for efficiently solving partial differential equations (PDEs), but their training depends on costly high‑fidelity data from numerical solvers, limiting applications in complex systems. We propose a self‑supervised neural operator (SNO) that generates accurate and diverse training data on the fly without numerical solvers. SNO consists of three parts: a physics‑informed sampler (PI‑sampler) based on Bayesian PINNs for efficient data generation, a function encoder (FE) for compact input‑output representations, and an encoder‑only Transformer for operator learning, mapping boundary/initial conditions, source terms, and geometries to PDE solutions. We validate SNO on 1D steady/unsteady nonlinear reaction‑diffusion equations, a 2D nonlinear PDE with varying geometries, and vortex‑induced vibration of a flexible cylinder in fluid dynamics. SNO achieves high accuracy in all cases, and lightweight finetuning (O(100) trainable variables) further improves predictions with only a few hundred steps. This work provides a new route toward pretrained foundation models as efficient PDE surrogates.

Abstract:
Fetal ultrasound standard plane classification is essential for reliable prenatal diagnosis but faces inherent challenges, including low tissue contrast, boundary ambiguity, and operator‑dependent image quality variations. To overcome these limitations, we propose a plug‑and‑play adaptive contrast adjustment module (ACAM), whose core design is inspired by the clinical practice of doctors adjusting image contrast to obtain clearer and more discriminative structural information. The module employs a shallow texture‑sensitive network to predict clinically plausible contrast parameters, transforms input images into multiple contrast‑enhanced views through differentiable mapping, and fuses them within downstream classifiers. Validated on a multi‑center dataset of 12,400 images across six anatomical categories, the module consistently improves performance across diverse models, with accuracy of lightweight models increasing by 2.02 percent, accuracy of traditional models increasing by 1.29 percent, and accuracy of state‑of‑the‑art models increasing by 1.15 percent. The innovation of the module lies in its content‑aware adaptation capability, replacing random preprocessing with physics‑informed transformations that align with sonographer workflows while improving robustness to imaging heterogeneity through multi‑view fusion. This approach effectively bridges low‑level image features with high‑level semantics, establishing a new paradigm for medical image analysis under real‑world image quality variations.

Abstract:
We propose an evolutionary Multi‑objective Optimization for Replica‑Exchange‑based Physics‑informed operator‑learning Networks (Morephy‑Net) to solve parametric partial differential equations (PDEs) in noisy data regimes, for both forward prediction and inverse identification. Existing physics‑informed neural networks and operator‑learning models (e.g., DeepONets and Fourier neural operators) often face three coupled challenges: (i) balancing data/operator and physics residual losses, (ii) maintaining robustness under noisy or sparse observations, and (iii) providing reliable uncertainty quantification. Morephy‑Net addresses these issues by integrating: (i) evolutionary multi‑objective optimization that treats data/operator and physics residual terms as separate objectives and searches the Pareto front, thereby avoiding ad hoc loss weighting; (ii) replica‑exchange stochastic gradient Langevin dynamics to enhance global exploration and stabilize training in non‑convex landscapes; and (iii) Bayesian uncertainty quantification obtained from stochastic sampling. We validate Morephy‑Net on representative forward and inverse problems, including the one‑dimensional Burgers equation and the time‑fractional mixed diffusion‑‑wave equation. The results demonstrate consistent improvements in accuracy, noise robustness, and calibrated uncertainty estimates over standard operator‑learning baselines.

Abstract:
I present a concise, first principles metrological framework for imaging dielectric biomaterials by probing the full phase space (Wigner) distribution of a quantum electromagnetic field. Building on a rigorous multilayer Maxwell and Cole Cole model for stratified tissue, my method (Quantum Phase space Tomography, QPST) couples analytical forward theory with quantum metrology and Bayesian inference. I prepare a structured quantum EM probe (e.g. a squeezed microwave pulse) that interacts with tissue and then perform full quantum state tomography of the outgoing field. The recovered Wigner quasi probability reveals subwavelength and non classical features lost in classical imaging. By projecting the measurement onto the analytically derived tissue response manifold, I recover key physiological parameters (e.g. layer thickness, dispersion). I further define a Dielectric Anaplasia Metric (DAM) that quantifies tissue microstructural heterogeneity (e.g. malignancy) via deviations in Cole Cole parameters. My design leverages state of the art quantum sensors (e.g. NV diamond magnetometers) and advanced inverse algorithms (physics informed neural networks, diffusion priors). Numerical examples demonstrate that QPST can non invasively map tissue permittivity with unprecedented sensitivity. This work bridges fundamental electromagnetic theory and emerging quantum technologies, promising a new paradigm for medical imaging.

Abstract:
Parameterized partial differential equations (PDEs) underpin the mathematical modeling of complex systems in diverse domains, including engineering, healthcare, and physics. A central challenge in using PDEs for real‑world applications is to accurately infer the parameters, particularly when the parameters exhibit non‑linear and spatiotemporal variations. Existing parameter estimation methods, such as sparse identification, physics‑informed neural networks (PINNs), and neural operators, struggle in such cases, especially with nonlinear dynamics, multiphysics interactions, or limited observations of the system response. To address these challenges, we introduce Neptune, a general‑purpose method capable of inferring parameter fields from sparse measurements of system responses. Neptune employs independent coordinate neural networks to continuously represent each parameter field in physical space or in state variables. Across various physical and biomedical problems, where direct parameter measurements are prohibitively expensive or unattainable, Neptune significantly outperforms existing methods, achieving robust parameter estimation from as few as 45 measurements, reducing parameter estimation errors by two orders of magnitude and dynamic response prediction errors by a factor of ten to baselines such as PINNs and neural operators. More importantly, it exhibits superior physical extrapolation capabilities, enabling reliable predictions in regimes far beyond the training data. By facilitating reliable and data‑efficient parameter inference, Neptune promises broad transformative impacts in engineering, healthcare, and beyond.

Abstract:
Accurate prediction of hydrogen sorption in clays, shales, and coals is vital for advancing underground hydrogen storage, natural hydrogen exploration, and radioactive waste containment. Traditional experimental methods, while foundational, are time‑consuming, error‑prone, and limited in capturing geological heterogeneity. This study introduces an adaptive physics‑informed neural network (PINN) framework with multi‑category feature engineering to enhance hydrogen sorption prediction. The framework integrates classical isotherm models with thermodynamic constraints to ensure physical consistency while leveraging deep learning flexibility. A comprehensive dataset consisting of 155 samples, which includes 50 clays, 60 shales, and 45 coals, was employed, incorporating diverse compositional properties and experimental conditions. Multi‑category feature engineering across seven categories captured complex sorption dynamics. The PINN employs deep residual networks with multi‑head attention, optimized via adaptive loss functions and Monte Carlo dropout for uncertainty quantification. K‑fold cross‑validation and hyperparameter optimization achieve significant accuracy (R2 = 0.979, RMSE = 0.045 mol per kg) with 67% faster convergence despite 15‑fold increased complexity. The framework demonstrates robust lithology‑specific performance across clay minerals (R2 = 0.981), shales (R2 = 0.971), and coals (R2 = 0.978), maintaining 85‑91% reliability scores. Interpretability analysis via SHAP, accumulated local effects, and Friedman's H‑statistics reveal that hydrogen adsorption capacity dominates predictions, while 86.7% of feature pairs exhibit strong interactions, validating the necessity of non‑linear modeling approaches. This adaptive physics‑informed framework accelerates site screening and enables risk‑informed decision‑making through robust uncertainty quantification.

Abstract:
We present PhISM, a physics‑informed deep learning architecture that learns without supervision to explicitly disentangle hyperspectral observations and model them with continuous basis functions. PhISM outperforms prior methods on several classification and regression benchmarks, requires limited labeled data, and provides additional insights thanks to interpretable latent representation.

Abstract:
Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations. In practice, one often employs stochastic gradient descent type algorithms to train the neural network. Therefore, the convergence guarantee of stochastic gradient descent is of fundamental importance. In this work, we establish the linear convergence of stochastic gradient descent / flow in training over‑parameterized two layer PINNs for a general class of activation functions in the sense of high probability. These results extend the existing result [18] in which gradient descent was analyzed. The challenge of the analysis lies in handling the dynamic randomness introduced by stochastic optimization methods. The key of the analysis lies in ensuring the positive definiteness of suitable Gram matrices during the training. The analysis sheds insight into the dynamics of the optimization process, and provides guarantees on the neural networks trained by stochastic algorithms.

Abstract:
Physics‑Informed Neural Networks (PINNs) present a transformative approach for smart grid modeling by integrating physical laws directly into learning frameworks, addressing critical challenges of data scarcity and physical consistency in conventional data‑driven methods. This paper evaluates PINNs' capabilities as surrogate models for smart grid dynamics, comparing their performance against XGBoost, Random Forest, and Linear Regression across three key experiments: interpolation, cross‑validation, and episodic trajectory prediction. By training PINNs exclusively through physics‑based loss functions (enforcing power balance, operational constraints, and grid stability) we demonstrate their superior generalization, outperforming data‑driven models in error reduction. Notably, PINNs maintain comparatively lower MAE in dynamic grid operations, reliably capturing state transitions in both random and expert‑driven control scenarios, while traditional models exhibit erratic performance. Despite slight degradation in extreme operational regimes, PINNs consistently enforce physical feasibility, proving vital for safety‑critical applications. Our results contribute to establishing PINNs as a paradigm‑shifting tool for smart grid surrogation, bridging data‑driven flexibility with first‑principles rigor. This work advances real‑time grid control and scalable digital twins, emphasizing the necessity of physics‑aware architectures in mission‑critical energy systems.

Abstract:
Accurate prediction of Lagrangian trajectories in turbulent flow remains challenging due to limited temporal information in transport functions. This paper shows that surrounding coherent motions sharing the same dynamics carry enough information to provide highly probable trajectories even from sparse temporal observations. The proposed coherent predictor builds on Lagrangian coherent structures (LCSs), the advective transport barriers that govern the cohesive motion of neighbouring particles. Coherent trajectories are quantified using a local segmentation with the finite‑time Lyapunov exponents (FTLE). The coherent predictor incorporates information from the particle's position history and neighbouring coherent velocity and acceleration into a novel cost function to predict its trajectory. The proposed cost function follows a physics‑informed approach where the position history acts as a data fidelity term and the coherent velocity and acceleration act as physics‑based regularisation constraints. We assess our proposed approach using both three‑dimensional (3D) synthetic and experimental data of the wake behind a smooth cylinder and two‑dimensional (2D) homogeneous isotropic turbulent (HIT) flow. The coherent predictor is deemed generic due to its consistent behaviour regardless of flow dimensions, Reynolds number, and flow topology. Our results show that the optimal cost function parameters can be modelled from the measurement uncertainties, giving lower prediction error and uncertainty than current methods. We see direct signatures of flow topology on the prediction error map, including the cylinder leading edge boundary layer, the sideward shear layers, and the vortex formation structures. These topologies are marked by high Lagrangian gradients and 3D directional motions.

Abstract:
Operator learning (OL) has emerged as a powerful tool in scientific machine learning (SciML) for approximating mappings between infinite‑dimensional functional spaces. One of its main applications is learning the solution operator of partial differential equations (PDEs). While much of the progress in this area has been driven by deep neural network‑based approaches such as Deep Operator Networks (DeepONet) and Fourier Neural Operator (FNO), recent work has begun to explore traditional machine learning methods for OL. In this work, we introduce polynomial chaos expansion (PCE) as an OL method. PCE has been widely used for uncertainty quantification (UQ) and has recently gained attention in the context of SciML. For OL, we establish a mathematical framework that enables PCE to approximate operators in both purely data‑driven and physics‑informed settings. The proposed framework reduces the task of learning the operator to solving a system of equations for the PCE coefficients. Moreover, the framework provides UQ by simply post‑processing the PCE coefficients, without any additional computational cost. We apply the proposed method to a diverse set of PDE problems to demonstrate its capabilities. Numerical results demonstrate the strong performance of the proposed method in both OL and UQ tasks, achieving excellent numerical accuracy and computational efficiency.

Abstract:
In semiconductor manufacturing, defect detection and localization are critical to ensuring product quality and yield. While X‑ray imaging is a reliable non‑destructive testing method, it is memory‑intensive and time‑consuming for large‑scale scanning, Magnetic Field Imaging (MFI) offers a more efficient means to localize regions of interest (ROI) for targeted X‑ray scanning. However, the limited availability of MFI datasets due to proprietary concerns presents a significant bottleneck for training machine learning (ML) models using MFI. To address this challenge, we consider an ML‑driven approach leveraging diffusion models with two physical constraints. We propose Physics Informed Generative Models for Magnetic Field Images (PI‑GenMFI) to generate synthetic MFI samples by integrating specific physical information. We generate MFI images for the most common defect types: power shorts. These synthetic images will serve as training data for ML algorithms designed to localize defect areas efficiently. To evaluate generated MFIs, we compare our model to SOTA generative models from both variational autoencoder (VAE) and diffusion methods. We present a domain expert evaluation to assess the generated samples. In addition, we present qualitative and quantitative evaluation using various metrics used for image generation and signal processing, showing promising results to optimize the defect localization process.

Abstract:
Multispectral imaging (MSI) captures data across multiple spectral bands, offering enhanced informational depth compared to standard RGB imaging and benefiting diverse fields such as agriculture, medical diagnostics, and industrial inspection. Conventional MSI systems, however, suffer from high cost, complexity, and limited performance in low‑light conditions. Moreover, data‑driven MSI methods depend heavily on large, labeled training datasets and struggle with generalization. In this work, we present a portable multispectral single‑pixel imaging (MS‑SPI) method that integrates a chip‑sized multispectral sensor for system miniaturization and leverages an untrained physics‑informed neural network (PINN) to reconstruct high‑quality spectral images without the need for labeled training data. The physics‑informed structure of the network enables the self‑corrected reconstruction of multispectral images directly with the input of raw measurements from the multispectral sensor. Our proof‑of‑concept prototype achieves the reconstruction of 12‑channel high‑quality spectral images at the sampling rate of 10%. We also experimentally validate its performance under varying sampling rate conditions, by comparing it with conventional compressive sensing algorithms. Furthermore, we demonstrate the application of this technique to an MSI‑based image segmentation task, in which spatial regions are discriminated according to their characteristic spectral signatures. This compact, high‑fidelity, and portable approach offers promising pathways to lightweight and cost‑effective spectral imaging on mobile platforms.

Abstract:
Predicting optoelectronic properties of large‑scale atomistic systems under realistic conditions is crucial for rational materials design, yet computationally prohibitive with first‑principles simulations. Recent neural network models have shown promise in overcoming these challenges, but typically require large datasets and lack physical interpretability. Physics‑inspired approximate models offer greater data efficiency and intuitive understanding, but often sacrifice accuracy and transferability. Here we present HAMSTER, a physics‑informed machine learning framework for predicting the quantum‑mechanical Hamiltonian of complex chemical systems. Starting from an approximate model encoding essential physical effects, HAMSTER captures the critical influence of dynamic environments on Hamiltonians using only few explicit first‑principles calculations. We demonstrate our approach on halide perovskites, achieving accurate prediction of optoelectronic properties across temperature and compositional variations, and scalability to systems containing tens of thousands of atoms. This work highlights the power of physics‑informed Hamiltonian learning for accurate and interpretable optoelectronic property prediction in large, complex systems.

Abstract:
We present a perspective on molecular machine learning (ML) in the field of chemical process engineering. Recently, molecular ML has demonstrated great potential in (i) providing highly accurate predictions for properties of pure components and their mixtures, and (ii) exploring the chemical space for new molecular structures. We review current state‑of‑the‑art molecular ML models and discuss research directions that promise further advancements. This includes ML methods, such as graph neural networks and transformers, which can be further advanced through the incorporation of physicochemical knowledge in a hybrid or physics‑informed fashion. Then, we consider leveraging molecular ML at the chemical process scale, which is highly desirable yet rather unexplored. We discuss how molecular ML can be integrated into process design and optimization formulations, promising to accelerate the identification of novel molecules and processes. To this end, it will be essential to create molecule and process design benchmarks and practically validate proposed candidates, possibly in collaboration with the chemical industry.

Abstract:
When solving time‑dependent partial differential equations(PDEs), traditional physics‑informed neural networks (PINNs) have inherent limitations: due to the lack of temporal causality, the network is forced to learn the later‑time control equations while fully capturing the initial conditions, resulting in the continuous accumulation of errors during the integration process. Meanwhile, the limited expressivity of a single network hinders its ability to capture diverse physical behaviors across multiple subdomains. To address these issues, we propose a domain‑decomposition‑based causal PINNs (DDC‑PINNs) framework. This framework enhances spatial representation through domain decomposition and employs a causal strategy to constrain the temporal learning sequence, thereby improving the accuracy and generalization ability of solutions to time‑varying problems. Within this framework, an approximate solution is first obtained through PINNs with domain decomposition. Subsequently, the time derivative term in the PDE is retained, while other solution‑dependent terms are replaced with this approximate solution, thereby simplifying the original PDEs into ordinary differential equations (ODEs). Finally, classical numerical methods for solving ODEs are employed to obtain the time‑dependent solution. DDC‑PINNs not only preserve the inherent computational efficiency and flexibility of PINNs but also effectively incorporate causality when solving time‑dependent PDEs. Numerical experiments verify the effectiveness of the proposed method.

Abstract:
Artificial intelligence (AI) has demonstrated considerable potential in the realm of medical imaging. However, the development of high‑performance AI models typically necessitates training on large‑scale, centralized datasets. This approach is confronted with significant challenges due to strict patient privacy regulations and legal restrictions on data sharing and utilization. These limitations hinder the development of large‑scale models in medical domains and impede continuous updates and training with new data. Federated Learning (FL), a privacy‑preserving distributed training framework, offers a new solution by enabling collaborative model development across fragmented medical datasets. In this survey, we review FL's contributions at two stages of the full‑stack medical analysis pipeline. First, in upstream tasks such as CT or MRI reconstruction, FL enables joint training of robust reconstruction networks on diverse, multi‑institutional datasets, alleviating data scarcity while preserving confidentiality. Second, in downstream clinical tasks like tumor diagnosis and segmentation, FL supports continuous model updating by allowing local fine‑tuning on new data without centralizing sensitive images. We comprehensively analyze FL implementations across the medical imaging pipeline, from physics‑informed reconstruction networks to diagnostic AI systems, highlighting innovations that improve communication efficiency, align heterogeneous data, and ensure secure parameter aggregation. Meanwhile, this paper provides an outlook on future research directions, aiming to serve as a valuable reference for the field's development.

Abstract:
Accurately quantifying long‑term risk probabilities in diverse stochastic systems is essential for safety‑critical control. However, existing sampling‑based and partial differential equation (PDE)‑based methods often struggle to handle complex varying dynamics. Physics‑informed neural networks learn surrogate mappings for risk probabilities from varying system parameters of fixed and finite dimensions, yet can not account for functional variations in system dynamics. To address these challenges, we introduce physics‑informed neural operator (PINO) methods to risk quantification problems, to learn mappings from varying functional system dynamics to corresponding risk probabilities. Specifically, we propose Neural Spline Operators (NeSO), a PINO framework that leverages B‑spline representations to improve training efficiency and achieve better initial and boundary condition enforcements, which are crucial for accurate risk quantification. We provide theoretical analysis demonstrating the universal approximation capability of NeSO. We also present two case studies, one with varying functional dynamics and another with high‑dimensional multi‑agent dynamics, to demonstrate the efficacy of NeSO and its significant online speed‑up over existing methods. The proposed framework and the accompanying universal approximation theorem are expected to be beneficial for other control or PDE‑related problems beyond risk quantification.

Abstract:
We present a novel analytical framework employing Physics‑Informed Neural Networks (PINNs) to constrain the cosmological constant Λ through the analysis of stellar orbits around the supermassive black hole (SMBH) Sgr A at the Galactic center. Focusing on the well‑observed S2 star, we use an inverse PINN (iPINN) architecture to infer orbital elements and estimate the total precession angle from astrometric data. By isolating the contribution from Λ, which is defined as the difference between the total precession and the Schwarzschild precession, we derive a stringent upper bound of Λ\leq 5.67 × 10^‑40, \mathrmm^‑2, which is approximately two orders of magnitude tighter than previous estimates obtained using similar data‑driven methods. Extension of our analysis to two additional long‑period S‑stars, S1 and S9, reveals that while the cosmological precession becomes relatively more prominent in such systems, limited orbital coverage introduces significant uncertainties in parameter estimation. Among the cases examined, the constraint derived from S2 remains the most robust. Our results highlight the potential of PINN‑based approaches for extracting physical insights from sparse or noisy astronomical data. Future applications to next‑generation observational data and further methodological improvements in machine learning are expected to refine the cosmological constraints and enable broader tests of gravitational theories.

Abstract:
Accurate subsurface reservoir pressure control is extremely challenging due to geological heterogeneity and multiphase fluid‑flow dynamics. Predicting behavior in this setting relies on high‑fidelity physics‑based simulations that are computationally expensive. Yet, the uncertain, heterogeneous properties that control these flows make it necessary to perform many of these expensive simulations, which is often prohibitive. To address these challenges, we introduce a physics‑informed machine learning workflow that couples a fully differentiable multiphase flow simulator, which is implemented in the DPFEHM framework with a convolutional neural network (CNN). The CNN learns to predict fluid extraction rates from heterogeneous permeability fields to enforce pressure limits at critical reservoir locations. By incorporating transient multiphase flow physics into the training process, our method enables more practical and accurate predictions for realistic injection‑extraction scenarios compare to previous works. To speed up training, we pretrain the model on single‑phase, steady‑state simulations and then fine‑tune it on full multiphase scenarios, which dramatically reduces the computational cost. We demonstrate that high‑accuracy training can be achieved with fewer than three thousand full‑physics multiphase flow simulations ‑‑ compared to previous estimates requiring up to ten million. This drastic reduction in the number of simulations is achieved by leveraging transfer learning from much less expensive single‑phase simulations.

Abstract:
The region of attraction is a key metric of the robustness of systems. This paper addresses the numerical solution of the generalized Zubov's equation, which produces a special Lyapunov function characterizing the robust region of attraction for perturbed systems. To handle the highly nonlinear characteristic of the generalized Zubov's equation, we propose a physics‑informed neural network framework that employs a policy iteration training scheme with rollout to approximate the viscosity solution. In addition to computing the optimal disturbance during the policy improvement process, we incorporate neural network‑generated value estimates as anchor points to facilitate the training procedure to prevent singularities in both low‑ and high‑dimensional systems. Numerical simulations validate the effectiveness of the proposed approach.

Abstract:
Graph Neural Networks (GNNs) have deeply modified the landscape of numerical simulations by demonstrating strong capabilities in approximating solutions of physical systems. However, their ability to extrapolate beyond their training domain (e.g. larger or structurally different graphs) remains uncertain. In this work, we establish that GNNs can serve purposes beyond their traditional role, and be exploited to generate numerical schemes, in conjunction with symbolic regression. First, we show numerically and theoretically that a GNN trained on a dataset consisting solely of two‑node graphs can extrapolate a first‑order Finite Volume (FV) scheme for the heat equation on out‑of‑distribution, unstructured meshes. Specifically, if a GNN achieves a loss \varepsilon on such a dataset, it implements the FV scheme with an error of \mathcalO(\varepsilon). Using symbolic regression, we show that the network effectively rediscovers the exact analytical formulation of the standard first‑order FV scheme. We then extend this approach to an unsupervised context: the GNN recovers the first‑order FV scheme using only a residual loss similar to Physics‑Informed Neural Networks (PINNs) with no access to ground‑truth data. Finally, we push the methodology further by considering higher‑order schemes: we train (i) a 2‑hop and (ii) a 2‑layers GNN using the same PINN loss, that autonomously discover (i) a second‑order correction term to the initial scheme using a 2‑hop stencil, and (ii) the classic second‑order midpoint scheme. These findings follows a recent paradigm in scientific computing: GNNs are not only strong approximators, but can be active contributors to the development of novel numerical methods.

Abstract:
Information asymmetry in financial markets, often amplified by strategically crafted corporate narratives, undermines the effectiveness of conventional textual analysis. We propose a novel multimodal framework for financial risk assessment that integrates textual sentiment with paralinguistic cues derived from executive vocal tract dynamics in earnings calls. Central to this framework is the Physics‑Informed Acoustic Model (PIAM), which applies nonlinear acoustics to robustly extract emotional signatures from raw teleconference sound subject to distortions such as signal clipping. Both acoustic and textual emotional states are projected onto an interpretable three‑dimensional Affective State Label (ASL) space‑Tension, Stability, and Arousal. Using a dataset of 1,795 earnings calls (approximately 1,800 hours), we construct features capturing dynamic shifts in executive affect between scripted presentation and spontaneous Q&A exchanges. Our key finding reveals a pronounced divergence in predictive capacity: while multimodal features do not forecast directional stock returns, they explain up to 43.8% of the out‑of‑sample variance in 30‑day realized volatility. Importantly, volatility predictions are strongly driven by emotional dynamics during executive transitions from scripted to spontaneous speech, particularly reduced textual stability and heightened acoustic instability from CFOs, and significant arousal variability from CEOs. An ablation study confirms that our multimodal approach substantially outperforms a financials‑only baseline, underscoring the complementary contributions of acoustic and textual modalities. By decoding latent markers of uncertainty from verifiable biometric signals, our methodology provides investors and regulators a powerful tool for enhancing market interpretability and identifying hidden corporate uncertainty.

Abstract:
We performed a detailed analysis of the magnetic structure and asymmetric eruption of a large (about 500 Mm) inverse S‑shaped filament partially located in AR 13229 on February 24, 2023. We linked the filament's pre‑eruptive magnetic configuration to its highly asymmetric eruption dynamics and the formation of a large‑scale coronal dimming in a weak‑field region (mean unsigned flux of about 5 G). To reconstruct the coronal magnetic field, we applied a physics‑informed neural network (PINN)‑based nonlinear force‑free field (NLFFF) extrapolation method to a pre‑eruption HMI vector magnetogram. The NLFFF extrapolation reveals a large‑scale magnetic flux rope (MFR) of about 500 Mm in length, consistent with the filament. We identified an extended MFR footprint to the east that connects to the J‑shaped flare ribbon, outlining where the coronal dimming began. Overlying strapping fields connect to the area into which the dimming and flare ribbon later expand. This configuration explains the formation of the dimming as a stationary flux rope and strapping flux dimming, with subsequent expansion driven by the growth of the MFR footprint through strapping‑strapping reconnection. Conversely, the western filament leg shows multiple anchor points and strong overlying magnetic fields, which suppressed the dimming and partially confined the eruption on that side. The reconstructed pre‑eruptive NLFFF configuration offers a clear physical explanation for the asymmetries seen in the eruption, flare geometry, and coronal dimming. This demonstrates that PINN‑based NLFFF extrapolation can effectively model large‑scale filaments extending into weak‑field regions, enhancing our understanding of complex solar eruptions.

Abstract:
The design of nonlinear superconducting quantum circuits often relies on time‑consuming iterative electromagnetic simulations requiring manual intervention. These interventions entail, for example, adjusting design variables such as resonator lengths or Josephson junction energies to meet target parameters such as mode frequencies, decay rates, and coupling strengths. Here, we present a method to efficiently automate the optimization of superconducting circuits, which significantly reduces the need for manual intervention. The method's efficiency arises from user‑defined, physics‑informed, nonlinear models that guide parameter updates toward the desired targets. Additionally, we provide a full implementation of our optimization method as an open‑source Python package, QDesignOptimizer. The package automates the design workflow by combining high‑accuracy electromagnetic simulations in Ansys HFSS and Energy Participation Ratio (pyEPR) analysis integrated with the design tool Qiskit‑Metal. Our implementation supports modular and flexible subsystem‑level analysis and is easily extensible to optimize for additional parameters. The method is not specific to superconducting circuits; as such, it can be applied to a range of nonlinear optimization problems across science and technology.

Abstract:
Physics‑Informed Neural Networks (PINNs) are becoming a popular method for solving PDEs, due to their mesh‑free nature and their ability to handle high‑dimensional problems where traditional numerical solvers often struggle. Despite their promise, the practical application of PINNs is still constrained by several fac‑ tors, a primary one being their often‑limited accuracy. This paper is dedicated to enhancing the accuracy of PINNs by introducing spectral‑prior guided multistage strategy. We propose two methods: Spectrum‑ Informed Multistage Physics‑Informed Neural Networks (SI‑MSPINNs) and Multistage Physics‑Informed Neural Networks with Spectrum Weighted Random Fourier Features (RFF‑MSPINNs). The SI‑MSPINNs integrate the core mechanism of Spectrum‑Informed Multistage Neural Network (SI‑MSNNs) and PINNs, in which we extract the Dominant Spectral Pattern (DSP) of residuals by the discrete Fourier transform. This DSP guides the network initialization to alleviate spectral bias, and gradually optimizes the resolution accuracy using a multistage strategy. The RFF‑MSPINNs combines random Fourier features with spectral weighting methods, dynamically adjusting the frequency sampling distribution based on the residual power spectral density, allowing the network to prioritize learning high‑energy physical modes. Through experimental verification of the Burgers equation and the Helmholtz equation, we show that both models significantly improve the accuracy of the original PINNs.

Abstract:
Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics‑informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis remains limited, especially regarding convergence guarantees. This work develops a general optimisation framework for energy minimisation problems arising from linear self‑adjoint elliptic PDEs, formulated over nonlinear but analytically tractable approximation spaces. The framework accommodates a natural split between linear and nonlinear parameters and supports hybrid optimisation strategies: linear variables are updated via linear solves or steepest descent, while nonlinear variables are handled using constrained projected descent. We establish both local and global convergence of the resulting algorithm under modular structural assumptions on the discrete energy functional, including differentiability, boundedness, regularity, and directional convexity. These assumptions are stated in an abstract form, allowing the framework to apply to a broad class of nonlinear approximation manifolds. In a companion paper [Magueresse, Badia (2025, arXiv:2508.17705)], we introduce a concrete instance of such a space based on overlapping free‑knot tensor‑product B‑splines, which satisfies the required assumptions and enables geometrically adaptive solvers with rigorous convergence guarantees.

Abstract:
This study proposes a Physics‑Informed Neural Network (PINN) framework to predict the low‑cycle fatigue (LCF) life of irradiated austenitic and ferritic/martensitic (F/M) steels used in nuclear reactors. These materials undergo cyclic loading, neutron irradiation, and elevated temperatures, leading to complex degradation mechanisms that are difficult to capture with conventional empirical or purely data‑driven models. The proposed PINN embeds fatigue‑life governing physical constraints into the loss function, enabling physically consistent learning while improving predictive accuracy, reliability, and generalizability. The model was trained on 495 strain‑controlled fatigue data points spanning irradiated and unirradiated conditions. Compared with traditional machine learning approaches, including Random Forest, Gradient Boosting, eXtreme Gradient Boosting, and conventional neural networks, the PINN demonstrated superior performance. SHapley Additive exPlanations (SHAP) analysis identified strain amplitude, irradiation dose, and test temperature as the dominant features, each exhibiting physically meaningful inverse correlations with fatigue life. Univariate and multivariate analyses revealed clear alloy‑specific degradation characteristics. Austenitic steels exhibited strong nonlinear coupling among strain amplitude, irradiation dose, and temperature, resulting in pronounced fatigue degradation under combined loading. In contrast, F/M steels demonstrated comparatively stable irradiation responses, including dose‑saturation behavior, but showed sensitivity to elevated temperatures beyond tempering thresholds. Overall, the proposed PINN framework serves as a reliable and interpretable tool for reactor‑relevant fatigue assessment, enabling performance evaluation for advanced nuclear applications.

Abstract:
We present FDTRImageEnhancer, an open‑source computational framework that improves thermal conductivity mapping from Frequency Domain ThermoReflectance (FDTR) phase data by integrating a physics‑based Gaussian convolution abstraction with microstructure‑aware deep learning. The Gaussian kernel models the spatial averaging effects of pump and probe beams, while k‑means clustering of high‑resolution structural images reduces the parameter space for inverse modeling. A physics‑informed neural network jointly minimizes phase‑data error and deviation from analytically recovered conductivity maps, enabling the detection of grain boundary thermal conductivity drops visually obscured in conventional FDTR inversions. Demonstrated on finite element‑generated synthetic data, the framework recovers bulk values within less than 0.5% error and qualitatively resolves grain boundary effects despite limited image resolution. Full Python code and datasets are provided for reproducibility, with the methodology readily adaptable to other inverse thermal transport problems.

Abstract:
Time‑dependent density functional theory (TDDFT) is a widely used method to investigate electron dynamics under external time‑dependent perturbations such as laser fields. In this work, we present a machine learning approach to accelerate electron dynamics simulations based on real time TDDFT using autoregressive neural operators as time‑propagators for the electron density. By leveraging physics‑informed constraints and featurization, and high‑resolution training data, our model achieves superior accuracy and computational speed compared to traditional numerical solvers. We demonstrate the effectiveness of our model on a class of one‑dimensional diatomic molecules under the influence of a range of laser parameters. This method has potential in enabling on‑the‑fly modeling of laser‑irradiated molecules and materials by utilizing fast machine learning predictions in a large space of varying experimental parameters of the laser.

Abstract:
Solving time‑dependent partial differential equations (PDEs) is fundamental to modeling critical phenomena across science and engineering. Physics‑Informed Neural Networks (PINNs) solve PDEs using deep learning. However, PINNs perform pointwise predictions that neglect the autoregressive property of dynamical systems, leading to instabilities and inaccurate predictions. We introduce Physics‑Informed Autoregressive Networks (PIANO) ‑‑ a framework that redesigns PINNs to model dynamical systems. PIANO operates autoregressively, explicitly conditioning future predictions on the past. It is trained through a self‑supervised rollout mechanism while enforcing physical constraints. We present a rigorous theoretical analysis demonstrating that PINNs suffer from temporal instability, while PIANO achieves stability through autoregressive modeling. Extensive experiments on challenging time‑dependent PDEs demonstrate that PIANO achieves state‑of‑the‑art performance, significantly improving accuracy and stability over existing methods. We further show that PIANO outperforms existing methods in weather forecasting.

Abstract:
For hyperbolic conservation laws, traditional methods and physics‑informed neural networks (PINNs) often encounter difficulties in capturing sharp discontinuities and maintaining temporal consistency. To address these challenges, we introduce a hybrid computational framework by coupling discontinuous Galerkin (DG) discretizations with a temporally progressive neural network architecture. Our method incorporates a structure‑preserving weak‑form loss ‑‑ combining DG residuals and Rankine‑Hugoniot jump conditions ‑‑ with a causality‑respecting progressive training strategy. The proposed framework trains neural networks sequentially across temporally decomposed subintervals, leveraging pseudo‑label supervision to ensure temporal coherence and solution continuity. This approach mitigates error accumulation and enhances the model's capacity to resolve shock waves and steep gradients without explicit limiters. Besides, a theoretical analysis establishes error bounds for the proposed framework, demonstrating convergence toward the physical solution under mesh refinement and regularized training. Numerical experiments on Burgers and Euler equations show that our method consistently outperforms standard PINNs, PINNs‑WE, and first‑order DG schemes in both accuracy and robustness, particularly in capturing shocks and steep gradients. These results highlight the promise of combining classical discretization techniques with machine learning to develop robust and accurate solvers for nonlinear hyperbolic systems.

Abstract:
Ultra‑high‑energy cosmic‑ray (UHECR) observatories require unbiased direction reconstruction to enable multi‑messenger astronomy with sparse, nanosecond‑scale radio pulses. Explicit likelihood methods often rely on simplified models, which may bias results and understate uncertainties. We introduce a simulation‑based inference pipeline that couples a physics‑informed graph neural network (GNN) to a normalizing‑flow posterior within the Learning the Universe Implicit Likelihood Inference framework. Each event is seeded by an analytic plane‑wavefront fit; the GNN refines this estimate by learning spatiotemporal correlations among antenna signals, and its frozen embedding conditions an eight‑block autoregressive flow that returns the full Bayesian posterior. Trained on about 8,000 realistic UHECR air‑shower simulations generated with the ZHAireS code, the posteriors are temperature‑calibrated to meet empirical coverage targets. We demonstrate a sub‑degree median angular resolution on test UHECR events, and find that the nominal 68% highest‑posterior‑density contours capture 71% \pm 2% of true arrival directions, indicating a mildly conservative uncertainty calibration. This approach provides physically interpretable reconstructions, well‑calibrated uncertainties, and rapid inference, making it ideally suited for upcoming experiments targeting highly inclined events, such as GRAND, AugerPrime Radio, and BEACON.

Abstract:
We present several key advances to the Physics and Equality Constrained Artificial Neural Networks (PECANN) framework, substantially improving its capacity to solve challenging partial differential equations (PDEs). Our enhancements broaden the framework's applicability and improve efficiency. First, we generalize the Augmented Lagrangian Method (ALM) to support multiple, independent penalty parameters for enforcing heterogeneous constraints. Second, we introduce a constraint aggregation technique to address inefficiencies associated with point‑wise enforcement. Third, we incorporate a single Fourier feature mapping to capture highly oscillatory solutions with multi‑scale features, where alternative methods often require multiple mappings or costlier architectures. Fourth, a novel time‑windowing strategy enables seamless long‑time evolution without relying on discrete time models. Fifth, and critically, we propose a conditionally adaptive penalty update (CAPU) strategy for ALM that accelerates the growth of Lagrange multipliers for constraints with larger violations, while enabling coordinated updates of multiple penalty parameters. CAPU accelerates the growth of Lagrange multipliers for selectively challenging constraints, enhancing constraint enforcement during training. We demonstrate the effectiveness of PECANN‑CAPU across diverse problems, including the transonic rarefaction problem, reversible scalar advection by a vortex, high‑wavenumber Helmholtz and Poisson's equations, and inverse heat source identification. The framework achieves competitive accuracy across all cases when compared with established methods and recent approaches based on Kolmogorov‑Arnold networks. Collectively, these advances improve the robustness, computational efficiency, and applicability of PECANN to demanding problems in scientific computing.

Abstract:
X‑ray phase contrast imaging significantly improves the visualization of structures with weak or uniform absorption, broadening its applications across a wide range of scientific disciplines. Propagation‑based phase contrast is particularly suitable for time‑ or dose‑critical in vivo/in situ/operando (tomography) experiments because it requires only a single intensity measurement. However, the phase information of the wave field is lost during the measurement and must be recovered. Conventional algebraic and iterative methods often rely on specific approximations or boundary conditions that may not be met by many samples or experimental setups. In addition, they require manual tuning of reconstruction parameters by experts, making them less adaptable for complex or variable conditions. Here we present a self‑learning approach for solving the inverse problem of phase retrieval in the near‑field regime of Fresnel theory using a single intensity measurement (hologram). A physics‑informed generative adversarial network is employed to reconstruct both the phase and absorbance of the unpropagated wave field in the sample plane from a single hologram. Unlike most deep learning approaches for phase retrieval, our approach does not require paired, unpaired, or simulated training data. This significantly broadens the applicability of our approach, as acquiring or generating suitable training data remains a major challenge due to the wide variability in sample types and experimental configurations. The algorithm demonstrates robust and consistent performance across diverse imaging conditions and sample types, delivering quantitative, high‑quality reconstructions for both simulated data and experimental datasets acquired at beamline P05 at PETRA III (DESY, Hamburg), operated by Helmholtz‑Zentrum Hereon. Furthermore, it enables the simultaneous retrieval of both phase and absorption information.

Abstract:
We propose an efficient hybrid least squares/gradient descent method to accelerate DeepONet training. Since the output of DeepONet can be viewed as linear with respect to the last layer parameters of the branch network, these parameters can be optimized using a least squares (LS) solve, and the remaining hidden layer parameters are updated by means of gradient descent form. However, building the LS system for all possible combinations of branch and trunk inputs yields a prohibitively large linear problem that is infeasible to solve directly. To address this issue, our method decomposes the large LS system into two smaller, more manageable subproblems \unicodex2014 one for the branch network and one for the trunk network \unicodex2014 and solves them separately. This method is generalized to a broader type of L^2 loss with a regularization term for the last layer parameters, including the case of unsupervised learning with physics‑informed loss.

Abstract:
In this work, a novel Eig‑PIELM framework is proposed that extends physics‑informed extreme learning machine for an efficient and accurate solution of linear eigenvalue problems. The method reformulates the governing differential equations into a compact algebraic system solvable in a single step. Boundary conditions are enforced exactly via an algebraic projection onto the boundary‑admissible subspace, eliminating the computational overhead of penalty parameters, and backpropagation while preserving the computational advantages of extreme learning machines. The proposed framework is mesh‑free and yields both eigenvalues and mode shapes simultaneously in one linear solve. The robustness and accuracy of the proposed framework is demonstrated through a range of benchmark problems. We believe that the mesh‑free nature, solution structure and accuracy of Eig‑PIELM makes it particularly valuable for parametric studies in mechanical, acoustic, and electromechanical systems where rapid frequency spectrum analysis is critical.

Abstract:
Map matching and registration are essential tasks in robotics for localisation and integration of multi‑session or multi‑robot data. Traditional methods rely on cameras or LiDARs to capture visual or geometric information but struggle in challenging conditions like smoke or dust. Magnetometers, on the other hand, detect magnetic fields, revealing features invisible to other sensors and remaining robust in such environments. In this paper, we introduce Mag‑Match, a novel method for extracting and describing features in 3D magnetic vector field maps to register different maps of the same area. Our feature descriptor, based on higher‑order derivatives of magnetic field maps, is invariant to global orientation, eliminating the need for gravity‑aligned mapping. To obtain these higher‑order derivatives map‑wide given point‑wise magnetometer data, we leverage a physics‑informed Gaussian Process to perform efficient and recursive probabilistic inference of both the magnetic field and its derivatives. We evaluate Mag‑Match in simulated and real‑world experiments against a SIFT‑based approach, demonstrating accurate map‑to‑map, robot‑to‑map, and robot‑to‑robot transformations ‑ even without initial gravitational alignment.

Abstract:
Sodium‑ion batteries are a cost‑effective and sustainable alternative to lithium‑ion systems for large‑scale energy storage. Hard carbon (HC) anodes, composed of disordered graphitic and amorphous domains, offer high capacity but exhibit complex, poorly understood ion transport behavior. In particular, the relationship between local microstructure and sodium mobility remains unresolved, hindering rational performance optimization. Here, we introduce a data‑driven framework that combines machine‑learned interatomic potentials with molecular dynamics simulations to systematically investigate sodium diffusion across a broad range of carbon densities and sodium loadings. By computing per‑ion structural descriptors, we identify the microscopic factors that govern ion transport. Unsupervised learning uncovers distinct diffusion modes, including hopping, clustering, and void trapping, while supervised analysis highlights tortuosity and NaNa coordination as primary determinants of mobility. Correlation mapping further connects these transport regimes to processing variables such as bulk density and sodium content. This physics‑informed approach establishes quantitative structure‑transport relationships that capture the heterogeneity of disordered carbon. Our findings deliver mechanistic insights into sodium‑ion dynamics and provide actionable design principles for engineering high‑performance HC anodes in next‑generation battery systems.

Abstract:
Guidance of generative models is typically achieved by modifying the probability flow vector field through the addition of a guidance field. In this paper, we instead propose the Source‑Guided Flow Matching (SGFM) framework, which modifies the source distribution directly while keeping the pre‑trained vector field intact. This reduces the guidance problem to a well‑defined problem of sampling from the source distribution. We theoretically show that SGFM recovers the desired target distribution exactly. Furthermore, we provide bounds on the Wasserstein error for the generated distribution when using an approximate sampler of the source distribution and an approximate vector field. The key benefit of our approach is that it allows the user to flexibly choose the sampling method depending on their specific problem. To illustrate this, we systematically compare different sampling methods and discuss conditions for asymptotically exact guidance. Moreover, our framework integrates well with optimal flow matching models since the straight transport map generated by the vector field is preserved. Experimental results on synthetic 2D benchmarks, physics‑informed generative tasks, and imaging inverse problems demonstrate the effectiveness and flexibility of the proposed framework.

Abstract:
Perceptualizing tool interactions with deformable structures in surgical procedures remains challenging, as unimodal visualization techniques often fail to capture the complexity of these interactions due to constraints such as occlusion and limited depth perception. This paper presents a novel approach to augment tool navigation in mixed reality environments by providing auditory representations of tool‑tissue dynamics, particularly for interactions with soft tissue. BioSonix, a physics‑informed design framework, utilizes tissue displacements in 3D space to compute excitation forces for a sound model encoding tissue properties such as stiffness and density. Biomechanical simulations were employed to model particle displacements resulting from tool‑tissue interactions, establishing a robust foundation for the method. An optimization approach was used to define configurations for capturing diverse interaction scenarios with varying tool trajectories. Experiments were conducted to validate the accuracy of the sound‑displacement mappings. Additionally, two user studies were performed: the first involved two clinical professionals (a neuroradiologist and a cardiologist), who confirmed the method's impact and achieved high task accuracy; the second included 22 biomedical experts, who demonstrated high discrimination accuracy in tissue differentiation and targeting tasks. The results revealed a strong correlation between tool‑tissue dynamics and their corresponding auditory profiles, highlighting the potential of these sound representations to enhance the intuitive understanding of complex interactions.

Abstract:
Silver released from TRISO fuel particles can migrate through the SiC layer and deposit on reactor components, posing radiation hazards and operational challenges. Despite numerous proposed mechanisms, the precise pathway of silver transport through intact 3C‑SiC remains unresolved. We present a physics‑informed model for estimating the effective diffusivity of silver in polycrystalline 3C‑SiC. Molecular dynamics (MD) simulations yield diffusivities for Σ3 and Σ9 grain boundaries (GBs), while literature values are used for other GB types and the bulk. These are combined using a bounds‑averaging approach accounting for distinct GB transport properties. Bayesian inference of experimental data provides credible intervals for effective Arrhenius parameters and reveals a correlation between activation energy and pre‑exponential factor. Although the homogenized model captures GB‑mediated transport mechanisms, it overpredicts silver diffusivity relative to experiments. To resolve this, a multiplicative correction based on reversible trapping at nano‑pores is introduced. It is derived from first principles and is shown to reproduce observed transport behavior. Sensitivity analysis identified trap desorption energy and Σ9 GB diffusivity as dominant factors influencing Ag transport. The resulting framework provides a mechanistic description of Ag transport suitable for integration into higher‑scale fuel performance models.

Abstract:
The Macroscopic Fundamental Diagram is a popular tool used to describe traffic dynamics in an aggregated way, with applications ranging from traffic control to incident analysis. However, estimating the MFD for a given network requires large numbers of loop detectors, which is not always available in practise. This article proposes a framework to alleviate the data scarcity challenge harnessing Meta‑Learning, a subcategory of Machine Learning that trains models to understand and adapt to new tasks on their own. We use Meta‑Learning to identify and exploit transferable patterns from data‑rich cities to cities where not enough data is available to estimate the MFD. The developed model is trained and tested by leveraging data from multiple cities and exploiting it to model the MFD of other cities with different shares of detectors and topological structures. The proposed Meta‑Learning framework is applied to an ad‑hoc Multi‑Task Physics‑Informed Neural Network, specifically designed to estimate the MFD. Results show an average MAE improvement in flow prediction of around 50% across cities (depending on the subset of loop detectors tested). The Meta‑Learning framework thus successfully generalises across diverse urban settings and improves performance on cities with limited data, demonstrating the potential of using Meta‑Learning when a limited number of detectors is available. We directly test this assumption by applying the Meta‑Learning outputs to unseen cities to simulate a real‑life application scenario and the wide applicability of the proposed methodology. Finally, the proposed framework is validated against traditional Transfer Learning approaches and tested with FitFun, a model for FD estimation from the literature, to prove its transferability.

Abstract:
Designing shape memory alloys (SMAs) that meet performance targets while remaining affordable and sustainable is a complex challenge. In this work, we focus on optimizing SMA compositions to achieve a desired martensitic start temperature (Ms) while minimizing cost. To do this, we use machine learning models as surrogate predictors and apply numerical optimization methods to search for suitable alloy combinations. We trained two types of machine learning models, a tree‑based ensemble and a neural network, using a dataset of experimentally characterized alloys and physics‑informed features. The tree‑based model was used with a derivative‑free optimizer (COBYLA), while the neural network, which provides gradient information, was paired with a gradient‑based optimizer (TRUST‑CONSTR). Our results show that while both models predict Ms with similar accuracy, the optimizer paired with the neural network finds better solutions more consistently. COBYLA often converged to suboptimal results, especially when the starting guess was far from the target. The TRUST‑CONSTR method showed more stable behavior and was better at reaching alloy compositions that met both objectives. This study demonstrates a practical approach to exploring new SMA compositions by combining physics‑informed data, machine learning models, and optimization algorithms. Although the scale of our dataset is smaller than simulation‑based efforts, the use of experimental data improves the reliability of the predictions. The approach can be extended to other materials where design trade‑offs must be made with limited data.

Abstract:
Reward machines (RMs) provide a structured way to specify non‑Markovian rewards in reinforcement learning (RL), thereby improving both expressiveness and programmability. Viewed more broadly, they separate what is known about the environment, captured by the reward mechanism, from what remains unknown and must be discovered through sampling. This separation supports techniques such as counterfactual experience generation and reward shaping, which reduce sample complexity and speed up learning. We introduce physics‑informed reward machines (pRMs), a symbolic machine designed to express complex learning objectives and reward structures for RL agents, thereby enabling more programmable, expressive, and efficient learning. We present RL algorithms capable of exploiting pRMs via counterfactual experiences and reward shaping. Our experimental results show that these techniques accelerate reward acquisition during the training phases of RL. We demonstrate the expressiveness and effectiveness of pRMs through experiments in both finite and continuous physical environments, illustrating that incorporating pRMs significantly improves learning efficiency across several control tasks.

Abstract:
Origami‑inspired structures provide unprecedented opportunities for creating lightweight, deployable systems with programmable mechanical responses. However, their design remains challenging due to complex nonlinear mechanics, multistability, and the need for precise control of deployment forces. Here, we present a physics‑informed neural network (PINN) framework for both forward prediction and inverse design of conical Kresling origami (CKO) without requiring pre‑collected training data. By embedding mechanical equilibrium equations directly into the learning process, the model predicts complete energy landscapes with high accuracy while minimizing non‑physical artifacts. The inverse design routine specifies both target stable‑state heights and separating energy barriers, enabling freeform programming of the entire energy curve. This capability is extended to hierarchical CKO assemblies, where sequential layer‑by‑layer deployment is achieved through programmed barrier magnitudes. Finite element simulations and experiments on physical prototypes validate the designed deployment sequences and barrier ratios, confirming the robustness of the approach. This work establishes a versatile, data‑free route for programming complex mechanical energy landscapes in origami‑inspired metamaterials, offering broad potential for deployable aerospace systems, morphing structures, and soft robotic actuators.

Abstract:
The solubility of CO2 in formation brines plays a critical role in the efficiency of carbon capture and storage (CCS) operations. It is strongly influenced by pressure, temperature, and brine composition. Various experimental studies and modeling approaches have been developed to estimate CO2 solubility under wide ranges of pressure, temperature, and salinities. This work makes three key contributions. First, we present an extensive literature review of experimental, theoretical, and simulation‑based approaches for measuring and predicting CO2 solubility across a wide range of conditions and also a discussion of how the different parameters affect solubility. Second, we introduce an improved set of temperature‑dependent Pitzer interaction parameters, yielding up to a 76% reduction in average absolute deviation compared to conventional values in the geochemical simulation software PHREEQC. Third, we develop a physics‑informed machine learning model that integrates thermodynamic intuition with data‑driven learning, achieving a 14% reduction in prediction error over the state‑of‑the‑art and up to 40% improvement at high salinities. Together, these advances provide a robust and accurate framework for predicting CO2 solubility, supporting more reliable CCS design and deployment.

Abstract:
Physics‑informed neural networks approach the approximation of differential equations by directly incorporating their structure and given conditions in a loss function. This enables conditions like, e.g., invariants to be easily added during the modelling phase. In addition, the approach can be considered as mesh free and can be utilised to compute solutions on arbitrary grids after the training phase. Therefore, physics‑informed neural networks are emerging as a promising alternative to solving differential equations with methods from numerical mathematics. However, their performance highly depends on a large variety of factors. In this paper, we systematically investigate and evaluate a core component of the approach, namely the training point distribution. We test two ordinary and two partial differential equations with five strategies for training data generation and shallow network architectures, with one and two hidden layers. In addition to common distributions, we introduce sine‑based training points, which are motivated by the construction of Chebyshev nodes. The results are challenged by using certain parameter combinations like, e.g., random and fixed‑seed weight initialisation for reproducibility. The results show the impact of the training point distributions on the solution accuracy and we find evidence that they are connected to the characteristics of the differential equation.

Abstract:
Transformer neural networks are increasingly used for physics‑based problems. In data‑driven PDE surrogates, training samples from varying boundary and initial conditions can cause erratic losses and spiky gradients; in physics‑informed neural networks (PINNs), stiff composite losses amplify this effect. We introduce Kourkoutas‑Beta, an Adam‑style optimizer where the fixed second‑moment discount beta2 is replaced by a layer‑wise dynamic value driven by a bounded ``sunspike'' ratio: the current pooled gradient norm divided by an exponential moving average (EMA) of past norms, squashed to the interval [0,1). Spikes lower beta2 toward beta2_min; calm phases keep it near beta2_max. Options include leaky‑AMSGrad (decay), trust‑region clipping (max_ratio), adaptive tiny terms, and several bias‑correction modes ``none'', ``beta2max'', ``exact'). With all features off and bias_correction=``none'', the method is exactly Adam. We test on four settings: (i) a Transformer PDE surrogate (Heat2D), (ii) a 3D PINN for heat conduction (Heat3D), (iii) a lightweight MLX synthetic task with jitter and rare‑trigger bursts, and (iv) a character‑level Transformer on 30 MB of enwik8 (small‑enwik8). Kourkoutas‑Beta improves stability and final loss versus fixed‑beta2 Adam. On small‑enwik8 it lowers bits‑per‑character by about 38% vs Adam‑0.95 and about 58% vs Adam‑0.999 over 10 seeds, with smaller variance. The method remains drop‑in, with runtime overhead comparable to Adam in testbeds A‑C and within single‑digit percent in testbed D. It preserves Adam‑style convergence guarantees while improving robustness under spiky gradients.

Abstract:
Dynamic control of soft continuum robots (SCRs) holds great potential for expanding their applications, but remains a challenging problem due to the high computational demands of accurate dynamic models. While data‑driven approaches like Koopman‑operator‑based methods have been proposed, they typically lack adaptability and cannot reconstruct the full robot shape, limiting their applicability. This work introduces a real‑time‑capable nonlinear model‑predictive control (MPC) framework for SCRs based on a domain‑decoupled physics‑informed neural network (DD‑PINN) with adaptable bending stiffness. The DD‑PINN serves as a surrogate for the dynamic Cosserat rod model with a speed‑up factor of 44000. It is also used within an unscented Kalman filter for estimating the model states and bending compliance from end‑effector position measurements. We implement a nonlinear evolutionary MPC running at 70 Hz on the GPU. In simulation, it demonstrates accurate tracking of dynamic trajectories and setpoint control with end‑effector position errors below 3 mm (2.3% of the actuator's length). In real‑world experiments, the controller achieves similar accuracy and accelerations up to 3.55 m/s2.

Abstract:
We present a hybrid surrogate model for electric vehicle parameter estimation and power consumption. We combine our novel architecture Spectral Parameter Operator built on a Fourier Neural Operator backbone for global context and a differentiable physics module in the forward pass. From speed and acceleration alone, it outputs time‑varying motor and regenerative braking efficiencies, as well as aerodynamic drag, rolling resistance, effective mass, and auxiliary power. These parameters drive a physics‑embedded estimate of battery power, eliminating any separate physics‑residual loss. The modular design lets representations converge to physically meaningful parameters that reflect the current state and condition of the vehicle. We evaluate on real‑world logs from a Tesla Model 3, Tesla Model S, and the Kia EV9. The surrogate achieves a mean absolute error of 0.2kW (about 1% of average traction power at highway speeds) for Tesla vehicles and about 0.8kW on the Kia EV9. The framework is interpretable, and it generalizes well to unseen conditions, and sampling rates, making it practical for path optimization, eco‑routing, on‑board diagnostics, and prognostics health management.

Abstract:
Traffic state estimation (TSE) fundamentally involves solving high‑dimensional spatiotemporal partial differential equations (PDEs) governing traffic flow dynamics from limited, noisy measurements. While Physics‑Informed Neural Networks (PINNs) enforce PDE constraints point‑wise, this paper adopts a physics‑informed deep operator network (PI‑DeepONet) framework that reformulates TSE as an operator learning problem. Our approach trains a parameterized neural operator that maps sparse input data to the full spatiotemporal traffic state field, governed by the traffic flow conservation law. Crucially, unlike PINNs that enforce PDE constraints point‑wise, PI‑DeepONet integrates traffic flow conservation model and the fundamental diagram directly into the operator learning process, ensuring physical consistency while capturing congestion propagation, spatial correlations, and temporal evolution. Experiments on the NGSIM dataset demonstrate superior performance over state‑of‑the‑art baselines. Further analysis reveals insights into optimal function generation strategies and branch network complexity. Additionally, the impact of input function generation methods and the number of functions on model performance is explored, highlighting the robustness and efficacy of proposed framework.

Abstract:
Electrochemical corrosion significantly reduces the durability of electrodes in water electrolyzers, adversely affecting hydrogen (H_2) production and cell efficiency. Current theoretical models inadequately assess corrosion behaviors in alkaline water electrolyzers. To address this, we developed a physics‑informed electrochemical corrosion model evaluating the corrosion characteristics of cathodes in alkaline systems, accounting for factors such as exchange current density (J_0), redox potential (E_0), Gibbs free energy of hydrogen adsorption (ΔG_\rm H), electrolyte concentration (C), system pressure (P), and temperature (T). The model calculates metrics including corrosion potential (E_\rm corr), corrosion current density (J_\rm corr), and corrosion rate (C_R). Our findings from potentiodynamic polarization indicate that gold (Au) shows the highest durability, while copper (Cu) and nickel (Ni) are promising cost‑effective alternatives. This work enhances the understanding of corrosion dynamics, contributing to the design of more efficient electrolyzer cells for hydrogen production.

Abstract:
We present a novel interdisciplinary framework that bridges synchronization theory and multi‑agent AI systems by adapting the Kuramoto model to describe the collective dynamics of heterogeneous AI agents engaged in complex task execution. By representing AI agents as coupled oscillators with both phase and amplitude dynamics, our model captures essential aspects of agent specialization, influence, and communication within networked systems. We introduce an order parameter to quantify the degree of coordination and synchronization, providing insights into how coupling strength, agent diversity, and network topology impact emergent collective behavior. Furthermore, we formalize a detailed correspondence between Chain‑of‑Thought prompting in AI reasoning and synchronization phenomena, unifying human‑like iterative problem solving with emergent group intelligence. Through extensive simulations on all‑to‑all and deterministic scale‑free networks, we demonstrate that increased coupling promotes robust synchronization despite heterogeneous agent capabilities, reflecting realistic collaborative AI scenarios. Our physics‑informed approach establishes a rigorous mathematical foundation for designing, analyzing, and optimizing scalable, adaptive, and interpretable multi‑agent AI systems. This work opens pathways for principled orchestration of agentic AI and lays the groundwork for future incorporation of learning dynamics and adaptive network architectures to further enhance system resilience and efficiency.

Abstract:
Tissue mechanics‑‑stiffness, density and impedance contrast‑‑are broadly informative biomarkers across diseases, yet routine CT, MRI, and B‑mode ultrasound rarely quantify them directly. While ultrasound tomography (UT) is intrinsically suited to in‑vivo biomechanical assessment by capturing transmitted and reflected wavefields, efficient and accurate full‑wave scattering models remain a bottleneck. Here, we introduce a generative neural physics framework that fuses generative models with physics‑informed partial differential equation (PDE) solvers to produce rapid, high‑fidelity 3D quantitative imaging of tissue mechanics. A compact neural surrogate for full‑wave propagation is trained on limited cross‑modality data, preserving physical accuracy while enabling efficient inversion. This enables, for the first time, accurate and efficient quantitative volumetric imaging of in vivo human breast and musculoskeletal tissues in under ten minutes, providing spatial maps of tissue mechanical properties not available from conventional reflection‑mode or standard UT reconstructions. The resulting images reveal biomechanical features in bone, muscle, fat, and glandular tissues, maintaining structural resolution comparable to 3T MRI while providing substantially greater sensitivity to disease‑related tissue mechanics.

Abstract:
Seismicity catalogs are larger than ever due to an explosion of techniques for enhanced earthquake detection and an abundance of high‑quality datasets. Bayesian inference is an appealing framework for locating earthquakes due to its ability to propagate and quantify uncertainty into the inversion results, but traditional methods do not scale well to high‑dimensional parameter spaces, making them unsuitable for double‑difference relocation where the number of parameters can reach the millions. Here we introduce SPIDER, a scalable Bayesian inference framework for double‑difference hypocenter relocation. SPIDER uses a physics‑informed neural network Eikonal solver together with a highly efficient sampler called Stochastic Gradient Langevin Dynamics to generate posterior samples jointly for entire seismicity catalogs. We show that traditional double‑difference relocation formulations neglect residual correlation between observations with common events, which biases uncertainty estimates. Our formulation is designed to whiten this residual correlation, and is readily parallelized over multiple GPUs for enhanced computational efficiency. We demonstrate the capabilities of SPIDER on a rigorous synthetic seismicity catalog and three real data catalogs from California and Japan. We introduce several ways to analyze high‑dimensional posterior distributions to aid in scientific interpretation and evaluation.

Abstract:
We present a framework that combines physics‑informed neural networks (PINNs) with Markov Chain Monte Carlo (MCMC) inference to constrain dynamical dark energy models using the Pantheon+ Type Ia supernova compilation. First, we train a physics‑informed neural network to learn the solution of the Friedmann equation and accurately reproduce the matter density term x_m(z) = Omega_m,0 (1+z)^3 across a range of Omega_m,0. For each of five two‑parameter equation‑of‑state (EoS) forms: Chevallier‑Polarski‑Linder (CPL), Barboza‑Alcaniz (BA), Jassal‑Bagla‑Padmanabhan (JBP), Linear‑z, and Logarithmic‑z, we derive the analytic dark energy factor x_de(z), embed the trained surrogate within a GPU‑accelerated likelihood pipeline, and sample the posterior of (h0, Omega_m,0, w0, wa, M0) using the emcee ensemble sampler with the full Pantheon+ covariance. All parameterizations remain consistent with a cosmological constant (w0 = ‑1, wa = 0) at the 95% credible level, with the tightest bounds from the CPL form. While the surrogate does not reduce computation time for a single run in simple models, it becomes advantageous for repeated analyses of the same EoS or for models with expensive likelihood evaluations, and can be shared as a reusable tool with different datasets within the training range of SNe redshifts. This flexibility makes the approach a scalable tool for future cosmological inference, especially in regimes where conventional ODE‑based methods are computationally prohibitive.

Abstract:
Accurate channel state information (CSI) is a critical bottleneck in modern wireless networks, with pilot overhead consuming 11% to 21% of transmission bandwidth and feedback delays causing severe throughput degradation under mobility. Addressing this requires rethinking how neural fields represent coherent wave phenomena. This work introduces neural Gaussian radio fields (\textcolorstanfordrednGRF), a physics‑informed framework that fundamentally reframes neural field design by replacing view‑dependent rasterization with direct complex‑valued aggregation in 3D space. This approach natively models wave superposition rather than visual occlusion. The architectural shift transforms the learning objective from function‑fitting to source‑recovery, a well‑posed inverse problem grounded in electromagnetic theory. While demonstrated for wireless channel estimation, the core principle of explicit primitive‑based fields with physics‑constrained aggregation extends naturally to any coherent wave‑based domain, including acoustic propagation, seismic imaging, and ultrasound reconstruction. Evaluations show that the inductive bias of \textcolorstanfordrednGRF achieves 10.9 dB higher prediction SNR than state‑of‑the‑art methods with 220× faster inference (1.1 ms vs. 242 ms), 18× lower measurement density, and 180× faster training. For large‑scale outdoor environments where implicit methods fail, \textcolorstanfordrednGRF achieves 28.32 dB SNR, demonstrating that structured representations supplemented by domain physics can fundamentally outperform generic deep learning architectures.

Abstract:
Nuclear coherent population transfer (NCPT) offers numerous potential applications, particularly in next‑generation nuclear clocks and nuclear batteries. However, the realization of high fidelity, fast operation, and low energy consumption in NCPT remains so far challenging. Here, we employ physics‑informed neural networks (PINNs) to the population transfer in an open three‑level nuclear system with spontaneous emission. The method embeds the system's control equations and boundary conditions into the loss function, thereby enabling the automatic learning of optimal laser pulse sequences that drive highly efficient population transfer. We take a short‑lived excited state of ^172\mathrmYb and a long‑lived state of ^229\mathrmTh as representative examples, and systematically compare the performance of the PINNs approach with three conventional control strategies. We show that PINNs can achieve higher transfer efficiency with smaller pulse areas and shorter durations across different lifetime regimes. Our results provide a new perspective to overcome the lifetime limitation and enhance the efficiency of nuclear state transfer.

Abstract:
This paper presents a data‑driven, nested Operator Inference (OpInf) approach for learning physics‑informed reduced‑order models (ROMs) from snapshot data of high‑dimensional dynamical systems. The approach exploits the inherent hierarchy within the reduced space to iteratively construct initial guesses for the OpInf learning problem that prioritize the interactions of the dominant modes. The initial guess computed for any target reduced dimension corresponds to a ROM with provably smaller or equal snapshot reconstruction error than with standard OpInf. Moreover, our nested OpInf algorithm can be warm‑started from previously learned models, enabling versatile application scenarios involving dynamic basis and model form updates. We demonstrate the performance of our algorithm on a cubic heat conduction problem, with nested OpInf achieving a four times smaller error than standard OpInf at a comparable offline time. Further, we apply nested OpInf to a large‑scale, parameterized model of the Greenland ice sheet where, despite model form approximation errors, it learns a ROM with, on average, 3% error and computational speed‑up factor above 19,000.

Abstract:
We propose an unsupervised anomaly detection approach based on a physics‑informed diffusion model for multivariate time series data. Over the past years, diffusion model has demonstrated its effectiveness in forecasting, imputation, generation, and anomaly detection in the time series domain. In this paper, we present a new approach for learning the physics‑dependent temporal distribution of multivariate time series data using a weighted physics‑informed loss during diffusion model training. A weighted physics‑informed loss is constructed using a static weight schedule. This approach enables a diffusion model to accurately approximate underlying data distribution, which can influence the unsupervised anomaly detection performance. Our experiments on synthetic and real‑world datasets show that physics‑informed training improves the F1 score in anomaly detection; it generates better data diversity and log‑likelihood. Our model outperforms baseline approaches, additionally, it surpasses prior physics‑informed work and purely data‑driven diffusion models on a synthetic dataset and one real‑world dataset while remaining competitive on others.

Abstract:
Blood flow imaging provides important information for hemodynamic behavior within the vascular system and plays an essential role in medical diagnosis and treatment planning. However, obtaining high‑quality flow images remains a significant challenge. In this work, we address the problem of denoising flow images that may suffer from artifacts due to short acquisition times or device‑induced errors. We formulate this task as an optimization problem, where the objective is to minimize the discrepancy between the modeled velocity field, constrained to satisfy the Navier‑Stokes equations, and the observed noisy velocity data. To solve this problem, we decompose it into two subproblems: a fluid subproblem and a geometry subproblem. The fluid subproblem leverages a Physics‑Informed Neural Network to reconstruct the velocity field from noisy observations, assuming a fixed domain. The geometry subproblem aims to infer the underlying flow region by optimizing a quasi‑conformal mapping that deforms a reference domain. These two subproblems are solved in an alternating Gauss‑Seidel fashion, iteratively refining both the velocity field and the domain. Upon convergence, the framework yields a high‑quality reconstruction of the flow image. We validate the proposed method through experiments on synthetic flow data in a converging channel geometry under varying levels of Gaussian noise, and on real‑like flow data in an aortic geometry with signal‑dependent noise. The results demonstrate the effectiveness and robustness of the approach. Additionally, ablation studies are conducted to assess the influence of key hyperparameters.

Abstract:
This study proposes a self‑optimization physics‑informed Fourier‑features randomized neural network (SO‑PIFRNN) framework, which significantly improves the numerical solving accuracy of PDEs through hyperparameter optimization mechanism. The framework employs a bi‑level optimization architecture: the outer‑level optimization utilizes a multi‑strategy collaborated particle swarm optimization (MSC‑PSO) algorithm to search for optimal hyperparameters of physics‑informed Fourier‑features randomized neural network, while the inner‑level optimization determines the output layer weights of the neural network via the least squares method. The core innovation of this study is embodied in the following three aspects: First, the Fourier basis function activation mechanism is introduced in the hidden layer of neural network, which significantly enhances the ability of the network to capture multi‑frequency components of the solution. Secondly, a novel derivative neural network method is proposed, which improves the calculation accuracy and efficiency of PIFRNN method. Finally, the MSC‑PSO algorithm of the hybrid optimization strategy is designed to improve the global search ability and convergence accuracy through the synergistic effect of dynamic parameter adjustment, elitist and mutation strategies. Through a series of numerical experiments, including multiscale equations in complex regions, high‑order equations, high‑dimensional equations and nonlinear equations, the validity of SO‑PIFRNN is verified. The experimental results affirm that SO‑PIFRNN exhibits superior approximation accuracy and frequency capture capability.

Abstract:
Accurate prediction of electronic band structures in two‑dimensional materials remains a fundamental challenge, with existing methods struggling to balance computational efficiency and physical accuracy. We present the Symmetry‑Constrained Multi‑Scale Physics‑Informed Neural Network (SCMS‑PINN) v35, which directly learns graphene band structures while rigorously enforcing crystallographic symmetries through a multi‑head architecture. Our approach introduces three specialized ResNet‑6 pathways ‑‑ K‑head for Dirac physics, M‑head for saddle points, and General head for smooth interpolation ‑‑ operating on 31 physics‑informed features extracted from k‑points. Progressive Dirac constraint scheduling systematically increases the weight parameter from 5.0 to 25.0, enabling hierarchical learning from global topology to local critical physics. Training on 10,000 k‑points over 300 epochs achieves 99.99% reduction in training loss (34.597 to 0.003) with validation loss of 0.0085. The model predicts Dirac point gaps within 30.3 μeV of theoretical zero and achieves average errors of 53.9 meV (valence) and 40.5 meV (conduction) across the Brillouin zone. All twelve C_6v operations are enforced through systematic averaging, guaranteeing exact symmetry preservation. This framework establishes a foundation for extending physics‑informed learning to broader two‑dimensional materials for accelerated discovery.

Abstract:
T2 mapping in fetal brain MRI has the potential to improve characterization of the developing brain, especially at mid‑field (0.55T), where T2 decay is slower. However, this is challenging as fetal MRI acquisition relies on multiple motion‑corrupted stacks of thick slices, requiring slice‑to‑volume reconstruction (SVR) to estimate a high‑resolution (HR) 3D volume. Currently, T2 mapping involves repeated acquisitions of these stacks at each echo time (TE), leading to long scan times and high sensitivity to motion. We tackle this challenge with a method that jointly reconstructs data across TEs, addressing severe motion. Our approach combines implicit neural representations with a physics‑informed regularization that models T2 decay, enabling information sharing across TEs while preserving anatomical and quantitative T2 fidelity. We demonstrate state‑of‑the‑art performance on simulated fetal brain and in vivo adult datasets with fetal‑like motion. We also present the first in vivo fetal T2 mapping results at 0.55T. Our study shows potential for reducing the number of stacks per TE in T2 mapping by leveraging anatomical redundancy.

Abstract:
Inverse scattering problems are critical in electromagnetic imaging and medical diagnostics but are challenged by their nonlinearity and diverse measurement scenarios. This paper proposes a physics‑informed deep contrast source inversion framework (DeepCSI) for fast and accurate medium reconstruction across various measurement conditions. Inspired by contrast source inversion (CSI) and neural operator methods, a residual multilayer perceptron (ResMLP) is employed to model current distributions in the region of interest under different transmitter excitations, effectively linearizing the nonlinear inverse scattering problem and significantly reducing the computational cost of traditional full‑waveform inversion. By modeling medium parameters as learnable tensors and utilizing a hybrid loss function that integrates state equation loss, data equation loss, and total variation regularization, DeepCSI establishes a fully differentiable framework for joint optimization of network parameters and medium properties. Compared with conventional methods, DeepCSI offers advantages in terms of simplicity and universal modeling capabilities for diverse measurement scenarios, including phase‑less and multi‑frequency observation. Simulations and experiments demonstrate that DeepCSI achieves high‑precision, robust reconstruction under full‑data, phaseless data, and multifrequency conditions, outperforming traditional CSI methods and providing an efficient and universal solution for complex inverse scattering problems.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs), yet they often fail to achieve accurate convergence in the H1 norm, especially in the presence of boundary approximation errors. In this work, we propose a novel method called Sobolev‑Stable Boundary Enforcement (SSBE), which redefines the boundary loss using Sobolev norms to incorporate boundary regularity directly into the training process. We provide rigorous theoretical analysis demonstrating that SSBE ensures bounded H1 error via a stability guarantee and derive generalization bounds that characterize its robustness under finite‑sample regimes. Extensive numerical experiments on linear and nonlinear PDEs, including Poisson, heat, and elliptic problems, show that SSBE consistently outperforms standard PINNs in terms of both relative L2 and H1 errors, even in high‑dimensional settings. The proposed approach offers a principled and practical solution for improving gradient fidelity and overall solution accuracy in neural network based PDE solvers.

Abstract:
Dynamic mode decomposition (DMD) is a data‑driven method for estimating the dynamics of a discrete dynamical system. This paper proposes a tensor‑based approach to DMD for applications in which the states can be viewed as tensors. Specifically, we use the \star_\boldsymbolM‑product framework for tensor decompositions which we demonstrate offers excellent compression compared to matrix‑based methods and can be implemented in a computationally efficient manner. We show how the proposed approach is connected to the traditional DMD and physics‑informed DMD frameworks. We give a computational framework for computing the tensor‑based DMD and detail the computational costs. We also give a randomized algorithm that enables efficient \star_\boldsymbolM‑DMD computations in the streaming setting. The numerical results show that the proposed method achieves equal or better accuracy for the same storage compared to the standard DMD on these examples and is more efficient to compute.

Abstract:
Solving partial differential equations (PDEs) efficiently and accurately remains a cornerstone challenge in science and engineering, especially for problems involving complex geometries and limited labeled data. We introduce a Physics‑ and Geometry‑ Aware Spatio‑Spectral Graph Neural Operator (πG‑Sp^2GNO) for learning the solution operators of time‑independent and time‑dependent PDEs. The proposed approach first improves upon the recently developed Sp^2GNO by enabling geometry awareness and subsequently exploits the governing physics to learn the underlying solution operator in a simulation‑free setup. While the spatio‑spectral structure present in the proposed architecture allows multiscale learning, two separate strategies for enabling geometry awareness is introduced in this paper. For time dependent problems, we also introduce a novel hybrid physics informed loss function that combines higher‑order time‑marching scheme with upscaled theory inspired stochastic projection scheme. This allows accurate integration of the physics‑information into the loss function. The performance of the proposed approach is illustrated on number of benchmark examples involving regular and complex domains, variation in geometry during inference, and time‑independent and time‑dependent problems. The results obtained illustrate the efficacy of the proposed approach as compared to the state‑of‑the‑art physics‑informed neural operator algorithms in the literature.

Abstract:
Traffic forecasting is essential for intelligent transportation systems. Accurate forecasting relies on continuous observations collected by traffic sensors. However, due to high deployment and maintenance costs, not all regions are equipped with such sensors. This paper aims to forecast for regions without traffic sensors, where the lack of historical traffic observations challenges the generalisability of existing models. We propose a model named GenCast, the core idea of which is to exploit external knowledge to compensate for the missing observations and to enhance generalisation. We integrate physics‑informed neural networks into GenCast, enabling physical principles to regularise the learning process. We introduce an external signal learning module to explore correlations between traffic states and external signals such as weather conditions, further improving model generalisability. Additionally, we design a spatial grouping module to filter localised features that hinder model generalisability. Extensive experiments show that GenCast consistently reduces forecasting errors on multiple real‑world datasets.

Abstract:
Physics‑informed neural networks (PINNs) have attracted considerable attention for their ability to integrate partial differential equation priors into deep learning frameworks; however, they often exhibit limited predictive accuracy when applied to complex problems. To address this issue, we propose LNN‑PINN, a physics‑informed neural network framework that incorporates a liquid residual gating architecture while preserving the original physics modeling and optimization pipeline to improve predictive accuracy. The method introduces a lightweight gating mechanism solely within the hidden‑layer mapping, keeping the sampling strategy, loss composition, and hyperparameter settings unchanged to ensure that improvements arise purely from architectural refinement. Across four benchmark problems, LNN‑PINN consistently reduced RMSE and MAE under identical training conditions, with absolute error plots further confirming its accuracy gains. Moreover, the framework demonstrates strong adaptability and stability across varying dimensions, boundary conditions, and operator characteristics. In summary, LNN‑PINN offers a concise and effective architectural enhancement for improving the predictive accuracy of physics‑informed neural networks in complex scientific and engineering problems.

Abstract:
We consider the evolution of linear matter perturbations in the context of the standard cosmological model (ΛCDM) and a phenomenological modified gravity model. We use the physics‑informed neural network (PINN) bundle method, which allows to integrate differential systems as an alternative to the traditional numerical method. We apply the PINN bundle method to the equation that describes the matter perturbation evolution, to compare its outcomes with recent data on structure growth, fσ_8. Unlike our previous works, we can calculate a bound on the error of this observable without using the numerical solution of the equation. For this, we use a method developed previously by ourselves to calculate an exact bound on the PINN‑based solution using only the outcomes of the network and its residual. On the other hand, the use of an updated data set allows us to obtain more stringent constraints on the plane Ω_m‑σ_8 than previous works.

Abstract:
Humans possess an exceptional ability to imagine 4D scenes, encompassing both motion and 3D geometry, from a single still image. This ability is rooted in our accumulated observations of similar scenes and an intuitive understanding of physics. In this paper, we aim to replicate this capacity in neural networks, specifically focusing on natural fluid imagery. Existing methods for this task typically employ simplistic 2D motion estimators to animate the image, leading to motion predictions that often defy physical principles, resulting in unrealistic animations. Our approach introduces a novel method for generating 4D scenes with physics‑consistent animation from a single image. We propose the use of a physics‑informed neural network that predicts motion for each surface point, guided by a loss term derived from fundamental physical principles, including the Navier‑Stokes equations. To capture appearance, we predict feature‑based 3D Gaussians from the input image and its estimated depth, which are then animated using the predicted motions and rendered from any desired camera perspective. Experimental results highlight the effectiveness of our method in producing physically plausible animations, showcasing significant performance improvements over existing methods. Our project page is https://physfluid.github.io/ .

Abstract:
Microwave Tomography (MWT) aims to reconstruct the dielectric properties of tissues from measured scattered electromagnetic fields. This inverse problem is highly nonlinear and ill‑posed, posing significant challenges for conventional optimization‑based methods, which, despite being grounded in physical models, often fail to recover fine structural details. Recent deep learning strategies, including end‑to‑end and post‑processing networks, have improved reconstruction quality but typically require large paired training datasets and may struggle to generalize. To overcome these limitations, we propose a physics‑informed hybrid framework that integrates diffusion models as learned regularization within a data‑consistency‑driven variational scheme. Specifically, we introduce Single‑Step Diffusion Regularization (SSD‑Reg), a novel approach that embeds diffusion priors into the iterative reconstruction process, enabling the recovery of complex anatomical structures without the need for paired data. SSD‑Reg maintains fidelity to both the governing physics and learned structural distributions, improving accuracy, stability, and robustness. Extensive experiments demonstrate that SSD‑Reg, implemented as a Plug‑and‑Play (PnP) module, provides a flexible and effective solution for tackling the ill‑posedness inherent in functional image reconstruction.

Abstract:
Hyperspectral imaging (HSI) is an advanced sensing modality that simultaneously captures spatial and spectral information, enabling non‑invasive, label‑free analysis of material, chemical, and biological properties. This Primer presents a comprehensive overview of HSI, from the underlying physical principles and sensor architectures to key steps in data acquisition, calibration, and correction. We summarize common data structures and highlight classical and modern analysis methods, including dimensionality reduction, classification, spectral unmixing, and AI‑driven techniques such as deep learning. Representative applications across Earth observation, precision agriculture, biomedicine, industrial inspection, cultural heritage, and security are also discussed, emphasizing HSI's ability to uncover sub‑visual features for advanced monitoring, diagnostics, and decision‑making. Persistent challenges, such as hardware trade‑offs, acquisition variability, and the complexity of high‑dimensional data, are examined alongside emerging solutions, including computational imaging, physics‑informed modeling, cross‑modal fusion, and self‑supervised learning. Best practices for dataset sharing, reproducibility, and metadata documentation are further highlighted to support transparency and reuse. Looking ahead, we explore future directions toward scalable, real‑time, and embedded HSI systems, driven by sensor miniaturization, self‑supervised learning, and foundation models. As HSI evolves into a general‑purpose, cross‑disciplinary platform, it holds promise for transformative applications in science, technology, and society.

Abstract:
Traffic state estimation (TSE) falls methodologically into three categories: model‑driven, data‑driven, and model‑data dual‑driven. Model‑driven TSE relies on macroscopic traffic flow models originated from hydrodynamics. Data‑driven TSE leverages historical sensing data and employs statistical models or machine learning methods to infer traffic state. Model‑data dual‑driven traffic state estimation attempts to harness the strengths of both aspects to achieve more accurate TSE. From the perspective of mathematical operator theory, TSE can be viewed as a type of operator that maps available measurements of inerested traffic state into unmeasured traffic state variables in real time. For the first time this paper proposes to study real‑time freeway TSE in the idea of physics‑informed deep operator network (PI‑DeepONet), which is an operator‑oriented architecture embedding traffic flow models based on deep neural networks. The paper has developed an extended architecture from the original PI‑DeepONet. The extended architecture is featured with: (1) the acceptance of 2‑D data input so as to support CNN‑based computations; (2) the introduction of a nonlinear expansion layer, an attention mechanism, and a MIMO mechanism; (3) dedicated neural network design for adaptive identification of traffic flow model parameters. A traffic state estimator built on the basis of this extended PI‑DeepONet architecture was evaluated with respect to a short freeway stretch of NGSIM and a large‑scale urban expressway in China, along with other four baseline TSE methods. The evaluation results demonstrated that this novel TSE method outperformed the baseline methods with high‑precision estimation results of flow and mean speed.

Abstract:
Rigorous error estimation is a fundamental topic in numerical analysis. With the increasing use of physics‑informed neural networks (PINNs) for solving partial differential equations, several approaches have been developed to quantify the associated prediction error. In this work, we build upon a semigroup‑based framework previously introduced by the authors for estimating the PINN error. While this estimator has so far been limited to academic examples ‑ due to the need to compute quantities related to input‑to‑state stability ‑ we extend its applicability to a significantly broader class of problems. This is accomplished by modifying the error bound and proposing numerical strategies to approximate the required stability parameters. The extended framework enables the certification of PINN predictions in more realistic scenarios, as demonstrated by a numerical study of Stokes flow around a cylinder.

Abstract:
Attitude control is a fundamental aspect of spacecraft operations. Model Predictive Control (MPC) has emerged as a powerful strategy for these tasks, relying on accurate models of the system dynamics to optimize control actions over a prediction horizon. In scenarios where physics models are incomplete, difficult to derive, or computationally expensive, machine learning offers a flexible alternative by learning the system behavior directly from data. However, purely data‑driven models often struggle with generalization and stability, especially when applied to inputs outside their training domain. To address these limitations, we investigate the benefits of incorporating Physics‑Informed Neural Networks (PINNs) into the learning of spacecraft attitude dynamics, comparing their performance with that of purely data‑driven approaches. Using a Real‑valued Non‑Volume Preserving (Real NVP) neural network architecture with a self‑attention mechanism, we trained several models on simulated data generated with the Basilisk simulator. Two training strategies were considered: a purely data‑driven baseline and a physics‑informed variant to improve robustness and stability. Our results demonstrate that the inclusion of physics‑based information significantly enhances the performance in terms of the mean relative error with the best architectures found by 27.08%. These advantages are particularly evident when the learned models are integrated into an MPC framework, where PINN‑based models consistently outperform their purely data‑driven counterparts in terms of control accuracy and robustness, and achieve improved settling times when compared to traditional MPC approaches, yielding improvements of up to 62%, when subject to observation noise and RWs friction.

Abstract:
Underwater explosion modeling faces a critical challenge of simultaneously resolving shock waves and gas‑liquid interfaces, as traditional methods struggle to balance accuracy and computational efficiency. To address this, we develop a physics‑informed neural network (PINN) framework featuring a dual‑network architecture, that one network learns flow‑field variables (pressure, density, velocity) from simulation data, while another network tracks the gas‑liquid interface despite lacking direct numerical solutions. Crucially, we introduce an interval‑constraint training strategy that penalizes interface deviations beyond grid spacing limits, paired with a physics‑preserving linear mapping of 1D spherical Euler equations to ensure consistency. Our results show that this approach accurately reconstructs spatiotemporal fields from coarse‑grid data, achieving superior computational efficiency over conventional CFD‑enabling rapid, mesh‑free blast‑load analysis for near/far‑field scenarios and extensibility to higher‑dimensional problems.

Abstract:
We present an evolved steerable version of the single‑tail Fish‑&‑Ribbon‑Inspired Small Swimming Harmonic roBot (FRISSHBot), a 59‑mg biologically inspired swimmer, which is driven by a new shape‑memory alloy (SMA)‑based bimorph actuator. The new FRISSHBot is controllable in the two‑dimensional (2D) space, which enabled the first demonstration of feedback‑controlled trajectory tracking of a single‑tail aquatic robot with onboard actuation at the subgram scale. These new capabilities are the result of a physics‑informed design with an enlarged head and shortened tail relative to those of the original platform. Enhanced by its design, this new platform achieves forward swimming speeds of up to 13.6 mm/s (0.38 Bl/s), which is over four times that of the original platform. Furthermore, when following 2D references in closed loop, the tested FRISSHBot prototype attains forward swimming speeds of up to 9.1 mm/s, root‑mean‑square (RMS) tracking errors as low as 2.6 mm, turning rates of up to 13.1 °/s, and turning radii as small as 10 mm.

Abstract:
In this paper, a physics‑informed multiresolution wavelet neural network (PIMWNN) method is proposed for solving partial differential equations (PDEs). This method uses the multiresolution wavelet neural network (MWNN) to approximate unknown functions, then substituting the MWNN into PDEs and training the MWNN by least‑squares algorithm. We apply the proposed method to various problems, including stationary/nonstationary advection, diffusion and advection‑diffusion problems, and linear/nonlinear time‑dependent problems. Numerical experiments show that the PIMWNN method can achieve higher accuracy and faster speed than Physics Informed Neural Networks (PINNs). Moreover, the PIMWNN method, being mesh‑free, can handle different boundary conditions easily and solve the time‑dependent problems efficiently. The proposed method is expected to solve the spectral bias problem in network training. These characteristics show the great potential of the PIMWNN method used in the field of numerical solving methods for PDEs.

Abstract:
Accurate and interpretable bearing fault classification is critical for ensuring the reliability of rotating machinery, particularly under variable operating conditions where domain shifts can significantly degrade model performance. This study proposes a physics‑informed multimodal convolutional neural network (CNN) with a late fusion architecture, integrating vibration and motor current signals alongside a dedicated physics‑based feature extraction branch. The model incorporates a novel physics‑informed loss function that penalizes physically implausible predictions based on characteristic bearing fault frequencies ‑ Ball Pass Frequency Outer (BPFO) and Ball Pass Frequency Inner (BPFI) ‑ derived from bearing geometry and shaft speed. Comprehensive experiments on the Paderborn University dataset demonstrate that the proposed physics‑informed approach consistently outperforms a non‑physics‑informed baseline, achieving higher accuracy, reduced false classifications, and improved robustness across multiple data splits. To address performance degradation under unseen operating conditions, three transfer learning (TL) strategies ‑ Target‑Specific Fine‑Tuning (TSFT), Layer‑Wise Adaptation Strategy (LAS), and Hybrid Feature Reuse (HFR) ‑ are evaluated. Results show that LAS yields the best generalization, with additional performance gains when combined with physics‑informed modeling. Validation on the KAIST bearing dataset confirms the framework's cross‑dataset applicability, achieving up to 98 percent accuracy. Statistical hypothesis testing further verifies significant improvements (p < 0.01) in classification performance. The proposed framework demonstrates the potential of integrating domain knowledge with data‑driven learning to achieve robust, interpretable, and generalizable fault diagnosis for real‑world industrial applications.

Abstract:
Chemical short‑range order (SRO) provides new opportunities for tuning alloy properties, but conventional computational thermodynamics frameworks such as CALPHAD, based on Bragg‑Williams mean‑field approximations, cannot properly describe SRO or order‑disorder transformations in multicomponent (\geq3) alloys. First‑principles approaches combined with the cluster variation method (CVM) or cluster expansion method (CEM) can capture SRO but suffer from high computational cost. Here we present a hybrid CVM‑CALPHAD framework with a thermodynamic solid solution model named as FYL‑CVM, enabled by the Fowler‑Yang‑Li (FYL) transform to reduce the number of variables required in free‑energy minimization. This achieves efficient modeling of SRO in multicomponent systems within the CALPHAD formalism. Benchmark tests on fcc AB binaries show that FYL‑CVM reproduces CVM phase diagrams with much higher efficiency, while non‑configurational contributions from vibrational, elastic, and electronic terms are also incorporated to capture their physical effects on order‑disorder boundaries. Applied to the Cu‑Au system, this method produces phase diagrams with experimental data in an efficient parameterization and elucidates the temperature‑composition dependence of SRO parameters via the SRO diagram. Its applicability to ternary alloys is also demonstrated for the Cu‑Au‑Ag system. Overall, this framework strikes a balance between accuracy and efficiency, extends CALPHAD to account for chemical SRO, and enables a comprehensive physics‑informed modeling of ordering phenomena. (This dissertation was submitted to the University of Virginia in 2023 as the author's doctoral research. For the original complete abstract, please refer to the PDF version.)

Abstract:
The Long Baseline Neutrino Facility (LBNF) at Fermilab will deliver a high‑intensity, multi‑megawatt neutrino beam to the Deep Underground Neutrino Experiment (DUNE), enabling precision tests of the three‑neutrino paradigm, CP violation searches, neutrino mass ordering determination, and supernova neutrino studies. In order to accelerate DUNE's physics reach and ensure robust beam operations, we propose an integrated AI‑driven framework with real‑time diagnostics and radiation‑hardened instrumentation. A physics‑informed digital twin is at the heart of this Real‑Time Beam Integrity Monitor. By reconstructing pion phase space from muon profiles and exploiting magnetic horn optic linearity, it enables spill‑by‑spill beam correction and flux stabilization. By using this approach, flux‑related systematics could be reduced from 5% to 1%, potentially accelerating the discovery of CP violations by four to six years. Complementing this, a US‑Japan R\&D effort will deploy a LGAD‑based muon monitor in the NuMI beamline. Time of Flight (ToF) measurements can be acquired with picosecond precision using this radiation‑hard system, enhancing sensitivity to horn chromatic effects. Simulations confirm strong responses to these effects. Machine learning models can predict beam quality and horn current with sub‑percent accuracy. This scalable, AI‑enabled strategy improves beam fidelity and reduces systematics, transforming high‑power accelerator operations.

Abstract:
Driven by recent advances in sensing and computing, deep reinforcement learning (DRL) technologies have shown great potential for addressing distribution system restoration (DSR) under uncertainty. However, their data‑intensive nature and reliance on the Markov Decision Process (MDP) assumption limit their ability to handle scenarios that require long‑term temporal dependencies or few‑shot and zero‑shot decision making. Emerging Decision Transformers (DTs), which leverage causal transformers for sequence modeling in DRL tasks, offer a promising alternative. However, their reliance on return‑to‑go (RTG) cloning and limited generalization capacity restricts their effectiveness in dynamic power system environments. To address these challenges, we introduce an innovative Dual‑Head Physics‑informed Graph Decision Transformer (DH‑PGDT) that integrates physical modeling, structural reasoning, and subgoal‑based guidance to enable scalable and robust DSR even in zero‑shot or few‑shot scenarios. DH‑PGDT features a dual‑head physics‑informed causal transformer architecture comprising Guidance Head, which generates subgoal representations, and Action Head, which uses these subgoals to generate actions independently of RTG. It also incorporates an operational constraint‑aware graph reasoning module that encodes power system topology and operational constraints to generate a confidence‑weighted action vector for refining DT trajectories. This design effectively improves generalization and enables robust adaptation to unseen scenarios. While this work focuses on DSR, the underlying computing model of the proposed PGDT is broadly applicable to sequential decision making across various power system operations and other complex engineering domains.

Abstract:
The measurement of black hole spin is considered one of the key problems in relativistic astrophysics. Existing methods, such as continuum fitting, X‑ray reflection spectroscopy and quasi‑periodic oscillation analysis, have systematic limitations in accuracy, interpretability and scalability. In this work, a hybrid approach is proposed in which theoretical models based on the Teukolsky formalism are integrated with Physics‑Informed Neural Networks (PINNs). A PINN model is developed to solve the linearized spin problem in the scalar case, with physical constraints directly embedded into the training process. Annotated data are not required; instead, the model is trained using the differential operator and boundary conditions as supervision. It is demonstrated that the PINN converges reliably, with residual loss values below 1e‑7 and a root mean squared error (RMSE) of the order of 1e‑6 (final approx 5.4 x 1e‑8). Benchmarking results indicate that the proposed method outperforms both classical and data‑driven machine learning approaches in terms of AUC and sensitivity, while also exhibiting superior interpretability, generalizability and adherence to physical principles, with moderate computational cost. Potential extensions include integration with general relativistic magnetohydrodynamics (GRMHD) solvers and application to real observational data. These findings support the viability of physics‑based machine learning as a robust framework for accurate and interpretable black hole spin estimation.

Abstract:
This study applied representation learning algorithms to satellite images and evaluated the learned latent spaces with classifications of various weather events. The algorithms investigated include the classical linear transformation, i.e., principal component analysis (PCA), state‑of‑the‑art deep learning method, i.e., convolutional autoencoder (CAE), and a residual network pre‑trained with large image datasets (PT). The experiment results indicated that the latent space learned by CAE consistently showed higher threat scores for all classification tasks. The classifications with PCA yielded high hit rates but also high false‑alarm rates. In addition, the PT performed exceptionally well at recognizing tropical cyclones but was inferior in other tasks. Further experiments suggested that representations learned from higher‑resolution datasets are superior in all classification tasks for deep‑learning algorithms, i.e., CAE and PT. We also found that smaller latent space sizes had minor impact on the classification task's hit rate. Still, a latent space dimension smaller than 128 caused a significantly higher false alarm rate. Though the CAE can learn latent spaces effectively and efficiently, the interpretation of the learned representation lacks direct connections to physical attributions. Therefore, developing a physics‑informed version of CAE can be a promising outlook for the current work.

Abstract:
This study proposes FTI‑PBSM (Fixed‑Time‑Increment Physics‑informed neural network‑Based Surrogate Model), a novel physics‑informed surrogate modeling framework designed for real‑time reconstruction of transient responses in time‑dependent Partial Differential Equations (PDEs) using only sparse, time‑dependent sensor measurements. Unlike conventional Physics‑Informed Neural Network (PINN)‑based models that rely on Automatic Differentiation (AD) over both spatial and temporal domains and require dedicated causal network architectures to impose temporal causality, the proposed approach entirely removes AD in the time direction. Instead, it leverages higher‑order numerical differentiation methods, such as the Central Difference, Adams‑Bashforth, and Backward Differentiation Formula, to explicitly impose temporal causality. This leads to a simplified model architecture with improved training stability, computational efficiency, and extrapolation capability. Furthermore, FTI‑PBSM is trained on sparse sensor measurements from multiple PDE cases generated by varying PDE coefficients, with the sensor data serving as model input. This enables the model to learn a parametric PDE family and generalize to unseen physical cases, accurately reconstructing full‑field transient solutions in real time. The proposed model is validated on four representative PDE problems‑the convection equation, diffusion‑reaction dynamics, Korteweg‑de Vries (KdV) equation, and Allen‑Cahn equation‑and demonstrates superior prediction accuracy and generalization performance compared to a causal PBSM, which is used as the baseline model, in both interpolation and extrapolation tasks. It also shows strong robustness to sensor noise and variations in training data size, while significantly reducing training time.

Abstract:
Accuracy in neural PDE solvers often breaks down not because of limited expressivity, but due to poor optimisation caused by ill‑conditioning, especially in multi‑fidelity and stiff problems. We study this issue in Physics‑Informed Extreme Learning Machines (PIELMs), a convex variant of neural PDE solvers, and show that asymptotic components in governing equations can produce highly ill‑conditioned activation matrices, severely limiting convergence. We introduce Shifted Gaussian Encoding, a simple yet effective activation filtering step that increases matrix rank and expressivity while preserving convexity. Our method extends the solvable range of Peclet numbers in steady advection‑diffusion equations by over two orders of magnitude, achieves up to six orders lower error on multi‑frequency function learning, and fits high‑fidelity image vectors more accurately and faster than deep networks with over a million parameters. This work highlights that conditioning, not depth, is often the bottleneck in scientific neural solvers and that simple architectural changes can unlock substantial gains.

Abstract:
Shockwave classification in shadowgraph imaging is challenging due to limited labeled data and complex flow structures. This study presents a hybrid framework that combines unsupervised autoencoder models with a fuzzy inference system to generate and interpret anomaly maps. Among the evaluated methods, the hybrid β‑VAE autoencoder with a fuzzy rule‑based system most effectively captured coherent shock features, integrating spatial context to enhance anomaly classification. The resulting approach enables interpretable, unsupervised classification of flow disruptions and lays the groundwork for real‑time, physics‑informed diagnostics in experimental and industrial fluid applications.

Abstract:
Multiscale organization is a hallmark of fluid turbulence in aerospace, energy, and transport systems. While quantum computing promises exponential speedups for solving the evolution equations governing flow fields, this potential is fundamentally hindered by the quantum state preparation bottleneck, the prohibitive cost of loading classical complex data into quantum states. Here, we overcome this barrier by introducing a physics‑informed, three‑stage geometric encoding method "turbuloscope", which efficiently generates turbulent fields relevant to high‑Reynolds‑number engineering flows. Rather than brute‑force data loading, our approach acts as a kaleidoscope, leveraging the multiscale structures of turbulence. We capture scale‑invariant self‑similarity via a hyperplane approximation in high‑dimensional feature space, and utilize the Hopf fibration to map quantum observables directly onto vortex tubes, the fundamental building blocks of turbulence that control mixing, drag, and heat transfer in mechanical systems. Remarkably, the algorithm requires no ancillary qubits, utilizes a linear‑depth quantum circuit, and scales logarithmically with the Reynolds number, an exponential speedup compared to classical methods. We demonstrate the power of this method by generating an instantaneous turbulent field at a high Reynolds number of 35,000 across over one billion grid points using only 30 qubits, reproducing Kolmogorov's 5/3 energy spectrum, tangled vortex structures, and strong intermittency. This asymptotically optimal approach not only signals a near‑term pathway to practical quantum advantage in engineering simulation, but establishes a scalable foundation for the quantum simulation of broad multiscale systems.

Abstract:
Accurately modeling and inferring solutions to time‑dependent partial differential equations (PDEs) over extended horizons remains a core challenge in scientific machine learning. Traditional full rollout (FR) methods, which predict entire trajectories in one pass, often fail to capture the causal dependencies and generalize poorly outside the training time horizon. Autoregressive (AR) approaches, evolving the system step by step, suffer from error accumulation, limiting long‑term accuracy. These shortcomings limit the long‑term accuracy and reliability of both strategies. To address these issues, we introduce the Physics‑Informed Time‑Integrated Deep Operator Network (PITI‑DeepONet), a dual‑output architecture trained via physics‑informed or hybrid physics‑ and data‑driven objectives to ensure stable, accurate long‑term evolution well beyond the training horizon. Instead of forecasting future states, the network learns the time‑derivative operator from the current state, integrating it using classical time‑stepping schemes to advance the solution in time. Additionally, the framework can leverage residual monitoring during inference to estimate prediction quality and detect when the system transitions outside the training domain. Applied to benchmark problems, PITI‑DeepONet demonstrates enhanced accuracy and stability over extended inference time horizons when compared to traditional methods. Mean relative \mathcalL_2 errors reduced by 84% (versus FR) and 79% (versus AR) for 1D heat equation; by 87% (versus FR) and 98% (versus AR) for the 1D Burgers equation; by 42% (versus FR) and 89% (versus AR) for the 2D Allen‑Cahn equation; and by 58% (vs. FR) and 61% (vs. AR) for the 1D Kuramoto‑Sivashinsky equation. By moving beyond classic FR and AR schemes, PITI‑DeepONet paves the way for more reliable, long‑term integration of complex, time‑dependent PDEs.

Abstract:
Mode conversion in non‑homogeneous elastic media makes it challenging to interpret physical properties accurately. Decomposing these modes correctly is crucial across various scientific areas. Recent machine learning approaches have been proposed to address this problem, utilizing the Helmholtz decomposition technique. In this paper, we investigate the capabilities of a physics‑informed neural network (PINN) in separating P and S modes by solving a scalar Poisson equation. This scalar formulation offers a dimensionally scalable reduction in computational cost compared to the traditional vector formulation. We verify the proposed method in both homogeneous and realistic non‑homogeneous elastic models as showcases. The obtained separated modes closely match those from conventional numerical techniques, while exhibiting reduced transverse wave leakage.

Abstract:
Resistance spot welding is the dominant joining process for the body‑in‑white in the automotive industry, where the weld nugget diameter is the key quality metric. Its measurement requires destructive testing, limiting the potential for efficient quality control. Physics‑informed neural networks were investigated as a promising tool to reconstruct internal process states from experimental data, enabling model‑based and non‑invasive quality assessment in aluminum spot welding. A major challenge is the integration of real‑world data into the network due to competing optimization objectives. To address this, we introduce two novel training strategies. First, experimental losses for dynamic displacement and nugget diameter are progressively included using a fading‑in function to prevent excessive optimization conflicts. We also implement a custom learning rate scheduler and early stopping based on a rolling window to counteract premature reduction due to increased loss magnitudes. Second, we introduce a conditional update of temperature‑dependent material parameters via a look‑up table, activated only after a loss threshold is reached to ensure physically meaningful temperatures. An axially symmetric two‑dimensional model was selected to represent the welding process accurately while maintaining computational efficiency. To reduce computational burden, the training strategies and model components were first systematically evaluated in one dimension, enabling controlled analysis of loss design and contact models. The two‑dimensional network predicts dynamic displacement and nugget growth within the experimental confidence interval, supports transferring welding stages from steel to aluminum, and demonstrates strong potential for fast, model‑based quality control in industrial applications.

Abstract:
Physics‑Informed Neural Network (PINN) is a deep learning framework that integrates the governing equations underlying data into a loss function. In this study, we consider the problem of estimating state variables and parameters in epidemiological models governed by ordinary differential equations using PINNs. In practice, not all trajectory data corresponding to the population described by models can be measured. Learning PINNs to estimate the unmeasured state variables and epidemiological parameters using partial measurements is challenging. Accordingly, we introduce the concept of algebraic observability of the state variables. Specifically, we propose augmenting the unmeasured data based on algebraic observability analysis. The validity of the proposed method is demonstrated through numerical experiments under three scenarios in the context of epidemiological modelling. Specifically, given noisy and partial measurements, the accuracy of unmeasured states and parameter estimation of the proposed method is shown to be higher than that of the conventional methods. The proposed method is also shown to be effective in practical scenarios, such as when the data corresponding to certain variables cannot be reconstructed from the measurements.

Abstract:
In this paper, we propose a computational framework for the optimal design of broadband absorbing materials composed of plasmonic nanoparticle arrays. This design problem poses several key challenges: (1) the complex multi‑particle interactions and high‑curvature geometries; (2) the requirement to achieve broadband frequency responses, including resonant regimes; (3) the complexity of shape derivative calculations; and (4) the non‑convexity of the optimization landscape. To systematically address these challenges, we employ three sequential strategies. First, we introduce a parameterized integral equation formulation that circumvents traditional shape derivative computations. Second, we develop a shape‑adaptive reduced basis method (RBM) that utilizes the eigenfunctions of the Neumann‑Poincaré operator for forward problems and their adjoint counterparts for adjoint problems, thereby addressing singularities and accelerating computations. Third, we propose a physics‑informed initialization strategy that estimates nanoparticle configurations under weak coupling assumptions, thereby improving the performance of gradient‑based optimization algorithms. The method's computational advantages are demonstrated through numerical experiments, which show accurate and efficient designs across various geometric configurations. Furthermore, the framework is flexible and extensible to other material systems and boundary conditions.

Abstract:
In this work, we employ physics‑informed neural operators to map pressure profiles from an input function space to the corresponding bubble radius responses. Our approach employs a two‑step DeepONet architecture. To address the intrinsic spectral bias of deep learning models, our model incorporates the Rowdy adaptive activation function, enhancing the representation of high‑frequency features. Moreover, we introduce the Kolmogorov‑Arnold network (KAN) based two‑step DeepOKAN model, which enhances interpretability (often lacking in conventional multilayer perceptron architectures) while efficiently capturing high‑frequency bubble dynamics without explicit utilization of activation functions in any form. We particularly investigate the use of spline basis functions in combination with radial basis functions (RBF) within our architecture, as they demonstrate superior performance in constructing a universal basis for approximating high‑frequency bubble dynamics compared to alternative formulations. Furthermore, we emphasize on the performance bottleneck of RBF while learning the high frequency bubble dynamics and showcase the advantage of using spline basis function for the trunk network in overcoming this inherent spectral bias. The model is systematically evaluated across three representative scenarios: (1) bubble dynamics governed by the Rayleigh‑Plesset equation with a single initial radius, (2) bubble dynamics governed by the Keller‑Miksis equation with a single initial radius, and (3) Keller‑Miksis dynamics with multiple initial radii. We also compare our results with state‑of‑the‑art neural operators, including Fourier Neural Operators, Wavelet Neural Operators, OFormer, and Convolutional Neural Operators. Our findings demonstrate that the two‑step DeepOKAN accurately captures both low‑ and high‑frequency behaviors, and offers a promising alternative to conventional numerical solvers.

Abstract:
Accurate modeling of scattering from three‑dimensional (3D) perfectly electrically conducting (PEC) targets at microwave frequencies constitutes a fundamental objective in computational electromagnetics, particularly for radar cross section (RCS) prediction and microwave scattering analysis. Classical solvers, such as the method of moments and the Multilevel Fast Multipole Algorithm (MLFMA), although provide high physical fidelity, they become costly under scenarios of repeated queries involving many incidence configurations or frequencies, whereas purely data‑driven surrogates often lack accuracy on geometrically complex targets. This paper proposes a U‑shaped physics‑informed artificial neural network (U‑PINet) for 3D microwave scattering analysis. Inspired by the near‑far field decomposition of MLFMA, U‑PINet combines a near‑field graph encoder, parameterized by learnable univariate basis functions, with a hierarchical multi‑scale fusion module organized on an octree partition. The proposed network is trained against a discretized residual of the electric‑field integral equation at surface collocation points, without requiring reference current labels. Experiments on canonical and geometrically complex 3D PEC targets, conducted under multiple frequency and polarization configurations and assessed through bistatic RCS reconstruction, showcase that U‑PINet outperforms representative physics‑informed baselines, and yields substantial runtime savings over the classical MLFMA solver under repeated‑query scenarios.

Abstract:
Location‑aware devices continuously generate massive volumes of trajectory data, creating demand for efficient compression. Line simplification is a common solution but typically assumes 2D trajectories and ignores time synchronization and motion continuity. We propose PILOT‑C, a novel trajectory compression framework that integrates frequency‑domain physics modeling with error‑bounded optimization. Unlike existing line simplification methods, PILOT‑C supports trajectories in arbitrary dimensions, including 3D, by compressing each spatial axis independently. Evaluated on four real‑world datasets, PILOT‑C achieves superior performance across multiple dimensions. In terms of compression ratio, PILOT‑C outperforms CISED‑W, the current state‑of‑the‑art SED‑based line simplification algorithm, by an average of 19.2%. For trajectory fidelity, PILOT‑C achieves an average of 32.6% reduction in error compared to CISED‑W. Additionally, PILOT‑C seamlessly extends to three‑dimensional trajectories while maintaining the same computational complexity, achieving a 49% improvement in compression ratios over SQUISH‑E, the most efficient line simplification algorithm on 3D datasets.

Abstract:
Physics‑informed neural networks (PINNs) have recently emerged as a popular approach for solving forward and inverse problems involving partial differential equations (PDEs). Compared to fully connected neural networks, PINNs based on convolutional neural networks offer advantages in the hard enforcement of boundary conditions and in reducing the computational cost of partial derivatives. However, the latter still struggles with slow convergence and even failure in some scenarios. In this study, we propose a matrix preconditioning method to improve the convergence of the latter. Specifically, we combine automatic differentiation with matrix coloring to compute the Jacobian matrix of the PDE system, which is used to construct the preconditioner via incomplete LU factorization. We subsequently use the preconditioner to scale the PDE residual in the loss function in order to reduce the condition number of the Jacobian matrix, which is key to improving the convergence of PINNs. To overcome the incompatibility between automatic differentiation and triangular solves in the preconditioning, we also design a framework based on the adjoint method to compute the gradients of the loss function with respect to the network parameters. By numerical experiments, we validate that the proposed method successfully and efficiently solves the multi‑scale problem and the high Reynolds number problem, in both of which PINNs fail to obtain satisfactory results.

Abstract:
Physics‑Informed Neural Networks (PINNs) show significant potential for solving inverse problems, especially when observations are limited and sparse, provided that the relevant physical equations are known. We use PINNs to estimate smooth velocity and pressure fields from synthetic 4D flow Magnetic Resonance Imaging (MRI) data. We analyze five non‑Newtonian dynamic 3D blood flow cases within a realistic aortic model, covering a range of hematocrit values from anemic to polycythemic conditions. To enhance state estimation results, we consider various design and training techniques for PINNs, including adaptive loss balancing, curriculum training, and a realistic measurement operator. Regarding blood rheology, the PINN approach accurately estimates viscosity globally and locally under peak systolic conditions. It also provides a clear pattern recognition for diastolic stages. Regarding mass conservation, PINN estimations effectively reproduce the bifurcation of flow through the different branches of the aorta, demonstrate an excellent representation of the non‑slip conditions at the walls, and accurately estimate pressure drops with relative errors below the 5% in the whole pressure field. We test our pressure drop estimations against the state of the art Virtual Work Energy Relative Pressure (vWERP) estimator, and we observe how our results outperform vWERP in terms of both accuracy and time resolution. Additionally, we find that the best results are achieved by computing the velocity field using the PINN, which is then integrated into the vWERP framework, leading to time super‑sampled and high‑order approximations, with a clinically admissible accuracy.

Abstract:
Accurate real‑time prediction of phase‑resolved ocean wave fields remains a critical yet largely unsolved problem, primarily due to the absence of practical data assimilation methods for reconstructing initial conditions from sparse or indirect wave measurements. While recent advances in supervised deep learning have shown potential for this purpose, they require large labelled datasets of ground truth wave data, which are infeasible to obtain in real‑world scenarios. To overcome this limitation, we propose a Physics‑Informed Neural Operator (PINO) framework for reconstructing spatially and temporally phase‑resolved, nonlinear ocean wave fields from sparse measurements, without the need for ground truth data during training. This is achieved by embedding residuals of the free surface boundary conditions of ocean gravity waves into the loss function of the PINO, constraining the solution space in a soft manner. After training, we validate our approach using highly realistic synthetic wave data and demonstrate the accurate reconstruction of nonlinear wave fields from both buoy time series and radar snapshots. Our results indicate that PINOs enable accurate, real‑time reconstruction and generalize robustly across a wide range of wave conditions, thereby paving the way for operational, data‑driven wave reconstruction and prediction in realistic marine environments.

Abstract:
High throughput experimentation tools, machine learning (ML) methods, and open material databases are radically changing the way new materials are discovered. From the experimentally driven approach in the past, we are moving quickly towards the artificial intelligence (AI) driven approach, realizing the 'inverse design' capabilities that allow the discovery of new materials given the desired properties. This review aims to discuss different principles of AI‑driven generative models that are applicable for materials discovery, including different materials representations available for this purpose. We will also highlight specific applications of generative models in designing new catalysts, semiconductors, polymers, or crystals while addressing challenges such as data scarcity, computational cost, interpretability, synthesizability, and dataset biases. Emerging approaches to overcome limitations and integrate AI with experimental workflows will be discussed, including multimodal models, physics informed architectures, and closed‑loop discovery systems. This review aims to provide insights for researchers aiming to harness AI's transformative potential in accelerating materials discovery for sustainability, healthcare, and energy innovation.

Abstract:
Accurately identifying the material composition of objects is a critical capability for AI robots powered by large language models (LLMs) to perform context‑aware manipulation. Radar technologies offer a promising sensing modality for material recognition task. When combined with deep learning, radar technologies have demonstrated strong potential in identifying the material of various objects. However, existing radar‑based solutions are often constrained to closed‑set object categories and typically require task‑specific data collection to train deep learning models, largely limiting their practical applicability. This raises an important question: Can we leverage the powerful reasoning capabilities of pre‑trained LLMs to directly infer material composition from raw radar signals? Answering this question is non‑trivial due to the inherent redundancy of radar signals and the fact that pre‑trained LLMs have no prior exposure to raw radar data during training. To address this, we introduce LLMaterial, the first study to investigate the feasibility of using LLM to identify materials directly from radar signals. First, we introduce a physics‑informed signal processing pipeline that distills high‑redundancy radar raw data into a set of compact intermediate parameters that encapsulate the material's intrinsic characteristics. Second, we adopt a retrieval‑augmented generation (RAG) strategy to provide the LLM with domain‑specific knowledge, enabling it to interpret and reason over the extracted intermediate parameters. Leveraging this integration, the LLM is empowered to perform step‑by‑step reasoning on the condensed radar features, achieving open‑set material recognition directly from raw radar signals. Preliminary results show that LLMaterial can effectively distinguish among a variety of common materials, highlighting its strong potential for real‑world material identification applications.

Abstract:
Object manipulation skills are necessary for robots operating in various daily‑life scenarios, ranging from warehouses to hospitals. They allow the robots to manipulate the given object to their desired arrangement in the cluttered environment. The existing approaches to solving object manipulations are either inefficient sampling based techniques, require expert demonstrations, or learn by trial and error, making them less ideal for practical scenarios. In this paper, we propose a novel, multimodal physics‑informed neural network (PINN) for solving object manipulation tasks. Our approach efficiently learns to solve the Eikonal equation without expert data and finds object manipulation trajectories fast in complex, cluttered environments. Our method is multimodal as it also reactively replans the robot's grasps during manipulation to achieve the desired object poses. We demonstrate our approach in both simulation and real‑world scenarios and compare it against state‑of‑the‑art baseline methods. The results indicate that our approach is effective across various objects, has efficient training compared to previous learning‑based methods, and demonstrates high performance in planning time, trajectory length, and success rates. Our demonstration videos can be found at https://youtu.be/FaQLkTV9knI.

Abstract:
Modern engineering systems are increasingly equipped with sensors for real‑time monitoring and decision‑making. However, the data collected by these sensors is often noisy and difficult to interpret, limiting its utility for control and diagnostics. In this work, we propose a physics‑informed denoising framework that integrates energy‑based model and Fisher score regularization to jointly reduce data noise and enforce physical consistency with a physics‑based model. The approach is first validated on benchmark problems, including the simple harmonic oscillator, Burgers' equation, and Laplace's equation, across varying noise levels. We then apply the denoising framework to real thermal emission data from laser powder bed fusion (LPBF) additive manufacturing experiments, using a trained Physics‑Informed Neural Network (PINN) surrogate model of the LPBF process to guide denoising. Results show that the proposed method outperforms baseline neural network denoisers, effectively reducing noise under a range of LPBF processing conditions. This physics‑guided denoising strategy enables robust, real‑time interpretation of low‑cost sensor data, facilitating predictive control and improved defect mitigation in additive manufacturing.

Abstract:
Optimization‑based PDE solvers that minimize scalar loss functions have gained increasing attention in recent years. These methods either define the loss directly over discrete variables, as in Optimizing a Discrete Loss (ODIL), or indirectly through a neural network surrogate, as in Physics‑Informed Neural Networks (PINNs). However, despite their promise, such methods often converge much more slowly than classical iterative solvers and are commonly regarded as inefficient. This work provides a theoretical insight, attributing the inefficiency to the use of the mean squared error (MSE) loss, which implicitly forms the normal equations, squares the condition number, and severely impairs optimization. To address this, we propose a novel Stabilized Gradient Residual (SGR) loss. By tuning a weight parameter, it flexibly modulates the condition number between the original system and its normal equations, while reducing to the MSE loss in the limiting case. We systematically benchmark the convergence behavior and optimization stability of the SGR loss within both the ODIL framework and PINNs‑employing either numerical or automatic differentiation‑and compare its performance against classical iterative solvers. Numerical experiments on a range of benchmark problems demonstrate that, within the ODIL framework, the proposed SGR loss achieves orders‑of‑magnitude faster convergence than the MSE loss. Further validation within the PINNs framework shows that, despite the high nonlinearity of neural networks, SGR consistently outperforms the MSE loss. These theoretical and empirical findings help bridge the performance gap between classical iterative solvers and optimization‑based solvers, highlighting the central role of loss conditioning, and provide key insights for the design of more efficient PDE solvers.

Abstract:
Physics‑Informed Neural Networks (PINNs) offer a powerful framework for solving PDEs by embedding physical laws into the learning process. However, when applied to domains with irregular boundaries, PINNs often suffer from instability and slow convergence, which stems from (1) inconsistent normalization due to geometric anisotropy, (2) inaccurate boundary enforcement, and (3) imbalanced loss term competition. A common workaround is to map the domain to a regular space. Yet, conventional mapping methods rely on case‑specific meshes, define Jacobians at pre‑specified fixed nodes, reformulate PDEs via the chain rule‑making them incompatible with modern automatic differentiation, tensor‑based frameworks. To bridge this gap, we propose JacobiNet, a learning‑based coordinate‑transformed PINN framework that unifies domain mapping and PDE solving within an end‑to‑end differentiable architecture. JacobiNet enables direct Jacobian computation via autograd, shares computation graph with downstream PINNs, thereby avoiding case‑specific meshing, explicit Jacobian computation/storage, and manual PDE reformulation while unlocking geometric‑editing operations. Separating physical modeling from geometric complexity, JacobiNet (1) addresses normalization challenges in the original anisotropic coordinates, (2) facilitates the hard enforcement of boundary conditions, and (3) mitigates the long‑standing imbalance among loss terms. Evaluated on various PDEs, JacobiNet reduces the relative L2 error from 0.11‑0.73 to 0.01‑0.09, achieving an average 15.6x improvement in accuracy. In vessel‑like domains with varying shapes, JacobiNet enables millisecond‑level mapping inference for unseen geometries, improves prediction accuracy by an average of 3.65x, while delivering over 10x speedup‑demonstrating strong generalization, accuracy, and efficiency.

Abstract:
Solving partial differential equations (PDEs) is an important yet challenging task in fluid mechanics. In this study, we embed an improved Fourier series into neural networks and propose a physics‑informed Fourier basis neural network (FBNN) by incorporating physical information to solve canonical PDEs in fluid mechanics. The results demonstrated that the proposed framework exhibits a strong nonlinear fitting capability and exceptional periodic modeling performance. In particular, our model shows significant advantages for the Burgers equation with discontinuous solutions and Helmholtz equation with strong periodicity. By directly introducing sparse distributed data to reconstruct the entire flow field, we further intuitively validated the direct superiority of FBNN over conventional artificial neural networks (ANN) as well as the benefits of incorporating physical information into the network. By adjusting the activation functions of networks and comparing with an ANN and conventional physics‑informed neural network, we proved that performance of the proposed FBNN architecture is not highly sensitive to the choice of activation functions. The nonlinear fitting capability of FBNN avoids excessive reliance on activation functions, thereby mitigating the risk of suboptimal outcomes or training failures stemming from unsuitable activation function choices.hese results highlightthe potential of PIFBNN as a powerful tool in computational fluid dynamics.

Abstract:
This paper introduces OptiGridML, a machine learning framework for discrete topology optimization in power grids. The task involves selecting substation breaker configurations that maximize cross‑region power exports, a problem typically formulated as a mixed‑integer program (MIP) that is NP‑hard and computationally intractable for large networks. OptiGridML replaces repeated MIP solves with a two‑stage neural architecture: a line‑graph neural network (LGNN) that approximates DC power flows for a given network topology, and a heterogeneous GNN (HeteroGNN) that predicts breaker states under structural and physical constraints. A physics‑informed consistency loss connects these components by enforcing Kirchhoff's law on predicted flows. Experiments on synthetic networks with up to 1,000 breakers show that OptiGridML achieves power export improvements of up to 18% over baseline topologies, while reducing inference time from hours to milliseconds. These results demonstrate the potential of structured, flow‑aware GNNs for accelerating combinatorial optimization in physical networked systems.

Abstract:
We propose a mesh‑free policy iteration framework based on physics‑informed neural networks (PINNs) for solving entropy‑regularized stochastic control problems. The method iteratively alternates between soft policy evaluation and improvement using automatic differentiation and neural approximation, without relying on spatial discretization. We present a detailed L^2 error analysis that decomposes the total approximation error into three sources: iteration error, policy network error, and PDE residual error. The proposed algorithm is validated with a range of challenging control tasks, including high‑dimensional linear‑quadratic regulation in 5D and 10D, as well as nonlinear systems such as pendulum and cartpole problems. Numerical results confirm the scalability, accuracy, and robustness of our approach across both linear and nonlinear benchmarks.

Abstract:
We propose a physics‑informed neural network policy iteration (PINN‑PI) framework for solving stochastic optimal control problems governed by second‑order Hamilton‑‑Jacobi‑‑Bellman (HJB) equations. At each iteration, a neural network is trained to approximate the value function by minimizing the residual of a linear PDE induced by a fixed policy. This linear structure enables systematic L^2 error control at each policy evaluation step, and allows us to derive explicit Lipschitz‑type bounds that quantify how value gradient errors propagate to the policy updates. This interpretability provides a theoretical basis for evaluating policy quality during training. Our method extends recent deterministic PINN‑based approaches to stochastic settings, inheriting the global exponential convergence guarantees of classical policy iteration under mild conditions. We demonstrate the effectiveness of our method on several benchmark problems, including stochastic cartpole, pendulum problems and high‑dimensional linear quadratic regulation (LQR) problems in up to 10D.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful class of mesh‑free numerical methods for solving partial differential equations (PDEs), particularly those involving complex geometries. In this work, we present an innovative Extended Interface Physics‑Informed Neural Network (XI‑PINN) framework specifically designed to solve parabolic moving interface problems. The proposed approach incorporates a level set function to characterize the interface, which can be obtained either directly or through a neural network solution. We conduct a rigorous a priori error analysis for the XI‑PINN method, providing error bounds for the approximation. Leveraging the Neural Tangent Kernel (NTK) theory, we further demonstrate that XI‑PINN achieves a faster training convergence rate compared to conventional PINN approaches. The method's versatility is further demonstrated by its application to the Oseen equations. We perform comprehensive numerical experiments to validate the efficacy, accuracy, and robustness of the proposed framework.

Abstract:
This work addresses the critical challenge of guaranteeing safety for complex dynamical systems where precise mathematical models are uncertain and data measurements are corrupted by noise. We develop a physics‑informed, direct data‑driven framework for synthesizing robust safety controllers (R‑SCs) for both discrete‑ and continuous‑time nonlinear polynomial systems that are subject to unknown‑but‑bounded disturbances. To do so, we introduce a notion of safety through robust control barrier certificates (R‑CBCs), which ensure avoidance of (potentially multiple) unsafe regions, offering a less conservative alternative to existing methods based on robust invariant sets. Our core innovation lies in integrating the fundamental physical principles with observed noisy data which drastically reduces data requirements, enabling robust safety analysis with significantly shorter trajectories, compared to purely data‑driven methods. To achieve this, the proposed synthesis procedure is formulated as a sum‑of‑squares (SOS) optimization program that systematically designs the R‑CBC and its associated R‑SC by leveraging both collected data and underlying physical laws. The efficacy of our framework is demonstrated on four benchmark systems, three discrete‑time and one continuous‑time nonlinear polynomial systems, confirming its ability to offer robust safety guarantees with reduced data demands.

Abstract:
The utilization of Deep Neural Networks (DNNs) in physical science and engineering applications has gained traction due to their capacity to learn intricate functions. While large datasets are crucial for training DNN models in fields like computer vision and natural language processing, obtaining such datasets for engineering applications is prohibitively expensive. Physics‑Informed Neural Networks (PINNs), a branch of Physics‑Informed Machine Learning (PIML), tackle this challenge by embedding physical principles within neural network architectures. PINNs have been extensively explored for solving diverse forward and inverse problems in fluid mechanics. Nonetheless, there is limited research on employing PINNs for flow reconstruction from sparse data under constrained computational resources. Earlier studies were focused on forward problems with well‑defined data. The present study attempts to develop models capable of reconstructing the flow field data from sparse datasets mirroring real‑world scenarios. This study focuses on two cases: (a) two‑dimensional (2D) unsteady laminar flow past a circular cylinder and (b) three‑dimensional (3D) unsteady turbulent flow past an ultra‑large container ship (ULCS). The first case compares the effectiveness of training methods like Standard PINN and Backward Compatible PINN (BC‑PINN) and explores the performance enhancements through systematic relaxation of physics constraints and dynamic weighting of loss function components. The second case highlights the capability of PINN‑based models to learn underlying physics from sparse data while accurately reconstructing the flow field for a highly turbulent flow.

Abstract:
Physics‑informed neural networks (PINNs) have attracted attention as an alternative approach to solve partial differential equations using a deep neural network (DNN). Their simplicity and capability allow them to solve inverse problems for many applications. Despite the versatility of PINNs, it remains challenging to reduce their training cost. Using a DNN pre‑trained with an arbitrary dataset with transfer learning or fine‑tuning is a potential solution. However, a pre‑trained model using a different geometry and flow condition than the target may not produce suitable results. This paper proposes a fine‑tuning approach for PINNs with coordinate transformation, modelling lid‑driven cavity flows with various shapes. We formulate the inverse problem, where the reference data inside the domain and wall boundary conditions are given. A pre‑trained PINN model with an arbitrary Reynolds number and shape is used to initialize a target DNN. To reconcile the reference shape with different targets, governing equations as a loss of the PINNs are given with coordinate transformation using a deformation gradient tensor. Numerical examples for various cavity flows with square, rectangular, shear deformed and inflated geometries demonstrate that the proposed fine‑tuning approach improves the training convergence compared with a randomly‑initialized model. A pre‑trained model with a similar geometry to the target further increases training efficiency. These findings are useful for real‑world applications such as modelling intra‑aneurysmal blood flows in clinical use.

Abstract:
Recent developments in AI techniques for space applications mirror the success achieved in terrestrial applications. Machine learning, which excels in data rich environments, is particularly well suited to space‑based computer vision applications, such as space optical attitude sensing. Of these sensors, digital sun sensors (DSS) are one of the most common and important sensors for spacecraft attitude determination. The main challenge in using the DSS for attitude estimation are sensor errors, which limit the overall achievable estimation accuracy. However, the traditional sun sensor calibration process is costly, slow, labor‑intensive and inefficient. These limitations motivate the use of AI techniques to enable more accurate and efficient DSS calibration. The objective of this work is to develop an end‑to‑end predictive calibration methodology for digital sun sensors to solve 2‑axis state estimates utilizing a sparse submanifold convolutional neural network (SSCNN). We find that the proposed framework can achieve state‑of‑the‑art performance on synthetic data with a mean accuracy of 0.005° for the two sun angle estimates. Furthermore, the model is highly capable of implicitly learning complex noise patterns and handling mixed noise types, thereby greatly improving the model robustness and accuracy to real‑world applications. The main contributions of this work are: (1) the first application (to our knowledge) of a CNN regression model to the problem of DSS predictive calibration, (2) the introduction of a fused end‑to‑end training approach for DSS calibration, (3) the creation of a publicly available physics‑informed synthetic dataset and simulation for DSS training images, and (4) the evaluation of the performance of the deep learning approach for various mask configurations.

Abstract:
Nonlinear partial differential equations (PDEs) are pivotal in modeling complex physical systems, yet traditional Physics‑Informed Neural Networks (PINNs) often struggle with unresolved residuals in critical spatiotemporal regions and violations of temporal causality. To address these limitations, we propose a novel Residual Guided Training strategy for Physics‑Informed Transformer via Generative Adversarial Networks (GAN). Our framework integrates a decoder‑only Transformer to inherently capture temporal correlations through autoregressive processing, coupled with a residual‑aware GAN that dynamically identifies and prioritizes high‑residual regions. By introducing a causal penalty term and an adaptive sampling mechanism, the method enforces temporal causality while refining accuracy in problematic domains. Extensive numerical experiments on the Allen‑Cahn, Klein‑Gordon, and Navier‑Stokes equations demonstrate significant improvements, achieving relative MSE reductions of up to three orders of magnitude compared to baseline methods. This work bridges the gap between deep learning and physics‑driven modeling, offering a robust solution for multiscale and time‑dependent PDE systems.

Abstract:
This paper presents an Output‑Recurrent Gated State Space Model (OR‑GSSM) for complex multiphase flows modeling and uncertainty quantification of exhaust vehicles during motion. By establishing the state‑space formulation of the gas‑liquid Navier‑Stokes equations applying semigroup theory and Galerkin projection, explicitly characterizing the dynamic coupling evolution between the velocity, pressure, and volume fraction fields. A novel Gated State Space Transition (GSST) unit is designed to learn parameterized transition and input matrices with adaptive timescales, enhancing physical interpretability and computational efficiency. The output recursion mechanism aligns with the numerical solution characteristics of state‑space equations, mitigating long‑term error accumulation and addressing training‑inference pattern mismatch issues inherent in teacher forcing and scheduled sampling. Validations on the underwater cone‑head and water‑exit hemisphere‑head vehicles demonstrate that: OR‑GSSM outperforms OR‑ConvLSTM and OR‑ConvGRU baselines in accuracy and computational efficiency through its physics‑informed adaptive state‑space unit design and parallel matrix operations; The output recursion mechanism ensures more stable training, better generalization, and higher prediction accuracy than teacher forcing and scheduled sampling; OR‑GSSM accurately captures the gas‑phase expansion, gas‑liquid mixing formation, backflow jet generation, bubble shedding, and entire water‑exit process, etc, showcasing outstanding modeling capability; Its uncertainty quantification effectively characterizes flow features and uncertainty distributions, validating prediction reliability. The proposed method resolves the accuracy‑real‑time trade‑off in traditional computational fluid dynamics, advancing machine learning for multiphase flow modeling and uncertainty quantification in exhaust vehicles.

Abstract:
Neural quantum states (NQS) provide flexible and compact wavefunction parameterizations for numerical studies of quantum many‑body physics. In particular, NQS aim to circumvent the exponential scaling of the Hilbert space by compressing quantum many‑body wavefunctions with a tractable amount of parameters. While inspired by deep learning, it remains unclear to what extent NQS share characteristics with neural networks used for standard machine learning tasks. We demonstrate that, in a simplified supervised setting, NQS exhibit the double descent phenomenon, a key feature of modern deep learning, where generalization worsens as network size increases before improving again in an overparameterized regime. Notably, we find the second descent to occur only for network sizes much larger than the Hilbert space dimension, i.e. network sizes that are out of reach for problems of practical interest. Within our setting, this observation places typical NQS in the underparameterized regime. We also observe that the optimal network size in the underparameterized regime depends on the number of unique training samples. While the double descent phenomenon does indeed translate to the NQS setting, potential practical consequences of our findings point more towards the need for symmetry‑aware, physics‑informed architecture design, rather than directly adopting machine learning heuristics.

Abstract:
The simultaneous production of four top quarks in association with a (W) boson at (\sqrts = 13) TeV is an rare SM process with a next‑to‑leading‑order (NLO) cross‑section of (6.6^+2.4_‑2.6 ab)\citesaiel. Identifying this process in the fully hadronic decay channel is particularly challenging due to overwhelming backgrounds from t\bart, t\bartW, t\bartZ, and triple‑top production processes. This study introduces a modified physics informed Neural Network, a hybrid graph neural network (GNN) enhancing event classification. The proposed model integrates Graph layers for particle‑level features, a custom Multi Layer Perceptron(MLP) based global stream with a quantum circuit and cross‑attention fusion to combine local and global representations. Physics‑informed Loss function enforce jet multiplicity constraints, derived from event decay dynamics. Benchmarked against conventional methods, the GNN achieves a signal significance (S/\sqrtS+B) of 0.174 and ROC‑AUC of 0.974, surpassing BDT's significance of 0.148 and ROC of 0.913, while Xgboost achieves a significance of 0.149 and ROC of 0.920. The classification models are trained on Monte Carlo (MC) simulations, with events normalized using cross‑section‑based reweighting to reflect their expected contributions in a dataset corresponding to 350\;fb^‑1 of integrated luminosity. This enhanced approach offers a framework for precision event selection at the LHC, leveraging high dimensional statistical learning and physics informed inference to tackle fundamental HEP challenges, aligning with ML developments.

Abstract:
We present a simulation‑based inference (SBI) framework to constrain the neutron star (NS) equation of state (EoS) from astrophysical observations of masses, radii and tidal deformabilities, using Neural posterior estimation (NPE) with Conditional Normalising Flows (CNF). To ensure that the model conforms with reality, physics‑informed constraints are embedded directly into the training loss. This enables efficient, likelihood‑free inference of full posterior distributions for key thermodynamic quantities‑including pressure, squared speed of sound, and the trace anomaly‑conditioned on observational data. Our models are trained on synthetic datasets generated from two agnostic EoS priors: polytropic parametrizations (PT) and gaussian process (GP) reconstructions. These datasets span various scenarios, including the presence or absence of tidal deformability information and observational noise. Across all settings, the method produces accurate and well‑calibrated posteriors, with uncertainties reduced when tidal deformability constraints are included. Furthermore, we find that the behavior of normalized predictive dispersions is strongly correlated with the maximum central density inside NSs, suggesting that the model can indirectly infer this physically meaningful quantity. The approach generalizes well across EoS families and accurately reconstructs derivative quantities such as the polytropic index, demonstrating its robustness and potential for probing dense matter in NS cores.

Abstract:
Metasurfaces are innovative planar optical structures capable of manipulating incident light properties. Accurate and computationally efficient modeling of such metasurfaces, particularly those with irregular geometries, remains a challenge for conventional solvers. In this work, we present a mesh‑free Physics‑Informed PointNet (PIPN) to model electromagnetic scattering from all‑dielectric metasurfaces that feature spatially varying nanopillars. Our approach uses the PointNet architecture to directly encode spatially varying material properties into the Physics‑Informed Machine Learning (PIML) framework. We demonstrate the generalization capability of our PIPN through evaluations on datasets; these datasets are generated with varying refractive indices representing common dielectric materials. Furthermore, the inclination angles are varied within each dataset, which represent expected manufacturing defects. Overall, our method provides a promising, mesh‑free framework for accurate and efficient modeling of complex optical structures represented by irregular geometries.

Abstract:
Wall‑pressure fluctuations beneath turbulent boundary layers drive noise and structural fatigue through interactions between fluid and structural modes. Conventional predictive models for the spectrum‑‑such as the widely accepted Goody model (AIAA Journal 42 (9), 2004, 1788‑‑1794)‑‑fail to capture the energetic growth in the low‑frequency range that occurs at high Reynolds number, while at the same time over‑predicting the variance. To address these shortcomings, two semi‑empirical models are proposed for the wall‑pressure spectrum in canonical turbulent boundary layers, pipes and channels for friction Reynolds numbers δ^+ ranging from 180 to 47 000. The models are based on consideration of two spectral components that represent the contributions to the wall pressure fluctuations from inner‑scale motions and outer‑scale motions. The first model expresses the pre‑multiplied spectrum as the sum of two overlapping log‑normal components: an inner‑scaled term that is δ^+‑invariant and an outer‑scaled term whose amplitude broadens smoothly with δ^+. Calibrated against large‑eddy simulations, direct numerical simulations, and recent high‑δ^+ pipe data, it reproduces the inner‑scaled peak and the emergence of an outer‑scaled peak at large δ^+. The second model, developed around newly available pipe data, uses theoretical arguments to prescribe the spectral shapes of the inner and outer components. Embedding the δ^+‑dependence in smooth asymptotic functions yields a formulation that varies continuously with δ^+ and generalises beyond the calibration range. Both models capture the full spectrum and recover the observed logarithmic growth of its variance, providing a compact, physics‑informed empirical representation for more accurate engineering predictions of wall‑pressure fluctuations.

Abstract:
The field of scientific machine learning, which originally utilized multilayer perceptrons (MLPs), is increasingly adopting Kolmogorov‑Arnold Networks (KANs) for data encoding. This shift is driven by the limitations of MLPs, including poor interpretability, fixed activation functions, and difficulty capturing localized or high‑frequency features. KANs address these issues with enhanced interpretability and flexibility, enabling more efficient modeling of complex nonlinear interactions and effectively overcoming the constraints associated with conventional MLP architectures. This review categorizes recent progress in KAN‑based models across three distinct perspectives: (i) data‑driven learning, (ii) physics‑informed modeling, and (iii) deep‑operator learning. Each perspective is examined through the lens of architectural design, training strategies, application efficacy, and comparative evaluation against MLP‑based counterparts. By benchmarking KANs against MLPs, we highlight consistent improvements in accuracy, convergence, and spectral representation, clarifying KANs' advantages in capturing complex dynamics while learning more effectively. In addition to reviewing recent literature, this work also presents several comparative evaluations that clarify central characteristics of KAN modeling and hint at their potential implications for real‑world applications. Finally, this review identifies critical challenges and open research questions in KAN development, particularly regarding computational efficiency, theoretical guarantees, hyperparameter tuning, and algorithm complexity. We also outline future research directions aimed at improving the robustness, scalability, and physical consistency of KAN‑based frameworks.

Abstract:
Physics‑informed holomorphic neural networks (PIHNNs) have recently emerged as efficient surrogate models for solving differential problems. By embedding the underlying problem structure into the network, PIHNNs require training only to satisfy boundary conditions, often resulting in significantly improved accuracy and computational efficiency compared to traditional physics‑informed neural networks (PINNs). In this work, we improve and extend the application of PIHNNs to two‑dimensional problems. First, we introduce a novel holomorphic network architecture based on the Kolmogorov‑Arnold representation (PIHKAN), which achieves higher accuracy with reduced model complexity. Second, we develop mathematical extensions that broaden the applicability of PIHNNs to a wider class of elliptic partial differential equations, including the Helmholtz equation. Finally, we propose a new method based on Laurent series theory that enables the application of holomorphic networks to multiply‑connected plane domains, thereby removing the previous limitation to simply‑connected geometries.

Abstract:
We propose a novel probabilistic framework, termed LVM‑GP, for uncertainty quantification in solving forward and inverse partial differential equations (PDEs) with noisy data. The core idea is to construct a stochastic mapping from the input to a high‑dimensional latent representation, enabling uncertainty‑aware prediction of the solution. Specifically, the architecture consists of a confidence‑aware encoder and a probabilistic decoder. The encoder implements a high‑dimensional latent variable model based on a Gaussian process (LVM‑GP), where the latent representation is constructed by interpolating between a learnable deterministic feature and a Gaussian process prior, with the interpolation strength adaptively controlled by a confidence function learned from data. The decoder defines a conditional Gaussian distribution over the solution field, where the mean is predicted by a neural operator applied to the latent representation, allowing the model to learn flexible function‑to‑function mapping. Moreover, physical laws are enforced as soft constraints in the loss function to ensure consistency with the underlying PDE structure. Compared to existing approaches such as Bayesian physics‑informed neural networks (B‑PINNs) and deep ensembles, the proposed framework can efficiently capture functional dependencies via merging a latent Gaussian process and neural operator, resulting in competitive predictive accuracy and robust uncertainty quantification. Numerical experiments demonstrate the effectiveness and reliability of the method.

Abstract:
In the rapidly evolving domain of autonomous systems, interaction among agents within a shared environment is both inevitable and essential for enhancing overall system capabilities. A key requirement in such multi‑agent systems is the ability of each agent to reliably predict the future positions of its nearest neighbors. Traditionally, graphs and graph theory have served as effective tools for modeling inter agent communication and relationships. While this approach is widely used, the present work proposes a novel method that leverages dynamic graphs in a forward looking manner. Specifically, the employment of EvolveGCN, a dynamic graph convolutional network, to forecast the evolution of inter‑agent relationships over time. To improve prediction accuracy and ensure physical plausibility, this research incorporates physics constrained loss functions based on the Clohessy‑Wiltshire equations of motion. This integrated approach enhances the reliability of future state estimations in multi‑agent scenarios.

Abstract:
Physics‑informed neural networks (PINNs) are numerical solvers that embed all the physical information of a system into the loss function of a neural network. In this way the learned solution accounts for data (if available), the governing differential equations, or any other constraint known of the physical problem. However, they face serious issues, notably their tendency to converge on trivial or misleading solutions. The latter occurs when, although the loss function reaches low values the model makes incorrect predictions. These difficulties become especially significant in differential equations involving multi‑scale behavior, such as rapidly varying terms and solutions exhibiting strong oscillatory behavior. To address these challenges, we introduce the Dynamical Boundary Constraint (DBC) algorithm, which imposes restrictions on the loss function based on prior training of the PINN. To demonstrate its applicability, we tested this approach on examples of different areas of physics.

Abstract:
Training neural networks can be challenging, especially as the complexity of the problem increases. Despite using wider or deeper networks, training them can be a tedious process, especially if a wrong choice of the hyperparameter is made. The learning rate is one of such crucial hyperparameters, which is usually kept static during the training process. Learning dynamics in complex systems often requires a more adaptive approach to the learning rate. This adaptability becomes crucial to effectively navigate varying gradients and optimize the learning process during the training process. In this paper, a dynamic learning rate scheduler (DLRS) algorithm is presented that adapts the learning rate based on the loss values calculated during the training process. Experiments are conducted on problems related to physics‑informed neural networks (PINNs) and image classification using multilayer perceptrons and convolutional neural networks, respectively. The results demonstrate that the proposed DLRS accelerates training and improves stability.

Abstract:
Accurately forecasting commodity demand remains a critical challenge due to volatile market dynamics, nonlinear dependencies, and the need for economically consistent predictions. This paper introduces PREIG, a novel deep learning framework tailored for commodity demand forecasting. The model uniquely integrates a Gated Recurrent Unit (GRU) architecture with physics‑informed neural network (PINN) principles by embedding a domain‑specific economic constraint: the negative elasticity between price and demand. This constraint is enforced through a customized loss function that penalizes violations of the physical rule, ensuring that model predictions remain interpretable and aligned with economic theory. To further enhance predictive performance and stability, PREIG incorporates a hybrid optimization strategy that couples NAdam and L‑BFGS with Population‑Based Training (POP). Experiments across multiple commodities datasets demonstrate that PREIG significantly outperforms traditional econometric models (ARIMA,GARCH) and deep learning baselines (BPNN,RNN) in both RMSE and MAPE. When compared with GRU,PREIG maintains good explainability while still performing well in prediction. By bridging domain knowledge, optimization theory and deep learning, PREIG provides a robust, interpretable, and scalable solution for high‑dimensional nonlinear time series forecasting in economy.

Abstract:
Physics‑informed neural networks and Physics‑informed DeepONet excel in solving partial differential equations; however, they often fail to converge for singularly perturbed problems. To address this, we propose two novel frameworks, Prandtl‑Van Dyke neural network(PVD‑Net) and its operator learning extension Prandtl‑Van Dyke Deep Operator Network (PVD‑ONet), which rely solely on governing equations without data. To address varying task‑specific requirements, both PVD‑Net and PVD‑ONet are developed in two distinct versions, tailored respectively for stability‑focused and high‑accuracy modeling. The leading‑order PVD‑Net adopts a two‑network architecture combined with Prandtl's matching condition, targeting stability‑prioritized scenarios. The high‑order PVD‑Net employs a five‑network design with Van Dyke's matching principle to capture fine‑scale boundary layer structures, making it ideal for high‑accuracy scenarios. PVD‑ONet generalizes PVD‑Net to the operator learning setting by assembling multiple DeepONet modules, directly mapping initial conditions to solution operators and enabling instant predictions for an entire family of boundary layer problems without retraining. Numerical experiments (second‑order equations with constant and variable coefficients, and internal layer problems) show that the proposed methods consistently outperform existing baselines. Moreover, beyond forward prediction, the proposed framework can be extended to inverse problems. It enables the inference of the scaling exponent governing boundary layer thickness from sparse data, providing potential for practical applications.

Abstract:
Neural networks have emerged as a powerful tool for modeling physical systems, offering the ability to learn complex representations from limited data while integrating foundational scientific knowledge. In particular, neuro‑symbolic approaches that combine data‑driven learning, the neuro, with symbolic equations and rules, the symbolic, address the tension between methods that are purely empirical, which risk straying from established physical principles, and traditional numerical solvers that demand complete geometric knowledge and can be prohibitively expensive for high‑fidelity simulations. In this work, we present a novel neuro‑symbolic framework for reconstructing and simulating elastic objects directly from sparse multi‑view image sequences, without requiring explicit geometric information. Specifically, we integrate a neural radiance field (NeRF) for object reconstruction with physics‑informed neural networks (PINN) that incorporate the governing partial differential equations of elasticity. In doing so, our method learns a spatiotemporal representation of deforming objects that leverages both image supervision and symbolic physical constraints. To handle complex boundary and initial conditions, which are traditionally confronted using finite element methods, boundary element methods, or sensor‑based measurements, we employ an energy‑constrained Physics‑Informed Neural Network architecture. This design enhances both simulation accuracy and the explainability of results.

Abstract:
Physics‑informed neural networks (PINNs) have plateaued at errors of 10^‑3‑10^‑4 for fourth‑order partial differential equations, creating a perceived precision ceiling that limits their adoption in engineering applications. We break through this barrier with a hybrid Fourier‑neural architecture for the Euler‑Bernoulli beam equation, achieving unprecedented L2 error of 1.94 × 10^‑7‑a 17‑fold improvement over standard PINNs and \(15‑500×\) better than traditional numerical methods. Our approach synergistically combines a truncated Fourier series capturing dominant modal behavior with a deep neural network providing adaptive residual corrections. A systematic harmonic optimization study revealed a counter‑intuitive discovery: exactly 10 harmonics yield optimal performance, with accuracy catastrophically degrading from 10^‑7 to 10^‑1 beyond this threshold. The two‑phase optimization strategy (Adam followed by L‑BFGS) and adaptive weight balancing enable stable ultra‑precision convergence. GPU‑accelerated implementation achieves sub‑30‑minute training despite fourth‑order derivative complexity. By addressing 12 critical gaps in existing approaches‑from architectural rigidity to optimization landscapes‑this work demonstrates that ultra‑precision is achievable through proper design, opening new paradigms for scientific computing where machine learning can match or exceed traditional numerical methods.

Abstract:
Multi‑scale PDE problems present significant challenges in scientific computing. While conventional MLP‑based deep learning methods exhibit spectral bias in resolving multi‑scale features, the physics‑informed Kolmogorov‑Arnold network (PIKAN) mitigates this issue through its novel architecture, demonstrating certain advantages. On the other hand, insights from the information bottleneck theory suggest that high‑resolution training points are essential for these hybrid methods to accurately capture multi‑scale behavior, although this requirement often leads to longer training times. To address this challenge, we propose a simple yet effective multi‑resolution training‑enhanced PIKAN framework, termed MR‑PIKAN, which trains the data‑physics hybrid model either sequentially or alternately across different resolutions. The proposed MR‑PIKAN is validated on various multi‑scale forward and inverse PDE problems. Numerical results indicate that this new training strategy effectively reduces computational costs without sacrificing accuracy, thereby enabling efficient solutions of complex multi‑scale PDEs in both forward and inverse settings.

Abstract:
Accurate prediction of future agent trajectories is a critical challenge for ensuring safe and efficient autonomous navigation, particularly in complex urban environments characterized by multiple plausible future scenarios. In this paper, we present a novel hybrid approach that integrates learning‑based with physics‑based constraints to address the multi‑modality inherent in trajectory prediction. Our method employs a variational Bayesian mixture model to effectively capture the diverse range of potential future behaviors, moving beyond traditional unimodal assumptions. Unlike prior approaches that predominantly treat trajectory prediction as a data‑driven regression task, our framework incorporates physical realism through sector‑specific boundary conditions and Model Predictive Control (MPC)‑based smoothing. These constraints ensure that predicted trajectories are not only data‑consistent but also physically plausible, adhering to kinematic and dynamic principles. Furthermore, our method produces interpretable and diverse trajectory predictions, enabling enhanced downstream decision‑making and planning in autonomous driving systems. We evaluate our approach on two benchmark datasets, demonstrating superior performance compared to existing methods. Comprehensive ablation studies validate the contributions of each component and highlight their synergistic impact on prediction accuracy and reliability. By balancing data‑driven insights with physics‑informed constraints, our approach offers a robust and scalable solution for navigating the uncertainties of real‑world urban environments.

Abstract:
Mathematical models in neural networks are powerful tools for solving complex differential equations and optimizing their parameters; that is, solving the forward and inverse problems, respectively. A forward problem predicts the output of a network for a given input by optimizing weights and biases. An inverse problem finds equation parameters or coefficients that effectively model the data. A Physics‑Informed Neural Network (PINN) can solve both problems. PINNs inject prior analytical information about the data into the cost function to improve model performance outside the training set boundaries. This also allows PINNs to efficiently solve problems with sparse data without overfitting by extrapolating the model to fit larger trends in the data. The prior information we implement is in the form of differential equations. Residuals are the differences between the left‑hand and right‑hand sides of corresponding differential equations; PINNs minimize these residuals to effectively solve the differential equation and take advantage of prior knowledge. In this way, the solution and parameters are embedded into the loss function and optimized, allowing both the weights of the neural network and the model parameters to be found simultaneously, solving both the forward and inverse problems in the process. In this paper, we will create PINNs with residuals of varying complexity, beginning with linear and quadratic models and then expanding to fit models for the heat equation and other complex differential equations. We will mainly use Python as the computing language, using the PyTorch library to aid us in our research.

Abstract:
Data used for training structural health monitoring (SHM) systems are expensive and often impractical to obtain, particularly labelled data. Population‑based SHM presents a potential solution to this issue by considering the available data across a population of structures. However, differences between structures will mean the training and testing distributions will differ; thus, conventional machine learning methods cannot be expected to generalise between structures. To address this issue, transfer learning (TL), can be used to leverage information across related domains. An important consideration is that the lack of labels in the target domain limits data‑based metrics to quantifying the discrepancy between the marginal distributions. Thus, a prerequisite for the application of typical unsupervised TL methods is to identify suitable source structures (domains), and a set of features, for which the conditional distributions are related to the target structure. Generally, the selection of domains and features is reliant on domain expertise; however, for complex mechanisms, such as the influence of damage on the dynamic response of a structure, this task is not trivial. In this paper, knowledge of physics is leveraged to select more similar features, the modal assurance criterion (MAC) is used to quantify the correspondence between the modes of healthy structures. The MAC is shown to have high correspondence with a supervised metric that measures joint‑distribution similarity, which is the primary indicator of whether a classifier will generalise between domains. The MAC is proposed as a measure for selecting a set of features that behave consistently across domains when subjected to damage, i.e. features with invariance in the conditional distributions. This approach is demonstrated on numerical and experimental case studies to verify its effectiveness in various applications.

Abstract:
Real‑time particle transverse momentum (p_T) estimation in high‑energy physics demands algorithms that are both efficient and accurate under strict hardware constraints. Static machine learning models degrade under high pileup and lack physics‑aware optimization, while generic graph neural networks (GNNs) often neglect domain structure critical for robust p_T regression. We propose a physics‑informed GNN framework that systematically encodes detector geometry and physical observables through four distinct graph construction strategies that systematically encode detector geometry and physical observables: station‑as‑node, feature‑as‑node, bending angle‑centric, and pseudorapidity (η)‑centric representations. This framework integrates these tailored graph structures with a novel Message Passing Layer (MPL), featuring intra‑message attention and gated updates, and domain‑specific loss functions incorporating p_T‑distribution priors. Our co‑design methodology yields superior accuracy‑efficiency trade‑offs compared to existing baselines. Extensive experiments on the CMS Trigger Dataset validate the approach: a station‑informed EdgeConv model achieves a state‑of‑the‑art MAE of 0.8525 with \ge55% fewer parameters than deep learning baselines, especially TabNet, while an η‑centric MPL configuration also demonstrates improved accuracy with comparable efficiency. These results establish the promise of physics‑guided GNNs for deployment in resource‑constrained trigger systems.

Abstract:
Phase‑field equations, mostly solved numerically, are known for capturing the mesoscale microstructural evolution of a material. However, such numerical solvers are computationally expensive as it needs to generate fine mesh systems to solve the complex Partial Differential Equations(PDEs) with good accuracy. Therefore, we propose an alternative approach of predicting the microstructural evolution subjected to periodic boundary conditions using Physics informed Neural Operators (PINOs). In this study, we have demonstrated the capability of PINO to predict the growth of θ^\prime precipitates in Al‑Cu alloys by learning the operator as well as by solving three coupled physics equations simultaneously. The coupling is of two second‑order Allen‑Cahn equation and one fourth‑order Cahn‑Hilliard equation. We also found that using Fourier derivatives(pseudo‑spectral method and Fourier extension) instead of Finite Difference Method improved the Cahn‑Hilliard equation loss by twelve orders of magnitude. Moreover, since differentiation is equivalent to multiplication in the Fourier domain, unlike Physics informed Neural Networks(PINNs), we can easily compute the fourth derivative of Cahn‑Hilliard equation without converting it to coupled second order derivative.

Abstract:
HyperDeepONets were introduced in Lee, Cho and Hwang [ICLR, 2023] as an alternative architecture for operator learning, in which a hypernetwork generates the weights for the trunk net of a DeepONet. While this improves expressivity, it incurs high memory and computational costs due to the large number of output parameters required. In this work we introduce, in the physics‑informed machine learning setting, a variation, PI‑LoRA‑HyperDeepONets, which leverage low‑rank adaptation (LoRA) to reduce complexity by decomposing the hypernetwork's output layer weight matrix into two smaller low‑rank matrices. This reduces the number of trainable parameters while introducing an extra regularization of the trunk networks' weights. Through extensive experiments on both ordinary and partial differential equations we show that PI‑LoRA‑HyperDeepONets achieve up to 70% reduction in parameters and consistently outperform regular HyperDeepONets in terms of predictive accuracy and generalization.

Abstract:
A fundamental requirement for full autonomy in mobile robots is accurate navigation even in situations where satellite navigation or cameras are unavailable. In such practical situations, relying only on inertial sensors will result in navigation solution drift due to the sensors' inherent noise and error terms. One of the emerging solutions to mitigate drift is to maneuver the robot in a snake‑like slithering motion to increase the inertial signal‑to‑noise ratio, allowing the regression of the mobile robot position. In this work, we propose MoRPI‑PINN as a physics‑informed neural network framework for accurate inertial‑based mobile robot navigation. By embedding physical laws and constraints into the training process, MoRPI‑PINN is capable of providing an accurate and robust navigation solution. Using real‑world experiments, we show accuracy improvements of over 85% compared to other approaches. MoRPI‑PINN is a lightweight approach that can be implemented even on edge devices and used in any typical mobile robot application.

Abstract:
High Q‑factor narrow‑band absorption exhibits high spectral selectivity enabling high‑sensitive photodetectors, sensors and thermal emitters. All‑dielectric metasurfaces are widely regarded as excellent candidates for giving rise to such narrow‑band absorption. However, designing metasurfaces with specific functionalities remains a challenging task both experimentally and computationally, which is why inverse design methods are increasingly being explored. Inverse design process is highly complex due to its non‑unique solutions and the higher dimensionality of the design space, making it challenging to precisely control the resonance wavelength, linewidth, and absorption intensity. In this paper, we present a novel hybrid methodology that integrates generative adversarial networks (GANs) (both classical and quantum) with physics‑informed neural networks (PINNs) for the inverse design of narrow‑band absorbing metasurfaces. By introducing a Fano‑shaped absorption spectrum equation into the PINN loss function, we enforce physical constraints on the resonance behavior, ensuring outputs that are both spectrally accurate and physically consistent. The study presents a comparison between a conventional GAN + PINN framework and a PINN augmented by a hybrid quantum‑classical GAN (QGAN). The findings indicate that the integrated PINN + QGAN model achieves faster convergence, requires 99.5% fewer training samples, and yields an order of magnitude lower MSE compared to conventional GANs. Remarkably, even though the training dataset only contains metasurfaces with Q‑factors on the order of 10^3, the model is able to generate highly asymmetric metasurface structures with Q‑factors exceeding 10^5. This study presents a novel framework that integrates quantum machine learning with physics‑based modeling, providing a promising method for quantum‑enhanced inverse design in nanophotonic systems.

Abstract:
This note presents two nontrivial, rotational equilibrium solutions to the spatial uniform gas pressure (isobaric) approximate model of Prosperetti in the inviscid case. Building on Gavrilov's work [GAFA 2019], we first establish the existence of equilibrium solutions with nontrivial (rotational) liquid flow. Second, we construct a nonspherically symmetric, horn‑torus‑shaped equilibrium bubble under mild spatial decay conditions of the liquid flow. In addition, we extend earlier results on the characterization of spherical equilibrium bubbles to the axisymmetric, purely azimuthal setting. Finally, we implement a numerical simulation of the equilibrium bubble shape using the Physics‑Informed Neural Network (PINN) approximation.

Abstract:
Physics‑informed neural networks (PINN) have achieved notable success in solving partial differential equations (PDE), yet solving the Navier‑Stokes equations (NSE) with complex boundary conditions remains a challenging task. In this paper, we introduce a novel Hybrid Boundary PINN (HB‑PINN) method that combines a pretrained network for efficient initialization with a boundary‑constrained mechanism. The HB‑PINN method features a primary network focused on inner domain points and a distance metric network that enhances predictions at the boundaries, ensuring accurate solutions for both boundary and interior regions. Comprehensive experiments have been conducted on the NSE under complex boundary conditions, including the 2D cylinder wake flow and the 2D blocked cavity flow with a segmented inlet. The proposed method achieves state‑of‑the‑art (SOTA) performance on these benchmark scenarios, demonstrating significantly improved accuracy over existing PINN‑based approaches.

Abstract:
O‑C shell mergers in massive stars are a site for producing the p nuclei by the γ process, but 1D stellar models rely on mixing length theory, which does not match the radial velocity profiles of 3D hydrodynamic simulations. We investigate how 3D macro physics informed mixing impacts the nucleosynthesis of p nuclei. We post‑process the O‑shell of the M_\mathrmZAMS = 15~\mathrmM_\odot, Z = 0.02 model from the NuGrid stellar data set. Applying a downturn to velocities at the boundary and increasing velocities across the shell as obtained in previous results, we find non‑linear, non‑monotonic increase in p‑nuclei production with a spread of 0.96 dex, and find that isotopic ratios can change. Reducing C‑shell ingestion rates as found in 3D simulations suppresses production, with spreads of 1.22‑1.84 dex across MLT and downturn scenarios. Applying dips to the diffusion profile to mimic quenching events also suppresses production, with a 0.51 dex spread. We analyze the impact of varying all photo‑disintegration rates of unstable n‑deficient isotopes from Se‑Po by a factor of 10 up and down. The nuclear physics variations for the MLT and downturn cases have a spread of 0.56‑0.78 dex. We also provide which reaction rates are correlated with the p nuclei, and find few correlations shared between mixing scenarios. Our results demonstrate that uncertainties in mixing arising from uncertain 3D macro physics are as significant as nuclear physics and are crucial for understanding p‑nuclei production during O‑C shell mergers quantitatively.

Abstract:
The equilibrium reconstruction of plasma is a core step in real‑time diagnostic tasks in fusion research. This paper explores a multi‑stage Physics‑Informed Neural Networks(PINNs) approach to solve the Grad‑Shafranov equation, achieving high‑precision solutions with an error magnitude of O(10^‑8) between the output of the second‑stage neural network and the analytical solution. Our results demonstrate that the multi‑stage PINNs provides a reliable tool for plasma equilibrium reconstruction.

Abstract:
We present a novel data‑driven approach for enhancing gradient reconstruction in unstructured finite volume methods for hyperbolic conservation laws, specifically for the 2D Euler equations. Our approach extends previous structured‑grid methodologies to unstructured meshes through a modified DeepONet architecture that incorporates local geometry in the neural network. The architecture employs local mesh topology to ensure rotation invariance, while also ensuring first‑order constraint on the learned operator. The training methodology incorporates physics‑informed regularization through entropy penalization, total variation diminishing penalization, and parameter regularization to ensure physically consistent solutions, particularly in shock‑dominated regions. The model is trained on high‑fidelity datasets solutions derived from sine waves and randomized piecewise constant initial conditions with periodic boundary conditions, enabling robust generalization to complex flow configurations or geometries. Validation test cases from the literature, including challenging geometry configuration, demonstrates substantial improvements in accuracy compared to traditional second‑order finite volume schemes. The method achieves gains of 20‑60% in solution accuracy while enhancing computational efficiency. A convergence study has been conveyed and reveal improved mesh convergence rates compared to the conventional solver. The proposed algorithm is faster and more accurate than the traditional second‑order finite volume solver, enabling high‑fidelity simulations on coarser grids while preserving the stability and conservation properties essential for hyperbolic conservation laws. This work is a part of a new generation of solvers that are built by combining Machine‑Learning (ML) tools with traditional numerical schemes, all while ensuring physical constraint on the results.

Abstract:
The development of ammonia‑methane (NH3‑CH4) combustion as a hydrogen‑carrier energy source faces major challenges such as significant NOx emissions, hindering its practical implementation. This paper examines how ethanol (C2H6O) and methanol (CH4O) additives influence formation pathways of NOx using ReaxFF molecular dynamics (MD) simulations at temperatures of 2,000 K and 3,000 K. Ten carefully designed fuel mixtures (C1‑C10) were evaluated across 0%, 5%, and 10% alcohol concentrations. The findings show that adding alcohol can effectively suppress NOx production, especially at elevated temperatures. At 3,000 K, 10% ethanol addition and 10% methanol addition reduced the production of NOx by approximately 39.6% and 30.1%, respectively, compared with the base fuel. This suppression is attributed to the charge redistribution and the redirection of nitrogen intermediates through stabilising pathways such as HNO, HNO2, and N2O. Simulation‑derived descriptors served as the training data for machine learning (ML) models, including Random Forest Regression (RFR), Support Vector Regression (SVR), Gradient Boosting Regression (GBR), and Fully Connected Neural Networks (FCNN). RFR achieved superior performance compared with other models with an R2 of 0.993 and mean absolute error (MAE) of 0.661. The trained ML models successfully predicted NOx emissions for both simulated alcohol ratios (0%, 5% and 12%), and non‑simulated alcohol ratios (2%, 7%, and 12%) which demonstrating how hybrid physics‑informed ML algorithms can extrapolate complex chemical behaviours, along with prediction errors under 5% for most extrapolated ethanol cases. The results showcase that the ReaxFF informed ML framework successfully serves as a basis for designing cleaner fuels and is capable of establishing a reliable structure for future predictive models in combustion chemistry.

Abstract:
Solving the wave equation is fundamental for geophysical applications. However, numerical solutions of the Helmholtz equation face significant computational and memory challenges. Therefore, we introduce a physics‑informed convolutional neural operator (PICNO) to solve the Helmholtz equation efficiently. The PICNO takes both the background wavefield corresponding to a homogeneous medium and the velocity model as input function space, generating the scattered wavefield as the output function space. Our workflow integrates PDE constraints directly into the training process, enabling the neural operator to not only fit the available data but also capture the underlying physics governing wave phenomena. PICNO allows for high‑resolution reasonably accurate predictions even with limited training samples, and it demonstrates significant improvements over a purely data‑driven convolutional neural operator (CNO), particularly in predicting high‑frequency wavefields. These features and improvements are important for waveform inversion down the road.

Abstract:
This work focuses on the behavior of stochastic gradient descent (SGD) in solving least‑squares regression with physics‑informed neural networks (PINNs). Past work on this topic has been based on the over‑parameterization regime, whose convergence may require the network width to increase vastly with the number of training samples. So, the theory derived from over‑parameterization may incur prohibitive computational costs and is far from practical experiments. We perform new optimization and generalization analysis for SGD in training two‑layer PINNs, making certain assumptions about the target function to avoid over‑parameterization. Given ε>0, we show that if the network width exceeds a threshold that depends only on ε and the problem, then the training loss and expected loss will decrease below O(ε).

Abstract:
This work presents predictive hydrodynamic simulations empowered by artificial intelligence (AI) for laser driven implosion experiments, taking the double‑cone ignition (DCI) scheme as an example. A Transformer‑based deep learning model MULTI‑Net is established to predict implosion features according to laser waveforms and target radius. A Physics‑Informed Decoder (PID) is proposed for high‑dimensional sampling, significantly reducing the prediction errors compared to Latin hypercube sampling. Applied to DCI experiments conducted on the SG‑II Upgrade facility, the MULTI‑Net model is able to predict the implosion dynamics measured by the x‑ray streak camera. It is found that an effective laser absorption factor about 65% is suitable for the one‑dimensional simulations of the DCI‑R10 experiments. For shot 33, the mean implosion velocity and collided plasma density reached 195 km/s and 117 g/cc, respectively. This study demonstrates a data‑driven AI framework that enhances the prediction ability of simulations for complicated laser fusion experiments.

Abstract:
Urban heat islands (UHIs) pose a critical challenge in densely populated cities and tropical climates where large amounts of energy are used to meet the cooling demand. To address this, Building and Construction Authority (BCA) of Singapore provides incentives for passive cooling such as using of solar‑reflective material in its Green Mark guidelines. Thus, understanding about its real‑world effectiveness in tropical urban environments is required. This study evaluated the effectiveness of solar‑reflective cool paint using a hybrid modelling framework combining a transient physical model and data driven model through field measurements. Several machine learning algorithms were compared including multiple‑linear regression (MLR), random forest regressor (RF), AdaBoost regressor (AB), extreme gradient boosting regressor (XGB), and TabPFN regressor (TPR). The results indicated that the transient physical model overestimated facade temperatures in the lower temperature ranges. The physics‑informed MLR achieved best performance with improved accuracy for pre‑cool paint (R2=0.96, RMSE=0.83C) and post‑cool paint (R2=0.95, RMSE=0.65C) scenarios, reducing RMSE by 26% and 44%, respectively. The hybrid model also effectively predicted hourly heat fluxes revealing substantial reductions in surface temperature and heat storage with increasing albedo. The maximum net heat flux q_net was reduced by about 30‑65 W/m2 in the post‑cool paint stage (albedo = 0.73) compared to the pre‑cool paint stage (albedo = 0.31). As albedo increases from 0.1 to 0.9, the sensitivity analysis predicts that the maximum daytime surface temperature will decrease by about 11C and the peak heat release of the net heat flux will decrease significantly from about 161 W/m2 to 27 W/m2.

Abstract:
Physics‑informed neural networks (PINNs) have gained prominence in recent years and are now effectively used in a number of applications. However, their performance remains unstable due to the complex landscape of the loss function. To address this issue, we reformulate PINN training as a nonconvex‑strongly concave saddle‑point problem. After establishing the theoretical foundation for this approach, we conduct an extensive experimental study, evaluating its effectiveness across various tasks and architectures. Our results demonstrate that the proposed method outperforms the current state‑of‑the‑art techniques.

Abstract:
The fundamental laws of physics are intrinsically geometric, dictating the evolution of systems through principles of symmetry and conservation. While modern machine learning offers powerful tools for modeling complex dynamics from data, common methods often ignore this underlying geometric fabric. Physics‑informed neural networks, for instance, can violate fundamental physical principles, leading to predictions that are unstable over long periods, particularly for high‑dimensional and chaotic systems. Here, we introduce Geometric Hamiltonian Neural Networks (GeoHNN), a framework that learns dynamics by explicitly encoding the geometric priors inherent to physical laws. Our approach enforces two fundamental structures: the Riemannian geometry of inertia, by parameterizing inertia matrices in their natural mathematical space of symmetric positive‑definite matrices, and the symplectic geometry of phase space, using a constrained autoencoder to ensure the preservation of phase space volume in a reduced latent space. We demonstrate through experiments on systems ranging from coupled oscillators to high‑dimensional deformable objects that GeoHNN significantly outperforms existing models. It achieves superior long‑term stability, accuracy, and energy conservation, confirming that embedding the geometry of physics is not just a theoretical appeal but a practical necessity for creating robust and generalizable models of the physical world.

Abstract:
We propose a mesh‑free policy iteration framework that combines classical dynamic programming with physics‑informed neural networks (PINNs) to solve high‑dimensional, nonconvex Hamilton‑‑Jacobi‑‑Isaacs (HJI) equations arising in stochastic differential games and robust control. The method alternates between solving linear second‑order PDEs under fixed feedback policies and updating the controls via pointwise minimax optimization using automatic differentiation. Under standard Lipschitz and uniform ellipticity assumptions, we prove that the value function iterates converge locally uniformly to the unique viscosity solution of the HJI equation. The analysis establishes equi‑Lipschitz regularity of the iterates, enabling provable stability and convergence without requiring convexity of the Hamiltonian. Numerical experiments demonstrate the accuracy and scalability of the method. In a two‑dimensional stochastic path‑planning game with a moving obstacle, our method matches finite‑difference benchmarks with relative L^2‑errors below %10^‑2%. In five‑ and ten‑dimensional publisher‑subscriber differential games with anisotropic noise, the proposed approach consistently outperforms direct PINN solvers, yielding smoother value functions and lower residuals. Our results suggest that integrating PINNs with policy iteration is a practical and theoretically grounded method for solving high‑dimensional, nonconvex HJI equations, with potential applications in robotics, finance, and multi‑agent reinforcement learning.

Abstract:
Gridization, the process of partitioning space into grids where users share similar channel characteristics, serves as a fundamental prerequisite for efficient large‑scale network optimization. However, existing methods like Geographical or Beam Space Gridization (GSG or BSG) are limited by reliance on unavailable location data or the flawed assumption that similar signal strengths imply similar channel properties. We propose Channel Space Gridization (CSG), a pioneering framework that unifies channel estimation and gridization for the first time. Formulated as a joint optimization problem, CSG uses only beam‑level reference signal received power (RSRP) to estimate Channel Angle Power Spectra (CAPS) and partition samples into grids with homogeneous channel characteristics. To perform CSG, we develop the CSG Autoencoder (CSG‑AE), featuring a trainable RSRP‑to‑CAPS encoder, a learnable sparse codebook quantizer, and a physics‑informed decoder based on the Localized Statistical Channel Model. On recognizing the limitations of naive training scheme, we propose a novel Pretraining‑Initialization‑Detached‑Asynchronous (PIDA) training scheme for CSG‑AE, ensuring stable and effective training by systematically addressing the common pitfalls of the naive training paradigm. Evaluations reveal that CSG‑AE excels in CAPS estimation accuracy and clustering quality on synthetic data. On real‑world datasets, it reduces Active Mean Absolute Error (MAE) by 30% and Overall MAE by 65% on RSRP prediction accuracy compared to salient baselines using the same data, while improving channel consistency, cluster sizes balance, and active ratio, advancing the development of gridization for large‑scale network optimization.

Abstract:
This letter develops a novel physics‑informed neural ordinary differential equations‑based framework to emulate the proprietary dynamics of the inverters ‑‑ essential for improved accuracy in grid dynamic simulations. In current industry practice, the original equipment manufacturers (OEMs) often do not disclose the exact internal controls and parameters of the inverters, posing significant challenges in performing accurate dynamic simulations and other relevant studies, such as gain tunings for stability analysis and controls. To address this, we propose a Physics‑Informed Latent Neural ODE Model (PI‑LNM) that integrates system physics with neural learning layers to capture the unmodeled behaviors of proprietary units. The proposed method is validated using a grid‑forming inverter (GFM) case study, demonstrating improved dynamic simulation accuracy over approaches that rely solely on data‑driven learning without physics‑based guidance.

Abstract:
Coastal communities increasingly face compound floods, where multiple drivers like storm surge, high tide, heavy rainfall, and river discharge occur together or in sequence to produce impacts far greater than any single driver alone. Traditional hydrodynamic models can provide accurate physics‑based simulations but require substantial computational resources for real‑time applications or risk assessments, while machine learning alternatives often sacrifice physical consistency for speed, producing unrealistic predictions during extreme events. This study addresses these challenges by developing ALPINE (All‑in‑one Physics Informed Neural Emulator), a physics‑informed neural network (PINN) framework to enforce complete shallow water dynamics in compound flood modeling. Unlike previous approaches that implement partial constraints, our framework simultaneously enforces mass conservation and both momentum equations, ensuring full adherence to Newton's laws throughout the prediction process. The model integrates a convolutional encoder‑decoder architecture with ConvLSTM temporal processing, trained using a composite loss function that balances data fidelity with physics‑based residuals. Using six historical storm events (four for training, one for validation, and one held‑out for unseen testing), we observe substantial improvements over baseline neural networks. ALPINE reduces domain‑averaged prediction errors and improves model skill metrics for water surface elevation and velocity components. Physics‑informed constraints prove most valuable during peak storm intensity, when multiple flood drivers interact and reliable predictions matter most. This approach yields a physically consistent emulator capable of supporting compound‑flood forecasting and large‑scale risk analyses while preserving physical realism essential for coastal emergency management.

Abstract:
We introduce MENO (''Matrix Exponential‑based Neural Operator''), a hybrid surrogate modeling framework for efficiently solving stiff systems of ordinary differential equations (ODEs) that exhibit a sparse nonlinear structure. In such systems, only a few variables contribute nonlinearly to the dynamics, while the majority influence the equations linearly. MENO exploits this property by decomposing the system into two components: the low‑dimensional nonlinear part is modeled using conventional neural operators, while the linear time‑varying subsystem is integrated using a novel neural matrix exponential formulation. This approach combines the exact solution of linear time‑invariant systems with learnable, time‑dependent graph‑based corrections applied to the linear operators. Unlike black‑box or soft‑constrained physics‑informed (PI) models, MENO embeds the governing equations directly into its architecture, ensuring physical consistency (e.g., steady states), improved robustness, and more efficient training. We validate MENO on three complex thermochemical systems: the POLLU atmospheric chemistry model, an oxygen mixture in thermochemical nonequilibrium, and a collisional‑radiative argon plasma in one‑ and two‑dimensional shock‑tube simulations. MENO achieves relative errors below 2% in trained zero‑dimensional settings and maintains good accuracy in extrapolatory multidimensional regimes. It also delivers substantial computational speedups, achieving up to 4 800× on GPU and 185× on CPU compared to standard implicit ODE solvers. Although intrusive by design, MENO's physics‑based architecture enables superior generalization and reliability, offering a scalable path for real‑time simulation of stiff reactive systems.

Abstract:
We apply a graybox machine‑learning framework to model and control a qubit undergoing Markovian and non‑Markovian dynamics from environmental noise. The approach combines physics‑informed equations with a lightweight transformer neural network based on the self‑attention mechanism. The model is trained on simulated data and learns an effective operator that predicts observables accurately, even in the presence of memory effects. We benchmark both non‑Gaussian random‑telegraph noise and Gaussian Ornstein‑Uhlenbeck noise and achieve low prediction errors even in challenging noise coupling regimes. Using the model as a dynamics emulator, we perform gradient‑based optimal control to identify pulse sequences implementing a universal set of single‑qubit gates, achieving fidelities above 99% for the lowest considered value of the coupling and remaining above 90% for the highest.

Abstract:
Traditional echocardiographic parameters such as ejection fraction (EF) and global longitudinal strain (GLS) have limitations in the early detection of cardiac dysfunction. EF often remains normal despite underlying pathology, and GLS is influenced by load conditions and vendor variability. There is a growing need for reproducible, interpretable, and operator‑independent parameters that capture subtle and global cardiac functional alterations. We introduce the Acoustic Index, a novel AI‑derived echocardiographic parameter designed to quantify cardiac dysfunction from standard ultrasound views. The model combines Extended Dynamic Mode Decomposition (EDMD) based on Koopman operator theory with a hybrid neural network that incorporates clinical metadata. Spatiotemporal dynamics are extracted from echocardiographic sequences to identify coherent motion patterns. These are weighted via attention mechanisms and fused with clinical data using manifold learning, resulting in a continuous score from 0 (low risk) to 1 (high risk). In a prospective cohort of 736 patients, encompassing various cardiac pathologies and normal controls, the Acoustic Index achieved an area under the curve (AUC) of 0.89 in an independent test set. Cross‑validation across five folds confirmed the robustness of the model, showing that both sensitivity and specificity exceeded 0.8 when evaluated on independent data. Threshold‑based analysis demonstrated stable trade‑offs between sensitivity and specificity, with optimal discrimination near this threshold. The Acoustic Index represents a physics‑informed, interpretable AI biomarker for cardiac function. It shows promise as a scalable, vendor‑independent tool for early detection, triage, and longitudinal monitoring. Future directions include external validation, longitudinal studies, and adaptation to disease‑specific classifiers.

Abstract:
Physics‑guided approaches offer a promising path toward accurate and generalisable impact identification in composite structures, especially when experimental data are sparse. This paper presents a hybrid framework for impact localisation and force estimation in composite plates, combining a data‑driven implementation of First‑Order Shear Deformation Theory (FSDT) with machine learning and uncertainty quantification. The structural configuration and material properties are inferred from dispersion relations, while boundary conditions are identified via modal characteristics to construct a low‑fidelity but physically consistent FSDT model. This model enables physics‑informed data augmentation for extrapolative localisation using supervised learning. Simultaneously, an adaptive regularisation scheme derived from the same model improves the robustness of impact force reconstruction. The framework also accounts for uncertainty by propagating localisation uncertainty through the force estimation process, producing probabilistic outputs. Validation on composite plate experiments confirms the framework's accuracy, robustness, and efficiency in reducing dependence on large training datasets. The proposed method offers a scalable and transferable solution for impact monitoring and structural health management in composite aerostructures.

Abstract:
Neural networks (NNs) have great potential in solving the ground state of various many‑body problems. However, several key challenges remain to be overcome before NNs can tackle problems and system sizes inaccessible with more established tools. Here, we present a general and efficient method for learning the NN representation of an arbitrary many‑body complex wave function from its N‑particle probability density and probability current density and successfully test on (non‑Abelian) fractional quantum Hall states and chiral BCS wavefunction. Having reached overlaps as large as 99.9%, we employ our neural wave function for pre‑training to effortlessly solve the fractional quantum Hall problem with Coulomb interactions and realistic Landau‑level mixing for as many as 25 particles and uncover distinctive features of the edge. Our work demonstrates efficient, scalable and accurate simulation of highly‑entangled quantum matter using general‑purpose deep NNs enhanced with physics‑informed initialization.

Abstract:
We discuss physics‑informed renormalisation group flows (PIRGs) for general operators. We show that operator PIRGs provide a comprehensive access to all correlation functions of the quantum field theory under investigation. The operator PIRGs can be seen as a completion of the PIRG‑approach, whose qualitative computational simplification and structural insights are now fully accessible for general applications. The potential of this setup is assessed within a simple analytic example of the zero‑dimensional ϕ^4‑theory for which the generating functions of the fundamental field are computed within a vertex expansion, using the one‑ to ten‑point functions.

Abstract:
Partial differential equations (PDEs) with near singular solutions pose significant challenges for traditional numerical methods, particularly in complex geometries where mesh generation and adaptive refinement become computationally expensive. While deep‑learning‑based approaches, such as Physics‑Informed Neural Networks (PINNs) and the Random Feature Method (RFM), offer mesh‑free alternatives, they often lack adaptive resolution in critical regions, limiting their accuracy for solutions with steep gradients or singularities. In this work, we propose the Adaptive Feature Capture Method (AFCM), a novel machine learning framework that adaptively redistributes neurons and collocation points in high‑gradient regions to enhance local expressive power. Inspired by adaptive moving mesh techniques, AFCM employs the gradient norm of an approximate solution as a monitor function to guide the reinitialization of feature function parameters. This ensures that partition hyperplanes and collocation points cluster where they are most needed, achieving higher resolution without increasing computational overhead. The AFCM extends the capabilities of RFM to handle PDEs with near‑singular solutions while preserving its mesh‑free efficiency. Numerical experiments demonstrate the method's effectiveness in accurately resolving near‑singular problems, even in complex geometries. By bridging the gap between adaptive mesh refinement and randomized neural networks, AFCM offers a robust and scalable approach for solving challenging PDEs in scientific and engineering applications.

Abstract:
This study presents a novel physics informed, data‑driven modeling framework for capturing the strongly nonlinear thermo‑viscoelastic behavior of soft materials exhibiting stress softening, with emphasis on the Mullins effect. Unlike previous approaches limited to quasi‑static or isothermal conditions, our model unifies rate dependence, temperature sensitivity, large strain cyclic loading, and evolving damage mechanisms. Thermodynamic admissibility is ensured via a custom loss function that embeds the Clausius Duhem inequality and explicitly constrains the damage variable for physically realistic softening. A Temporal Convolutional Network is trained on high fidelity experimental data across multiple temperatures, strain rates, and stretch levels, enabling the model to capture rich thermomechanical coupling and history dependence. The framework generalizes to unseen thermo mechanical conditions, higher strain rates, and larger deformations, and remains robust to input noise. Validation against finite element simulations using Abaqus/Explicit demonstrates excellent agreement under cyclic loading and damage evolution, confirming the surrogate models effectiveness for advanced simulation workflows.

Abstract:
We propose a general hybrid physics‑informed machine learning framework for modeling nonlinear, history‑dependent viscoelastic behavior under multiaxial cyclic loading. The approach is built on a generalized internal state variable‑based visco‑hyperelastic constitutive formulation, where stress is decomposed into volumetric, isochoric hyperelastic, and isochoric viscoelastic components. Gaussian Process Regression (GPR) models the equilibrium response, while Recurrent Neural Networks (RNNs) with Long Short‑Term Memory (LSTM) units capture time‑dependent viscoelastic effects. Physical constraints, including objectivity, material symmetry, and thermodynamic consistency, are enforced to ensure physically valid predictions. After developing the general form of the surrogate model based on tensor integrity bases and response functions, we employed the nonlinear Holzapfel differential viscoelastic model to generate training data. Two datasets, one for short‑term and another for long‑term relaxation, are constructed to span a wide range of material memory characteristics. The model is trained and tested under diverse multiaxial loading conditions, including different stretch levels applied independently in the longitudinal and transverse directions, varying strain rates, and both tension and compression states, even beyond the training domain. Energy dissipation is explicitly analyzed at different strain rates for both datasets to verify thermodynamic consistency through the second law. The results show that the proposed framework accurately captures complex, nonlinear, and rate‑dependent material responses. Moreover, it demonstrates strong robustness to synthetic noise, enabling generalizable and physically consistent predictions under realistic and variable loading scenarios.

Abstract:
Simulating hair dynamics that generalize across arbitrary hairstyles, body shapes, and motions is a critical challenge. Our novel two‑stage neural solution is the first to leverage Transformer‑based architectures for such a broad generalization. We propose a Transformer‑powered static network that predicts static draped shapes for any hairstyle, effectively resolving hair‑body penetrations and preserving hair fidelity. Subsequently, a dynamic network with a novel cross‑attention mechanism fuses static hair features with kinematic input to generate expressive dynamics and complex secondary motions. This dynamic network also allows for efficient fine‑tuning of challenging motion sequences, such as abrupt head movements. Our method offers real‑time inference for both static single‑frame drapes and dynamic drapes over pose sequences. Our method demonstrates high‑fidelity and generalizable dynamic hair across various styles, guided by physics‑informed losses, and can resolve penetrations even for complex, unseen long hairstyles, highlighting its broad generalization.

Abstract:
Accurate characterization of quantum systems is essential for the development of quantum technologies, particularly in the noisy intermediate‑scale quantum (NISQ) era. While traditional methods for Hamiltonian learning and noise characterization often require extensive measurements and scale poorly with system size, machine learning approaches offer promising alternatives. In this work, we extend the inverse physics‑informed neural network (referred to as PINNverse) framework to open quantum systems governed by Lindblad master equations. By incorporating both coherent and dissipative dynamics into the neural network training, our method enables simultaneous identification of Hamiltonian parameters and decay rates from noisy experimental data. We demonstrate the effectiveness and robustness of the approach through numerical simulations of two‑qubit open systems. Our results show that PINNverse provides a scalable and noise‑resilient framework for quantum system identification, with potential applications in quantum control and error mitigation.

Abstract:
Digital Twins (DTs) are virtual representations of physical systems synchronized in real time through Internet of Things (IoT) sensors and computational models. In industrial applications, DTs enable predictive maintenance, fault diagnosis, and process optimization. This paper explores the mathematical foundations of DTs, hybrid modeling techniques, including Physics Informed Neural Networks (PINNs), and their implementation in industrial scenarios. We present key applications, computational tools, and future research directions.

Abstract:
Many physical systems are described by partial differential equations (PDEs), and solving these equations and estimating their coefficients or boundary conditions (BCs) from observational data play a crucial role in understanding the associated phenomena. Recently, a machine learning approach known as physics‑informed neural network, which solves PDEs using neural networks by minimizing the sum of residuals from the PDEs, BCs, and data, has gained significant attention in the scientific community. In this study, we investigate a physics‑informed linear model (PILM) that uses linear combinations of basis functions to represent solutions, thereby enabling an analytical representation of optimal solutions. The PILM was formulated and verified for illustrative forward and inverse problems including cases with uncertain BCs. Furthermore, the PILM was applied to estimate crustal strain rates using geodetic data. Specifically, physical regularization that enforces elastic equilibrium on the velocity fields was compared with mathematical regularization that imposes smoothness constraints. From a Bayesian perspective, mathematical regularization exhibited superior performance. The PILM provides an analytically solvable framework applicable to linear forward and inverse problems, underdetermined systems, and physical regularization.

Abstract:
Training deep neural networks for scientific computing remains computationally expensive due to the slow formation of diverse feature representations in early training stages. Recent studies identify a staircase phenomenon in training dynamics, where loss decreases are closely correlated with increases in \varepsilon‑rank, reflecting the effective number of linearly independent neuron functions. Motivated by this observation, this work proposes a structured first‑layer initialization (SFLI) pre‑training method to enhance the diversity of neural features at initialization by constructing \varepsilon‑linearly independent neurons in the input layer. We present systematic initialization schemes compatible with various activation functions and integrate the strategy into multiple neural architectures, including modified multi‑layer perceptrons and physics‑informed residual adaptive networks. Extensive numerical experiments on function approximation and PDE benchmarks, demonstrate that SFLI significantly improves the initial \varepsilon‑rank, accelerates convergence, mitigates spectral bias, and enhances prediction accuracy. With the help of SILP, we only need to add one line of code to conventional existing algorithms.

Abstract:
The development of advanced packaging is essential in the semiconductor manufacturing industry. However, non‑destructive testing (NDT) of advanced packaging becomes increasingly challenging due to the depth and complexity of the layers involved. In such a scenario, Magnetic field imaging (MFI) enables the imaging of magnetic fields generated by currents. For MFI to be effective in NDT, the magnetic fields must be converted into current density. This conversion has typically relied solely on a Fast Fourier Transform (FFT) for magnetic field inversion; however, the existing approach does not consider eddy current effects or image misalignment in the test setup. In this paper, we present a spatial‑physics informed model (SPIM) designed for a 3D spiral sample scanned using Superconducting QUantum Interference Device (SQUID) microscopy. The SPIM encompasses three key components: i) magnetic image enhancement by aligning all the "sharp" wire field signals to mitigate the eddy current effect using both in‑phase (I‑channel) and quadrature‑phase (Q‑channel) images; (ii) magnetic image alignment that addresses skew effects caused by any misalignment of the scanning SQUID microscope relative to the wire segments; and (iii) an inversion method for converting magnetic fields to magnetic currents by integrating the Biot‑Savart Law with FFT. The results show that the SPIM improves I‑channel sharpness by 0.3% and reduces Q‑channel sharpness by 25%. Also, we were able to remove rotational and skew misalignments of 0.30 in a real image. Overall, SPIM highlights the potential of combining spatial analysis with physics‑driven models in practical applications.

Abstract:
This paper presents a neural network (NN)‑based solver for an integro‑differential equation that models shrinkage‑induced fragmentation. The proposed method directly maps input parameters to the corresponding probability density function without numerically solving the governing equation, thereby significantly reducing computational costs. Specifically, it enables efficient evaluation of the density function in Monte Carlo simulations while maintaining accuracy comparable to or even exceeding that of conventional finite difference schemes. Validatation on synthetic data demonstrates both the method's computational efficiency and predictive reliability. This study establishes a foundation for the data‑driven inverse analysis of fragmentation and suggests the potential for extending the framework beyond pre‑specified model structures.

Abstract:
Stirred tanks are vital in chemical and biotechnological processes, particularly as bioreactors. Although computational fluid dynamics (CFD) is widely used to model the flow in stirred tanks, its high computational cost‑especially in multi‑query scenarios for process design and optimization‑drives the need for efficient data‑driven surrogate models. However, acquiring sufficiently large datasets can be costly. Physics‑informed neural networks (PINNs) offer a promising solution to reduce data requirements while maintaining accuracy by embedding underlying physics into neural network (NN) training. This study quantifies the data requirements of vanilla PINNs for developing surrogate models of a flow field in a 2D stirred tank. We compare these requirements with classical supervised neural networks and boundary‑informed neural networks (BINNs). Our findings demonstrate that surrogate models can achieve prediction errors around 3% across Reynolds numbers from 50 to 5000 using as few as six datapoints. Moreover, employing an approximation of the velocity profile in place of real data labels leads to prediction errors of around 2.5%. These results indicate that even with limited or approximate datasets, PINNs can be effectively trained to deliver high accuracy comparable to high‑fidelity data.

Abstract:
The complexity of armed conflicts is expressed in the number of fatalities that may span several orders of magnitude. This study presents a minimalist, physics‑informed approach to estimating the likelihood of extreme conflict fatalities at the country level of analysis using Bayesian modeling and energy‑based dynamics. Leveraging the Boltzmann distribution to construct a Dirichlet prior, we formulate a predictive measure that captures the underlying entropy and energy states of conflict severity. By analyzing a dataset of 112 countries in conflict, we support the predictive power of the proposed measure. The findings suggest that extreme conflict events may be better understood through a minimal but theoretically grounded approach.

Abstract:
We propose a transfer learning framework for sound source reconstruction in Near‑field Acoustic Holography (NAH), which adapts a well‑trained data‑driven model from one type of sound source to another using a physics‑informed procedure. The framework comprises two stages: (1) supervised pre‑training of a complex‑valued convolutional neural network (CV‑CNN) on a large dataset, and (2) purely physics‑informed fine‑tuning on a single data sample based on the Kirchhoff‑Helmholtz integral. This method follows the principles of transfer learning by enabling generalization across different datasets through physics‑informed adaptation. The effectiveness of the approach is validated by transferring a pre‑trained model from a rectangular plate dataset to a violin top plate dataset, where it shows improved reconstruction accuracy compared to the pre‑trained model and delivers performance comparable to that of Compressive‑Equivalent Source Method (C‑ESM). Furthermore, for successful modes, the fine‑tuned model outperforms both the pre‑trained model and C‑ESM in accuracy.

Abstract:
Stellar flare forecasting represents a critical frontier in astrophysics, offering profound insights into stellar activity mechanisms and exoplanetary habitability assessments. Yet the inherent unpredictability of flare activity, rooted in stellar diversity and evolutionary stages, underpins the field's core challenges: (1) sparse, incomplete, noisy lightcurve data from traditional observations; (2) ineffective multi‑scale flare evolution capture via single representations; (3) poor physical interpretability in data‑driven models lacking physics‑informed priors. To address these challenges, we propose StellarF, a physics‑informed framework synergizing general Al with astrophysical domain knowledge via three core components: a unified preprocessing pipeline for lightcurve refinement (missing‑value imputation, temporal patch partitioning, adaptive sample filtering); a Low‑Rank Adaptation (LoRA)‑finetuned large language model (LLM) backbone enhanced by first‑order difference augmentation, flare statistical information, and flare historical record modules for multimodal fusion instead of only simple representations; and a novel physics‑informed loss embedding a minimum rising rate prior, appended to the cross‑entropy loss, to align with flare physics. Extensive experiments on Kepler and TESS datasets show StellarF achieves state‑of‑the‑art performance across key metrics, setting new benchmarks for flare forecasting. This work bridges general AI with astrophysics, offering a practical, physically interpretable paradigm for transient event forecasting in time‑domain astronomy.

Abstract:
Semiconductor manufacturing relies heavily on film deposition processes, such as Chemical Vapor Deposition and Physical Vapor Deposition. These complex processes require precise control to achieve film uniformity, proper adhesion, and desired functionality. Recent advancements in Physics‑Informed Neural Networks (PINNs), an innovative machine learning (ML) approach, have shown significant promise in addressing challenges related to process control, quality assurance, and predictive modeling within semiconductor film deposition and other manufacturing domains. This paper provides a comprehensive review of ML applications targeted at semiconductor film deposition processes. Through a thematic analysis, we identify key trends, existing limitations, and research gaps, offering insights into both the advantages and constraints of current methodologies. Our structured analysis aims to highlight the potential integration of these ML techniques to enhance interpretability, accuracy, and robustness in film deposition processes. Additionally, we examine state‑of‑the‑art PINN methods, discussing strategies for embedding physical knowledge, governing laws, and partial differential equations into advanced neural network architectures tailored for semiconductor manufacturing. Based on this detailed review, we propose novel research directions that integrate the strengths of PINNs to significantly advance film deposition processes. The contributions of this study include establishing a clear pathway for future research in integrating physics‑informed ML frameworks, addressing existing methodological gaps, and ultimately improving precision, scalability, and operational efficiency within semiconductor manufacturing.

Abstract:
System inference for nonlinear dynamic models, represented by ordinary differential equations (ODEs), remains a significant challenge in many fields, particularly when the data are noisy, sparse, or partially observable. In this paper, we propose a Simulation‑based Generative Model for Imperfect Data (SiGMoID) that enables precise and robust inference for dynamic systems. The proposed approach integrates two key methods: (1) physics‑informed neural networks with hyper‑networks that constructs an ODE solver, and (2) Wasserstein generative adversarial networks that estimates ODE parameters by effectively capturing noisy data distributions. We demonstrate that SiGMoID quantifies data noise, estimates system parameters, and infers unobserved system components. Its effectiveness is validated validated through realistic experimental examples, showcasing its broad applicability in various domains, from scientific research to engineered systems, and enabling the discovery of full system dynamics.

Abstract:
Physics‑informed machine learning frameworks such as Physics‑Informed Neural Networks (PINNs) and Physics‑Informed Extreme Learning Machines (PI‑ELMs) have shown great promise for solving partial differential equations (PDEs) but struggle with localized sharp gradients and singularly perturbed regimes, PINNs due to spectral bias and PI‑ELMs due to their single‑shot, non‑adaptive formulation. We propose the Kernel‑Adaptive Physics‑Informed Extreme Learning Machine (KAPI‑ELM), which performs Bayesian optimization over a low‑dimensional, physically interpretable hyperparameter space governing the distribution of Radial Basis Function (RBF) centers and widths. This converts high‑dimensional weight optimization into a low‑dimensional distributional search, enabling targeted kernel refinement in regions with sharp gradients while also improving baseline solutions in smooth‑flow regimes by tuning RBF supports. KAPI‑ELM is validated on benchmark forward and inverse problems (1D convection‑diffusion and 2D Poisson) involving PDEs with sharp gradients. It accurately resolves steep layers, improves smooth‑solution fidelity, and recovers physical parameters robustly, matching or surpassing advanced methods such as the extended Theory of Functional Connections (X‑TFC) with nearly an order of magnitude fewer tunable parameters. An extension to nonlinear problems is demonstrated by a curriculum‑based solution of the steady Navier‑Stokes equations via successive linearizations, yielding stable solutions for benchmark lid‑driven cavity flow up to Re=100. These results indicate that KAPI‑ELM provides an efficient and unified approach for forward and inverse PDEs, particularly in challenging sharp‑gradient regimes.

Abstract:
This paper presents a novel framework for whole‑body torque control of humanoid robots without joint torque sensors, designed for systems with electric motors and high‑ratio harmonic drives. The approach integrates Physics‑Informed Neural Networks (PINNs) for friction modeling and Unscented Kalman Filtering (UKF) for joint torque estimation, within a real‑time torque control architecture. PINNs estimate nonlinear static and dynamic friction from joint and motor velocity readings, capturing effects like motor actuation without joint movement. The UKF utilizes PINN‑based friction estimates as direct measurement inputs, improving torque estimation robustness. Experimental validation on the ergoCub humanoid robot demonstrates improved torque tracking accuracy, enhanced energy efficiency, and superior disturbance rejection compared to the state‑of‑the‑art Recursive Newton‑Euler Algorithm (RNEA), using a dynamic balancing experiment. The framework's scalability is shown by consistent performance across robots with similar hardware but different friction characteristics, without re‑identification. Furthermore, a comparative analysis with position control highlights the advantages of the proposed torque control approach. The results establish the method as a scalable and practical solution for sensorless torque control in humanoid robots, ensuring torque tracking, adaptability, and stability in dynamic environments.

Abstract:
Machine learning (ML) techniques have recently gained significant attention for solving compliance minimization (CM) problems. However, these methods typically provide poor feature boundaries, are very expensive, and lack a systematic mechanism to control the design complexity. Herein, we address these limitations by proposing a mesh‑free and simultaneous framework based on physics‑informed Gaussian processes (GPs). In our approach, we parameterize the design and state variables with GP priors which have independent kernels but share a multi‑output neural network (NN) as their mean function. The architecture of this NN is based on Parametric Grid Convolutional Attention Networks (PGCANs) which not only mitigate spectral bias issues, but also provide an interpretable mechanism to control design complexity. We estimate all the parameters of our GP‑based representations by simultaneously minimizing the compliance, total potential energy, and residual of volume fraction constraint. Importantly, our loss function exclude all data‑based residuals as GPs automatically satisfy them. We also develop computational schemes based on curriculum training and numerical integration to increase the efficiency and robustness of our approach which is shown to (1) produce super‑resolution topologies with fast convergence, (2) achieve comparable compliance and less gray area fraction compared to traditional numerical methods, (3) provide control over fine‑scale features, and (4) outperform competing ML‑based methods.

Abstract:
This paper introduces a framework based on physics‑informed neural networks (PINNs) for addressing key challenges in nonlinear lattices, including solution approximation, bifurcation diagram construction, and linear stability analysis. We first employ PINNs to approximate solutions of nonlinear systems arising from lattice models, using the Levenberg‑Marquardt algorithm to optimize network weights for greater accuracy. To enhance computational efficiency in high‑dimensional settings, we integrate a stochastic sampling strategy. We then extend the method by coupling PINNs with a continuation approach to compute snaking bifurcation diagrams, incorporating an auxiliary equation to effectively track successive solution branches. For linear stability analysis, we adapt PINNs to compute eigenvectors, introducing output constraints to enforce positivity, in line with Sturm‑Liouville theory. Numerical experiments are conducted on the discrete Allen‑Cahn equation with cubic and quintic nonlinearities in one to five spatial dimensions. The results demonstrate that the proposed approach achieves accuracy comparable to, or better than, traditional numerical methods, especially in high‑dimensional regimes where computational resources are a limiting factor. These findings highlight the potential of neural networks as scalable and efficient tools for the study of complex nonlinear lattice systems.

Abstract:
Accurate estimation of Remaining Useful Life (RUL) and State of Health (SOH) is essential for Prognostics and Health Management (PHM) across a wide range of industrial applications. We propose a novel framework ‑‑ Reinforced Graph‑Based Physics‑Informed Neural Networks Enhanced with Dynamic Weights (RGPD) ‑‑ that combines physics‑based supervision with advanced spatio‑temporal learning. Graph Convolutional Recurrent Networks (GCRNs) embed graph‑convolutional filters within recurrent units to capture how node representations evolve over time. Graph Attention Convolution (GATConv) leverages a self‑attention mechanism to compute learnable, edge‑wise attention coefficients, dynamically weighting neighbor contributions for adaptive spatial aggregation. A Soft Actor‑Critic (SAC) module is positioned between the Temporal Attention Unit (TAU) and GCRN to further improve the spatio‑temporal learning. This module improves attention and prediction accuracy by dynamically scaling hidden representations to minimize noise and highlight informative features. To identify the most relevant physical constraints in each area, Q‑learning agents dynamically assign weights to physics‑informed loss terms, improving generalization across real‑time industrial systems and reducing the need for manual tuning. In both RUL and SOH estimation tasks, the proposed method consistently outperforms state‑of‑the‑art models, demonstrating strong robustness and predictive accuracy across varied degradation patterns across three diverse industrial benchmark datasets.

Abstract:
We introduce the Energy Dissipation Rate guided Adaptive Sampling (EDRAS) strategy, a novel method that substantially enhances the performance of Physics‑Informed Neural Networks (PINNs) in solving thermodynamically consistent partial differential equations (PDEs) over arbitrary domains. EDRAS leverages the local energy dissipation rate density as a guiding metric to identify and adaptively re‑sample critical collocation points from both the interior and boundary of the computational domain. This dynamical sampling approach improves the accuracy of residual‑based PINNs by aligning the training process with the underlying physical structure of the system. In this study, we demonstrate the effectiveness of EDRAS using the Allen‑Cahn phase field model in irregular geometries, achieving up to a sixfold reduction in the relative mean square error compared to traditional residual‑based adaptive refinement (RAR) methods. Moreover, we compare EDRAS with other residual‑based adaptive sampling approaches and show that EDRAS is not only computationally more efficient but also more likely to identify high‑impact collocation points. Through numerical solutions of the Allen‑Cahn equation with both static (Neumann) and dynamic boundary conditions in 2D disk‑ and ellipse‑shaped domains solved using PINN coupled with EDRAS, we gain significant insights into how dynamic boundary conditions influence bulk phase evolution and thermodynamic behavior. The proposed approach offers an effective, physically informed enhancement to PINN frameworks for solving thermodynamically consistent models, making PINN a robust and versatile computational tool for investigating complex thermodynamic processes in arbitrary geometries.

Abstract:
We present the first foundational AI model for universal physics simulation that learns physical laws directly from boundary‑condition data without requiring a priori equation encoding. Traditional physics‑informed neural networks (PINNs) and finite‑difference methods necessitate explicit mathematical formulation of governing equations, fundamentally limiting their generalizability and discovery potential. Our sketch‑guided diffusion transformer approach reimagines computational physics by treating simulation as a conditional generation problem, where spatial boundary conditions guide the synthesis of physically accurate steady‑state solutions. By leveraging enhanced diffusion transformer architectures with novel spatial relationship encoding, our model achieves direct boundary‑to‑equilibrium mapping and is generalizable to diverse physics domains. Unlike sequential time‑stepping methods that accumulate errors over iterations, our approach bypasses temporal integration entirely, directly generating steady‑state solutions with SSIM > 0.8 while maintaining sub‑pixel boundary accuracy. Our data‑informed approach enables physics discovery through learned representations analyzable via Layer‑wise Relevance Propagation (LRP), revealing emergent physical relationships without predetermined mathematical constraints. This work represents a paradigm shift from AI‑accelerated physics to AI‑discovered physics, establishing the first truly universal physics simulation framework.

Abstract:
Wire‑arc directed energy deposition (DED) has emerged as a promising additive manufacturing (AM) technology for large‑scale structural engineering applications. However, the complex thermal dynamics inherent to the process present challenges in ensuring structural integrity and mechanical properties of fabricated thick walls and plates. While finite element method (FEM) simulations have been conventionally employed to predict thermal history during deposition, their computational demand remains prohibitively high for actual large‑scale applications. Given the necessity of multiple repetitive simulations for heat management and the determination of an optimal printing strategy, FEM simulation quickly becomes entirely infeasible. Instead, advancements have been made in using trained neural networks as surrogate models for rapid prediction. However, traditional data‑driven approaches necessitate large amounts of relevant and verifiable external data, during the training and validation of the neural network. Regarding large‑scale wire‑arc DED, none of these data sources are readily available in quantities sufficient for an accurate surrogate. The introduction of physics‑informed neural networks (PINNs) has opened up an alternative simulation strategy by leveraging the existing physical knowledge of the phenomena with advanced machine learning methods. Despite their theoretical advantages, PINNs have seen limited application in the context of large‑scale wire‑arc DED for structural engineering. This study investigates the scalability of PINNs, focusing on efficient collocation points sampling, a critical factor controlling both the training time and model performance. Results show PINNs can reduce computational time and effort by up to 98.6%, while maintaining the desired accuracy and offering "super‑resolution". Future directions for enhancing PINN performance in metal AM are discussed.

Abstract:
We introduce MVPinn, a Physics‑Informed Neural Network (PINN) approach tailored for solving the Milne‑Eddington (ME) inversion problem, specifically applied to spectropolarimetric observations from the Big Bear Solar Observatory's Near‑InfraRed Imaging Spectropolarimeter (BBSO/NIRIS) at the Fe I 1.56 μm lines. Traditional ME inversion methods, though widely used, are computationally intensive, sensitive to noise, and often struggle to accurately capture complex profile asymmetries resulting from gradients in magnetic field strength, orientation, and line‑of‑sight velocities. By embedding the ME radiative transfer equations directly into the neural network training as physics‑informed constraints, our MVPinn method robustly and efficiently retrieves magnetic field parameters, significantly outperforming traditional inversion methods in accuracy, noise resilience, and the ability to handle asymmetric and weak polarization signals. After training, MVPinn infers one magnetogram in about 15 seconds, compared to tens of minutes required by traditional ME inversion on high‑resolution spectropolarimetric data. Quantitative comparisons demonstrate excellent agreement with well‑established magnetic field measurements from the SDO/HMI and Hinode/SOT‑SP instruments, with correlation coefficients of approximately 90%. In particular, MVPINN aligns better with Hinode/SOT‑SP data, indicating some saturation of HMI data at high magnetic strengths. We further analyze the physical significance of profile asymmetries and the limitations inherent in the ME model assumption. Our results illustrate the potential of physics‑informed machine learning methods in high‑spatial‑temporal solar observations, preparing for more sophisticated, real‑time magnetic field analysis essential for current and next‑generation solar telescopes and space weather monitoring.

Abstract:
Efficient cooling is vital for the performance and reliability of modern systems such as electronics, nuclear reactors, and industrial equipment. Jet impingement cooling is widely used for its high local heat transfer rates. Accurate estimation of convective heat transfer coefficient (CHTC) is essential for design, simulation, and control of thermal systems. However, estimating spatially varying CHTCs from limited and noisy temperature data poses a challenging inverse problem. This study presents a physics‑informed neural network (PINN) framework to estimate both averaged and spatially varying CHTCs at the fluid‑solid interface in a jet impingement setup at Reynolds number 5000. The model uses sparse and noisy temperature data from within the solid and embeds the transient heat conduction equation along with boundary and initial conditions into its loss function. This enables inference of unknown boundary parameters without explicit modeling of the fluid domain. Validation is performed using synthetic temperature data from high‑fidelity conjugate heat transfer (CHT) simulations. The framework is tested under various additive Gaussian noise levels (up to 30 percent) and sampling rates 0.25 to 4.0 per second. For noise levels up to 10% and sampling rates of 0.5 per second or higher, estimated CHTCs match CHT‑derived benchmarks with relative errors below 8 percent. Even under high‑noise scenarios, the framework maintains predictive accuracy when time resolution is sufficient. These results highlight the method's robustness to noise and sparse data, offering a scalable alternative to traditional inverse methods, experimental measurements, or full CHT modeling for estimating boundary thermal parameters in real‑world cooling applications.

Abstract:
Turbulent fluid flows are among the most computationally demanding problems in science, requiring enormous computational resources that become prohibitive at high flow speeds. Physics‑informed neural networks (PINNs) represent a radically different approach that trains neural networks directly from physical equations rather than data, offering the potential for continuous, mesh‑free solutions. Here we show that appropriately designed PINNs can successfully simulate fully turbulent flows in both two and three dimensions, directly learning solutions to the fundamental fluid equations without traditional computational grids or training data. Our approach combines several algorithmic innovations including adaptive network architectures, causal training, and advanced optimization methods to overcome the inherent challenges of learning chaotic dynamics. Through rigorous validation on challenging turbulence problems, we demonstrate that PINNs accurately reproduce key flow statistics including energy spectra, kinetic energy, enstrophy, and Reynolds stresses. Our results demonstrate that neural equation solvers can handle complex chaotic systems, opening new possibilities for continuous turbulence modeling that transcends traditional computational limitations.

Abstract:
Physics‑informed machine learning (PIML) is an emerging framework that integrates physical knowledge into machine learning models. This physical prior often takes the form of a partial differential equation (PDE) system that the regression function must satisfy. In the first part of this dissertation, we analyze the statistical properties of PIML methods. In particular, we study the properties of physics‑informed neural networks (PINNs) in terms of approximation, consistency, overfitting, and convergence. We then show how PIML problems can be framed as kernel methods, making it possible to apply the tools of kernel ridge regression to better understand their behavior. In addition, we use this kernel formulation to develop novel physics‑informed algorithms and implement them efficiently on GPUs. The second part explores industrial applications in forecasting energy signals during atypical periods. We present results from the Smarter Mobility challenge on electric vehicle charging occupancy and examine the impact of mobility on electricity demand. Finally, we introduce a physics‑constrained framework for designing and enforcing constraints in time series, applying it to load forecasting and tourism forecasting in various countries.

Abstract:
Traditional numerical methods often struggle with the complexity and scale of modeling pollutant transport across vast and dynamic oceanic domains. This paper introduces a Physics‑Informed Neural Network (PINN) framework to simulate the dispersion of pollutants governed by the 2D advection‑diffusion equation. The model achieves physically consistent predictions by embedding physical laws and fitting to noisy synthetic data, generated via a finite difference method (FDM), directly into the neural network training process. This approach addresses challenges such as non‑linear dynamics and the enforcement of boundary and initial conditions. Synthetic data sets, augmented with varying noise levels, are used to capture real‑world variability. The training incorporates a hybrid loss function including PDE residuals, boundary/initial condition conformity, and a weighted data fit term. The approach takes advantage of the Julia language scientific computing ecosystem for high‑performance simulations, offering a scalable and flexible alternative to traditional solvers

Abstract:
Active flow control is applied to improve the aerodynamic performance of a NACA0018 airfoil operating as a single‑bladed vertical axis wind turbine (VAWT). Results computed by wall‑resolved large‑eddy simulations (LES) highlight the detrimental effects of the dynamic stall vortex (DSV) and trailing‑edge vortex (TEV) on turbine efficiency, primarily through increased drag and energy loss. The proposed flow control strategy effectively delays flow separation and suppresses large‑scale vortex formation, particularly at moderate actuation frequencies. The control parameters are grounded in bi‑global stability and finite‑time resolvent analyses. These techniques identify the excitation of coupling modes between shear layer and wake instabilities as a mechanism for promoting flow reattachment and preventing vorticity accumulation, ultimately leading to enhanced torque production. The control strategy is energy‑efficient, consuming only 1% of the turbine's output power while yielding substantial aerodynamic performance gains. These findings demonstrate the promise of physics‑informed active flow control in mitigating dynamic stall and advancing the design of next‑generation VAWTs.

Abstract:
Physics‑informed neural networks (PINNs) have proven to be a promising method for the rapid solving of partial differential equations (PDEs) in both forward and inverse problems. However, due to the smoothness assumption of functions approximated by general neural networks, PINNs are prone to spectral bias and numerical instability and suffer from reduced accuracy when solving PDEs with sharp spatial transitions or fast temporal evolution. To address this limitation, a discontinuity‑aware physics‑informed neural network (DPINN) method is proposed. It incorporates an adaptive Fourier‑feature embedding layer to mitigate spectral bias and capture steep gradients, a discontinuity‑aware network that generalizes the Kolmogorov representation theorem to the discontinuous regime for the modeling of shock‑wave properties, mesh transformation to accelerate convergence across complex geometries, and learnable local artificial viscosity to stabilize the algorithm near discontinuities. In numerical experiments regarding the inviscid Burgers' equation, Riemann problems, and transonic and supersonic airfoil flows, DPINN demonstrated superior accuracy in capturing discontinuities compared to existing methods.

Abstract:
Traditional physics‑informed neural networks (PINNs) do not guarantee strict constraint satisfaction. This is problematic in engineering systems where minor violations of governing laws can degrade the reliability and consistency of model predictions. In this work, we introduce KKT‑Hardnet, a neural network architecture that enforces linear and nonlinear equality and inequality constraints up to machine precision. It leverages a differentiable projection onto the feasible region by solving Karush‑Kuhn‑Tucker (KKT) conditions of a distance minimization problem. Furthermore, we reformulate the nonlinear KKT conditions via a log‑exponential transformation to construct a sparse system with linear and exponential terms. We apply KKT‑Hardnet to nonconvex pooling problem and a real‑world chemical process simulation. Compared to multilayer perceptrons and PINNs, KKT‑Hardnet achieves strict constraint satisfaction. It also circumvents the need to balance data and physics residuals in PINN training. This enables the integration of domain knowledge into machine learning towards reliable hybrid modeling of complex systems.

Abstract:
Physics‑informed neural networks have shown promise in solving partial differential equations (PDEs) by integrating physical constraints into neural network training, but their performance is sensitive to the sampling of points. Based on the impressive performance of quasi Monte‑Carlo methods in high dimensional problems, this paper proposes Quasi‑Random Physics‑Informed Neural Networks (QRPINNs), which use low‑discrepancy sequences for sampling instead of random points directly from the domain. Theoretically, QRPINNs have been proven to have a better convergence rate than PINNs. Empirically, experiments demonstrate that QRPINNs significantly outperform PINNs and some representative adaptive sampling methods, especially in high‑dimensional PDEs. Furthermore, combining QRPINNs with adaptive sampling can further improve the performance.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical constraints into the loss function. However, standard optimizers such as Adam often struggle to balance competing loss terms, particularly in stiff or ill‑conditioned systems. In this work, we propose a PDE‑aware optimizer that adapts parameter updates based on the variance of per‑sample PDE residual gradients. This method addresses gradient misalignment without incurring the heavy computational costs of second‑order optimizers such as SOAP. We benchmark the PDE‑aware optimizer against Adam and SOAP on 1D Burgers', Allen‑Cahn and Korteweg‑de Vries(KdV) equations. Across both PDEs, the PDE‑aware optimizer achieves smoother convergence and lower absolute errors, particularly in regions with sharp gradients. Our results demonstrate the effectiveness of PDE residual‑aware adaptivity in enhancing stability in PINNs training. While promising, further scaling on larger architectures and hardware accelerators remains an important direction for future research.

Abstract:
As ground‑based gravitational‑wave (GW) detectors improve in sensitivity, gravitational‑wave background (GWB) signals will progressively become detectable. Currently, searches for the GWB model the signal as a power law; however, deviations from this model will be relevant at increased sensitivity. Therefore, to prepare for the range of potentially detectable GWB signals, we propose an interpolation model implemented through a transdimensional reversible‑jump Markov chain Monte Carlo algorithm. This interpolation model foregoes a specific physics‑informed model (of which there are a great many) in favor of a flexible model that can accurately recover a broad range of potential signals. In this paper, we employ this framework for an array of GWB applications. We present three dimensionless fractional GW energy density injections and recoveries as examples of the capabilities of this spline interpolation model. We further demonstrate how our model can be implemented for hierarchical GW analysis on Ω_\rm GW.

Abstract:
Accurate computational modeling of damage and fracture remains a central challenge in solid mechanics. The finite element method (FEM) is widely used for numerical modeling of fracture problems; however, classical damage models without gradient regularization yield mesh‑dependent and usually inaccurate predictions. The use of gradient damage with FEM improves numerical robustness but introduces significant mathematical and numerical implementation complexities. Physics‑informed neural networks (PINNs) can encode the governing partial differential equations, boundary conditions, and constitutive models into the loss functions, offering a new method for fracture modeling. Prior applications of PINNs have been limited to small‑strain problems and have incorporated gradient damage formulation without a critical evaluation of its necessity. Since PINNs in their basic form are meshless, this work presents a PINN framework for modeling fracture in elastomers undergoing large deformation without the gradient damage formulation. The PINN implementation here does not require training data and utilizes the collocation method to formulate physics‑informed loss functions. We have validated the PINN's predictions for various defect configurations using benchmark solutions obtained from FEM with gradient damage formulation. The crack paths obtained using the PINN are approximately insensitive to the collocation point distribution. This study offers new insights into the feasibility of using PINNs without gradient damage and suggests a simplified and efficient computational modeling strategy for fracture problems. The PINN's performance has been evaluated through systematic variations in key neural network parameters to provide an assessment and guidance for future applications. The results motivate the extension of PINN‑based approaches to a broader class of materials and damage models in mechanics.

Abstract:
Data driven approaches have the potential to make modeling complex, nonlinear physical phenomena significantly more computationally tractable. For example, computational modeling of fracture is a core challenge where machine learning techniques have the potential to provide a much needed speedup that would enable progress in areas such as mutli‑scale modeling and uncertainty quantification. Currently, phase field modeling (PFM) of fracture is one such approach that offers a convenient variational formulation to model crack nucleation, branching and propagation. To date, machine learning techniques have shown promise in approximating PFM simulations. However, most studies rely on overly simple benchmarks that do not reflect the true complexity of the fracture processes where PFM excels as a method. To address this gap, we introduce a challenging dataset based on PFM simulations designed to benchmark and advance ML methods for fracture modeling. This dataset includes three energy decomposition methods, two boundary conditions, and 1,000 random initial crack configurations for a total of 6,000 simulations. Each sample contains 100 time steps capturing the temporal evolution of the crack field. Alongside this dataset, we also implement and evaluate Physics Informed Neural Networks (PINN), Fourier Neural Operators (FNO) and UNet models as baselines, and explore the impact of ensembling strategies on prediction accuracy. With this combination of our dataset and baseline models drawn from the literature we aim to provide a standardized and challenging benchmark for evaluating machine learning approaches to solid mechanics. Our results highlight both the promise and limitations of popular current models, and demonstrate the utility of this dataset as a testbed for advancing machine learning in fracture mechanics research.

Abstract:
Accurately modeling malware propagation is essential for designing effective cybersecurity defenses, particularly against adaptive threats that evolve in real time. While traditional epidemiological models and recent neural approaches offer useful foundations, they often fail to fully capture the nonlinear feedback mechanisms present in real‑world networks. In this work, we apply scientific machine learning to malware modeling by evaluating three approaches: classical Ordinary Differential Equations (ODEs), Universal Differential Equations (UDEs), and Neural ODEs. Using data from the Code Red worm outbreak, we show that the UDE approach substantially reduces prediction error compared to both traditional and neural baselines by 44%, while preserving interpretability. We introduce a symbolic recovery method that transforms the learned neural feedback into explicit mathematical expressions, revealing suppression mechanisms such as network saturation, security response, and malware variant evolution. Our results demonstrate that hybrid physics‑informed models can outperform both purely analytical and purely neural approaches, offering improved predictive accuracy and deeper insight into the dynamics of malware spread. These findings support the development of early warning systems, efficient outbreak response strategies, and targeted cyber defense interventions.

Abstract:
Physics‑Informed Neural Networks (PINNs) are increasingly used to approximate solutions of partial differential equations (PDEs), particularly in high dimensions. In real‑world settings, data are often noisy, making it crucial to understand when a predictor can still achieve low empirical risk. Yet, little is known about the conditions under which a PINN can do so effectively. We analyse PINNs applied to the Hamilton‑‑Jacobi‑‑Bellman (HJB) PDE and establish a lower bound on the network size required for the supervised PINN empirical risk to fall below the variance of noisy supervision labels. Specifically, if a predictor achieves empirical risk O(η) below σ^2 (the variance of the supervision data), then necessarily d_N\log d_N\gtrsim N_s η^2, where N_s is the number of samples and d_N the number of trainable parameters. A similar constraint holds in the fully unsupervised PINN setting when boundary labels are noisy. Thus, simply increasing the number of noisy supervision labels does not offer a ``free lunch'' in reducing empirical risk. We also give empirical studies on the HJB PDE, the Poisson PDE and the the Navier‑Stokes PDE set to produce the Taylor‑Green solutions. In these experiments we demonstrate that PINNs indeed need to be beyond a threshold model size for them to train to errors below σ^2. These results provide a quantitative foundation for understanding parameter requirements when training PINNs in the presence of noisy data.

Abstract:
This paper presents a physics‑informed neural network (PINN) for modeling first‑order Ambisonic (FOA) room impulse responses (RIRs). PINNs have demonstrated promising performance in sound field interpolation by combining the powerful modeling capability of neural networks and the physical principles of sound propagation. In room acoustics, PINNs have typically been trained to represent the sound pressure measured by omnidirectional microphones where the wave equation or its frequency‑domain counterpart, i.e., the Helmholtz equation, is leveraged. Meanwhile, FOA RIRs additionally provide spatial characteristics and are useful for immersive audio generation with a wide range of applications. In this paper, we extend the PINN framework to model FOA RIRs. We derive two physics‑informed priors for FOA RIRs based on the correspondence between the particle velocity and the (X, Y, Z)‑channels of FOA. These priors associate the predicted W‑channel and other channels through their partial derivatives and impose the physically feasible relationship on the four channels. Our experiments confirm the effectiveness of the proposed method compared with a neural network without the physics‑informed prior.

Abstract:
Machine learning has emerged as a transformative tool for solving differential equations (DEs), yet prevailing methodologies remain constrained by dual limitations: data‑driven methods demand costly labeled datasets while model‑driven techniques face efficiency‑accuracy trade‑offs. We present the Mathematical Artificial Data (MAD) framework, a new paradigm that integrates physical laws with data‑driven learning to facilitate large‑scale operator discovery. By exploiting DEs' intrinsic mathematical structure to generate physics‑embedded analytical solutions and associated synthetic data, MAD fundamentally eliminates dependence on experimental or simulated training data. This enables computationally efficient operator learning across multi‑parameter systems while maintaining mathematical rigor. Through numerical demonstrations spanning 2D parametric problems where both the boundary values and source term are functions, we showcase MAD's generalizability and superior efficiency/accuracy across various DE scenarios. This physics‑embedded‑data‑driven framework and its capacity to handle complex parameter spaces gives it the potential to become a universal paradigm for physics‑informed machine intelligence in scientific computing.

Abstract:
State estimation for nonlinear dynamical systems is a critical challenge in control and engineering applications, particularly when only partial and noisy measurements are available. This paper introduces a novel Adaptive Physics‑Informed Neural Network‑based Observer (PINN‑Obs) for accurate state estimation in nonlinear systems. Unlike traditional model‑based observers, which require explicit system transformations or linearization, the proposed framework directly integrates system dynamics and sensor data into a physics‑informed learning process. The observer adaptively learns an optimal gain matrix, ensuring convergence of the estimated states to the true system states. A rigorous theoretical analysis establishes formal convergence guarantees, demonstrating that the proposed approach achieves uniform error minimization under mild observability conditions. The effectiveness of PINN‑Obs is validated through extensive numerical simulations on diverse nonlinear systems, including an induction motor model, a satellite motion system, and benchmark academic examples. Comparative experimental studies against existing observer designs highlight its superior accuracy, robustness, and adaptability.

Abstract:
We present Kurucz‑a1, a physics‑informed neural network (PINN) that emulates 1D stellar atmosphere models under Local Thermodynamic Equilibrium (LTE), addressing a critical bottleneck in differentiable stellar spectroscopy. By incorporating hydrostatic equilibrium as a physical constraint during training, Kurucz‑a1 creates a differentiable atmospheric structure solver that maintains physical consistency while achieving computational efficiency. Kurucz‑a1 can achieve superior hydrostatic equilibrium and more consistent with the solar observed spectra compared to ATLAS‑12 itself, demonstrating the advantages of modern optimization techniques. Combined with modern differentiable radiative transfer codes, this approach enables data‑driven optimization of universal physical parameters across diverse stellar populations‑a capability essential for next‑generation stellar astrophysics.

Abstract:
Modern power systems face significant challenges in state estimation and real‑time monitoring, particularly regarding response speed and accuracy under faulty conditions or cyber‑attacks. This paper proposes a hybrid approach using physics‑informed neural networks (PINNs) to enhance the accuracy and robustness, of power system state estimation. By embedding physical laws into the neural network architecture, PINNs improve estimation accuracy for transmission grid applications under both normal and faulty conditions, while also showing potential in addressing security concerns such as data manipulation attacks. Experimental results show that the proposed approach outperforms traditional machine learning models, achieving up to 83% higher accuracy on unseen subsets of the training dataset and 65% better performance on entirely new, unrelated datasets. Experiments also show that during a data manipulation attack against a critical bus in a system, the PINN can be up to 93% more accurate than an equivalent neural network.

Abstract:
Cardiac diseases are among the leading causes of morbidity and mortality worldwide, which requires accurate and timely diagnostic strategies. In this study, we introduce an innovative approach that combines deep learning image registration with physics‑informed regularization to predict the biomechanical properties of moving cardiac tissues and extract features for disease classification. We utilize the energy strain formulation of Neo‑Hookean material to model cardiac tissue deformations, optimizing the deformation field while ensuring its physical and biomechanical coherence. This explainable approach not only improves image registration accuracy, but also provides insights into the underlying biomechanical processes of the cardiac tissues. Evaluation on the Automated Cardiac Diagnosis Challenge (ACDC) dataset achieved Dice scores of 0.945 for the left ventricular cavity, 0.908 for the right ventricular cavity, and 0.905 for the myocardium. Subsequently, we estimate the local strains within the moving heart and extract a detailed set of features used for cardiovascular disease classification. We evaluated five classification algorithms, Logistic Regression, Multi‑Layer Perceptron, Support Vector Classifier, Random Forest, and Nearest Neighbour, and identified the most relevant features using a feature selection algorithm. The best performing classifier obtained a classification accuracy of 98% in the training set and 100% in the test set of the ACDC dataset. By integrating explainable artificial intelligence, this method empowers clinicians with a transparent understanding of the model's predictions based on cardiac mechanics, while also significantly improving the accuracy and reliability of cardiac disease diagnosis, paving the way for more personalized and effective patient care.

Abstract:
In this paper we study chaotic behavior in the forced dissipative non‑linear Schrödinger equation, so called the Bekki‑Nozaki equation. Chaotic systems are often seen in a strong sensitivity to initial conditions,leading to error accumulation over time when traditional numerical methods are applied. To address this difficulty, we employ Physics‑Informed Neural Networks(PINNs), a mesh‑free deep learning framework. PINNs mitigate error accumulation in chaotic systems by solving partial differential equations without discretizing the computational domain. We demonstrate that PINNs successfully reproduce chaotic behavior of the Bekki‑Nozaki equation. The results of the inverse analysis indicate a correlation between the governing equation's predictability and its chaotic nature of the solution.

Abstract:
We propose a physics‑informed machine learning framework called P‑DivGNN to reconstruct local stress fields at the micro‑scale, in the context of multi‑scale simulation given a periodic micro‑structure mesh and mean, macro‑scale, stress values. This method is based in representing a periodic micro‑structure as a graph, combined with a message passing graph neural network. We are able to retrieve local stress field distributions, providing average stress values produced by a mean field reduced order model (ROM) or Finite Element (FE) simulation at the macro‑scale. The prediction of local stress fields are of utmost importance considering fracture analysis or the definition of local fatigue criteria. Our model incorporates physical constraints during training to constraint local stress field equilibrium state and employs a periodic graph representation to enforce periodic boundary conditions. The benefits of the proposed physics‑informed GNN are evaluated considering linear and non linear hyperelastic responses applied to varying geometries. In the non‑linear hyperelastic case, the proposed method achieves significant computational speed‑ups compared to FE simulation, making it particularly attractive for large‑scale applications.

Abstract:
Characterizing quantum flakes is a critical step in quantum hardware engineering because the quality of these flakes directly influences qubit performance. Although computer vision methods for identifying two‑dimensional quantum flakes have emerged, they still face significant challenges in estimating flake thickness. These challenges include limited data, poor generalization, sensitivity to domain shifts, and a lack of physical interpretability. In this paper, we introduce one of the first Physics‑informed Adaptation Learning approaches to overcome these obstacles. We focus on two main issues, i.e., data scarcity and generalization. First, we propose a new synthetic data generation framework that produces diverse quantum flake samples across various materials and configurations, reducing the need for time‑consuming manual collection. Second, we present φ‑Adapt, a physics‑informed adaptation method that bridges the performance gap between models trained on synthetic data and those deployed in real‑world settings. Experimental results show that our approach achieves state‑of‑the‑art performance on multiple benchmarks, outperforming existing methods. Our proposed approach advances the integration of physics‑based modeling and domain adaptation. It also addresses a critical gap in leveraging synthesized data for real‑world 2D material analysis, offering impactful tools for deep learning and materials science communities.

Abstract:
This paper establishes an explicit L^2‑estimate for weak solutions u to linear elliptic equations in divergence form with general coefficients and external source term f, stating that the L^2‑norm of u over U is bounded by a constant multiple of the L^2‑norm of f over U. In contrast to classical approaches based on compactness arguments, the proposed method, which employs a divergence‑free transformation method, provides a computable and explicit constant C>0. The L^2‑estimate remains robust even when there is no zero‑order term, and the analysis further demonstrates that the constant C>0 decreases as the diffusion coefficient or the zero‑order term increases. These quantitative results provide a rigorous foundation for applications such as a posteriori error estimates in Physics‑Informed Neural Networks (PINNs), where explicit error bounds are essential.

Abstract:
Acoustic data provide scientific and engineering insights in fields ranging from bioacoustics and communications to ocean and earth sciences. In this review, we survey recent advances and the transformative potential of machine learning (ML) in acoustics, including deep learning (DL). Using the Python high‑level programming language, we demonstrate a broad collection of ML techniques to detect and find patterns for classification, regression, and generation in acoustics data automatically. We have ML examples including acoustic data classification, generative modeling for spatial audio, and physics‑informed neural networks. This work includes AcousticsML, a set of practical Jupyter notebook examples on GitHub demonstrating ML benefits and encouraging researchers and practitioners to apply reproducible data‑driven approaches to acoustic challenges.

Abstract:
Predicting the failure and plasticity of solids remains a longstanding challenge, with broad implications for materials design and functional reliability. Disordered solids like metallic glasses can fail either abruptly or gradually without clear precursors, and the mechanical response depends strongly on composition, thermal history and deformation protocol ‑‑ impeding generalizable modeling. While deep learning methods offer predictive power, they often rely on numerous input parameters, hindering interpretability, methodology advancement and practical deployment. Here, we propose a macroscopic, physically grounded approach that uses plastic strain accumulation in the elastic regime to robustly predict deformation and yield. This method reduces complexity and improves interpretability, offering a practical alternative for disordered materials. For the Cu‑Zr‑(Al) metallic glasses prepared with varied annealing, we identify two limiting regimes of plastic strain growth: power‑law in poorly annealed and exponential in well‑annealed samples. A physics‑informed framework with Bayesian inference extracts growth parameters from stress‑strain data within ～5% strain, enabling early prediction of bulk response and yield point, well before the failure. The predictive performance improves with annealing, and bulk plasticity correlates with the microscopic plastic activity from scattered to growth near yielding. This work presents a physically interpretable and experimentally relevant framework for predicting plasticity and failure in metallic glasses from early mechanical response, offering both theoretical insights and practical tools for material characterization and design.

Abstract:
The Lane‑Emden equation, a nonlinear second‑order ordinary differential equation, plays a fundamental role in theoretical physics and astrophysics, particularly in modeling the structure of stellar interiors. Also referred to as the polytropic differential equation, it describes the behavior of self‑gravitating polytropic spheres. In this study, we present a novel approach to the solution of the eigenvalue problem which arises when considering the Lane‑Emden equation for n = 0, 1, 2, 3, 4 using Physics‑Informed Neural Networks (PINNs). The novelty of this work is that, we not only solve the Lane‑Emden equation via PINNS but we also determine the eigenvalue, r, which is the stellar radius. Hyperparameter tuning was conducted using Bayesian optimization in the Optuna framework to identify optimal values for the number of hidden layers, number of neurons, activation function, optimizer, and learning rate for each value of n. The results show that, for n = 0, 1, PINNs achieve near‑exact agreement with theoretical eigenvalues (errors < 0.000806%). While for more nonlinear cases, n = 2, 3 and n=4, PINNs yield errors below 0.0009% and 0.05% respectively, validating their robustness.

Abstract:
We study the problem of learning neural network models for Ordinary Differential Equations (ODEs) with parametric uncertainties. Such neural network models capture the solution to the ODE over a given set of parameters, initial conditions, and range of times. Physics‑Informed Neural Networks (PINNs) have emerged as a promising approach for learning such models that combine data‑driven deep learning with symbolic physics models in a principled manner. However, the accuracy of PINNs degrade when they are used to solve an entire family of initial value problems characterized by varying parameters and initial conditions. In this paper, we combine symbolic differentiation and Taylor series methods to propose a class of higher‑order models for capturing the solutions to ODEs. These models combine neural networks and symbolic terms: they use higher order Lie derivatives and a Taylor series expansion obtained symbolically, with the remainder term modeled as a neural network. The key insight is that the remainder term can itself be modeled as a solution to a first‑order ODE. We show how the use of these higher order PINNs can improve accuracy using interesting, but challenging ODE benchmarks. We also show that the resulting model can be quite useful for situations such as controlling uncertain physical systems modeled as ODEs.

Abstract:
Despite the success of deep learning methods in quantum chemistry, their representational capacity is most often confined to neutral, closed‑shell molecules. However, real‑world chemical systems often exhibit complex characteristics, including varying charges, spins, and environments. We introduce OrbitAll, a geometry‑ and physics‑informed deep learning framework that can represent all molecular systems with electronic structure information. OrbitAll utilizes spin‑polarized orbital features from the underlying quantum mechanical method, and combines it with graph neural networks satisfying SE(3)‑equivariance. The resulting framework can represent and process any molecular system with arbitrary charges, spins, and environmental effects. OrbitAll demonstrates superior performance and generalization on predicting charged, open‑shell, and solvated molecules, while also robustly extrapolating to molecules significantly larger than the training data by leveraging a physics‑informed architecture. OrbitAll achieves chemical accuracy using 10 times fewer training data than competing AI models, with a speedup of approximately 10^3 ‑ 10^4 compared to density functional theory.

Abstract:
Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical techniques such as finite elements and finite differences. In this paper, we introduce a new class of Physics‑Informed Neural Networks (PINNs) trained using discontinuous Galerkin finite element methods. Unlike standard collocation‑based PINNs that rely on pointwise gradient evaluations and Monte Carlo quadrature, our approach computes the loss functional using finite element interpolation and integration. This avoids costly pointwise derivative computations, particularly advantageous for elliptic PDEs requiring second‑order derivatives, and inherits key stability and accuracy benefits from the finite element framework. We present a convergence analysis based on variational arguments and support our theoretical findings with numerical experiments that demonstrate improved efficiency and robustness.

Abstract:
Over the past decade, an abundance of information from neutron‑star observations, nuclear experiments and theory has transformed our efforts to elucidate the properties of dense matter. However, at high densities relevant to the cores of neutron stars, substantial uncertainty about the dense matter equation of state (EoS) remains. In this work, we present a semiparametric EoS framework aimed at better integrating knowledge across these domains in astrophysical inference. We use a Meta‑model at low densities, and Gaussian Process extensions at high densities. Comparisons between our semiparametric framework to fully nonparametric EoS representations show that imposing nuclear theoretical and experimental constraints through the Meta‑model up to nuclear saturation density results in constraints on the pressure up to twice nuclear saturation density. We show that our Gaussian Process trained on EoS models with nucleonic, hyperonic, and quark compositions extends the range of EoS explored at high density compared to a piecewise polytropic extension schema, under the requirements of causality of matter and of supporting the existence of heavy pulsars. We find that maximum TOV masses above 3.2 M_\odot can be supported by causal EoS compatible with nuclear constraints at low densities. We then combine information from existing observations of heavy pulsar masses, gravitational waves from binary neutron star mergers, and X‑ray pulse profile modeling of millisecond pulsars within a Bayesian inference scheme using our semiparametric EoS prior. With current astrophysical observations, we find a favored pressure at two times nuclear saturation density of P(2ρ_\rm nuc) = 1.98^+2.13_‑1.08×10^34 dyn/cm^2, a radius of a 1.4 M_\odot neutron star value of R_1.4 = 11.4^+0.98_‑0.60\;km, and M_\rm max = 2.31_‑0.23^+0.35 M_\odot at the 90% credible level.

Abstract:
We present a novel approach to compute three‑dimensional Magnetohydrodynamic equilibria by parametrizing Fourier modes with artificial neural networks and compare it to equilibria computed by conventional solvers. The full nonlinear global force residual across the volume in real space is then minimized with first order optimizers. Already,we observe competitive computational cost to arrive at the same minimum residuals computed by existing codes. With increased computational cost,lower minima of the residual are achieved by the neural networks,establishing a new lower bound for the force residual. We use minimally complex neural networks,and we expect significant improvements for solving not only single equilibria with neural networks,but also for computing neural network models valid over continuous distributions of equilibria.

Abstract:
Proton exchange membrane (PEM) electrolyzers are pivotal for sustainable hydrogen production, yet their long‑term performance is hindered by membrane degradation, which poses reliability and safety challenges. Therefore, accurate modeling of this degradation is essential for optimizing durability and performance. To address these concerns, traditional physics‑based models have been developed, offering interpretability but requiring numerous parameters that are often difficult to measure and calibrate. Conversely, data‑driven approaches, such as machine learning, offer flexibility but may lack physical consistency and generalizability. To address these limitations, this study presents the first application of Physics‑Informed Neural Networks (PINNs) to model membrane degradation in PEM electrolyzers. The proposed PINN framework couples two ordinary differential equations, one modeling membrane thinning via a first‑order degradation law and another governing the time evolution of the cell voltage under membrane degradation. Results demonstrate that the PINN accurately captures the long‑term system's degradation dynamics while preserving physical interpretability with limited noisy data. Consequently, this work introduces a novel hybrid modeling approach for estimating and understanding membrane degradation mechanisms in PEM electrolyzers, offering a foundation for more robust predictive tools in electrochemical system diagnostics.

Abstract:
The industrialization of catalytic processes hinges on the availability of reliable kinetic models for design, optimization, and control. Traditional mechanistic models demand extensive domain expertise, while many data‑driven approaches often lack interpretability and fail to enforce physical consistency. To overcome these limitations, we propose the Physics‑Informed Automated Discovery of Kinetics (PI‑ADoK) framework. By integrating physical constraints directly into a symbolic regression approach, PI‑ADoK narrows the search space and substantially reduces the number of experiments required for model convergence. Additionally, the framework incorporates a robust uncertainty quantification strategy via the Metropolis‑Hastings algorithm, which propagates parameter uncertainty to yield credible prediction intervals. Benchmarking our method against conventional approaches across several catalytic case studies demonstrates that PI‑ADoK not only enhances model fidelity but also lowers the experimental burden, highlighting its potential for efficient and reliable kinetic model discovery in chemical reaction engineering.

Abstract:
Earthquake‑related phenomena such as seismic waves and crustal deformation impact broad regions, requiring large‑scale modeling with careful treatment of artificial outer boundaries. Physics‑informed neural networks (PINNs) have been applied to analyze wavefront propagation, acoustic and elastic waveform propagations, and crustal deformation in semi‑infinite domains. In this study, we investigated the capability of PINNs for modeling earthquake crustal deformation in 3‑D structures. To improve modeling accuracy, four neural networks were constructed to represent the displacement and stress fields in two subdomains divided by a fault surface and its extension. Forward simulations exhibited high accuracy for internal deformation but yielded errors for rigid motions, underscoring the inherent difficulty in constraining static deformation at an infinite distance. In the inversion analysis, fault slip distributions were estimated using surface observational data. Application to real data from the 2008 Iwate‑Miyagi inland earthquake showed a fault slip consistent with previous studies, despite underestimation of the magnitude. This study demonstrates the capability of PINNs to analyze 3‑D crustal deformation, thereby offering a flexible approach for large‑scale earthquake modeling using real‑world observations and crustal structures.

Abstract:
We present a hybrid machine learning framework that combines Physics‑Informed Neural Operators (PINOs) with score‑based generative diffusion models to simulate the full spatio‑temporal evolution of two‑dimensional, incompressible, resistive magnetohydrodynamic (MHD) turbulence across a broad range of Reynolds numbers (\mathrmRe). The framework leverages the equation‑constrained generalization capabilities of PINOs to predict coherent, low‑frequency dynamics, while a conditional diffusion model stochastically corrects high‑frequency residuals, enabling accurate modeling of fully developed turbulence. Trained on a comprehensive ensemble of high‑fidelity simulations with \mathrmRe \in \100, 250, 500, 750, 1000, 3000, 10000\, the approach achieves state‑of‑the‑art accuracy in regimes previously inaccessible to deterministic surrogates. At \mathrmRe=1000 and 3000, the model faithfully reconstructs the full spectral energy distributions of both velocity and magnetic fields late into the simulation, capturing non‑Gaussian statistics, intermittent structures, and cross‑field correlations with high fidelity. At extreme turbulence levels (\mathrmRe=10000), it remains the first surrogate capable of recovering the high‑wavenumber evolution of the magnetic field, preserving large‑scale morphology and enabling statistically meaningful predictions.

Abstract:
Accurate and scalable surrogate models for AC power flow are essential for real‑time grid monitoring, contingency analysis, and decision support in increasingly dynamic and inverter‑dominated power systems. However, most existing surrogates fall short of practical deployment due to their limited capacity to capture long‑range nonlinear dependencies in meshed transmission networks and their weak enforcement of physical laws. These models often require extensive hyperparameter tuning, exhibit poor generalization under topology changes or large load swings, and typically do not quantify uncertainty or scale well beyond a few hundred buses. To address these challenges, this paper proposes a gated graph neural network (GGNN) surrogate for AC power‑flow estimation under topological uncertainty. The model is trained across multiple IEEE benchmark networks of varying size and complexity, each incorporating randomized line contingencies and up to 40% load variation. To improve robustness and generalization, we explore both conventional supervised learning and physics‑informed self‑supervised training strategies. Comparative evaluations show that the proposed GGNN consistently outperforms prior GNN‑based surrogates, achieving predictions closely aligned with Newton‑‑Raphson solutions. By embedding operational constraints directly into the architecture and loss function, the model ensures physical consistency and delivers a lightweight, accurate, and scalable tool for real‑time grid operations.

Abstract:
In this paper, we propose SubLoRA, a rank determination method for Low‑Rank Adaptation (LoRA) based on submodular function maximization. In contrast to prior approaches, such as AdaLoRA, that rely on first‑order (linearized) approximations of the loss function, SubLoRA utilizes second‑order information to capture the potentially complex loss landscape by incorporating the Hessian matrix. We show that the linearization becomes inaccurate and ill‑conditioned when the LoRA parameters have been well optimized, motivating the need for a more reliable and nuanced second‑order formulation. To this end, we reformulate the rank determination problem as a combinatorial optimization problem with a quadratic objective. However, solving this problem exactly is NP‑hard in general. To overcome the computational challenge, we introduce a submodular function maximization framework and devise a greedy algorithm with approximation guarantees. We derive a sufficient and necessary condition under which the rank‑determination objective becomes submodular, and construct a closed‑form projection of the Hessian matrix that satisfies this condition while maintaining computational efficiency. Our method combines solid theoretical foundations, second‑order accuracy, and practical computational efficiency. We further extend SubLoRA to a joint optimization setting, alternating between LoRA parameter updates and rank determination under a rank budget constraint. Extensive experiments on fine‑tuning physics‑informed neural networks (PINNs) for solving partial differential equations (PDEs) demonstrate the effectiveness of our approach. Results show that SubLoRA outperforms existing methods in both rank determination and joint training performance.

Abstract:
Training physics‑informed neural networks (PINNs) for forward problems often suffers from severe convergence issues, hindering the propagation of information from regions where the desired solution is well‑defined. Haitsiukevich and Ilin (2023) proposed an ensemble approach that extends the active training domain of each PINN based on i) ensemble consensus and ii) vicinity to (pseudo‑)labeled points, thus ensuring that the information from the initial condition successfully propagates to the interior of the computational domain. In this work, we suggest replacing the ensemble by a Bayesian PINN, and consensus by an evaluation of the PINN's posterior variance. Our experiments show that this mathematically principled approach outperforms the ensemble on a set of benchmark problems and is competitive with PINN ensembles trained with combinations of Adam and LBFGS.

Abstract:
The integration of Scientific Machine Learning (SciML) techniques with uncertainty quantification (UQ) represents a rapidly evolving frontier in computational science. This work advances Physics‑Informed Neural Networks (PINNs) by incorporating probabilistic frameworks to effectively model uncertainty in complex systems. Our approach enhances the representation of uncertainty in forward problems by combining generative modeling techniques with PINNs. This integration enables in a systematic fashion uncertainty control while maintaining the predictive accuracy of the model. We demonstrate the utility of this method through applications to random differential equations and random partial differential equations (PDEs).

Abstract:
While digital twins (DT) hold promise for providing real‑time insights into complex energy assets, much of the current literature either does not offer a clear framework for information exchange between the model and the asset, lacks key features needed for real‑time implementation, or gives limited attention to model uncertainty. Here, we aim to solve these gaps by proposing a variational digital twin (VDT) framework that augments standard neural architectures with a single Bayesian output layer. This lightweight addition, along with a novel VDT updating algorithm, lets a twin update in seconds on commodity GPUs while producing calibrated uncertainty bounds that can inform experiment design, control algorithms, and model reliability. The VDT is evaluated on four energy‑sector problems. For critical‑heat‑flux prediction, uncertainty‑driven active learning reaches R2 = 0.98 using 47 % fewer experiments and one‑third the training time of random sampling. A three‑year renewable‑generation twin maintains R2 > 0.95 for solar output and curbs error growth for volatile wind forecasts via monthly updates that process only one month of data at a time. A nuclear reactor transient cooldown twin reconstructs thermocouple signals with R2 > 0.99 and preserves accuracy after 50 % sensor loss, demonstrating robustness to degraded instrumentation. Finally, a physics‑informed Li‑ion battery twin, retrained after every ten discharges, lowers voltage mean‑squared error by an order of magnitude relative to the best static model while adapting its credible intervals as the cell approaches end‑of‑life. These results demonstrate that combining modest Bayesian augmentation with efficient update schemes turns conventional surrogates into uncertainty‑aware, data‑efficient, and computationally tractable DTs, paving the way for dependable models across industrial and scientific energy systems.

Abstract:
Accurate dynamics modeling is essential for quadrotors to achieve precise trajectory tracking in various applications. Traditional physical knowledge‑driven modeling methods face substantial limitations in unknown environments characterized by variable payloads, wind disturbances, and external perturbations. On the other hand, data‑driven modeling methods suffer from poor generalization when handling out‑of‑distribution (OoD) data, restricting their effectiveness in unknown scenarios. To address these challenges, we introduce the Physics‑Informed Wind‑Adaptive Network (PI‑WAN), which combines knowledge‑driven and data‑driven modeling methods by embedding physical constraints directly into the training process for robust quadrotor dynamics learning. Specifically, PI‑WAN employs a Temporal Convolutional Network (TCN) architecture that efficiently captures temporal dependencies from historical flight data, while a physics‑informed loss function applies physical principles to improve model generalization and robustness across previously unseen conditions. By incorporating real‑time prediction results into a model predictive control (MPC) framework, we achieve improvements in closed‑loop tracking performance. Comprehensive simulations and real‑world flight experiments demonstrate that our approach outperforms baseline methods in terms of prediction accuracy, tracking precision, and robustness to unknown environments.

Abstract:
Spin‑lattice relaxation time (T_1) is an important biomarker in cardiac parametric mapping for characterizing myocardial tissue and diagnosing cardiomyopathies. Conventional Modified Look‑Locker Inversion Recovery (MOLLI) acquires 11 breath‑hold baseline images with interleaved rest periods to ensure mapping accuracy. However, prolonged scanning can be challenging for patients with poor breathholds, often leading to motion artifacts that degrade image quality. In addition, T_1 mapping requires voxel‑wise nonlinear fitting to a signal recovery model involving an iterative estimation process. Recent studies have proposed deep‑learning approaches for rapid T_1 mapping using shortened sequences to reduce acquisition time for patient comfort. Nevertheless, existing methods overlook important physics constraints, limiting interpretability and generalization. In this work, we present an accelerated, end‑to‑end T_1 mapping framework leveraging Physics‑Informed Neural Ordinary Differential Equations (ODEs) to model temporal dynamics and address these challenges. Our method achieves high‑accuracy T_1 estimation from a sparse subset of baseline images and ensures efficient null index estimation at test time. Specifically, we develop a continuous‑time LSTM‑ODE model to enable selective Look‑Locker (LL) data acquisition with arbitrary time lags. Experimental results show superior performance in T_1 estimation for both native and post‑contrast sequences and demonstrate the strong benefit of our physics‑based formulation over direct data‑driven T_1 priors.

Abstract:
The discovery of two‑dimensional (2D) materials with tailored properties is critical to meet the increasing demands of high‑performance applications across flexible electronics, optoelectronics, catalysis, and energy storage. However, current 2D material databases are constrained by limited scale and compositional diversity. In this study, we introduce a scalable active learning workflow that integrates deep neural networks with density functional theory (DFT) calculations to efficiently explore a vast set of candidate structures. These structures are generated through physics‑informed elemental substitution strategies, enabling broad and systematic discovery of stable 2D materials. Through six iterative screening cycles, we established the creation of the Monolayer 2D Materials Database (ML2DDB), which contains 242,546 DFT‑validated stable structures‑an order‑of‑magnitude increase over the largest known 2D materials databases. In particular, the number of ternary and quaternary compounds showed the most significant increase. Combining this database with a generative diffusion model, we demonstrated effective structure generation under specified chemistry and symmetry constraints. This work accomplished an organically interconnected loop of 2D material data expansion and application, which provides a new paradigm for the discovery of new materials.

Abstract:
Underwater object detection is crucial for autonomous navigation, environmental monitoring, and marine exploration, but it is severely hampered by light attenuation, turbidity, and occlusion. Current methods balance accuracy and computational efficiency, but they have trouble deploying in real‑time under low visibility conditions. Through the integration of physics‑informed augmentation techniques with the YOLOv12 architecture, this study advances underwater detection. With Residual ELAN blocks to preserve structural features in turbid waters and Area Attention to maintain large receptive fields for occluded objects while reducing computational complexity. Underwater optical properties are addressed by domain‑specific augmentations such as turbulence adaptive blurring, biologically grounded occlusion simulation, and spectral HSV transformations for color distortion. Extensive tests on four difficult datasets show state‑of‑the‑art performance, with Brackish data registering 98.30% mAP at 142 FPS. YOLOv12 improves occlusion robustness by 18.9%, small‑object recall by 22.4%, and detection precision by up to 7.94% compared to previous models. The crucial role of augmentation strategy is validated by ablation studies. This work offers a precise and effective solution for conservation and underwater robotics applications.

Abstract:
We report a novel physics‑informed neural framework for reconstructing unsteady fluid‑structure interactions (FSI) from sparse, single‑phase observations of the flow. Our approach combines a modal surface model with coordinate neural representations of the fluid and solid states, constrained by the fluid's governing equations and interface conditions. Using only off‑body Lagrangian particle tracks and a moving‑wall boundary condition, the method infers both flow fields and structural motion. It does not require a constitutive model for the solid or measurements of surface position, although including these can improve performance. We demonstrate the approach numerically on two canonical FSI benchmarks: vortex‑induced oscillations of a 2D flapping plate and pulse‑wave propagation in a 3D flexible pipe. We also demonstrate it on flow around a swimming fish. In all cases, the framework achieves accurate reconstructions of flow states and structural deformations despite acute data sparsity near the moving interface. A key result is that reconstructions remain robust to over‑parameterization. This work extends physics‑informed neural networks to coupled fluid‑structure dynamics learned from single‑phase observations, and it provides a pathway toward quantitative FSI analysis when flow measurements are sparse and structural measurements are asynchronous or unavailable.

Abstract:
The current contribution develops a Variational Physics‑Informed Neural Network (VPINN)‑based framework for the analysis and design of multiphase architected solids. The elaborated VPINN methodology is based on the Petrov‑Galerkin approach, with a deep neural network acting as trial function and local polynomials as test functions. For the analysis, a Galerkin Boundary Element Method (GBEM) scheme is developed to generate the mechanical field data, employing solely domain boundary information. The VPINN methodology is complemented by an adaptive, tree‑based integration scheme for the evaluation of the weak‑form integrals. Different double‑phase material architectures are considered, with the VPINNs demonstrating their ability to capture the deformation fields with considerable accuracy. Moreover, the performance enhancement by the incorporation of additional semi‑analytical information at auxiliary internal points is analyzed. Tree‑based integration schemes are shown to be capable of robustly capturing inner material discontinuities upon substantial computational cost reductions. The results suggest that the proposed VPINN formulation offers comparative advantages in the modeling of multiphase architected materials compared to classical PINN formulations. The analysis paves the way for the development of variational physics‑informed computational models for the mechanical analysis of complex architected multiphase materials and structures.

Abstract:
We introduce MRF‑DiPh, a novel physics informed denoising diffusion approach for multiparametric tissue mapping from highly accelerated, transient‑state quantitative MRI acquisitions like Magnetic Resonance Fingerprinting (MRF). Our method is derived from a proximal splitting formulation, incorporating a pretrained denoising diffusion model as an effective image prior to regularize the MRF inverse problem. Further, during reconstruction it simultaneously enforces two key physical constraints: (1) k‑space measurement consistency and (2) adherence to the Bloch response model. Numerical experiments on in‑vivo brain scans data show that MRF‑DiPh outperforms deep learning and compressed sensing MRF baselines, providing more accurate parameter maps while better preserving measurement fidelity and physical model consistency‑critical for solving reliably inverse problems in medical imaging.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a promising approach for solving partial differential equations (PDEs) by embedding the governing physics into the loss function associated with a deep neural network. In this work, a Quantum PINNs (QPINN) framework is proposed to solve two‑dimensional (2D) time‑dependent Maxwell's equations. Our approach utilizes a parametrized quantum circuit in conjunction with the classical neural network architecture and enforces physical laws, including a global energy conservation principle, during training. A quantum simulation library, TorQ, was developed to efficiently compute circuit outputs and derivatives by leveraging GPU acceleration based on PyTorch, enabling end‑to‑end training of the QPINN. The method was evaluated on two 2D electromagnetic wave propagation problems: one in free space (vacuum) and the other has an added dielectric medium. Multiple quantum circuit ansätze, input scales, and an added loss term were compared in a thorough ablation study. Furthermore, recent techniques to enhance PINN convergence, including random Fourier feature embeddings and adaptive time weighting, have been incorporated. Our results demonstrate that the QPINN achieves accuracy comparable to, and even greater than, the classical PINN baseline, while using a significantly smaller number of trainable parameters. This study also shows that adding an energy conservation term to the loss stabilizes training and improves the physical fidelity of the solution in the lossless free‑space case. This added term helps mitigate a new kind of barren plateau (BP) related phenomenon ‑ ``black hole'' (BH) loss landscape for the quantum experiments in that scenario. By optimizing the quantum‑circuit ansatz and embedding energy‑conservation constraints, our QPINN achieves up to a 19% higher accuracy on 2D Maxwell benchmark problems compared to a classical PINN.

Abstract:
Physics‑informed neural networks (PINNs) offer a flexible way to solve partial differential equations (PDEs) with machine learning, yet they still fall well short of the machine‑precision accuracy many scientific tasks demand. In this work, we investigate whether the precision ceiling comes from the ill‑conditioning of the PDEs or from the typical multi‑layer perceptron (MLP) architecture. We introduce the Barycentric Weight Layer (BWLer), which models the PDE solution through barycentric polynomial interpolation. A BWLer can be added on top of an existing MLP (a BWLer‑hat) or replace it completely (explicit BWLer), cleanly separating how we represent the solution from how we take derivatives for the PDE loss. Using BWLer, we identify fundamental precision limitations within the MLP: on a simple 1‑D interpolation task, even MLPs with O(1e5) parameters stall around 1e‑8 RMSE ‑‑ about eight orders above float64 machine precision ‑‑ before any PDE terms are added. In PDE learning, adding a BWLer lifts this ceiling and exposes a tradeoff between achievable accuracy and the conditioning of the PDE loss. For linear PDEs we fully characterize this tradeoff with an explicit error decomposition and navigate it during training with spectral derivatives and preconditioning. Across five benchmark PDEs, adding a BWLer on top of an MLP improves RMSE by up to 30x for convection, 10x for reaction, and 1800x for wave equations while remaining compatible with first‑order optimizers. Replacing the MLP entirely lets an explicit BWLer reach near‑machine‑precision on convection, reaction, and wave problems (up to 10 billion times better than prior results) and match the performance of standard PINNs on stiff Burgers' and irregular‑geometry Poisson problems. Together, these findings point to a practical path for combining the flexibility of PINNs with the precision of classical spectral solvers.

Abstract:
Traditional elastic wavefield separation methods, while accurate, often demand substantial computational resources, especially for large geological models or 3D scenarios. Purely data‑driven neural network approaches can be more efficient, but may fail to generalize and maintain physical consistency due to the absence of explicit physical constraints. Here, we propose a physics‑informed conditional diffusion model for elastic wavefield separation that seamlessly integrates domain‑specific physics equations into both the training and inference stages of the reverse diffusion process. Conditioned on full elastic wavefields and subsurface P‑ and S‑wave velocity profiles, our method directly predicts clean P‑wave modes while enforcing Laplacian separation constraints through physics‑guided loss and sampling corrections. Numerical experiments on diverse scenarios yield the separation results that closely match conventional numerical solutions but at a reduced cost, confirming the effectiveness and generalizability of our approach.

Abstract:
Person re‑identification (ReID) across aerial and ground vantage points has become crucial for large‑scale surveillance and public safety applications. Although significant progress has been made in ground‑only scenarios, bridging the aerial‑ground domain gap remains a formidable challenge due to extreme viewpoint differences, scale variations, and occlusions. Building upon the achievements of the AG‑ReID 2023 Challenge, this paper introduces the AG‑VPReID 2025 Challenge ‑ the first large‑scale video‑based competition focused on high‑altitude (80‑120m) aerial‑ground ReID. Constructed on the new AG‑VPReID dataset with 3,027 identities, over 13,500 tracklets, and approximately 3.7 million frames captured from UAVs, CCTV, and wearable cameras, the challenge featured four international teams. These teams developed solutions ranging from multi‑stream architectures to transformer‑based temporal reasoning and physics‑informed modeling. The leading approach, X‑TFCLIP from UAM, attained 72.28% Rank‑1 accuracy in the aerial‑to‑ground ReID setting and 70.77% in the ground‑to‑aerial ReID setting, surpassing existing baselines while highlighting the dataset's complexity. For additional details, please refer to the official website at https://agvpreid25.github.io.

Abstract:
Accurate image alignment is essential for monitoring crack evolution in structural health monitoring (SHM), particularly under real‑world conditions involving perspective distortion, occlusion, and low contrast. However, traditional feature detectors such as SIFT and SURF, which rely on Gaussian‑based scale spaces, tend to suppress high‑frequency edges, making them unsuitable for thin crack localization. Lightweight binary alternatives like ORB and BRISK, while computationally efficient, often suffer from poor keypoint repeatability on textured or shadowed surfaces. This study presents a physics‑informed alignment framework that adapts the open KAZE architecture to SHM‑specific challenges. By utilizing nonlinear anisotropic diffusion to construct a crack‑preserving scale space, and integrating RANSAC‑based homography estimation, the framework enables accurate geometric correction without the need for training, parameter tuning, or prior calibration. The method is validated on time‑lapse images of masonry and concrete acquired via handheld smartphone under varied field conditions, including shadow interference, cropping, oblique viewing angles, and surface clutter. Compared to classical detectors, the proposed framework reduces crack area and spine length errors by up to 70 percent and 90 percent, respectively, while maintaining sub‑5 percent alignment error in key metrics. Unsupervised, interpretable, and computationally lightweight, this approach supports scalable deployment via UAVs and mobile platforms. By tailoring nonlinear scale‑space modeling to SHM image alignment, this work offers a robust and physically grounded alternative to conventional techniques for tracking real‑world crack evolution.

Abstract:
In the realm of computational fluid dynamics, traditional numerical methods, which heavily rely on discretization, typically necessitate the formulation of partial differential equations (PDEs) in conservative form to accurately capture shocks and other discontinuities in compressible flows. Conversely, utilizing non‑conservative forms often introduces significant errors near these discontinuities or results in smeared shocks. This dependency poses a considerable limitation, particularly as many PDEs encountered in complex physical phenomena, such as multi‑phase flows, are inherently non‑conservative. This inherent non‑conservativity restricts the direct applicability of standard numerical solvers designed for conservative forms. This work aims to thoroughly investigate the sensitivity of Physics‑Informed Neural Networks (PINNs) to the choice of PDE formulation (conservative vs. non‑conservative) when solving problems involving shocks and discontinuities. We have conducted this investigation across a range of benchmark problems, specifically the Burgers equation and both steady and unsteady Euler equations, to provide a comprehensive understanding of PINNs capabilities in this critical area.

Abstract:
When using reinforcement learning (RL) to tackle physical control tasks, inductive biases that encode physics priors can help improve sample efficiency during training and enhance generalization in testing. However, the current practice of incorporating these helpful physics‑informed inductive biases inevitably runs into significant manual labor and domain expertise, making them prohibitive for general users. This work explores a symbolic approach to distill physics‑informed inductive biases into RL agents, where the physics priors are expressed in a domain‑specific language (DSL) that is human‑readable and naturally explainable. Yet, the DSL priors do not translate directly into an implementable policy due to partial and noisy observations and additional physical constraints in navigation tasks. To address this gap, we develop a physics‑informed program‑guided RL (PiPRL) framework with applications to indoor navigation. PiPRL adopts a hierarchical and modularized neuro‑symbolic integration, where a meta symbolic program receives semantically meaningful features from a neural perception module, which form the bases for symbolic programming that encodes physics priors and guides the RL process of a low‑level neural controller. Extensive experiments demonstrate that PiPRL consistently outperforms purely symbolic or neural policies and reduces training time by over 26% with the help of the program‑based inductive biases.

Abstract:
Distributed acoustic sensing (DAS) has attracted considerable attention across various fields and artificial intelligence (AI) technology plays an important role in DAS applications to realize event recognition and denoising. Existing AI models require real‑world data (RWD), whether labeled or not, for training, which is contradictory to the fact of limited available event data in real‑world scenarios. Here, a physics‑informed DAS neural network paradigm is proposed, which does not need real‑world events data for training. By physically modeling target events and the constraints of real world and DAS system, physical functions are derived to train a generative network for generation of DAS events data. DAS debackground net is trained by using the generated DAS events data to eliminate background noise in DAS data. The effectiveness of the proposed paradigm is verified in event identification application based on a public dataset of DAS spatiotemporal data and in belt conveyor fault monitoring application based on DAS time‑frequency data, and achieved comparable or better performance than data‑driven networks trained with RWD. Owing to the introduction of physical information and capability of background noise removal, the paradigm demonstrates generalization in same application on different sites. A fault diagnosis accuracy of 91.8% is achieved in belt conveyor field with networks which transferred from simulation test site without any fault events data of test site and field for training. The proposed paradigm is a prospective solution to address significant obstacles of data acquisition and intense noise in practical DAS applications and explore more potential fields for DAS.

Abstract:
Flexoelectricity, the coupling between strain gradients and electric polarization, poses significant computational challenges due to its governing fourth‑order partial differential equations that require C1‑continuous solutions. To address these issues, we propose a physics‑informed neural network (PINN) framework grounded in an energy‑based formulation that treats both forward and inverse problems within a unified architecture. The forward problem is recast as a saddle‑point optimization of the total potential energy, solved via the deep energy method (DEM), which circumvents the direct computation of high‑order derivatives. For the inverse problem of identifying unknown flexoelectric coefficients from sparse measurements, we introduce an additional variational loss that enforces stationarity with respect to the electric potential, ensuring robust and stable parameter inference. The framework integrates finite element‑based numerical quadrature for stable energy evaluation and employs hard constraints to rigorously enforce boundary conditions. Numerical results for both direct and converse flexoelectric effects show excellent agreement with mixed‑FEM solutions, and the inverse model accurately recovers material parameters from limited data. This study establishes a unified, mesh‑compatible, and scalable PINN approach for high‑order electromechanical problems, offering a promising alternative to traditional simulation techniques.

Abstract:
Trackerless freehand ultrasound reconstruction aims to reconstruct 3D volumes from sequences of 2D ultrasound images without relying on external tracking systems. By eliminating the need for optical or electromagnetic trackers, this approach offers a low‑cost, portable, and widely deployable alternative to more expensive volumetric ultrasound imaging systems, particularly valuable in resource‑constrained clinical settings. However, predicting long‑distance transformations and handling complex probe trajectories remain challenging. The TUS‑REC2024 Challenge establishes the first benchmark for trackerless 3D freehand ultrasound reconstruction by providing a large publicly available dataset, along with a baseline model and a rigorous evaluation framework. By the submission deadline, the Challenge had attracted 43 registered teams, of which 6 teams submitted 21 valid dockerized solutions. The submitted methods span a wide range of approaches, including the state space model, the recurrent model, the registration‑driven volume refinement, the attention mechanism, and the physics‑informed model. This paper provides a comprehensive background introduction and literature review in the field, presents an overview of the challenge design and dataset, and offers a comparative analysis of submitted methods across multiple evaluation metrics. These analyses highlight both the progress and the current limitations of state‑of‑the‑art approaches in this domain and provide insights for future research directions. All data and code are publicly available to facilitate ongoing development and reproducibility. As a live and evolving benchmark, it is designed to be continuously iterated and improved. The Challenge was held at MICCAI 2024 and is organised again at MICCAI 2025, reflecting its sustained commitment to advancing this field.

Abstract:
Elucidating the biomechanical behavior of the myocardium is crucial for understanding cardiac physiology, but cannot be directly inferred from clinical imaging and typically requires finite element (FE) simulations. However, conventional FE methods are computationally expensive and often fail to reproduce observed cardiac motions. We propose IMC‑PINN‑FE, a physics‑informed neural network (PINN) framework that integrates imaged motion consistency (IMC) with FE modeling for patient‑specific left ventricular (LV) biomechanics. Cardiac motion is first estimated from MRI or echocardiography using either a pre‑trained attention‑based network or an unsupervised cyclic‑regularized network, followed by extraction of motion modes. IMC‑PINN‑FE then rapidly estimates myocardial stiffness and active tension by fitting clinical pressure measurements, accelerating computation from hours to seconds compared to traditional inverse FE. Based on these parameters, it performs FE modeling across the cardiac cycle at 75x speedup. Through motion constraints, it matches imaged displacements more accurately, improving average Dice from 0.849 to 0.927, while preserving realistic pressure‑volume behavior. IMC‑PINN‑FE advances previous PINN‑FE models by introducing back‑computation of material properties and better motion fidelity. Using motion from a single subject to reconstruct shape modes also avoids the need for large datasets and improves patient specificity. IMC‑PINN‑FE offers a robust and efficient approach for rapid, personalized, and image‑consistent cardiac biomechanical modeling.

Abstract:
Efficient simulation of Laser Powder Bed Fusion (LPBF) is crucial for process prediction due to the lasting issue of high computational cost associated with traditional numerical methods such as finite element analysis (FEA). While a Physics‑Informed Neural Network (PINN) can predict solution fields with small training data and enables the generalization of new process parameters via transfer learning, it suffers from accuracy degradation in time‑dependent problems due to the accumulation of residual and the difficulty in capturing the steep spatial and temporal gradients inherent in the LPBF process. To overcome this issue, this study develops an efficient modeling framework, FEA‑Regulated Physics‑Informed Neural Network (FEA‑PINN), to accelerate the prediction of melt pool dynamics phenomena in an LPBF process while maintaining the FEA accuracy. The innovation of FEA‑PINN manifested itself in two aspects. First, a novel strategy has been developed within the PINN model to capture the dynamic phase change of powder‑liquid‑solid, enabling the tracking of material status during laser melting. The model further incorporates temperature‑dependent material properties, phase change behavior of the powder bed, Marangoni convection, and natural convection within the melt pool. Second, the FEA‑PINN framework integrates corrective FEA simulations during inference to enforce physical consistency, reduce error drift, and capture the steep gradients. A comparative analysis shows that FEA‑PINN achieves accuracy comparable to FEA while significantly reducing computational cost. The framework has been validated against benchmark FEA data for single‑track scanning in LPBF.

Abstract:
Physics‑informed Neural Networks (PINNs) have emerged as an efficient way to learn surrogate neural solvers of PDEs by embedding the physical model in the loss function and minimizing its residuals using automatic differentiation at so‑called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements. In this paper, we propose a new quadrature method for approximating definite integrals based on the hessian of the considered function, and that we leverage to guide the selection of the collocation points during the training process of PINNs.

Abstract:
Reliable multiscale models of thrombosis require platelet‑scale fidelity at organ‑scale cost, a gap that scientific machine learning has the potential to narrow. We train a DeepONet surrogate on platelet dynamics generated with LAMMPS for platelets spanning ten elastic moduli and capillary numbers (0.07 ‑ 0.77). The network takes in input the wall shear stress, bond stiffness, time, and initial particle coordinates and returns the full three‑dimensional deformation of the membrane. Mean‑squared‑error minimization with Adam and adaptive learning‑rate decay yields a median displacement error below 1%, a 90th percentile below 3%, and a worst case below 4% over the entire calibrated range while accelerating computation by four to five orders of magnitude. Leave‑extremes‑out retraining shows graceful extrapolation: the held‑out stiffest and most compliant platelets retain sub‑3% median error and an 8% maximum. Error peaks coincide with transient membrane self‑contact, suggesting improvements via graph neural trunks and physics‑informed torque regularization. These results classify the surrogate as high‑fidelity and position it for seamless coupling with continuum CFD, enabling platelet‑resolved hemodynamic simulations in patient‑specific geometries and opening new avenues for predictive thrombosis modeling.

Abstract:
We develop a principled framework for discovering causal structure in partial differential equations (PDEs) using physics‑informed neural networks and counterfactual perturbations. Unlike classical residual minimization or sparse regression methods, our approach quantifies operator‑level necessity through functional interventions on the governing dynamics. We introduce causal sensitivity indices and structural deviation metrics to assess the influence of candidate differential operators within neural surrogates. Theoretically, we prove exact recovery of the causal operator support under restricted isometry or mutual coherence conditions, with residual bounds guaranteeing identifiability. Empirically, we validate the framework on both synthetic and real‑world datasets across climate dynamics, tumor diffusion, and ocean flows. Our method consistently recovers governing operators even under noise, redundancy, and data scarcity, outperforming standard PINNs and DeepONets in structural fidelity. This work positions causal PDE discovery as a tractable and interpretable inference task grounded in structural causal models and variational residual analysis.

Abstract:
The helium I line intensity ratio (LIR) method is used to measure the electron density (n_e) and temperature (T_e) of fusion‑relevant plasmas. Although the collisional‑radiative model (CRM) has been used to predict n_e and T_e, recent studies have shown that machine learning approaches can provide better measurements if a sufficient dataset for training is available. This study investigates a hybrid neural network approach that combines CRM‑ and experiment‑based models. Although the CRM‑based model alone exhibited negative transfer in most cases, the ensemble model modestly improved the prediction accuracy of T_e. Notably, in data‑limited scenarios, the CRM‑based model outperformed the others for T_e prediction, highlighting its potential for applications with constrained diagnostic access.

Abstract:
Physics‑informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by computational limitations, PINNs are typically optimized using a finite set of points, which poses significant challenges in guaranteeing their convergence and accuracy. In this study, we proposed a new weighting scheme that will adaptively change the weights to the loss functions from isolated points to their continuous neighborhood regions. The empirical results show that our weighting scheme can reduce the relative L^2 errors to a lower value.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a promising computational framework for solving differential equations by integrating deep learning with physical constraints. However, their application in gas turbines is still in its early stages, requiring further refinement and standardization for wider adoption. This survey provides a comprehensive review of PINNs in Industrial Gas Turbines (IGTs) research, highlighting their contributions to the analysis of aerodynamic and aeromechanical phenomena, as well as their applications in flow field reconstruction, fatigue evaluation, and flutter prediction, and reviews recent advancements in accuracy, computational efficiency, and hybrid modelling strategies. In addition, it explores key research efforts, implementation challenges, and future directions aimed at improving the robustness and scalability of PINNs.

Abstract:
We investigate the high‑precision training of Physics‑Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer‑assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier‑Stokes equations, we obtain a solution whose loss is 4 digits smaller than that obtained in \citewang2023asymptotic with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher‑dimensional problems.

Abstract:
The growing reliance on power electronics introduces new challenges requiring detailed time‑domain analyses with fast and accurate circuit simulation tools. Currently, commercial time‑domain simulation software are mainly relying on physics‑based methods to simulate power electronics. Recent work showed that data‑driven and physics‑informed learning methods can increase simulation speed with limited compromise on accuracy, but many challenges remain before deployment in commercial tools can be possible. In this paper, we propose a physics‑informed bidirectional long‑short term memory neural network (BiLSTM‑PINN) model to simulate the time‑domain response of a closed‑loop dc‑dc boost converter for various operating points, parameters, and perturbations. A physics‑informed fully‑connected neural network (FCNN) and a BiLSTM are also trained to establish a comparison. The three methods are then compared using step‑response tests to assess their performance and limitations in terms of accuracy. The results show that the BiLSTM‑PINN and BiLSTM models outperform the FCNN model by more than 9 and 4.5 times, respectively, in terms of median RMSE. Their standard deviation values are more than 2.6 and 1.7 smaller than the FCNN's, making them also more consistent. Those results illustrate that the proposed BiLSTM‑PINN is a potential alternative to other physics‑based or data‑driven methods for power electronics simulations.

Abstract:
This paper describes a sound source localization (SSL) technique that combines an α‑stable model for the observed signal with a neural network‑based approach for modeling steering vectors. Specifically, a physics‑informed neural network, referred to as Neural Steerer, is used to interpolate measured steering vectors (SVs) on a fixed microphone array. This allows for a more robust estimation of the so‑called α‑stable spatial measure, which represents the most plausible direction of arrival (DOA) of a target signal. As an α‑stable model for the non‑Gaussian case (α \in (0, 2)) theoretically defines a unique spatial measure, we choose to leverage it to account for residual reconstruction error of the Neural Steerer in the downstream tasks. The objective scores indicate that our proposed technique outperforms state‑of‑the‑art methods in the case of multiple sound sources.

Abstract:
Multi‑query applications such as parameter estimation, uncertainty quantification and design optimization for parameterized PDE systems are expensive due to the high computational cost of high‑fidelity simulations. Reduced/Latent state dynamics approaches for parameterized PDEs offer a viable method where high‑fidelity data and machine learning techniques are used to reduce the system's dimensionality and estimate the dynamics of low‑dimensional reduced states. These reduced state dynamics approaches rely on high‑quality data and struggle with highly sparse spatiotemporal noisy measurements typically obtained from experiments. Furthermore, there is no guarantee that these models satisfy governing physical conservation laws, especially for parameters that are not a part of the model learning process. In this article, we propose a reduced state dynamics approach, which we refer to as ECLEIRS, that satisfies conservation laws exactly even for parameters unseen in the model training process. ECLEIRS is demonstrated for two applications: 1) obtaining clean solution signals from sparse and noisy measurements of parametric systems, and 2) predicting dynamics for unseen system parameters. We compare ECLEIRS with other reduced state dynamics approaches, those that do not enforce any physical constraints and those with physics‑informed loss functions, for three shock‑propagation problems: 1‑D advection, 1‑D Burgers and 2‑D Euler equations. The numerical experiments conducted in this study demonstrate that ECLEIRS provides the most accurate prediction of dynamics for unseen parameters even in the presence of highly sparse and noisy data. We also demonstrate that ECLEIRS yields solutions and fluxes that satisfy the governing conservation law up to machine precision for unseen parameters, while the other methods yield much higher errors and do not satisfy conservation laws.

Abstract:
Physics‑informed deep learning has achieved remarkable progress by embedding geometric priors, such as Hamiltonian symmetries and variational principles, into neural networks, enabling structure‑preserving models that extrapolate with high accuracy. However, in systems with dissipation and holonomic constraints, ubiquitous in legged locomotion and multibody robotics, the canonical symplectic form becomes degenerate, undermining the very invariants that guarantee stability and long‑term prediction. In this work, we tackle this foundational limitation by introducing Presymplectification Networks (PSNs), the first framework to learn the symplectification lift via Dirac structures, restoring a non‑degenerate symplectic geometry by embedding constrained systems into a higher‑dimensional manifold. Our architecture combines a recurrent encoder with a flow‑matching objective to learn the augmented phase‑space dynamics end‑to‑end. We then attach a lightweight Symplectic Network (SympNet) to forecast constrained trajectories while preserving energy, momentum, and constraint satisfaction. We demonstrate our method on the dynamics of the ANYmal quadruped robot, a challenging contact‑rich, multibody system. To the best of our knowledge, this is the first framework that effectively bridges the gap between constrained, dissipative mechanical systems and symplectic learning, unlocking a whole new class of geometric machine learning models, grounded in first principles yet adaptable from data.

Abstract:
Modeling viscoelastic behavior is crucial in engineering and biomechanics, where materials undergo time‑dependent deformations, including stress relaxation, creep buckling and biological tissue development. Traditional numerical methods, like the finite element method, often require explicit meshing, artificial perturbations or embedding customised programs to capture these phenomena, adding computational complexity. In this study, we develop an energy‑based physics‑informed neural network (PINN) framework using an incremental approach to model viscoelastic creep, stress relaxation, buckling, and growth‑induced morphogenesis. Physics consistency is ensured by training neural networks to minimize the systems potential energy functional, implicitly satisfying equilibrium and constitutive laws. We demonstrate that this framework can naturally capture creep buckling without pre‑imposed imperfections, leveraging inherent training dynamics to trigger instabilities. Furthermore, we extend our framework to biological tissue growth and morphogenesis, predicting both uniform expansion and differential growth‑induced buckling in cylindrical structures. Results show that the energy‑based PINN effectively predicts viscoelastic instabilities, post‑buckling evolution and tissue morphological evolution, offering a promising alternative to traditional methods. This study demonstrates that PINN can be a flexible robust tool for modeling complex, time‑dependent material behavior, opening possible applications in structural engineering, soft materials, and tissue development.

Abstract:
Connected automated vehicles (CAVs) cruising control strategies have been extensively studied at the microscopic level. CAV controllers sense and react to traffic both upstream and downstream, yet most macroscopic models still assume locality, where the desired speed only depends on local density. The nonlocal macroscopic traffic flow models that explicitly capture the ``look ahead'' and ``look behind'' nonlocal CAV dynamics remain underexplored. In this paper, we propose a Physics‑informed Neural Network framework to directly learn a macroscopic non‑local flow model from a generic looking‑ahead looking‑behind vehicle motion model, which bridges the micro‑macro modeling gap. We reconstruct macroscopic traffic states from synthetic CAV trajectories generated by the proposed microscopic control designs, and then learn a non‑local traffic flow model that embeds a non‑local conservation law to capture the resulting look‑ahead look‑behind dynamics. To analyze how CAV control parameters affect nonlocal traffic flow, we conduct high‑fidelity driving simulator experiments to collect human drivers' trajectory data with varying downstream and upstream visibility, which serves as a baseline for tuning CAV control gains. Our analysis validates that the learned non‑local flow model predicts CAV traffic dynamics more accurately than local models, and the fundamental diagram exhibits far less scatter in the speed ‑ density relation. We further show that the looking‑ahead/looking‑behind control gains mainly reshape the non‑local kernels, while the macroscopic speed and non‑local density relation mainly depends on the desired speed function choice of the CAV controller. Our results provide a systematic approach for learning non‑local macroscopic traffic‑flow models directly from generic CAV control designs.

Abstract:
In this work, we develop interface‑gated physics‑informed neural networks (IG‑PINNs) to solve elliptic interface equations. In IG‑PINNs, we use a fully connected neural network to capture the smooth behavior across the entire domain. In each subdomain separated by the interface, an interface‑gated network is utilized to provide corrections at the interface. In the architectural design of the interface‑gated network, we introduce a gating mechanism and a level‑set function derived from the interface. This design enables the interface‑gated network to effectively handle discontinuous jumps across the interface. Some numerical experiments have confirmed the effectiveness of the IG‑PINNs, demonstrating higher accuracy compared with PINNs, interface PINNs (I‑PINNs) and multi‑domain PINNs (M‑PINNs).

Abstract:
Neural ray tracing (RT) has emerged as a promising paradigm for channel modeling by combining physical propagation principles with neural networks. It enables high modeling accuracy and efficiency. However, current neural RT methods face two key limitations: constrained generalization capability due to strong spatial dependence, and weak adherence to electromagnetic laws. In this paper, we propose GeNeRT, a Generalizable Neural RT framework with enhanced generalization, accuracy and efficiency. GeNeRT supports both intra‑scenario spatial transferability and inter‑scenario zero‑shot generalization. By incorporating Fresnel‑inspired neural network design, it also achieves higher accuracy in multipath component (MPC) prediction. Furthermore, a GPU‑tensorized acceleration strategy is introduced to improve runtime efficiency. Extensive experiments conducted in outdoor scenarios demonstrate that GeNeRT generalizes well across untrained regions within a scenario and entirely unseen environments, and achieves superior accuracy in MPC prediction compared to baselines. Moreover, it outperforms Wireless Insite in runtime efficiency, particularly in multi‑transmitter settings. Ablation experiments validate the effectiveness of the network architecture and training strategy in capturing physical principles of ray‑surface interactions.

Abstract:
Quantifying and propagating modeling uncertainties is crucial for reliability analysis, robust optimization, and other model‑based algorithmic processes in engineering design and control. Now, physics‑informed machine learning (PIML) methods have emerged in recent years as a new alternative to traditional computational modeling and surrogate modeling methods, offering a balance between computing efficiency, modeling accuracy, and interpretability. However, their ability to predict and propagate modeling uncertainties remains mostly unexplored. In this paper, a promising class of auto‑differentiable hybrid PIML architectures that combine partial physics and neural networks or ANNs (for input transformation or adaptive parameter estimation) is integrated with Bayesian Neural networks (replacing the ANNs); this is done with the goal to explore whether BNNs can successfully provision uncertainty propagation capabilities in the PIML architectures as well, further supported by the auto‑differentiability of these architectures. A two‑stage training process is used to alleviate the challenges traditionally encountered in training probabilistic ML models. The resulting BNN‑integrated PIML architecture is evaluated on an analytical benchmark problem and flight experiments data for a fixed‑wing RC aircraft, with prediction performance observed to be slightly worse or at par with purely data‑driven ML and original PIML models. Moreover, Monte Carlo sampling of probabilistic BNN weights was found to be most effective in propagating uncertainty in the BNN‑integrated PIML architectures.

Abstract:
Accurate inverse dynamics models are essential tools for controlling industrial robots. Recent research combines neural network regression with inverse dynamics formulations of the Newton‑Euler and the Euler‑Lagrange equations of motion, resulting in so‑called Newtonian neural networks and Lagrangian neural networks, respectively. These physics‑informed models seek to identify unknowns in the analytical equations from data. Despite their potential, current literature lacks guidance on choosing between Lagrangian and Newtonian networks. In this study, we show that when motor torques are estimated instead of directly measuring joint torques, Lagrangian networks prove less effective compared to Newtonian networks as they do not explicitly model dissipative torques. The performance of these models is compared to neural network regression on data of a MABI MAX 100 industrial robot.

Abstract:
Retired electric vehicle batteries offer immense potential to support low‑carbon energy systems, but uncertainties in their degradation behavior and data inaccessibilities under second‑life use pose major barriers to safe and scalable deployment. This work proposes a Physics‑Informed Mixture of Experts (PIMOE) network that computes battery degradation trajectories using partial, field‑accessible signals in a single cycle. PIMOE leverages an adaptive multi‑degradation prediction module to classify degradation modes using expert weight synthesis underpinned by capacity‑voltage and relaxation data, producing latent degradation trend embeddings. These are input to a use‑dependent recurrent network for long‑term trajectory prediction. Validated on 207 batteries across 77 use conditions and 67,902 cycles, PIMOE achieves an average mean absolute percentage (MAPE) errors of 0.88% with a 0.43 ms inference time. Compared to the state‑of‑the‑art Informer and PatchTST, it reduces computational time and MAPE by 50%, respectively. Compatible with random state of charge region sampling, PIMOE supports 150‑cycle forecasts with 1.50% average and 6.26% maximum MAPE, and operates effectively even with pruned 5MB training data. Broadly, PIMOE framework offers a deployable, history‑free solution for battery degradation trajectory computation, redefining how second‑life energy storage systems are assessed, optimized, and integrated into the sustainable energy landscape.

Abstract:
In this paper, the physics informed neural networks (PINNs) is employed for the numerical simulation of heat transfer involving a moving source. To reduce the computational effort, a new training method is proposed that uses a continuous time‑stepping through transfer learning. Within this, the time interval is divided into smaller intervals and a single network is initialized. On this single network each time interval is trained with the initial condition for (n+1)th as the solution obtained at nth time increment. Thus, this framework enables the computation of large temporal intervals without increasing the complexity of the network itself. The proposed framework is used to estimate the temperature distribution in a homogeneous medium with a moving heat source. The results from the proposed framework is compared with traditional finite element method and a good agreement is seen.

Abstract:
Fluid‑like materials are ubiquitous, spanning from living biological tissues to geological formations, and across scales ranging from micrometers to kilometers. Inferring their rheological properties remains a major challenge, particularly when traditional rheometry fails to capture their complex, three‑dimensional, and often heterogeneous behavior. This difficulty is exacerbated by system size, boundary conditions, and other material‑specific physical, chemical, or thermal constraints. In this work, we explore whether rheological laws can be inferred directly from flow observations. We propose a physics‑informed neural network (PINN) framework designed to learn constitutive viscoplastic laws from velocity field data alone. Our method uses a neural network to interpolate the velocity field, enabling the computation of velocity gradients via automatic differentiation. These gradients are used to estimate the residuals of the governing conservation laws, which implicitly depend on the unknown rheology. We jointly optimize both the constitutive model and the velocity field representation by minimizing the physical residuals and discrepancies from observed data. We validate our approach on synthetic velocity fields generated from numerical simulations using Herschel‑Bulkley, Carreau and Panastasiou models under various flow conditions. The algorithm reliably infers rheological parameters, even in the presence of significant noise. We analyze the dependence of inference performance on flow geometry and sampling, highlighting the importance of shear rate distribution in the dataset. Finally, we explore preliminary strategies for model‑agnostic inference via embedded model selection, demonstrating the potential of PINNs for identifying the most suitable rheological law from candidate models.

Abstract:
This paper proposes a rank inspired neural network (RINN) to tackle the initialization sensitivity issue of physics informed extreme learning machines (PIELM) when numerically solving partial differential equations (PDEs). Unlike PIELM which randomly initializes the parameters of its hidden layers, RINN incorporates a preconditioning stage. In this stage, covariance‑driven regularization is employed to optimize the orthogonality of the basis functions generated by the last hidden layer. The key innovation lies in minimizing the off‑diagonal elements of the covariance matrix derived from the hidden‑layer output. By doing so, pairwise orthogonality constraints across collocation points are enforced which effectively enhances both the numerical stability and the approximation ability of the optimized function space.The RINN algorithm unfolds in two sequential stages. First, it conducts a non‑linear optimization process to orthogonalize the basis functions. Subsequently, it solves the PDE constraints using linear least‑squares method. Extensive numerical experiments demonstrate that RINN significantly reduces performance variability due to parameter initialization compared to PIELM. Incorporating an early stopping mechanism based on PDE loss further improves stability, ensuring consistently high accuracy across diverse initialization settings.

Abstract:
Random Feature Methods (RFMs) and their variants such as extreme learning machine finite‑basis physics‑informed neural networks (ELM‑FBPINNs) offer a scalable approach for solving partial differential equations (PDEs) by using localized, overlapping and randomly initialized neural network basis functions to approximate the PDE solution and training them to minimize PDE residuals through solving structured least‑squares problems. This combination leverages the approximation power of randomized neural networks and the parallelism of domain decomposition. However, the resulting least‑squares systems are often severely ill‑conditioned, due to local redundancy among random basis functions, which significantly affects the convergence of standard solvers. In this work, we introduce a block rank‑revealing QR (RRQR) filtering and preconditioning strategy that operates directly on the structured least‑squares problem. First, local RRQR factorizations identify and remove redundant basis functions while preserving numerically informative ones, reducing problem size, and improving conditioning. Second, we use these factorizations to construct a right preconditioner for the global problem which preserves block‑sparsity and numerical stability. Third, we derive deterministic bounds of the condition number of the preconditioned system, with probabilistic refinements for small overlaps. We validate our approach on challenging, multi‑scale PDE problems in 1D, 2D, and (2+1)D, demonstrating reductions in condition numbers by up to eleven orders of magnitude, LSQR convergence speedups by factors of 10‑1000, and higher accuracy than both unpreconditioned and additive Schwarz‑preconditioned baselines, all at significantly lower memory and computational cost. These results establish RRQR‑based preconditioning as a scalable, accurate, and efficient enhancement for RFM‑based PDE solvers.

Abstract:
Physics‑informed neural operators have emerged as a powerful paradigm for solving parametric partial differential equations (PDEs), particularly in the aerospace field, enabling the learning of solution operators that generalize across parameter spaces. However, existing methods either suffer from limited expressiveness due to fixed basis/coefficient designs, or face computational challenges due to the high dimensionality of the parameter‑to‑weight mapping space. We present LFR‑PINO, a novel physics‑informed neural operator that introduces two key innovations: (1) a layered hypernetwork architecture that enables specialized parameter generation for each network layer, and (2) a frequency‑domain reduction strategy that significantly reduces parameter count while preserving essential spectral features. This design enables efficient learning of a universal PDE solver through pre‑training, capable of directly handling new equations while allowing optional fine‑tuning for enhanced precision. The effectiveness of this approach is demonstrated through comprehensive experiments on four representative PDE problems, where LFR‑PINO achieves 22.8%‑68.7% error reduction compared to state‑of‑the‑art baselines. Notably, frequency‑domain reduction strategy reduces memory usage by 28.6%‑69.3% compared to Hyper‑PINNs while maintaining solution accuracy, striking an optimal balance between computational efficiency and solution fidelity.

Abstract:
The UT GraphCast Hindcast Dataset from 1979 to 2024 is a comprehensive global weather forecast archive generated using the Google DeepMind GraphCast Operational model. Developed by researchers at The University of Texas at Austin under the WCRP umbrella, this dataset provides daily 15 day deterministic forecasts at 00UTC on an approximately 25 km global grid for a 45 year period. GraphCast is a physics informed graph neural network that was trained on ECMWF ERA5 reanalysis. It predicts more than a dozen key atmospheric and surface variables on 37 vertical levels, delivering a full medium range forecast in under one minute on modern hardware.

Abstract:
The prediction of crystal properties is essential for understanding structure‑property relationships and accelerating the discovery of functional materials. However, conventional approaches relying on experimental measurements or density functional theory (DFT) calculations are often resource‑intensive, limiting their scalability. Machine learning (ML) models offer a promising alternative by learning complex structure‑property relationships from data, enabling faster predictions. Yet, existing ML models often rely on labeled data, adopt representations that poorly capture essential structural characteristics, and lack integration with physical principles‑‑factors that limit their generalizability and interpretability. Here, we introduce CLOUD (Crystal Language mOdel for Unified and Differentiable materials modeling), a transformer‑based framework trained on a novel Symmetry‑Consistent Ordered Parameter Encoding (SCOPE) that encodes crystal symmetry, Wyckoff positions, and composition in a compact, coordinate‑free string representation. Pre‑trained on over six million crystal structures, CLOUD is fine‑tuned on multiple downstream tasks and achieves competitive performance in predicting a wide range of material properties, demonstrating strong scaling performance. Furthermore, as proof of concept of differentiable materials modeling, CLOUD is applied to predict the phonon internal energy and heat capacity, which integrates the Debye model to preserve thermodynamic consistency. The CLOUD‑DEBYE framework enforces thermodynamic consistency and enables temperature‑dependent property prediction without requiring additional data. These results demonstrate the potential of CLOUD as a scalable and physics‑informed foundation model for crystalline materials, unifying symmetry‑consistent representations with physically grounded learning for property prediction and materials discovery.

Abstract:
Physics‑informed neural networks (PINNs) offer a powerful approach to solving partial differential equations (PDEs), which are ubiquitous in the quantitative sciences. Applied to both forward and inverse problems across various scientific domains, PINNs have recently emerged as a valuable tool in the field of scientific machine learning. A key aspect of their training is that the data ‑‑ spatio‑temporal points sampled from the PDE's input domain ‑‑ are readily available. Influence functions, a tool from the field of explainable AI (XAI), approximate the effect of individual training points on the model, enhancing interpretability. In the present work, we explore the application of influence function‑based sampling approaches for the training data. Our results indicate that such targeted resampling based on data attribution methods has the potential to enhance prediction accuracy in physics‑informed neural networks, demonstrating a practical application of an XAI method in PINN training.

Abstract:
We propose S^2GPT‑PINN, a sparse and small model for solving parametric partial differential equations (PDEs). Similar to Small Language Models (SLMs), S^2GPT‑PINN is tailored to domain‑specific (families of) PDEs and characterized by its compact architecture and minimal computational power. Leveraging a small amount of extremely high quality data via a mathematically rigorous greedy algorithm that is enabled by the large full‑order models, S^2GPT‑PINN relies on orders of magnitude less parameters than PINNs to achieve extremely high efficiency via two levels of customizations. The first is knowledge distillation via task‑specific activation functions that are transferred from Pre‑Trained PINNs. The second is a judicious down‑sampling when calculating the physics‑informed loss of the network compressing the number of data sites by orders of magnitude to the size of the small model.

Abstract:
Accurate free‑energy estimation is essential in molecular simulation, yet the periodic boundary conditions (PBC) commonly used in computer simulations have rarely been explicitly exploited. Equilibrium methods such as umbrella sampling, metadynamics, and adaptive biasing force require extensive sampling, while non‑equilibrium pulling with Jarzynski's equality suffers from poor convergence due to exponential averaging. Here, we introduce a physics‑informed, score‑based diffusion framework: by mapping PBC simulations onto a Brownian particle in a periodic potential, we derive the Fokker‑Planck steady‑state score that directly encodes free‑energy gradients. A neural network is trained on non‑equilibrium trajectories to learn this score, providing a principled scheme to efficiently reconstruct the potential of mean force (PMF). On benchmark periodic potentials and small‑molecule membrane permeation, our method is up to one order of magnitude more efficient than umbrella sampling.

Abstract:
Physics‑informed neural networks (PINNs) are extensively used to represent various physical systems across multiple scientific domains. The same can be said for cardiac electrophysiology, wherein fully‑connected neural networks (FCNNs) have been employed to predict the evolution of an action potential in a 2D space following the two‑parameter phenomenological Aliev‑Panfilov (AP) model. In this paper, the training behaviour of PINNs is investigated to determine optimal hyperparameters to predict the electrophysiological activity of the myocardium in 3D according to the AP model, with the inclusion of boundary and material parameters. An FCNN architecture is employed with the governing partial differential equations in their strong form, which are scaled consistently with normalization of network inputs. The finite element (FE) method is used to generate training data for the network. Numerical examples with varying spatial dimensions and parameterizations are generated using the trained models. The network predicted fields for both the action potential and the recovery variable are compared with the respective FE simulations. Network losses are weighed with individual scalar values. Their effect on training and prediction is studied to arrive at a method of controlling losses during training.

Abstract:
Multimodal time series forecasting is foundational in various fields, such as utilizing satellite imagery and numerical data for predicting typhoons in climate science. However, existing multimodal approaches primarily focus on utilizing text data to help time series forecasting, leaving the visual data in existing time series datasets untouched. Furthermore, it is challenging for models to effectively capture the physical information embedded in visual data, such as satellite imagery's temporal and geospatial context, which extends beyond images themselves. To address this gap, we propose physics‑informed positional encoding (PIPE), a lightweight method that embeds physical information into vision language models (VLMs). PIPE introduces two key innovations: (1) a physics‑informed positional indexing scheme for mapping physics to positional IDs, and (2) a variant‑frequency positional encoding mechanism for encoding frequency information of physical variables and sequential order of tokens within the embedding space. By preserving both the physical information and sequential order information, PIPE significantly improves multimodal alignment and forecasting accuracy. Through the experiments on the most representative and the largest open‑sourced satellite image dataset, PIPE achieves state‑of‑the‑art performance in both deep learning forecasting and climate domain methods, demonstrating superiority across benchmarks, including a 12% improvement in typhoon intensity forecasting over prior works. Our code is provided in the supplementary material.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a transformative framework for addressing operator learning and inverse problems involving the Korteweg‑de Vries (KdV) equation for internal solitary waves. By integrating physical constraints with data‑driven optimization, PINNs overcome the critical challenges of parameter unmeasurability in the KdV equation for internal solitary waves in two‑layer fluid systems. This work addresses two problems: (1) Operator learning constructs a mapping from parameters to solutions, enabling wave evolution predictions from unknown parameters. Comparative studies demonstrate prediction errors as low as 10^‑4 when using 1000 training points. (2) Inverse problem solving leverages sparse and potentially noisy observational data with physics‑regularized constraints to invert nonlinear coefficients successfully. Compared to conventional approaches, this end‑to‑end differentiable paradigm unifies operator learning and inverse problem‑solving while overcoming mesh discretization errors and high‑dimensional parameter space iteration costs. The method shows effectiveness for internal wave problems in stratified fluids, providing both accurate forward modeling and robust parameter inversion capabilities, even under noise.

Abstract:
Accurately estimating spatially heterogeneous elasticity parameters, particularly Young's modulus and Poisson's ratio, from noisy displacement measurements remains significantly challenging in inverse elasticity problems. Existing inverse estimation techniques are often limited by instability, pronounced sensitivity to measurement noise, and difficulty in recovering absolute‑scale Young's modulus. This work presents a novel Inverse Elasticity Physics‑Informed Neural Network (IE‑PINN) specifically designed to robustly reconstruct heterogeneous distributions of elasticity parameters from noisy displacement data based on linear elasticity physics. IE‑PINN integrates three distinct neural network architectures dedicated to separately modeling displacement fields, strain fields, and elasticity distributions, thereby significantly enhancing stability and accuracy against measurement noise. Additionally, a two‑phase estimation strategy is introduced: the first phase recovers relative spatial distributions of Young's modulus and Poisson's ratio, and the second phase calibrates the absolute scale of Young's modulus using imposed loading boundary conditions. Additional methodological innovations, including positional encoding, sine activation functions, and a sequential pretraining protocol, further enhance the model's performance and robustness. Extensive numerical experiments demonstrate that IE‑PINN effectively overcomes critical limitations encountered by existing methods, delivering accurate absolute‑scale elasticity estimations even under severe noise conditions. This advancement holds substantial potential for clinical imaging diagnostics and mechanical characterization, where measurements typically encounter substantial noise.

Abstract:
Analysis of nonlinear autonomous systems typically involves estimating domains of attraction, which have been a topic of extensive research interest for decades. Despite that, accurately estimating domains of attraction for nonlinear systems remains a challenging task, where existing methods are conservative or limited to low‑dimensional systems. The estimation becomes even more challenging when accounting for state constraints. In this work, we propose a framework to accurately estimate safe (state‑constrained) domains of attraction for discrete‑time autonomous nonlinear systems. In establishing this framework, we first derive a new Zubov equation, whose solution corresponds to the exact safe domain of attraction. The solution to the aforementioned Zubov equation is shown to be unique and continuous over the whole state space. We then present a physics‑informed approach to approximating the solution of the Zubov equation using neural networks. To obtain certifiable estimates of the domain of attraction from the neural network approximate solutions, we propose a verification framework that can be implemented using standard verification tools (e.g., α,\!β‑CROWN and dReal). To illustrate its effectiveness, we demonstrate our approach through numerical examples concerning nonlinear systems with state constraints.

Abstract:
We propose a hybrid physics‑informed machine learning framework to approximate invariant manifolds (IMs) of discrete‑time dynamical systems driven by exogenous autonomous dynamics (exosystems). Such systems appear in applications ranging from control theory to modeling collective multi‑agent behavior (e.g., bird flocks, traffic dynamics) under hierarchical leadership. The IM learning problem is formulated as solving nonlinear functional equations derived from the invariance equation, expressing the manifold as a relationship between exogenous and system states. The proposed approach combines polynomial series with shallow neural networks, leveraging their complementary strengths. We focus on low‑ to medium‑dimensional manifolds where polynomial expansions remain tractable. Near equilibrium, polynomial series provide interpretability and convergence, while farther away neural networks capture global structure through their universal approximation capability. A continuity penalty enforces consistency between both representations at their interface, and training is performed using analytically derived derivatives within the Levenberg‑Marquardt scheme. Naturally, depending on the dimensionality of the input‑driven system, one may also employ a purely neural network‑based IM approximation, for which we also establish a universal approximation theorem based on certain assumptions on system dynamics. The framework is evaluated on two benchmark problems: an enzymatic bioreactor and a leader‑follower car‑following model. We analyze convergence, approximation accuracy, and computational cost, and compare standalone neural networks, polynomial expansions, and the hybrid method. Results show that the hybrid approach achieves superior accuracy compared to standalone schemes.

Abstract:
This survey provides a comprehensive overview of the emerging field of world models grounded in the foundation of acoustic physical information. It examines the theoretical underpinnings, essential methodological frameworks, and recent technological advancements in leveraging acoustic signals for high‑fidelity environmental perception, causal physical reasoning, and predictive simulation of dynamic events. The survey explains how acoustic signals, as direct carriers of mechanical wave energy from physical events, encode rich, latent information about material properties, internal geometric structures, and complex interaction dynamics. Specifically, this survey establishes the theoretical foundation by explaining how fundamental physical laws govern the encoding of physical information within acoustic signals. It then reviews the core methodological pillars, including Physics‑Informed Neural Networks (PINNs), generative models, and self‑supervised multimodal learning frameworks. Furthermore, the survey details the significant applications of acoustic world models in robotics, autonomous driving, healthcare, and finance. Finally, it systematically outlines the important technical and ethical challenges while proposing a concrete roadmap for future research directions toward robust, causal, uncertainty‑aware, and responsible acoustic intelligence. These elements collectively point to a research pathway towards embodied active acoustic intelligence, empowering AI systems to construct an internal "intuitive physics" engine through sound.

Abstract:
Urban systems are typical examples of complex systems, where the integration of physics‑based modeling with artificial intelligence (AI) presents a promising paradigm for enhancing predictive accuracy, interpretability, and decision‑making. In this context, AI excels at capturing complex, nonlinear relationships, while physics‑based models ensure consistency with real‑world laws and provide interpretable insights. We provide a comprehensive review of physics‑informed AI methods in urban applications. The proposed taxonomy categorizes existing approaches into three paradigms ‑ Physics‑Integrated AI, Physics‑AI Hybrid Ensemble, and AI‑Integrated Physics ‑ and further details seven representative methods. This classification clarifies the varying degrees and directions of physics‑AI integration, guiding the selection and development of appropriate methods based on application needs and data availability. We systematically examine their applications across eight key urban domains: energy, environment, economy, transportation, information, public services, emergency management, and the urban system as a whole. Our analysis highlights how these methodologies leverage physical laws and data‑driven models to address urban challenges, enhancing system reliability, efficiency, and adaptability. By synthesizing existing methodologies and their urban applications, we identify critical gaps and outline future research directions, paving the way toward next‑generation intelligent urban system modeling.

Abstract:
Human activity intensity prediction is crucial to many location‑based services. Despite tremendous progress in modeling dynamics of human activity, most existing methods overlook physical constraints of spatial interaction, leading to uninterpretable spatial correlations and over‑smoothing phenomenon. To address these limitations, this work proposes a physics‑informed deep learning framework, namely Gravity‑informed Spatiotemporal Transformer (Gravityformer) by integrating the universal law of gravitation to refine transformer attention. Specifically, it (1) estimates two spatially explicit mass parameters based on spatiotemporal embedding feature, (2) models the spatial interaction in end‑to‑end neural network using proposed adaptive gravity model to learn the physical constraint, and (3) utilizes the learned spatial interaction to guide and mitigate the over‑smoothing phenomenon in transformer attention. Moreover, a parallel spatiotemporal graph convolution transformer is proposed for achieving a balance between coupled spatial and temporal learning. Systematic experiments on six real‑world large‑scale activity datasets demonstrate the quantitative and qualitative superiority of our model over state‑of‑the‑art benchmarks. Additionally, the learned gravity attention matrix can be not only disentangled and interpreted based on geographical laws, but also improved the generalization in zero‑shot cross‑region inference. This work provides a novel insight into integrating physical laws with deep learning for spatiotemporal prediction.

Abstract:
Inference and prediction under partial knowledge of a physical system is challenging, particularly when multiple confounding sources influence the measured response. Explicitly accounting for these influences in physics‑based models is often infeasible due to epistemic uncertainty, cost, or time constraints, resulting in models that fail to accurately describe the behavior of the system. On the other hand, data‑driven machine learning models such as variational autoencoders are not guaranteed to identify a parsimonious representation. As a result, they can suffer from poor generalization performance and reconstruction accuracy in the regime of limited and noisy data. We propose a physics‑informed variational autoencoder architecture that combines the interpretability of physics‑based models with the flexibility of data‑driven models. To promote disentanglement of the known physics and confounding influences, the latent space is partitioned into physically meaningful variables that parametrize a physics‑based model, and data‑driven variables that capture variability in the domain and class of the physical system. The encoder is coupled with a decoder that integrates physics‑based and data‑driven components, and constrained by an adversarial training objective that prevents the data‑driven components from overriding the known physics, ensuring that the physics‑grounded latent variables remain interpretable. We demonstrate that the model is able to disentangle features of the input signal and separate the known physics from confounding influences using supervision in the form of class and domain observables. The model is evaluated on a series of synthetic case studies relevant to engineering structures, demonstrating the feasibility of the proposed approach.

Abstract:
We present a unified theoretical framework for analyzing the stability and consistency of Physics‑Informed Neural Networks (PINNs), grounded in operator coercivity, variational formulations, and non‑asymptotic perturbation theory. PINNs approximate solutions to partial differential equations (PDEs) by minimizing residual losses over sampled collocation and boundary points. We formalize both operator‑level and variational notions of consistency, proving that residual minimization in Sobolev norms leads to convergence in energy and uniform norms under mild regularity. Deterministic stability bounds quantify how bounded perturbations to the network outputs propagate through the full composite loss, while probabilistic concentration results via McDiarmid's inequality yield sample complexity guarantees for residual‑based generalization. A unified generalization bound links residual consistency, projection error, and perturbation sensitivity. Empirical results on elliptic, parabolic, and nonlinear PDEs confirm the predictive accuracy of our theoretical bounds across regimes. The framework identifies key structural principles, such as operator coercivity, activation smoothness, and sampling admissibility, that underlie robust and generalizable PINN training, offering principled guidance for the design and analysis of PDE‑informed learning systems.

Abstract:
Magnetar phenomena are likely intertwined with the location and structure of magnetospheric currents. General‑relativistic effects are important in shaping the force‑free equilibria describing static configurations, though most studies have quantified their impact only in cases of axial symmetry. Using a novel methodology based on physics‑informed neural networks, fully three‑dimensional configurations of varying stellar compactness are constructed. Realistic profiles for surface currents, qualitatively capturing the geometry of observed hotspots, are applied as boundary conditions to deduce the amount of free energy available to fuel outburst activity. It is found that the lowest‑energy solution branches permit only a \approx 30% excess relative to current‑starved solutions in axisymmetric cases with global twists, regardless of compactness, reducing to \approx 5% in 3D models with localised spots. Accounting for redshift reductions to their inferred dipole moments from timing data, explaining magnetar burst energetics therefore becomes more difficult unless the field hosts non‑negligible multipoles. Discussions on other aspects of magnetar phenomena are also provided.

Abstract:
We present a dual‑guided framework for reconstructing unsteady incompressible flow fields using sparse observations. The approach combines optimized sensor placement with a physics‑informed guided generative model. Sensor locations are selected using mutual information theory applied to a reduced‑order model of the flow, enabling efficient identification of high‑information observation points with minimal computational cost. These sensors, once selected, provide targeted observations that guide a denoising diffusion probabilistic model conditioned by physical constraints. Extensive experiments on 2D laminar cylinder wake flows demonstrate that under sparse sensing conditions, the structured sensor layouts fail to capture key flow dynamics, yielding high reconstruction errors. In contrast, our optimized sensor placement strategy achieves accurate reconstructions with L2 errors as low as 0.05, even with a limited number of sensors, confirming the effectiveness of the proposed approach in data‑limited regimes. When the number of sensors is higher than a threshold, however, both methods perform comparably. Our dual‑guided approach bridges reduced order model‑based sensor position optimization with modern generative modeling, providing accurate, physics‑consistent reconstruction from sparse data for scientific machine‑learning problems.

Abstract:
Electroencephalography (EEG) is extensively employed in medical diagnostics and brain‑computer interface (BCI) applications due to its non‑invasive nature and high temporal resolution. However, EEG analysis faces significant challenges, including noise, nonstationarity, and inter‑subject variability, which hinder its clinical utility. Traditional neural networks often lack integration with biophysical knowledge, limiting their interpretability, robustness, and potential for medical translation. To address these limitations, this study introduces NeuroPhysNet, a novel Physics‑Informed Neural Network (PINN) framework tailored for EEG signal analysis and motor imagery classification in medical contexts. NeuroPhysNet incorporates the FitzHugh‑Nagumo model, embedding neurodynamical principles to constrain predictions and enhance model robustness. Evaluated on the BCIC‑IV‑2a dataset, the framework achieved superior accuracy and generalization compared to conventional methods, especially in data‑limited and cross‑subject scenarios, which are common in clinical settings. By effectively integrating biophysical insights with data‑driven techniques, NeuroPhysNet not only advances BCI applications but also holds significant promise for enhancing the precision and reliability of clinical diagnostics, such as motor disorder assessments and neurorehabilitation planning.

Abstract:
In this work, a physics‑informed neural networks (PINNs) based algorithm is used for simulation of nonlinear 1D and 2D Burgers' type models. This scheme relies on a neural network built to approximate the problem solution and use a trial function that meets the initial data and boundary criteria. First of all, a brief mathematical formulation of the problem and the structure of PINNs, including the neural network architecture, loss construction, and training methodology is described. Finally, the algorithm is demonstrated with five test problems involving variations of the 1D coupled, 2D single and 2D coupled Burgers' models. We compare the PINN‑based solutions with exact results to assess accuracy and convergence of the developed algorithm. The results demonstrate that PINNs may faithfully replicate nonlinear PDE solutions and offer competitive performance in terms of inaccuracy and flexibility. This work demonstrates the potential of PINNs as a reliable approach to solving complex time‑dependent PDEs.

Abstract:
In the modern context of power systems, rapid, scalable, and physically plausible power flow predictions are essential for ensuring the grid's safe and efficient operation. While traditional numerical methods have proven robust, they require extensive computation to maintain physical fidelity under dynamic or contingency conditions. In contrast, recent advancements in artificial intelligence (AI) have significantly improved computational speed; however, they often fail to enforce fundamental physical laws during real‑world contingencies, resulting in physically implausible predictions. In this work, we introduce KCLNet, a physics‑informed graph neural network that incorporates Kirchhoff's Current Law as a hard constraint via hyperplane projections. KCLNet attains competitive prediction accuracy while ensuring zero KCL violations, thereby delivering reliable and physically consistent power flow predictions critical to secure the operation of modern smart grids.

Abstract:
Physics‑informed Neural Motion Planners (PiNMPs) provide a data‑efficient framework for solving the Eikonal Partial Differential Equation (PDE) and representing the cost‑to‑go function for motion planning. However, their scalability remains limited by spectral bias and the complex loss landscape of PDE‑driven training. Domain decomposition mitigates these issues by dividing the environment into smaller subdomains, but existing methods enforce continuity only at individual spatial points. While effective for function approximation, these methods fail to capture the spatial connectivity required for motion planning, where the cost‑to‑go function depends on both the start and goal coordinates rather than a single query point. We propose Finite Basis Neural Time Fields (FB‑NTFields), a novel neural field representation for scalable cost‑to‑go estimation. Instead of enforcing continuity in output space, FB‑NTFields construct a latent space representation, computing the cost‑to‑go as a distance between the latent embeddings of start and goal coordinates. This enables global spatial coherence while integrating domain decomposition, ensuring efficient large‑scale motion planning. We validate FB‑NTFields in complex synthetic and real‑world scenarios, demonstrating substantial improvements over existing PiNMPs. Finally, we deploy our method on a Unitree B1 quadruped robot, successfully navigating indoor environments. The supplementary videos can be found at https://youtu.be/OpRuCbLNOwM.

Abstract:
Estimating composition dependent diffusion coefficients in multicomponent alloys was a longstanding challenge due to limitations in experimental methods. In this study, we have first demonstrated a strategic design of producing only three diffusion couples to estimate all types, that is tracer, intrinsic, and interdiffusion coefficients at the Kirkendall marker planes. This establishes a systematic variation of diffusion coefficients with composition in a very wide composition range of the NiCoFeCrMn system in comparison to the data available on impurity diffusion coefficients in pure elements and tracer diffusion coefficients at the equiatomic composition estimated by the radiotracer method. Following, a physics‑informed Neural Network based numerical inverse method is developed to extract composition‑dependent diffusivities over the whole composition range of the diffusion couples.

Abstract:
Accurate vessel trajectory prediction is crucial for navigational safety, route optimization, traffic management, search and rescue operations, and autonomous navigation. Traditional data‑driven models lack real‑world physical constraints, leading to forecasts that disobey vessel motion dynamics, such as in scenarios with limited or noisy data where sudden course changes or speed variations occur due to external factors. To address this limitation, we propose a Physics‑Informed Neural Network (PINN) approach for trajectory prediction that integrates a streamlined kinematic model for vessel motion into the neural network training process via a first‑ and second‑order, finite difference physics‑based loss function. This loss function, discretized using the first‑order forward Euler method, Heun's second‑order approximation, and refined with a midpoint approximation based on Taylor series expansion, enforces fidelity to fundamental physical principles by penalizing deviations from expected kinematic behavior. We evaluated PINN using real‑world AIS datasets that cover diverse maritime conditions and compared it with state‑of‑the‑art models. Our results demonstrate that the proposed method reduces average displacement errors by up to 32% across models and datasets while maintaining physical consistency. These results enhance model reliability and adherence to mission‑critical maritime activities, where precision translates into better situational awareness in the oceans.

Abstract:
Free‑flow road networks, such as suburban highways, are increasingly experiencing traffic congestion due to growing commuter inflow and limited infrastructure. Traditional control mechanisms, such as traffic signals or local heuristics, are ineffective or infeasible in these high‑speed, signal‑free environments. We introduce self‑regulating cars, a reinforcement learning‑based traffic control protocol that dynamically modulates vehicle speeds to optimize throughput and prevent congestion, without requiring new physical infrastructure. Our approach integrates classical traffic flow theory, gap acceptance models, and microscopic simulation into a physics‑informed RL framework. By abstracting roads into super‑segments, the agent captures emergent flow dynamics and learns robust speed modulation policies from instantaneous traffic observations. Evaluated in the high‑fidelity PTV Vissim simulator on a real‑world highway network, our method improves total throughput by 5%, reduces average delay by 13%, and decreases total stops by 3% compared to the no‑control setting. It also achieves smoother, congestion‑resistant flow while generalizing across varied traffic patterns, demonstrating its potential for scalable, ML‑driven traffic management.

Abstract:
This study enhances the application of Physics‑Informed Neural Networks (PINNs) for modeling discontinuous solutions in both hydrodynamics and relativistic hydrodynamics. Conventional PINNs, trained with partial differential equation residuals, frequently face convergence issues and lower accuracy near discontinuities. To address these issues, we build on the recently proposed locally linearized PINNs (LLPINNs), which improve shock detection by modifying the Jacobian matrix resulting from the linearization of the equations, only in regions where the velocity field exhibits strong compression. However, the original LLPINN framework required a priori knowledge of shock velocities, limiting its practical utility. We present a generalized LLPINN method that dynamically computes shock speeds using neighboring states and applies jump conditions through entropy constraints. Additionally, we introduce locally Roe PINNs (LRPINNs), which incorporate an approximate Roe Riemann solver to improve shock resolution and conservation properties across discontinuities. These methods are adapted to two‑dimensional Riemann problems by using a divergence‑based shock detection combined with dimensional splitting, delivering precise solutions. Compared to a high‑order weighted essentially non‑oscillatory solver, our method produces sharper shock transitions but smoother solutions in areas with small‑scale vortex structures. Future research will aim to improve the resolution of these small‑scale features without compromising the model's ability to accurately capture shocks.

Abstract:
Accurate real‑time estimation of human movement dynamics, including internal joint moments and muscle forces, is essential for applications in clinical diagnostics and sports performance monitoring. Inertial measurement units (IMUs) provide a minimally intrusive solution for capturing motion data, particularly when used in sparse sensor configurations. However, current real‑time methods rely on supervised learning, where a ground truth dataset needs to be measured with laboratory measurement systems, such as optical motion capture. These systems are known to introduce measurement and processing errors and often fail to generalize to real‑world or previously unseen movements, necessitating new data collection efforts that are time‑consuming and impractical. To overcome these limitations, we propose SSPINNpose, a self‑supervised, physics‑informed neural network that estimates joint kinematics and kinetics directly from IMU data, without requiring ground truth labels for training. We run the network output through a physics model of the human body to optimize physical plausibility and generate virtual measurement data. Using this virtual sensor data, the network is trained directly on the measured sensor data instead of a ground truth. When compared to optical motion capture, SSPINNpose is able to accurately estimate joint angles and joint moments at an RMSD of 8.7 deg and 4.9 BWBH%, respectively, for walking and running at speeds up to 4.9 m/s at a latency of 3.5 ms. Furthermore, the framework demonstrates robustness across sparse sensor configurations and can infer the anatomical locations of the sensors. These results underscore the potential of SSPINNpose as a scalable and adaptable solution for real‑time biomechanical analysis in both laboratory and field environments.

Abstract:
The relative balance between physics and data within any physics‑informed machine learner is an important modelling consideration to ensure that the benefits of both physics and data‑based approaches are maximised. An over reliance on physical knowledge can be detrimental, particularly when the physics‑based component of a model may not accurately represent the true underlying system. An underutilisation of physical knowledge potentially wastes a valuable resource, along with benefits in model interpretability and reduced demand for expensive data collection. Achieving an optimal physics‑data balance is a challenging aspect of model design, particularly if the level varies through time; for example, one might have a physical approximation, only valid within particular regimes, or a physical phenomenon may be known to only occur when given conditions are met (e.g. at high temperatures). This paper develops novel, physically‑informed, change‑point kernels for Gaussian processes, capable of dynamically varying the reliance upon available physical knowledge. A high level of control is granted to a user, allowing for the definition of conditions in which they believe a phenomena should occur and the rate at which the knowledge should be phased in and out of a model. In circumstances where users may be less certain, the switching reliance upon physical knowledge may be automatically learned and recovered from the model in an interpretable and intuitive manner. Variation of the modelled noise based on the physical phenomena occurring is also implemented to provide a more representative capture of uncertainty alongside predictions. The capabilities of the new kernel structures are explored through the use of two engineering case studies: the directional wind loading of a cable‑stayed bridge and the prediction of aircraft wing strain during in‑flight manoeuvring.

Abstract:
We propose transformed Diffsuion‑Wave fractional Physics‑Informed Neural Networks (tDWfPINNs) for efficiently solving time‑fractional diffusion‑wave equations with fractional order Î±\in(1,2). Conventional numerical methods for these equations often compromise the mesh‑free advantage of Physics‑Informed Neural Networks (PINNs) or impose high computational costs when computing fractional derivatives. The proposed method avoids first‑order derivative calculations at quadrature points by introducing an integrand transformation technique, significantly reducing computational costs associated with fractional derivative evaluation while preserving accuracy. We conduct a comprehensive comparative analysis applying this integrand transformation in conjunction with both Monte Carlo integration and Gauss‑Jacobi quadrature schemes across various time‑fractional PDEs. Our results demonstrate that tDWfPINNs achieve superior computational efficiency without sacrificing accuracy. Furthermore, we incorporate the proposed approach into adaptive sampling approaches such as the residual‑based adaptive distribution (RAD) for the time‑fractional Burgers equation with order Î±\in(1,2), which exhibits complex solution dynamics. The experiments show that the Gauss‑Jacobi method typically outperforms the Monte Carlo approach; however, careful consideration is required when selecting the number of quadrature points. Overall, the proposed tDWfPINNs offer a significant advancement in the numerical solution of time‑fractional diffusion‑wave equations, providing an accurate and scalable mesh‑free alternative for challenging fractional models.

Abstract:
Studying physics‑informed neural networks (PINNs) for modeling partial differential equations to solve the acoustic wave field has produced promising results for simple geometries in two‑dimensional domains. One option is to compute the time‑harmonic wave field using the Helmholtz equation. Compared to existing numerical models, the physics‑informed neural networks forward problem has to overcome several topics related to the convergence of the optimization toward the "true" solution. The topics reach from considering the physical dimensionality (from 2D to 3D), the modeling of realistic sources (from a self‑similar source to a realistic confined point source), the modeling of sound‑hard (Neumann) boundary conditions, and the modeling of the full wave field by considering the complex solution quantities. Within this contribution, we study 3D room acoustic cases at low frequency, varying the source definition and the number of boundary condition sets and using a complex speed of sound model to account for some degree of absorption. We assess the convergence behavior by looking at the loss landscape of the PINN architecture, the L^2 error compared to a finite element reference simulation for each network architecture and configuration. The convergence studies showed that at least six training points per wavelength are necessary for accurate training and subsequent predictions of the PINN. The developments are part of an initiative aiming to model the low‑frequency behavior of room acoustics, including absorbers.

Abstract:
High‑quality power flow datasets are essential for training machine learning models in power systems. However, security and privacy concerns restrict access to real‑world data, making statistically accurate and physically consistent synthetic datasets a viable alternative. We develop a diffusion model for generating synthetic power flow datasets from real‑world power grids that both replicate the statistical properties of the real‑world data and ensure AC power flow feasibility. To enforce the constraints, we incorporate gradient guidance based on the power flow constraints to steer diffusion sampling toward feasible samples. For computational efficiency, we further leverage insights from the fast decoupled power flow method and propose a variable decoupling strategy for the training and sampling of the diffusion model. These solutions lead to a physics‑informed diffusion model, generating power flow datasets that outperform those from the standard diffusion in terms of feasibility and statistical similarity, as shown in experiments across IEEE benchmark systems.

Abstract:
Understanding the grain morphology, orientation distribution, and crystal structure of nanocrystals is essential for optimizing the mechanical and physical properties of functional materials. Synchrotron X‑ray Laue microdiffraction is a powerful technique for characterizing crystal structures and orientation mapping using focused X‑rays. However, when grain sizes are smaller than the beam size, mixed peaks in the Laue pattern from neighboring grains limit the resolution of grain morphology mapping. We propose a physics‑informed machine learning (PIML) approach that combines a CNN feature extractor with a physics‑informed filtering algorithm to overcome the spatial resolution limits of X‑rays, achieving nanoscale resolution for grain mapping. Our PIML method successfully resolves the grain size, orientation distribution, and morphology of Au nanocrystals through synchrotron microdiffraction scans, showing good agreement with electron backscatter diffraction results. This PIML‑assisted synchrotron microdiffraction analysis can be generalized to other diffraction‑based probes, enabling the characterization of nanosized structures with micron‑sized probes.

Abstract:
Understanding and reasoning about dynamics governed by physical laws through visual observation, akin to human capabilities in the real world, poses significant challenges. Currently, object‑centric dynamic simulation methods, which emulate human behavior, have achieved notable progress but overlook two critical aspects: 1) the integration of physical knowledge into models. Humans gain physical insights by observing the world and apply this knowledge to accurately reason about various dynamic scenarios; 2) the validation of model adaptability across diverse scenarios. Real‑world dynamics, especially those involving fluids and objects, demand models that not only capture object interactions but also simulate fluid flow characteristics. To address these gaps, we introduce SlotPi, a slot‑based physics‑informed object‑centric reasoning model. SlotPi integrates a physical module based on Hamiltonian principles with a spatio‑temporal prediction module for dynamic forecasting. Our experiments highlight the model's strengths in tasks such as prediction and Visual Question Answering (VQA) on benchmark and fluid datasets. Furthermore, we have created a real‑world dataset encompassing object interactions, fluid dynamics, and fluid‑object interactions, on which we validated our model's capabilities. The model's robust performance across all datasets underscores its strong adaptability, laying a foundation for developing more advanced world models.

Abstract:
Hamiltonian learning (HL), enabling precise estimation of system parameters and underlying dynamics, plays a critical role in characterizing quantum systems. However, conventional HL methods face challenges in noise robustness and resource efficiency, especially under limited measurements. In this work, we present Inverse Physics‑Informed Neural Networks for Hamiltonian Learning (iPINN‑HL), an approach that incorporates the Schrödinger equation as a soft constraint via a loss function penalty into the ML procedure. This formulation allows the model to integrate both observational data and known physical laws to infer Hamiltonian parameters with greater accuracy and resource efficiency. We benchmark iPINN‑HL against a deep‑neural‑network‑based quantum state tomography method (denoted as DNN‑HL) and demonstrate its effectiveness across several different scenarios, including one‑dimensional spin chains, cross‑resonance gate calibration, crosstalk identification, and real‑time compensation to parameter drift. Our results show that iPINN‑HL can approach the Heisenberg limit and exhibits robustness to noises, while outperforming DNN‑HL in accuracy and resource efficiency. Therefore, iPINN‑HL is a powerful and flexible framework for quantum system characterization for practical tasks.

Abstract:
In recent years, with the advancements in machine learning and neural networks, algorithms using physics‑informed neural networks (PINNs) to solve PDEs have gained widespread applications. While these algorithms are well‑suited for a wide range of equations, they often exhibit suboptimal performance when applied to equations with large local gradients, resulting in substantial localized errors. To address this issue, this paper proposes an adaptive PINN algorithm designed to improve accuracy in such cases. The core idea of the algorithm is to adaptively adjust the distribution of collocation points based on the recovery‑type a‑posterior error of the current numerical solution, enabling a better approximation of the true solution. This approach is inspired by the adaptive finite element method. By combining the recovery‑type a‑posteriori estimator, a gradient‑recovery estimator commonly used in the adaptive finite element method (FEM) with PINNs, we introduce the Recovery‑type a‑posteriori estimator enhanced adaptive PINN (R‑PINN) and compare its performance with a typical adaptive PINN algorithm, FI‑PINN. Our results demonstrate that R‑PINN achieves faster convergence with fewer adaptive points and significantly outperforms in the cases with multiple regions of large errors than FI‑PINN. Notably, our method is a hybrid numerical approach for solving partial differential equations, integrating adaptive FEM with PINNs.

Abstract:
Solving and optimising Partial Differential Equations (PDEs) in geometrically parameterised domains often requires iterative methods, leading to high computational and time complexities. One potential solution is to learn a direct mapping from the parameters to the PDE solution. Two prominent methods for this are Data‑driven Non‑Intrusive Reduced Order Models (DROMs) and Parametrised Physics Informed Neural Networks (PPINNs). However, their accuracy tends to degrade as the number of geometric parameters increases. To address this, we propose adopting Generative Models to create new geometries, effectively reducing the number of parameters, and improving the performance of DROMs and PPINNs. The first section briefly reviews the general theory of Generative Models and provides some examples, whereas the second focusses on their application to geometries with fixed or variable points, emphasising their integration with DROMs and PPINNs. DROMs trained on geometries generated by these models demonstrate enhanced accuracy due to reduced parameter dimensionality. For PPINNs, we introduce a methodology that leverages Generative Models to reduce the parameter dimensions and improve convergence. This approach is tested on a Poisson equation defined over deformed Stanford Bunny domains.

Abstract:
We design strategies in nonlinear geometric analysis to temper the effects of adversarial learning for sufficiently smooth data of numerical method‑type dynamics in encoder‑decoder methods, variational and deterministic, through the use of geometric flow regularization. We augment latent spaces with geometric flows to control structure, relying on adaptations of curvature and Ricci flow. All of our flows are solved using physics‑informed learning. Traditional geometric meaning is traded for computing ability, but we maintain key geometric invariants, the primary of which are maintained, intrinsically‑low structure, nontriviality due to sufficient lower bounds on curvature, distortion of volume element, that develop quality in the inference stage. We instill representations that are canonical, smooth, curvature‑aware, geodesic‑aware, and non‑topologically void or sparse. The primary bottleneck of a Ricci curvature flow is that Ricci curvature is high order, thus expensive to compute, so we will attempt to overcome this with properly justified proxies. Our primary contributions are fourfold. We develop a loss based on Gaussian curvature using closed path circulation integration for surfaces, bypassing automatic differentiation of the Christoffel symbols through use of Stokes' theorem. We invent a new parametric flow valid under a Taylor expansion derived from the Gauss equation. We develop two strategies based on time differentiation of functionals, one with a special case of scalar curvature for conformally‑changed metrics, and another with harmonic maps, their energy, and induced metrics. Our losses are diminished analytically and mostly heuristic but maintain overall integral latent structure. We showcase that curvature flows and the formulation of geometric structure in intermediary encoded settings enhance learning and overall zero‑shot and adversarial fidelity.

Abstract:
Quantitative magnetic resonance imaging (qMRI) provides tissue‑specific parameters vital for clinical diagnosis. Although simultaneous multi‑parametric qMRI (MP‑qMRI) technologies enhance imaging efficiency, robustly reconstructing qMRI from highly undersampled, high‑dimensional measurements remains a significant challenge. This difficulty arises primarily because current reconstruction methods that rely solely on a single prior or physics‑informed model to solve the highly ill‑posed inverse problem, which often leads to suboptimal results. To overcome this limitation, we propose LoREIN, a novel unsupervised and dual‑prior‑integrated framework for accelerated 3D MP‑qMRI reconstruction. Technically, LoREIN incorporates both low‑rank prior and continuity prior via low‑rank representation (LRR) and implicit neural representation (INR), respectively, to enhance reconstruction fidelity. The powerful continuous representation of INR enables the estimation of optimal spatial bases within the low‑rank subspace, facilitating high‑fidelity reconstruction of weighted images. Simultaneously, the predicted multi‑contrast weighted images provide essential structural and quantitative guidance, further enhancing the reconstruction accuracy of quantitative parameter maps. Furthermore, our work introduces a zero‑shot learning paradigm with broad potential in complex spatiotemporal and high‑dimensional image reconstruction tasks, further advancing the field of medical imaging.

Abstract:
The equation of state (EoS) is a needed input to determine the neutron‑star global properties and to relate them. It is thus important to provide consistent and unified EoSs to avoid possible biases in the analyses coming from the use of inconsistent EoSs. We propose a numerical tool, CUTER, allowing the user to consistently match a nuclear‑physics informed crust to an arbitrary higher density EoS. We present here the second version of this tool, CUTER v2. Two functionalities are available with the CUTER v2 tool, allowing the user to reconstruct either the whole (outer and inner) crust, or the outer crust only. We show that the code, that has been tested and validated for use by the astrophysical community, is able to efficiently perform both tasks, allowing the computation of neutron‑star global properties in a consistent way.

Abstract:
The solution of partial differential equations (PDES) on irregular domains has long been a subject of significant research interest. In this work, we present an approach utilizing physics‑informed neural networks (PINNs) to achieve irregular‑to‑regular domain mapping. Thus we can use finite difference method and physics‑informed convolutional neural networks to solve PDEs on rectangular grids instead of the original irregular boundary. Structured grids on irregular domains are obtained by inverse mapping. We demonstrate PINN's versatile capability to produce customized structured grids tailored to diverse computational requirements, thereby significantly facilitating PDEs solving.

Abstract:
Differential equations are involved in modeling many engineering problems. Many efforts have been devoted to solving differential equations. Due to the flexibility of neural networks, Physics Informed Neural Networks (PINNs) have recently been proposed to solve complex differential equations and have demonstrated superior performance in many applications. While the L2 loss function is usually a default choice in PINNs, it has been shown that the corresponding numerical solution is incorrect and unstable for some complex equations. In this work, we propose a new PINNs framework named Kernel Packet accelerated PINNs (KP‑PINNs), which gives a new expression of the loss function using the reproducing kernel Hilbert space (RKHS) norm and uses the Kernel Packet (KP) method to accelerate the computation. Theoretical results show that KP‑PINNs can be stable across various differential equations. Numerical experiments illustrate that KP‑PINNs can solve differential equations effectively and efficiently. This framework provides a promising direction for improving the stability and accuracy of PINNs‑based solvers in scientific computing.

Abstract:
We propose an efficient thermodynamics‑informed latent space dynamics identification (tLaSDI) framework for the reduced‑order modeling of parametric nonlinear dynamical systems. This framework integrates autoencoders for dimensionality reduction with newly developed parametric GENERIC formalism‑informed neural networks (pGFINNs), which enable efficient learning of parametric latent dynamics while preserving key thermodynamic principles such as free energy conservation and entropy generation across the parameter space. To further enhance model performance, a physics‑informed active learning strategy is incorporated, leveraging a greedy, residual‑based error indicator to adaptively sample informative training data, outperforming uniform sampling at equivalent computational cost. Numerical experiments on the Burgers' equation and the 1D/1V Vlasov‑Poisson equation demonstrate that the proposed method achieves up to 3,528x speed‑up with 1‑3% relative errors, and significant reduction in training (50‑90%) and inference (57‑61%) cost. Moreover, the learned latent space dynamics reveal the underlying thermodynamic behavior of the system, offering valuable insights into the physical‑space dynamics.

Abstract:
Industrial processes must be robust and adaptable, as environments and tasks are often unpredictable, while operational errors remain costly and difficult to detect. AI‑based control systems offer a path forward, yet typically depend on supervised learning with extensive labelled datasets, which limits their ability to generalize across variable and data‑scarce industrial settings. Foundation models could enable broader reasoning and knowledge integration, but rarely deliver the quantitative precision demanded by engineering applications. Here, we introduceControl and Interpretation of Production via Hybrid Expertise and Reasoning (CIPHER): a vision‑language‑action (VLA) model framework aiming to replicate human‑like reasoning for industrial control, instantiated in a commercial‑grade 3D printer. It integrates a process expert, a regression model enabling quantitative characterization of system states required for engineering tasks. CIPHER also incorporates retrieval‑augmented generation to access external expert knowledge and support physics‑informed, chain‑of‑thought reasoning. This hybrid architecture exhibits strong generalization to out‑of‑distribution tasks. It interprets visual or textual inputs from process monitoring, explains its decisions, and autonomously generates precise machine instructions, without requiring explicit annotations. CIPHER thus lays the foundations for autonomous systems that act with precision, reason with context, and communicate decisions transparently, supporting safe and trusted deployment in industrial settings.

Abstract:
This paper conducts a preliminary study to investigate the feasibility of a physics‑informed extreme learning machine (PIELM) for solving the Terzaghi consolidation equation and interpreting the coefficient of consolidation of soil from piezocone penetration tests (CPTu). In the PIELM framework, the target solution is approximated by a single‑layer feed‑forward extreme learning machine (ELM) network, instead of the deep neural networks typically employed in physics‑informed neural networks (PINNs). Physical laws and measured data are integrated into a loss vector, which is minimized via least squares methods during ELM training. As a result, training efficiency is significantly improved by avoiding the gradient‑descent optimisation commonly used in PINNs. The performance of PIELM is evaluated using three forward‑problem case studies. Notably, a time‑stepping strategy is incorporated into the PIELM framework to alleviate sharp gradients caused by inconsistent initial and boundary conditions. This paper further applies PIELM to estimate the soil consolidation coefficient, given that initial distributions of excess water pressure are often unavailable in CPTu dissipation tests (conducted following the pauses of penetration). By combining physical laws (excluding initial conditions) with measured data (i.e., excess pore‑water pressure at the probe surface), the results demonstrate that PIELM is an effective tool for interpreting CPTu dissipation tests, owing to its ability to fuse data with physical constraints. This study contributes to the interpretation of consolidation coefficients from CPTu dissipation tests, particularly in scenarios where initial distributions of excess water pressure are not prior‑known.

Abstract:
Accurate characterization of temperature‑dependent thermoelectric properties (TEPs), such as thermal conductivity and the Seebeck coefficient, is essential for reliable modeling and efficient design of thermoelectric devices. However, their nonlinear temperature dependence and coupled transport behavior make both forward simulation and inverse identification difficult, particularly under sparse measurement conditions. In this study, we develop a physics‑informed machine learning approach that employs physics‑informed neural networks (PINN) for solving forward and inverse problems in thermoelectric systems, and neural operators (PINO) to enable generalization across diverse material systems. The PINN enables field reconstruction and material property inference by embedding governing transport equations into the loss function, while the PINO generalizes this inference capability across diverse materials without retraining. Trained on simulated data for 20 p‑type materials and evaluated on 60 unseen materials, the PINO model demonstrates accurate and label‑free inference of TEPs using only sparse field data. The proposed framework offers a scalable, generalizable, and data‑efficient approach for thermoelectric property identification, paving the way for high‑throughput screening and inverse design of advanced thermoelectric materials.

Abstract:
Subseasonal‑to‑seasonal (S2S) forecasting, which predicts climate conditions from several weeks to months in advance, represents a critical frontier for agricultural planning, energy management, and disaster preparedness. However, it remains one of the most challenging problems in atmospheric science, due to the chaotic dynamics of atmospheric systems and complex interactions across multiple scales. Current approaches often fail to explicitly model underlying physical processes and teleconnections that are crucial at S2S timescales. We introduce TelePiT, a novel deep learning architecture that enhances global S2S forecasting through integrated multi‑scale physics and teleconnection awareness. Our approach consists of three key components: (1) Spherical Harmonic Embedding, which accurately encodes global atmospheric variables onto spherical geometry; (2) Multi‑Scale Physics‑Informed Neural ODE, which explicitly captures atmospheric physical processes across multiple learnable frequency bands; (3) Teleconnection‑Aware Transformer, which models critical global climate interactions through explicitly modeling teleconnection patterns into the self‑attention. Extensive experiments demonstrate that TelePiT significantly outperforms state‑of‑the‑art data‑driven baselines and operational numerical weather prediction systems across all forecast horizons, marking a significant advance toward reliable S2S forecasting.

Abstract:
Accurate and efficient modeling of soft‑tissue interactions is fundamental for advancing surgical simulation, surgical robotics, and model‑based surgical automation. To achieve real‑time latency, classical Finite Element Method (FEM) solvers are often replaced with neural approximations; however, naively training such models in a fully data‑driven manner without incorporating physical priors frequently leads to poor generalization and physically implausible predictions. We present a novel physics‑informed neural simulation framework that enables real‑time prediction of soft‑tissue deformations under complex single‑ and multi‑grasper interactions. Our approach integrates Kelvinlet‑based analytical priors with large‑scale FEM data, capturing both linear and nonlinear tissue responses. This hybrid design improves predictive accuracy and physical plausibility across diverse neural architectures while maintaining the low‑latency performance required for interactive applications. We validate our method on challenging surgical manipulation tasks involving standard laparoscopic grasping tools, demonstrating substantial improvements in deformation fidelity and temporal stability over existing baselines. These results establish Kelvinlet‑augmented learning as a principled and computationally efficient paradigm for real‑time, physics‑aware soft‑tissue simulation in surgical AI.

Abstract:
This paper revisits Followerstopper, a phase‑space‑based control system that had demonstrated its ability to mitigate emergent traffic jams due to stop‑and‑go traffic during rush hour in the mixed‑autonomy setting. Followerstopper was deployed on an autonomous vehicle. The controller attenuates the emanant traffic waves by regulating its velocity according to the relative distance and velocity of the leader car. While regulating the velocity, the controller also prevents the collision of the ego vehicle with the lead vehicle within the range specified by the controller's design parameter. The controller design is based on a configurable quadratic curve on relative distance‑relative velocity phase‑space that allows the transition of the regulated velocity from (i) no modification of input, (ii) decelerating to match the leader's velocity (iii) braking to avoid any imminent collision. In this paper, we explore the phase‑space properties of Followerstopper and provide a detailed description of a nonlinear control law that regulates the reference input to Followerstopper within the physics‑informed boundaries. We also provide a new discussion on the nominal control law that regulates the reference speed to Followerstopper to avoid unrealistic and unsafe acceleration.

Abstract:
We introduce scalable machine learning models to accurately predict two key quantum transport properties, the transmission coefficient T(E) and average local density of states (Average‑LDOS) in two‑dimensional (2D) hexagonal materials with magnetic disorder. Using a tight binding Hamiltonian combined with the Non‑Equilibrium Green's Function (NEGF) formalism, we generate a large dataset of over 400,000 unique configurations across graphene, germanene, silicene, and stanene nanoribbons with varying geometries, impurity concentrations, and energy levels. A central contribution of this work is the development of a geometrydriven, physically interpretable feature space that enables the models to generalize across material types and device sizes. Random Forest regression and classification models are evaluated in terms of accuracy, stability, and extrapolation ability. Regression consistently outperforms classification in capturing continuous transport behavior on in‑domain data. However, extrapolation performance degrades significantly, revealing the limitations of tree‑based models in unseen regimes. This study highlights both the potential and constraints of scalable ML models for quantum transport prediction and motivates future research into physics‑informed or graph‑based learning architectures for improved generalization in spintronic and nanoelectronic device design.

Abstract:
Physics‑informed Kolmogorov‑Arnold Networks (PIKANs), and in particular their Chebyshev‑based variants (cPIKANs), have recently emerged as promising models for solving partial differential equations (PDEs). However, their training dynamics and convergence behavior remain largely unexplored both theoretically and numerically. In this work, we aim to advance the theoretical understanding of cPIKANs by analyzing them using Neural Tangent Kernel (NTK) theory. Our objective is to discern the evolution of kernel structure throughout gradient‑based training and its subsequent impact on learning efficiency. We first derive the NTK of standard cKANs in a supervised setting, and then extend the analysis to the physics‑informed context. We analyze the spectral properties of NTK matrices, specifically their eigenvalue distributions and spectral bias, for four representative PDEs: the steady‑state Helmholtz equation, transient diffusion and Allen‑Cahn equations, and forced vibrations governed by the Euler‑Bernoulli beam equation. We also conduct an investigation into the impact of various optimization strategies, e.g., first‑order, second‑order, and hybrid approaches, on the evolution of the NTK and the resulting learning dynamics. Results indicate a tractable behavior for NTK in the context of cPIKANs, which exposes learning dynamics that standard physics‑informed neural networks (PINNs) cannot capture. Spectral trends also reveal when domain decomposition improves training, directly linking kernel behavior to convergence rates under different setups. To the best of our knowledge, this is the first systematic NTK study of cPIKANs, providing theoretical insight that clarifies and predicts their empirical performance.

Abstract:
Accurate driving cycle construction is crucial for vehicle design, fuel economy analysis, and environmental impact assessments. A generative Physics‑Informed Expected SARSA‑Monte Carlo (PIESMC) approach that constructs representative driving cycles by capturing transient dynamics, acceleration, deceleration, idling, and road grade transitions while ensuring model fidelity is introduced. Leveraging a physics‑informed reinforcement learning framework with Monte Carlo sampling, PIESMC delivers efficient cycle construction with reduced computational cost. Experimental evaluations on two real‑world datasets demonstrate that PIESMC replicates key kinematic and energy metrics, achieving up to a 57.3% reduction in cumulative kinematic fragment errors compared to the Micro‑trip‑based (MTB) method and a 10.5% reduction relative to the Markov‑chain‑based (MCB) method. Moreover, it is nearly an order of magnitude faster than conventional techniques. Analyses of vehicle‑specific power distributions and wavelet‑transformed frequency content further confirm its ability to reproduce experimental central tendencies and variability.

Abstract:
Scientific machine learning (SciML) provides a structured approach to integrating physical knowledge into data‑driven modeling, offering significant potential for advancing hydrological research. In recent years, multiple methodological families have emerged, including physics‑informed machine learning, physics‑guided machine learning, hybrid physics‑machine learning, and data‑driven physics discovery. Within each of these families, a proliferation of heterogeneous approaches has developed independently, often without conceptual coordination. This fragmentation complicates the assessment of methodological novelty and makes it difficult to identify where meaningful advances can still be made in the absence of a unified conceptual framework. This review, the first focused overview of SciML in hydrology, addresses these limitations by proposing a unified methodological framework for each SciML family, bringing together representative contributions into a coherent structure that fosters conceptual clarity and supports cumulative progress in hydrological modeling. Finally, we highlight the limitations and future opportunities of each unified family to guide systematic research in hydrology, where these methods remain underutilized.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a powerful meshless tool for topology optimization, capable of simultaneously determining optimal topologies and physical solutions. However, conventional PINNs rely on density‑based topology descriptions, which necessitate manual interpolation and limit their applicability to complex geometries. To address this, we propose Lagrangian topology‑conscious PINNs (LT‑PINNs), a novel framework for boundary‑focused engineering optimization. By parameterizing the control variables of topology boundary curves as learnable parameters, LT‑PINNs eliminate the need for manual interpolation and enable precise boundary determination. We further introduce specialized boundary condition loss function and topology loss function to ensure sharp and accurate boundary representations, even for intricate topologies. The accuracy and robustness of LT‑PINNs are validated via two types of partial differential equations (PDEs), including elastic equation with Dirichlet boundary conditions and Laplace's equation with Neumann boundary conditions. Furthermore, we demonstrate effectiveness of LT‑PINNs on more complex time‑dependent and time‑independent flow problems without relying on measurement data, and showcase their engineering application potential in flow velocity rearrangement, transforming a uniform upstream velocity into a sine‑shaped downstream profile. The results demonstrate (1) LT‑PINNs achieve substantial reductions in relative L2 errors compared with the state‑of‑art density topology‑oriented PINNs (DT‑PINNs), (2) LT‑PINNs can handle arbitrary boundary conditions, making them suitable for a wide range of PDEs, and (3) LT‑PINNs can infer clear topology boundaries without manual interpolation, especially for complex topologies.

Abstract:
The modeling and control of single‑phase flow systems governed by Partial Differential Equations (PDEs) present challenges, especially under transient conditions. In this work, we extend the Physics‑Informed Neural Nets for Control (PINC) framework, originally proposed to modeling and control of Ordinary Differential Equations (ODE) without the need of any labeled data, to the PDE case, particularly to single‑phase incompressible and compressible flows, integrating neural networks with physical conservation laws. The PINC model for PDEs is structured into two stages: a steady‑state network, which learns equilibrium solutions for a wide range of control inputs, and a transient network, which captures dynamic responses under time‑varying boundary conditions. We propose a simplifying assumption that reduces the dimensionality of the spatial coordinate regarding the initial condition, allowing the efficient training of the PINC network. This simplification enables the derivation of optimal control policies using Model Predictive Control (MPC). We validate our approach through numerical experiments, demonstrating that the PINC model, which is trained exclusively using physical laws, i.e., without labeled data, accurately represents flow dynamics and enables real‑time control applications. The results highlight the PINC's capability to efficiently approximate PDE solutions without requiring iterative solvers, making it a promising alternative for fluid flow monitoring and optimization in engineering applications.

Abstract:
Partial differential equations (PDEs) serve as the cornerstone of mathematical physics. In recent years, Physics‑Informed Neural Networks (PINNs) have significantly reduced the dependence on large datasets by embedding physical laws directly into the training of neural networks. However, when dealing with complex problems, the accuracy of PINNs still has room for improvement. To address this issue, we introduce the Over‑PINNs framework, which leverages automatic differentiation (AD) to generate higher‑order auxiliary equations that impose additional physical constraints. These equations are incorporated as extra loss terms in the training process, effectively enhancing the model's ability to capture physical information through an "overdetermined" approach. Numerical results illustrate that this method exhibits strong versatility in solving various types of PDEs. It achieves a significant improvement in solution accuracy without incurring substantial additional computational costs.

Abstract:
Neural networks have been applied to control problems, typically by combining data, differential equation residuals, and objective costs in the training loss or by incorporating auxiliary architectural components. Instead, we propose a streamlined approach that decouples the control problem from the training process, rendering these additional layers of complexity unnecessary. In particular, our analysis and computational experiments demonstrate that a simple neural operator architecture, such as DeepONet, coupled with an unconstrained optimization routine, can solve tracking‑type partial differential equation (PDE) constrained control problems with a single physics‑informed training phase and a subsequent optimization phase. We achieve this by adding a penalty term to the cost function based on the differential equation residual to penalize deviations from the PDE constraint. This allows gradient computations with respect to the control using automatic differentiation through the trained neural operator within an iterative optimization routine, while satisfying the PDE constraints. Once trained, the same neural operator can be reused across different tracking targets without retraining. We benchmark our method on scalar elliptic (Poisson's equation), nonlinear transport (viscous Burgers' equation), and flow (Stokes equation) control problems. For the Poisson and Burgers problems, we compare against adjoint‑based solvers: for the time‑dependent Burgers problem, the approach achieves competitive accuracy with iteration times up to four times faster, while for the linear Poisson problem, the adjoint method retains superior accuracy, suggesting the approach is best suited to nonlinear and time‑dependent settings. For the flow control problem, we verify the feasibility of the optimized control through a reference forward solver.

Abstract:
Stochastic thermodynamics investigates energetic and entropic bounds in small systems. Foundational results, e.g., the first and second laws, predominantly rely on the Markov (memoryless) assumption. Although physicists recognise that the Markov assumption is questionable in real experimental setups, extending stochastic thermodynamics to general non‑Markov systems has proven challenging. Fundamentally, it has been elusive how to model memory‑dependent non‑Gaussian fluctuations consistently with thermodynamic laws. Here we establish a general theory of stochastic thermodynamics for classical non‑Markov jump processes. We introduce a key technique, called the Fourier embedding, which converts non‑Markov jump processes into Markovian field dynamics of auxiliary Fourier modes. This yields necessary and sufficient conditions for time‑reversal symmetry and enables the derivation of the second law for a broad class of strong‑memory dynamics that admit the Fourier embedding. We demonstrate the power of our framework by presenting two novel non‑Markov models: (i) a history‑dependent two‑level system and (ii) a history‑dependent random walk. Our work accommodates diverse non‑Markov dynamics in realistic experimental settings and offers a guiding principle for physics‑informed modelling of history‑dependent fluctuations.

Abstract:
From characterizing the speed of a thermal system's response to computing natural modes of vibration, eigenvalue analysis is ubiquitous in engineering. In spite of this, eigenvalue problems have received relatively little treatment compared to standard forward and inverse problems in the physics‑informed machine learning literature. In particular, neural network discretizations of solutions to eigenvalue problems have seen only a handful of studies. Owing to their nonlinearity, neural network discretizations prevent the conversion of the continuous eigenvalue differential equation into a standard discrete eigenvalue problem. In this setting, eigenvalue analysis requires more specialized techniques. Using a neural network discretization of the eigenfunction, we show that a variational form of the eigenvalue problem called the "Rayleigh quotient" in tandem with a Gram‑Schmidt orthogonalization procedure is a particularly simple and robust approach to find the eigenvalues and their corresponding eigenfunctions. This method is shown to be useful for finding sets of harmonic functions on irregular domains, parametric and nonlinear eigenproblems, and high‑dimensional eigenanalysis. We also discuss the utility of harmonic functions as a spectral basis for approximating solutions to partial differential equations. Through various examples from engineering mechanics, the combination of the Rayleigh quotient objective, Gram‑Schmidt procedure, and the neural network discretization of the eigenfunction is shown to offer unique advantages for handling continuous eigenvalue problems.

Abstract:
BridgeNet is a novel hybrid framework that integrates convolutional neural networks with physics‑informed neural networks to efficiently solve non‑linear, high‑dimensional Fokker‑Planck equations (FPEs). Traditional PINNs, which typically rely on fully connected architectures, often struggle to capture complex spatial hierarchies and enforce intricate boundary conditions. In contrast, BridgeNet leverages adaptive CNN layers for effective local feature extraction and incorporates a dynamically weighted loss function that rigorously enforces physical constraints. Extensive numerical experiments across various test cases demonstrate that BridgeNet not only achieves significantly lower error metrics and faster convergence compared to conventional PINN approaches but also maintains robust stability in high‑dimensional settings. This work represents a substantial advancement in computational physics, offering a scalable and accurate solution methodology with promising applications in fields ranging from financial mathematics to complex system dynamics.

Abstract:
Accurate epidemic forecasting is critical for informing public health decisions and timely interventions. While Physics‑Informed Neural Networks have shown promise in various scientific domains, their potential application to real‑time epidemic forecasting remains underexplored. Here, we present SIR‑INN, a hybrid forecasting framework that integrates the mechanistic structure of the classical Susceptible‑Infectious‑Recovered (SIR) model into a neural network architecture. Trained once on synthetic epidemic scenarios, the model is able to generalize across epidemic conditions without retraining. From limited and noisy observations, SIR‑INN infers key transmission parameters via Markov chain Monte Carlo, generating probabilistic short‑ and long‑term forecasts. We validate SIR‑INN using national influenza data from the Italian National Institute of Health in the 2023‑2024 and 2024‑2025 seasons. The model performs competitively with current state‑of‑the‑art approaches, particularly in terms of Weighted Interval Score. It shows accurate predictive performance in nearly all phases of the outbreak, with improved accuracy observed for the 2024‑2025 influenza season. Credible uncertainty intervals are consistently maintained, while coverage metrics highlight room for improvement in uncertainty calibration. SIR‑INN offers a computationally efficient, transparent, and generalizable solution for epidemic forecasting, appropriately leveraging the framework's hybrid design. Its ability to provide real‑time predictions of epidemic dynamics, together with uncertainty quantification, makes it a promising tool for real‑world epidemic forecasting.

Abstract:
We introduce a physics‑informed neural network (PINNs) framework for modelling the spatial distribution of chromodynamic fields induced by quark‑antiquark pairs, based on lattice Monte Carlo simulations. In contrast to conventional neural networks, PINNs incorporate physical laws‑expressed here as differential equations governing type‑II superconductivity‑directly into the training objective. By embedding these equations into the loss function, we guide the network to learn physically consistent solutions. Adopting an inverse problem approach, we extract the parameters of the superconducting equations from lattice QCD data and subsequently solve them. To accommodate physical boundary conditions, we recast the system into an integro‑differential form and extend the analysis within the fractional PINNs framework. The accuracy of the reconstructed field distribution is assessed via relative L_2‑error norms. We further extract physical observables such as the string tension and the mean width of the flux tube, offering quantitative insight into the confinement mechanism. This method enables the reconstruction of colour field profiles as functions of quark‑antiquark separation without recourse to predefined parametric models. Our results illuminate aspects of the dual Meissner effect and highlight the promise of data‑driven strategies in addressing non‑perturbative challenges in quantum chromodynamics.

Abstract:
Physical intelligence ‑‑ anticipating and shaping the world from partial, multisensory observations ‑‑ is critical for next‑generation world models. We propose FOLIAGE, a physics‑informed multimodal world model for unbounded accretive surface growth. In its Action‑Perception loop, a unified context encoder maps images, mesh connectivity, and point clouds to a shared latent state. A physics‑aware predictor, conditioned on physical control actions, advances this latent state in time to align with the target latent of the surface, yielding a Modality‑Agnostic Growth Embedding (MAGE) that interfaces with critic heads for downstream objectives. FOLIAGE's Accretive Graph Network (AGN) captures dynamic connectivity through Age Positional Encoding and Energy‑Gated Message‑Passing. Geometry‑Correspondence Fusion and Cross‑Patch Masking enhance MAGE's expressiveness, while Hierarchical Pooling balances global context with local dynamics. We create SURF‑GARDEN, a world model learning platform comprising a Counterfactual Physics Simulator, a Multimodal Correspondence Extractor, and Evolution Tracing, which generates 7,200 diverse surface‑growth sequences. SURF‑BENCH, our physical‑intelligence evaluation suite, evaluates six core tasks ‑‑ topology recognition, inverse material estimation, growth‑stage classification, latent roll‑out, cross‑modal retrieval, and dense correspondence ‑‑ and four stress tests ‑‑ sensor dropout, zero‑shot modality transfer, long‑horizon prediction, and physics ablation ‑‑ to probe resilience. FOLIAGE outperforms specialized baselines while remaining robust across dynamic environments, establishing a new world‑model based, multimodal pathway to physical intelligence.

Abstract:
Traffic congestion and collisions represent significant economic, environmental, and social challenges worldwide. Traditional traffic management approaches have shown limited success in addressing these complex, dynamic problems. To address the current research gaps, three potential tools are developed: a comprehensive 3D city‑wide simulation environment that integrates both macroscopic and microscopic traffic dynamics; a collision model; and a reinforcement learning framework with custom reward functions prioritizing safety over efficiency. Unity game engine‑based simulation is used for direct collision modeling. A custom reward enabled reinforcement learning method, proximal policy optimization (PPO) model, yields substantial improvements over baseline results, reducing the number of serious collisions, number of vehicle‑vehicle collisions, and total distance travelled by over 3 times the baseline values. The model also improves fuel efficiency by 39% and reduces carbon emissions by 88%. Results establish feasibility for city‑wide 3D traffic simulation applications incorporating the vision‑zero safety principles of the Department of Transportation, including physics‑informed, adaptable, realistic collision modeling, as well as appropriate reward modeling for real‑world traffic signal light control towards reducing collisions, optimizing traffic flow and reducing greenhouse emissions.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful tool for solving differential equations by integrating physical laws into the learning process. This work leverages PINNs to simulate gravitational collapse, a critical phenomenon in astrophysics and cosmology. We introduce the Schrödinger‑Poisson informed neural network (SPINN) which solve nonlinear Schrödinger‑Poisson (SP) equations to simulate the gravitational collapse of Fuzzy Dark Matter (FDM) in both 1D and 3D settings. Results demonstrate accurate predictions of key metrics such as mass conservation, density profiles, and structure suppression, validating against known analytical or numerical benchmarks. This work highlights the potential of PINNs for efficient, possibly scalable modeling of FDM and other astrophysical systems, overcoming the challenges faced by traditional numerical solvers due to the non‑linearity of the involved equations and the necessity to resolve multi‑scale phenomena especially resolving the fine wave features of FDM on cosmological scales.

Abstract:
We propose a novel, lightweight, and physically inspired approach to modeling the dynamics of parallel distributed‑memory programs. Inspired by the Kuramoto model, we represent MPI processes as coupled oscillators with topology‑aware interactions, custom coupling potentials, and stochastic noise. The resulting system of nonlinear ordinary differential equations opens a path to modeling key performance phenomena of parallel programs, including synchronization, delay propagation and decay, bottlenecks, and self‑desynchronization. This paper introduces interaction potentials to describe memory‑ and compute‑bound workloads and employs multiple quantitative metrics ‑‑ such as an order parameter, synchronization entropy, phase gradients, and phase differences ‑‑ to evaluate phase coherence and disruption. We also investigate the role of local noise and show that moderate noise can accelerate resynchronization in scalable applications. Our simulations align qualitatively with MPI trace data, showing the potential of physics‑informed abstractions to predict performance patterns, which offers a new perspective for performance modeling and software‑hardware co‑design in parallel computing.

Abstract:
We propose a new approach to vision‑based dexterous grasp translation, which aims to transfer grasp intent across robotic hands with differing morphologies. Given a visual observation of a source hand grasping an object, our goal is to synthesize a functionally equivalent grasp for a target hand without requiring paired demonstrations or hand‑specific simulations. We frame this problem as a stochastic transport between grasp distributions using the Schrödinger Bridge formalism. Our method learns to map between source and target latent grasp spaces via score and flow matching, conditioned on visual observations. To guide this translation, we introduce physics‑informed cost functions that encode alignment in base pose, contact maps, wrench space, and manipulability. Experiments across diverse hand‑object pairs demonstrate our approach generates stable, physically grounded grasps with strong generalization. This work enables semantic grasp transfer for heterogeneous manipulators and bridges vision‑based grasping with probabilistic generative modeling. Additional details at https://grasp2grasp.github.io/

Abstract:
We investigate the Barrow‑Tsallis Holographic Dark Energy (BTHDE) model using both traditional Markov Chain Monte Carlo (MCMC) methods and a Bayesian Physics‑Informed Neural Network (PINN) framework, employing a range of cosmological observations. Our analysis incorporates data from Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), CMB lensing, Cosmic Chronometers (CC), and the Pantheon+ Type Ia supernova compilation. We focus on constraining the Hubble constant H_0 , the nonextensive entropy index q , the Barrow exponent Δ, and the Granda‑Oliveros parameters α and β, along with the total neutrino mass Σm_ν. The Bayesian PINN approach yields more precise constraints than MCMC, particularly for β, and tighter upper bounds on Σm_ν. The inferred values of H_0 from both methods lie between those from Planck 2018 and SH_0ES (R22), alleviating the Hubble tension to within 1.3σ‑2.1σ depending on the dataset combination. Notably, the Bayesian PINN achieves consistent results across CC and Pantheon+ datasets, while maintaining physical consistency via embedded differential constraints. The combination of CMB and late‑time probes leads to the most stringent constraints, with Σm_ν< 0.114 eV and H_0 = 70.6 \pm 1.35 km/s/Mpc. These findings suggest that the BTHDE model provides a viable framework for addressing cosmological tensions and probing modified entropy scenarios, while highlighting the complementary strengths of machine learning and traditional Bayesian inference in cosmological modeling.

Abstract:
A central problem in machine learning is often formulated as follows: Given a dataset \(x_j, y_j)\_j=1^M, which is a sample drawn from an unknown probability distribution, the goal is to construct a functional model f such that f(x) \approx y for any (x, y) drawn from the same distribution. Neural networks and kernel‑based methods are commonly employed for this task due to their capacity for fast and parallel computation. The approximation capabilities, or expressive power, of these methods have been extensively studied over the past 35 years. In this paper, we will present examples of key ideas in this area found in the literature. We will discuss emerging trends in machine learning including the role of shallow/deep networks, approximation on manifolds, physics‑informed neural surrogates, neural operators, and transformer architectures. Despite function approximation being a fundamental problem in machine learning, approximation theory does not play a central role in the theoretical foundations of the field. One unfortunate consequence of this disconnect is that it is often unclear how well trained models will generalize to unseen or unlabeled data. In this review, we examine some of the shortcomings of the current machine learning framework and explore the reasons for the gap between approximation theory and machine learning practice. We will then introduce our novel research to achieve function approximation on unknown manifolds without the need to learn specific manifold features, such as the eigen‑decomposition of the Laplace‑Beltrami operator or atlas construction. In many machine learning problems, particularly classification tasks, the labels y_j are drawn from a finite set of values.

Abstract:
Sonar imaging is fundamental to underwater exploration, with critical applications in defense, navigation, and marine research. Shadow regions, in particular, provide essential cues for object detection and classification, yet existing studies primarily focus on highlight‑based analysis, leaving shadow‑based classification underexplored. To bridge this gap, we propose a Context‑adaptive sonar image classification framework that leverages advanced image processing techniques to extract and integrate discriminative shadow and highlight features. Our framework introduces a novel shadow‑specific classifier and adaptive shadow segmentation, enabling effective classification based on the dominant region. This approach ensures optimal feature representation, improving robustness against noise and occlusions. In addition, we introduce a Region‑aware denoising model that enhances sonar image quality by preserving critical structural details while suppressing noise. This model incorporates an explainability‑driven optimization strategy, ensuring that denoising is guided by feature importance, thereby improving interpretability and classification reliability. Furthermore, we present S3Simulator+, an extended dataset incorporating naval mine scenarios with physics‑informed noise specifically tailored for the underwater sonar domain, fostering the development of robust AI models. By combining novel classification strategies with an enhanced dataset, our work addresses key challenges in sonar image analysis, contributing to the advancement of autonomous underwater perception.

Abstract:
We introduce WARP (Weight‑space Adaptive Recurrent Prediction), a simple yet powerful model that unifies weight‑space learning with linear recurrence to redefine sequence modeling. Unlike conventional recurrent neural networks (RNNs) which collapse temporal dynamics into fixed‑dimensional hidden states, WARP explicitly parametrizes its hidden state as the weights and biases of a distinct auxiliary neural network, and uses input differences to drive its recurrence. This brain‑inspired formulation enables efficient gradient‑free adaptation of the auxiliary network at test‑time, in‑context learning abilities, and seamless integration of domain‑specific physical priors. Empirical validation shows that WARP matches or surpasses state‑of‑the‑art baselines on diverse classification tasks, featuring in the top three in 4 out of 6 real‑world challenging datasets. Furthermore, extensive experiments across sequential image completion, multivariate time series forecasting, and dynamical system reconstruction demonstrate its expressiveness and generalisation capabilities. Remarkably, a physics‑informed variant of our model outperforms the next best model by more than 10x. Ablation studies confirm the architectural necessity of key components, solidifying weight‑space linear RNNs as a transformative paradigm for adaptive machine intelligence.

Abstract:
We introduce a Physics‑Informed Neural Networks(PINN) to solve a relativistic Burgers equation in the exterior domain of a Schwarzschild black hole. Our main contribution is a PINN architecture that is able to simulate shock wave formations in such curved spacetime, by training a shock‑aware network block and introducing a Godunov‑inspired residuals in the loss function. We validate our method with numerical experiments with different kinds of initial conditions. We show its ability to reproduce both smooth and discontinuous solutions in the context of general relativity.

Abstract:
Data‑driven soft sensors have been widely applied in complex industrial processes. However, the interpretable spatio‑temporal features extraction by soft sensors remains a challenge. In this light, this work introduces a novel method termed spatio‑temporal consistent and interpretable model (STCIM). First, temporal and spatial features are captured and aligned by a far topological spatio‑temporal consistency extraction block. Then, the features are mapped into an interpretable latent space for further prediction by explicitly giving physical meanings to latent variables. The efficacy of the proposed STCIM is demonstrated through the modeling of two generated datasets and a real‑life dataset of coal‑fired power plants. The corresponding experiments show: 1) The generalization of STCIM outperforms other methods, especially in different operation situations. 2) The far topological spatio‑temporal consistency is vital for feature alignment. 3) The hyper‑parameters of physics‑informed interpretable latent space loss decide the performance of STCIM.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a powerful tool for solving forward and inverse problems involving partial differential equations (PDEs) by incorporating physical laws into the training process. However, the performance of PINNs is often hindered in real‑world scenarios involving noisy observational data and missing physics, particularly in inverse problems. In this work, we propose an iterative multi‑objective PINN ensemble Kalman filter (iPINNER) framework that improves the robustness and accuracy of PINNs in both forward and inverse problems by using the ensemble Kalman filter and the non‑dominated sorting genetic algorithm III (NSGA‑III). Specifically, NSGA‑III is used as a multi‑objective optimizer that can generate various ensemble members of PINNs along the optimal Pareto front, while accounting the model uncertainty in the solution space. These ensemble members are then utilized within the EnKF to assimilate noisy observational data. The EnKF's analysis is subsequently used to refine the data loss component for retraining the PINNs, thereby iteratively updating their parameters. The iterative procedure generates improved solutions to the PDEs. The proposed method is tested on two benchmark problems: the one‑dimensional viscous Burgers equation and the time‑fractional mixed diffusion‑wave equation (TFMDWE). The numerical results show it outperforms standard PINNs in handling noisy data and missing physics.

Abstract:
Alternating Current Optimal Power Flow (AC‑OPF) aims to optimize generator power outputs by utilizing the non‑linear relationships between voltage magnitudes and phase angles in a power system. However, current AC‑OPF solvers struggle to effectively represent the complex relationship between variable distributions in the constraint space and their corresponding optimal solutions. This limitation in constraint modeling restricts the system's ability to develop diverse knowledge representations. Additionally, modeling the power grid solely based on spatial topology further limits the integration of additional prior knowledge, such as temporal information. To overcome these challenges, we propose DDA‑PIGCN (Dynamic Domain Adaptation‑Driven Physics‑Informed Graph Convolutional Network), a new method designed to address constraint‑related issues and build a graph‑based learning framework that incorporates spatiotemporal features. DDA‑PIGCN improves consistency optimization for features with varying long‑range dependencies by applying multi‑layer, hard physics‑informed constraints. It also uses a dynamic domain adaptation learning mechanism that iteratively updates and refines key state variables under predefined constraints, enabling precise constraint verification. Moreover, it captures spatiotemporal dependencies between generators and loads by leveraging the physical structure of the power grid, allowing for deep integration of topological information across time and space. Extensive comparative and ablation studies show that DDA‑PIGCN delivers strong performance across several IEEE standard test cases (such as case9, case30, and case300), achieving mean absolute errors (MAE) from 0.0011 to 0.0624 and constraint satisfaction rates between 99.6% and 100%, establishing it as a reliable and efficient AC‑OPF solver.

Abstract:
Physics‑informed neural networks (PINNs) offer a powerful framework for seismic wavefield modeling, yet they typically require time‑consuming retraining when applied to different velocity models. Moreover, their training can suffer from slow convergence due to the complexity of of the wavefield solution. To address these challenges, we introduce a latent diffusion‑based strategy for rapid and effective PINN initialization. First, we train multiple PINNs to represent frequency‑domain scattered wavefields for various velocity models, then flatten each trained network's parameters into a one‑dimensional vector, creating a comprehensive parameter dataset. Next, we employ an autoencoder to learn latent representations of these parameter vectors, capturing essential patterns across diverse PINN's parameters. We then train a conditional diffusion model to store the distribution of these latent vectors, with the corresponding velocity models serving as conditions. Once trained, this diffusion model can generate latent vectors corresponding to new velocity models, which are subsequently decoded by the autoencoder into complete PINN parameters. Experimental results indicate that our method significantly accelerates training and maintains high accuracy across in‑distribution and out‑of‑distribution velocity scenarios.

Abstract:
Artificial intelligence (AI) is reshaping inverse design in manufacturing, enabling high‑performance discovery in materials, products, and processes. However, purely data‑driven approaches often struggle in realistic manufacturing settings characterized by sparse data, high‑dimensional design spaces, and complex constraints. This perspective proposes an integrated framework built on three complementary pillars: domain knowledge to establish physically meaningful objectives and constraints while removing variables with limited relevance, physics‑informed machine learning to enhance generalization under limited or biased data, and large language model‑based interfaces to support intuitive, human‑centered interaction. Using injection molding as an illustrative example, we demonstrate how these components can operate in practice and conclude by highlighting key challenges for applying such approaches in realistic manufacturing environments.

Abstract:
Physics‑informed neural networks (PINN) have been widely used in computational physics to solve partial differential equations (PDEs). In this study, we propose an energy‑embedding‑based physics‑informed neural network method for solving the one‑dimensional time‑independent Schrödinger equation to obtain ground‑ and excited‑state wave functions, as well as energy eigenvalues by incorporating an embedding layer to generate process‑driven data. The method demonstrates high accuracy for several well‑known potentials, such as the infinite potential well, harmonic oscillator potential, Woods‑Saxon potential, and double‑well potential. Further validation shows that the method also performs well in solving the radial Coulomb potential equation, showcasing its adaptability and extensibility. The proposed approach can be extended to solve other partial differential equations beyond the Schrödinger equation and holds promise for applications in high‑dimensional quantum systems.

Abstract:
Long‑term forecasting of chaotic systems remains a fundamental challenge due to the intrinsic sensitivity to initial conditions and the complex geometry of strange attractors. Conventional approaches, such as reservoir computing, typically require training data that incorporates long‑term continuous dynamical behavior to comprehensively capture system dynamics. While advanced deep sequence models can capture transient dynamics within the training data, they often struggle to maintain predictive stability and dynamical coherence over extended horizons. Here, we propose PhyxMamba, a framework that integrates a Mamba‑based state‑space model with physics‑informed principles to forecast long‑term behavior of chaotic systems given short‑term historical observations on their state evolution. We first reconstruct the attractor manifold with time‑delay embeddings to extract global dynamical features. After that, we introduce a generative training scheme that enables Mamba to replicate the physical process. It is further augmented by multi‑patch prediction and attractor geometry regularization for physical constraints, enhancing predictive accuracy and preserving key statistical properties of systems. Extensive experiments on simulated and real‑world chaotic systems demonstrate that PhyxMamba delivers superior forecasting accuracy and faithfully captures essential statistics from short‑term historical observations.

Abstract:
The local structure of a protein strongly impacts its function and interactions with other molecules. Therefore, a concise, informative representation of a local protein environment is essential for modeling and designing proteins and biomolecular interactions. However, these environments' extensive structural and chemical variability makes them challenging to model, and such representations remain under‑explored. In this work, we propose a novel representation for a local protein environment derived from the intermediate features of atomistic foundation models (AFMs). We demonstrate that this embedding effectively captures both local structure (e.g., secondary motifs), and chemical features (e.g., amino‑acid identity and protonation state). We further show that the AFM‑derived representation space exhibits meaningful structure, enabling the construction of data‑driven priors over the distribution of biomolecular environments. Finally, in the context of biomolecular NMR spectroscopy, we demonstrate that the proposed representations enable a first‑of‑its‑kind physics‑informed chemical shift predictor that achieves state‑of‑the‑art accuracy. Our results demonstrate the surprising effectiveness of atomistic foundation models and their emergent representations for protein modeling beyond traditional molecular simulations. We believe this will open new lines of work in constructing effective functional representations for protein environments.

Abstract:
Physics‑informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed time‑dependent reaction‑advection‑diffusion equations exhibit internal moving transition layers with sharp gradients, and thus the standard PINN becomes ineffective. In this work, we propose a deep asymptotic expansion (DAE) method, which is inspired by asymptotic analysis and leverages deep learning to approximate the smooth part of the expansion. We first derive the governing equations for transition layers, which are then solved using PINN. Numerical experiments show that the DAE outperforms the standard PINN, gPINN, and PINN with adaptive sampling. We also show its robustness with respect to training point distributions, network architectures, and random seeds.

Abstract:
Tasks that involve complex interactions between objects with unknown dynamics make planning before execution difficult. These tasks require agents to iteratively improve their actions after actively exploring causes and effects in the environment. For these type of tasks, we propose Causal‑PIK, a method that leverages Bayesian optimization to reason about causal interactions via a Physics‑Informed Kernel to help guide efficient search for the best next action. Experimental results on Virtual Tools and PHYRE physical reasoning benchmarks show that Causal‑PIK outperforms state‑of‑the‑art results, requiring fewer actions to reach the goal. We also compare Causal‑PIK to human studies, including results from a new user study we conducted on the PHYRE benchmark. We find that Causal‑PIK remains competitive on tasks that are very challenging, even for human problem‑solvers.

Abstract:
Physics Informed Neural Networks (PINNs) have been emerging as a powerful computational tool for solving differential equations. However, the applicability of these models is still in its initial stages and requires more standardization to gain wider popularity. Through this survey, we present a comprehensive overview of PINNs approaches exploring various aspects related to their architecture, variants, areas of application, real‑world use cases, challenges, and so on. Even though existing surveys can be identified, they fail to provide a comprehensive view as they primarily focus on either different application scenarios or limit their study to a superficial level. This survey attempts to bridge the gap in the existing literature by presenting a detailed analysis of all these factors combined with recent advancements and state‑of‑the‑art research in PINNs. Additionally, we discuss prevalent challenges in PINNs implementation and present some of the future research directions as well. The overall contributions of the survey can be summarised into three sections: A detailed overview of PINNs architecture and variants, a performance analysis of PINNs on different equations and application domains highlighting their features. Finally, we present a detailed discussion of current issues and future research directions.

Abstract:
Models of the fluid‑structure interaction (FSI) model for the air puff test were analysed. Using Abaqus, the air puff test is applied to eyes with varying biomechanical parameters, such as material properties, corneal thickness, and radius. A reduced order model of the air puff (a turbulent impinging jet) has been acquired to decrease simulation time from 48 hours for the FSI model to approximately 12 minutes for the finite element analysis (FEA) model alone. To further accelerate simulations and improve model accuracy, Physics‑Informed Neural Networks (PINNs) will be integrated with the reduced‑order model. This hybrid approach will help expand the model to a larger dataset, enhancing intraocular pressure (IOP) estimation accuracy and the corneal material properties algorithm through inverse FEA. Additionally, a neural network (NN) framework with embedded Gaussian‑modulated waveforms is proposed to model the pressure and deformation distributions on the corneal surface as functions of spatial and temporal parameters. By learning the relationship between corneal biomechanical inputs such as Corneal Central Thickness (CCT), Intraocular Pressure (IOP), and baseline properties (Mu), and the governing coefficients of pressure and deformation, the network accurately reconstructs the result that matches well with the high‑fidelity CFD data. This approach can quickly capture the distribution of pressure and deformation. It can also provide insights into the distinct spatial and temporal dynamics of pressure and deformation, giving a more comprehensive understanding of fluid‑structure interaction phenomena in the air puff test.

Abstract:
Efficient thermal and power management in modern multiprocessor systems‑on‑chip (MPSoCs) demands accurate power consumption estimation. One of the state‑of‑the‑art approaches, Alternative Blind Power Identification (ABPI), theoretically eliminates the dependence on steady‑state temperatures, addressing a major shortcoming of previous approaches. However, ABPI performance has remained unverified in actual hardware implementations. In this study, we conduct the first empirical validation of ABPI on commercial hardware using the NVIDIA Jetson Xavier AGX platform. Our findings reveal that, while ABPI provides computational efficiency and independence from steady‑state temperature, it exhibits considerable accuracy deficiencies in real‑world scenarios. To overcome these limitations, we introduce a novel approach that integrates Custom Physics‑Informed Neural Networks (CPINNs) with the underlying thermal model of ABPI. Our approach employs a specialized loss function that harmonizes physical principles with data‑driven learning, complemented by multi‑objective genetic algorithm optimization to balance estimation accuracy and computational cost. In experimental validation, CPINN‑ABPI achieves a reduction of 84.7% CPU and 73.9% GPU in the mean absolute error (MAE) relative to ABPI, with the weighted mean absolute percentage error (WMAPE) improving from 47%‑‑81% to ～12%. The method maintains real‑time performance with 195.3~μs of inference time, with similar 85%‑‑99% accuracy gains across heterogeneous SoCs.

Abstract:
Modeling physical systems in a generative manner offers several advantages, including the ability to handle partial observations, generate diverse solutions, and address both forward and inverse problems. Recently, diffusion models have gained increasing attention in the modeling of physical systems, particularly those governed by partial differential equations (PDEs). However, diffusion models only access noisy data \boldsymbolx_t at intermediate steps, making it infeasible to directly enforce constraints on the clean sample \boldsymbolx_0 at each noisy level. As a workaround, constraints are typically applied to the expectation of clean samples \mathbbE[\boldsymbolx_0|\boldsymbolx_t], which is estimated using the learned score network. However, imposing PDE constraints on the expectation does not strictly represent the one on the true clean data, known as Jensen's Gap. This gap creates a trade‑off: enforcing PDE constraints may come at the cost of reduced accuracy in generative modeling. To address this, we propose a simple yet effective post‑hoc distillation approach, where PDE constraints are not injected directly into the diffusion process, but instead enforced during a post‑hoc distillation stage. We term our method as Physics‑Informed Distillation of Diffusion Models (PIDDM). This distillation not only facilitates single‑step generation with improved PDE satisfaction, but also support both forward and inverse problem solving and reconstruction from randomly partial observation. Extensive experiments across various PDE benchmarks demonstrate that PIDDM significantly improves PDE satisfaction over several recent and competitive baselines, such as PIDM, DiffusionPDE, and ECI‑sampling, with less computation overhead. Our approach can shed light on more efficient and effective strategies for incorporating physical constraints into diffusion models.

Abstract:
Fractional physics‑informed neural networks (fPINNs) have been successfully introduced in [Pang, Lu and Karniadakis, SIAM J. Sci. Comput. 41 (2019) A2603‑A2626], which observe relative errors of 10^‑3 \, ～ \, 10^‑4 for the subdiffusion equations. However their high‑precision (multiprecision) numerical solution remains challenging, due to the limited regularity of the subdiffusion model caused by the nonlocal operator. To fill in the gap, we present the multistage fPINNs based on traditional multistage PINNs [Wang and Lai, J. Comput. Phys. 504 (2024) 112865]. Numerical experiments show that the relative errors improve to 10^‑7 \, ～ \, 10^‑8 for the subdiffusion equations on uniform or nouniform meshes.

Abstract:
Averaging techniques such as Ruppert‑‑Polyak averaging and exponential movering averaging (EMA) are powerful approaches to accelerate optimization procedures of stochastic gradient descent (SGD) optimization methods such as the popular ADAM optimizer. However, depending on the specific optimization problem under consideration, the type and the parameters for the averaging need to be adjusted to achieve the smallest optimization error. In this work we propose an averaging approach, which we refer to as parallel averaged ADAM (PADAM), in which we compute parallely different averaged variants of ADAM and during the training process dynamically select the variant with the smallest optimization error. A central feature of this approach is that this procedure requires no more gradient evaluations than the usual ADAM optimizer as each of the averaged trajectories relies on the same underlying ADAM trajectory and thus on the same underlying gradients. We test the proposed PADAM optimizer in 13 stochastic optimization and deep neural network (DNN) learning problems and compare its performance with known optimizers from the literature such as standard SGD, momentum SGD, Adam with and without EMA, and ADAMW. In particular, we apply the compared optimizers to physics‑informed neural network, deep Galerkin, deep backward stochastic differential equation and deep Kolmogorov approximations for boundary value partial differential equation problems from scientific machine learning, as well as to DNN approximations for optimal control and optimal stopping problems. In nearly all of the considered examples PADAM achieves, sometimes among others and sometimes exclusively, essentially the smallest optimization error. This work thus strongly suggest to consider PADAM for scientific machine learning problems and also motivates further research for adaptive averaging procedures within the training of DNNs.

Abstract:
Due to divergence instability, the accuracy of low‑order conforming finite element methods for nearly incompressible elasticity equations deteriorates as the Lamé coefficient λ\to\infty, or equivalently as the Poisson ratio ν\to1/2. This phenomenon, known as locking or non‑robustness, remains not fully understood despite extensive investigation. In this work, we illustrate first that an analogous instability arises when applying the popular Physics‑Informed Neural Networks (PINNs) to nearly incompressible elasticity problems, leading to significant loss of accuracy and convergence difficulties. Then, to overcome this challenge, we propose a robust decomposition‑based PINN framework that reformulates the elasticity equations into balanced subsystems, thereby eliminating the ill‑conditioning that causes locking. Our approach simultaneously solves the forward and inverse problems to recover both the decomposed field variables and the associated external conditions. We will also perform a convergence analysis to further enhance the reliability of the proposed approach. Moreover, through various numerical experiments, including constant, variable and parametric Lamé coefficients, we illustrate the efficiency of the proposed methodology.

Abstract:
We propose a physics‑informed neural networks (PINNs) framework to solve the infinite‑horizon optimal control problem of nonlinear systems. In particular, since PINNs are generally able to solve a class of partial differential equations (PDEs), they can be employed to learn the value function of the infinite‑horizon optimal control problem via solving the associated steady‑state Hamilton‑Jacobi‑Bellman (HJB) equation. However, an issue here is that the steady‑state HJB equation generally yields multiple solutions; hence if PINNs are directly employed to it, they may end up approximating a solution that is different from the optimal value function of the problem. We tackle this by instead applying PINNs to a finite‑horizon variant of the steady‑state HJB that has a unique solution, and which uniformly approximates the optimal value function as the horizon increases. An algorithm to verify if the chosen horizon is large enough is also given, as well as a method to extend it ‑‑ with reduced computations and robustness to approximation errors ‑‑ in case it is not. Unlike many existing methods, the proposed technique works well with non‑polynomial basis functions, does not require prior knowledge of a stabilizing controller, and does not perform iterative policy evaluations. Simulations are performed, which verify and clarify theoretical findings.

Abstract:
In the era of AI, neural networks have become increasingly popular for modeling, inference, and prediction, largely due to their potential for universal approximation. With the proliferation of such deep learning models, a question arises: are leaner statistical methods still relevant? To shed insight on this question, we employ the mechanistic nonlinear ordinary differential equation (ODE) inverse problem as a testbed, using the physics‑informed neural network (PINN) as a representative of the deep learning paradigm and manifold‑constrained Gaussian process inference (MAGI) as a representative of statistically principled methods. Through case studies involving the SEIR model from epidemiology and the Lorenz model from chaotic dynamics, we demonstrate that statistical methods are far from obsolete, especially when working with sparse and noisy observations. On tasks such as parameter inference and trajectory reconstruction, statistically principled methods consistently achieve lower bias and variance, while using far fewer parameters and requiring less hyperparameter tuning. Statistical methods can also decisively outperform deep learning models on out‑of‑sample future prediction, where the absence of relevant data often leads overparameterized models astray. Additionally, we find that statistically principled approaches are more robust to accumulation of numerical imprecision and can represent the underlying system more faithfully to the true governing ODEs.

Abstract:
In compressible fluid flow, reconstructing shocks, discontinuities, rarefactions, and their interactions from sparse measurements is an important inverse problem with practical applications. Moreover, physics‑informed machine learning has recently become an increasingly popular approach for performing reconstructions tasks. In this work we explore a machine learning recipe, known as GraphGPS, for reconstructing canonical compressible flows known as 3D Riemann problems from sparse observations, in a physics‑informed manner. The GraphGPS framework combines the benefits of positional encodings, local message‑passing of graphs, and global contextual awareness, and we explore the latter two components through an ablation study. Furthermore, we modify the aggregation step of message‑passing such that it is aware of shocks and discontinuities, resulting in sharper reconstructions of these features. Additionally, we modify message‑passing such that information flows strictly from known nodes only, which results in computational savings, better training convergence, and no degradation of reconstruction accuracy. We also show that the GraphGPS framework outperforms numerous machine learning benchmarks.

Abstract:
Natural‑gradient methods markedly accelerate the training of Physics‑Informed Neural Networks (PINNs), yet their Gauss‑‑Newton update must be solved in the parameter space, incurring a prohibitive O(n^3) time complexity, where n is the number of network trainable weights. We show that exactly the same step can instead be formulated in a generally smaller residual space of size m = \sum_γ N_γ d_γ, where each residual class γ (e.g. PDE interior, boundary, initial data) contributes N_γ collocation points of output dimension d_γ. Building on this insight, we introduce Dual Natural Gradient Descent (D‑NGD). D‑NGD computes the Gauss‑‑Newton step in residual space, augments it with a geodesic‑acceleration correction at negligible extra cost, and provides both a dense direct solver for modest m and a Nystrom‑preconditioned conjugate‑gradient solver for larger m. Experimentally, D‑NGD scales second‑order PINN optimization to networks with up to 12.8 million parameters, delivers one‑ to three‑order‑of‑magnitude lower final error L^2 than first‑order methods (Adam, SGD) and quasi‑Newton methods, and ‑‑ crucially ‑‑ enables natural‑gradient training of PINNs at this scale on a single GPU.

Abstract:
A search for the violation of the charge‑parity (CP) symmetry in the production of top quarks in association with Z bosons is presented, using events with at least three charged leptons and additional jets. The search is performed in a sample of proton‑proton collision data collected by the CMS experiment at the CERN LHC in 2016‑2018 at a center‑of‑mass energy of 13 TeV and in 2022 at 13.6 TeV, corresponding to a total integrated luminosity of 173 fb^‑1. For the first time in this final state, observables that are odd under the CP transformation are employed. Also for the first time, physics‑informed machine‑learning techniques are used to construct these observables. While for standard model (SM) processes the distributions of these observables are predicted to be symmetric around zero, CP‑violating modifications of the SM would introduce asymmetries. Two CP‑odd operators \mathcalO_\texttW^\textI and \mathcalO_\texttZ^\textI in the SM effective field theory are considered that may modify the interactions between top quarks and electroweak bosons. The obtained results are consistent with the SM prediction within two standard deviations, and exclusion limits on the associated Wilson coefficients of ‑2.7 \lt c_\texttW^\textI \lt 2.5 and ‑0.2 \lt c_\texttZ^\textI \lt 2.0 are set at 95% confidence level. The largest discrepancy is observed in c_\texttZ^\textI where data is consistent with positive values, with an observed local significance with respect to the SM hypothesis of 2.5 standard deviations, when only linear terms are considered.

Abstract:
Thermally induced laser noise poses a critical limitation to the sensitivity of quantum sensor arrays employing ultra‑stable amplified lasers, primarily stemming from nonlinear gain‑temperature coupling effects in tapered amplifiers (TAs). To address this challenge, we present a robust intelligent control strategy that synergistically integrates an encoder‑decoder physics‑informed gated recurrent unit (PI‑GRU) network with a model predictive control (MPC) framework. Our methodology incorporates physical soft constraints into the neural network architecture, yielding a predictive model with enhanced physical consistency that demonstrates robust extrapolation capabilities beyond the training data distribution. Leveraging the PI‑GRU model's accurate multi‑step predictive performance, we implement a hierarchical parallel MPC architecture capable of real‑time thermal instability compensation. This hybrid approach achieves cross‑domain consistent thermal stabilization in TAs under diverse laser power operations. Remarkably, while trained exclusively on low‑power operational data, our system demonstrates exceptional generalization, improving prediction accuracy by 58.2% and temperature stability by 69.1% in previously unseen high‑power operating regimes, as experimentally validated. The novel synchronization of physics‑informed neural networks with advanced MPC frameworks presented in this work establishes a groundbreaking paradigm for addressing robustness challenges in cross‑domain predictive control applications, overcoming conventional modeling limitations.

Abstract:
Despite the remarkable progress of physics‑informed neural networks (PINNs) in scientific computing, they continue to face challenges when solving hydrodynamic problems with multiple discontinuities. In this work, we propose Separation‑Transfer Physics Informed Neural Networks (ST‑PINNs) to address such problems. By sequentially resolving discontinuities from strong to weak and leveraging transfer learning during training, ST‑PINNs significantly reduce the problem complexity and enhance solution accuracy. To the best of our knowledge, this is the first study to apply a PINNs‑based approach to the two‑dimensional unsteady planar shock refraction problem, offering new insights into the application of PINNs to complex shock‑interface interactions. Numerical experiments demonstrate that ST‑PINNs more accurately capture sharp discontinuities and substantially reduce solution errors in hydrodynamic problems involving multiple discontinuities.

Abstract:
Data‑driven discovery of dynamics in biological systems allows for better observation and characterization of processes, such as calcium signaling in cell culture. Recent advancements in techniques allow the exploration of previously unattainable insights of dynamical systems, such as the Sparse Identification of Non‑Linear Dynamics (SINDy), overcoming the limitations of more classic methodologies. The latter requires some prior knowledge of an effective library of candidate terms, which is not realistic for a real case study. Using inspiration from fields like traffic density estimation and control theory, we propose a methodology for characterization and performance analysis of calcium delivery in a family of cells. In this work, we compare the performance of the Constrained Regularized Least‑Squares Method (CRLSM) and Physics‑Informed Neural Networks (PINN) for system identification and parameter discovery for governing ordinary differential equations (ODEs). The CRLSM achieves a fairly good parameter estimate and a good data fit when using the learned parameters in the Consensus problem. On the other hand, despite the initial hypothesis, PINNs fail to match the CRLSM performance and, under the current configuration, do not provide fair parameter estimation. However, we have only studied a limited number of PINN architectures, and it is expected that additional hyperparameter tuning, as well as uncertainty quantification, could significantly improve the performance in future works.

Abstract:
We propose FMEnets, a physics‑informed machine learning framework for the design and analysis of non‑ideal plug flow reactors. FMEnets integrates the fundamental governing equations (Navier‑Stokes for fluid flow, material balance for reactive species transport, and energy balance for temperature distribution) into a unified multi‑scale network model. The framework is composed of three interconnected sub‑networks with independent optimizers that enable both forward and inverse problem‑solving. In the forward mode, FMEnets predicts velocity, pressure, species concentrations, and temperature profiles using only inlet and outlet information. In the inverse mode, FMEnets utilizes sparse multi‑residence‑time measurements to simultaneously infer unknown kinetic parameters and states. FMEnets can be implemented either as FME‑PINNs, which employ conventional multilayer perceptrons, or as FME‑KANs, based on Kolmogorov‑Arnold Networks. Comprehensive ablation studies highlight the critical role of the FMEnets architecture in achieving accurate predictions. Specifically, FME‑KANs are more robust to noise than FME‑PINNs, although both representations are comparable in accuracy and speed in noise‑free conditions. The proposed framework is applied to three different sets of reaction scenarios and is compared with finite element simulations. FMEnets effectively captures the complex interactions, achieving relative errors less than 2.5% for the unknown kinetic parameters. The new network framework not only provides a computationally efficient alternative for reactor design and optimization, but also opens new avenues for integrating empirical correlations, limited and noisy experimental data, and fundamental physical equations to guide reactor design.

Abstract:
Jet flavour tagging enables the identification of jets originating from heavy‑flavour quarks in proton‑proton collisions at the Large Hadron Collider, playing a critical role in its physics programmes. This paper presents GN2, a transformer‑based flavour tagging algorithm deployed by the ATLAS Collaboration that represents a different methodology compared to previous approaches. Designed to classify jets based on the flavour of their constituent particles, GN2 processes low‑level tracking information in an end‑to‑end architecture and incorporates physics‑informed auxiliary training objectives to enhance both interpretability and performance. Its performance is validated in both simulation and collision data. The measured c‑jet (light‑jet) rejection in data is improved by a factor of 3.5 (1.8) for a 70% b‑jet tagging efficiency, compared to the previous algorithm. GN2 provides substantial benefits for physics analyses involving heavy‑flavour jets, such as measurements of Higgs boson pair production and the couplings of bottom and charm quarks to the Higgs boson, and demonstrates the impact of advanced machine learning methods in experimental particle physics.

Abstract:
We present a novel formulation for modeling phase‑field fracture propagation based on the Integrated Finite Element Neural Network (IFENN) framework. IFENN is a hybrid solver scheme that utilizes neural networks as PDE solvers within FEM, preserving accuracy via residual minimization while achieving speed‑up via swift network predictions and reduction of the size of system of equations in coupled problems. In this work, we introduce a radically new formulation of IFENN in which the phase‑field variable is calculated using physics‑informed convolutional networks (PICNNs), while the equilibrium equation is still solved using FEM to maintain the solver robustness. Unlike conventional approaches, which rely on sequence or time‑dependent models, we eliminate the need to include temporal features in the training setup and inference stage. Instead, we show that it is sufficient to learn only the spatial coupling between the strain energy density and the phase‑field variable in the vicinity of the fracture process zone, and utilize this information along the advancing crack simulation. We train a single CNN in a purely physics‑based, unsupervised manner on just two load increments from a single‑notch tension problem, with a total training time of only 5 minutes. Following this exceptionally minimal and fast training, we show that the same PICNN can (when embedded within IFENN) model crack propagation in a very wide range of unseen scenarios, including arbitrarily rectangular domains, single and multiple interacting cracks, varying mesh densities, and arbitrary loading paths. The proposed formulation delivers breakthroughs that address many of the limitations in the existing literature of hybrid modeling, introducing a new paradigm for the development of generalizable, physics‑consistent hybrid models that are applicable to fracture and other coupled problems.

Abstract:
Recent advances in generative AI offer promising solutions for synthetic data generation but often rely on large datasets for effective training. To address this limitation, we propose a novel generative model that learns from limited data by incorporating physical constraints to enhance performance. Specifically, we extend the VAE architecture by incorporating physical models in the generative process, enabling it to capture underlying dynamics more effectively. While physical models provide valuable insights, they struggle to capture complex temporal dependencies present in real‑world data. To bridge this gap, we introduce a discrepancy term to account for unmodeled dynamics, represented within a latent Gaussian Process VAE (GPVAE). Furthermore, we apply regularization to ensure the generated data aligns closely with observed data, enhancing both the diversity and accuracy of the synthetic samples. The proposed method is applied to indoor temperature data, achieving state‑of‑the‑art performance. Additionally, we demonstrate that PIGPVAE can produce realistic samples beyond the observed distribution, highlighting its robustness and usefulness under distribution shifts.

Abstract:
Uncertainty quantification (UQ) in scientific machine learning is increasingly critical as neural networks are widely adopted to tackle complex problems across diverse scientific disciplines. For physics‑informed neural networks (PINNs), a prominent model in scientific machine learning, uncertainty is typically quantified using Bayesian or dropout methods. However, both approaches suffer from a fundamental limitation: the prior distribution or dropout rate required to construct honest confidence sets cannot be determined without additional information. In this paper, we propose a novel method within the framework of extended fiducial inference (EFI) to provide rigorous uncertainty quantification for PINNs. The proposed method leverages a narrow‑neck hyper‑network to learn the parameters of the PINN and quantify their uncertainty based on imputed random errors in the observations. This approach overcomes the limitations of Bayesian and dropout methods, enabling the construction of honest confidence sets based solely on observed data. This advancement represents a significant breakthrough for PINNs, greatly enhancing their reliability, interpretability, and applicability to real‑world scientific and engineering challenges. Moreover, it establishes a new theoretical framework for EFI, extending its application to large‑scale models, eliminating the need for sparse hyper‑networks, and significantly improving the automaticity and robustness of statistical inference.

Abstract:
Physics‑informed neural networks (PINNs), owing to their mesh‑free nature, offer a powerful approach for solving high‑dimensional partial differential equations (PDEs) in complex geometries, including irregular domains. This capability effectively circumvents the challenges of mesh generation that traditional numerical methods face in high‑dimensional or geometrically intricate settings. While recent studies have extended PINNs to manifolds, the theoretical foundations remain scarce. Existing theoretical analyses of PINNs in Euclidean space often rely on smoothness assumptions for the solutions. However, recent empirical evidence indicates that PINNs may struggle to approximate solutions with low regularity, such as those arising from nonlinear hyperbolic equations. In this paper, we develop a framework for PINNs tailored to the efficient approximation of weak solutions, particularly nonlinear hyperbolic equations defined on manifolds. We introduce a novel weak PINN (wPINN) formulation on manifolds that leverages the well‑posedness theory to approximate entropy solutions of geometry‑compatible hyperbolic conservation laws on manifolds. Employing tools from approximation theory, we establish a convergence analysis of the algorithm, including an analysis of approximation errors for time‑dependent entropy solutions. This analysis provides insight into the accumulation of approximation errors over long time horizons. Notably, the network complexity depends only on the intrinsic dimension, independent of the ambient space dimension. Our results match the minimax rate in the d‑dimensional Euclidean space, demonstrating that PINNs can alleviate the curse of dimensionality in the context of low‑dimensional manifolds. Finally, we validate the performance of the proposed wPINN framework through numerical experiments, confirming its ability to efficiently approximate entropy solutions on manifolds.

Abstract:
In this study, agglomerate breakage in homogeneous isotropic turbulence is investigated using particle‑resolved direct numerical simulations. Single agglomerates composed of 500 monodisperse spherical particles are considered, and their interaction with the turbulent flow is resolved through an immersed boundary method coupled with a soft‑sphere discrete element model. A range of Reynolds numbers and cohesion levels is examined to assess their influence on the breakup behavior. Detailed insights into the underlying breakage mechanisms are provided through the analysis of local flow structures and fluid stresses. Strain‑dominated regions are identified as the primary contributors to the onset and propagation of particle erosion. The benefits of the particle‑resolved simulation framework in capturing these physical processes in detail are demonstrated. The predicted fragment size distributions and breakup modes are analyzed leading to the outcome that erosion‑driven breakage is the dominating mechanism. The time evolution of the fragment number and the main agglomerate structure is quantified. The breakage rate is evaluated and its dependence on the modified adhesion number is established, showing a power‑law decay that agrees with general trends reported in the literature. In addition, the analysis of the fragment ejection direction reveals a strong alignment with the local deformation plane spanned by the most extensional and compressive strain‑rate eigenvectors, indicating that breakage results from the interplay between flow stretching and compression. The results contribute to the development of physics‑informed breakup kernels for use in efficient but less‑detailed simulation approaches such as point‑particle Euler‑‑Lagrange predictions with agglomerates represented by effective spheres or Euler‑‑Euler simulations.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a promising approach for solving complex fluid dynamics problems, yet their application to fluid‑structure interaction (FSI) problems with moving boundaries remains largely unexplored. This work addresses the critical challenge of modeling FSI systems with moving interfaces, where traditional unified PINN architectures struggle to capture the distinct physics governing fluid and structural domains simultaneously. We present an innovative Eulerian‑Lagrangian PINN architecture that integrates immersed boundary method (IBM) principles to solve FSI problems with moving boundary conditions. Our approach fundamentally departs from conventional unified architectures by introducing domain‑specific neural networks: an Eulerian network for fluid dynamics and a Lagrangian network for structural interfaces, coupled through physics‑based constraints. Additionally, we incorporate learnable B‑spline activation functions with SiLU to capture both localized high‑gradient features near interfaces and global flow patterns. Empirical studies on a 2D cavity flow problem involving a moving solid structure show that while baseline unified PINNs achieve reasonable velocity predictions, they suffer from substantial pressure errors (12.9%) in structural regions. Our Eulerian‑Lagrangian architecture with learnable activations (EL‑L) achieves better performance across all metrics, improving accuracy by 24.1‑91.4% and particularly reducing pressure errors from 12.9% to 2.39%. These results demonstrate that domain decomposition aligned with physical principles, combined with locality‑aware activation functions, is essential for accurate FSI modeling within the PINN framework.

Abstract:
Magnetic resonance (MR) tagging is an imaging technique for noninvasively tracking tissue motion in vivo by creating a visible pattern of magnetization saturation (tags) that deforms with the tissue. Due to longitudinal relaxation and progression to steady‑state, the tags and tissue brightnesses change over time, which makes tracking with optical flow methods error‑prone. Although Fourier methods can alleviate these problems, they are also sensitive to brightness changes as well as spectral spreading due to motion. To address these problems, we introduce the brightness‑invariant tracking estimation (BRITE) technique for tagged MRI. BRITE disentangles the anatomy from the tag pattern in the observed tagged image sequence and simultaneously estimates the Lagrangian motion. The inherent ill‑posedness of this problem is addressed by leveraging the expressive power of denoising diffusion probabilistic models to represent the probabilistic distribution of the underlying anatomy and the flexibility of physics‑informed neural networks to estimate biologically‑plausible motion. A set of tagged MR images of a gel phantom was acquired with various tag periods and imaging flip angles to demonstrate the impact of brightness variations and to validate our method. The results show that BRITE achieves more accurate motion and strain estimates as compared to other state of the art methods, while also being resistant to tag fading.

Abstract:
Understanding and predicting human emotional and physiological states using wearable sensors has important applications in stress monitoring, mental health assessment, and affective computing. This study presents a novel Multi‑Task Physics‑Informed Neural Network (PINN) that performs Electrodermal Activity (EDA) prediction and emotion classification simultaneously, using the publicly available WESAD dataset. The model integrates psychological self‑report features (PANAS and SAM) with a physics‑inspired differential equation representing EDA dynamics, enforcing biophysically grounded constraints through a custom loss function. This loss combines EDA regression, emotion classification, and a physics residual term for improved interpretability. The architecture supports dual outputs for both tasks and is trained under a unified multi‑task framework. Evaluated using 5‑fold cross‑validation, the model achieves an average EDA RMSE of 0.0362, Pearson correlation of 0.9919, and F1‑score of 94.08 percent. These results outperform classical models such as SVR and XGBoost, as well as ablated variants like emotion‑only and EDA‑only models. In addition, the learned physical parameters including decay rate (alpha_0), emotional sensitivity (beta), and time scaling (gamma) are interpretable and stable across folds, aligning with known principles of human physiology. This work is the first to introduce a multi‑task PINN framework for wearable emotion recognition, offering improved performance, generalizability, and model transparency. The proposed system provides a foundation for future interpretable and multimodal applications in healthcare and human‑computer interaction.

Abstract:
Dimensionality reduction represents a crucial step in extracting meaningful insights from Molecular Dynamics (MD) simulations. Conventional approaches, including linear methods such as principal component analysis as well as various autoencoder architectures, typically operate under the assumption of independent and identically distributed data, disregarding the sequential nature of MD simulations. Here, we introduce a physics‑informed representation learning framework that leverages Gaussian Processes combined with variational autoencoders to exploit the temporal dependencies inherent in MD data. Time‑dependent kernel functions‑‑such as the Matérn kernel‑‑directly impose the temporal correlation structure of the input coordinates onto a low‑dimensional space, preserving Markovianity in the reduced representation while faithfully capturing the essential dynamics. Using a three‑dimensional toy model, we demonstrate that this approach can successfully identify and separate dynamically distinct states that are geometrically indistinguishable due to hidden degrees of freedom. Applying the framework to a 50\,μs‑long MD trajectory of T4 lysozyme, we uncover dynamically distinct conformational substates that previous analyses failed to resolve, revealing functional relationships that become apparent only when temporal correlations are taken into account. This time‑aware perspective provides a promising framework for understanding complex biomolecular systems, in which conventional collective variables fail to capture the full dynamical picture.

Abstract:
Despite the widely recognized success of residual connections in modern neural networks, their design principles remain largely heuristic. This paper introduces KITINet (Kinetics Theory Inspired Network), a novel architecture that reinterprets feature propagation through the lens of non‑equilibrium particle dynamics and partial differential equation (PDE) simulation. At its core, we propose a residual module that models feature updates as the stochastic evolution of a particle system, numerically simulated via a discretized solver for the Boltzmann transport equation (BTE). This formulation mimics particle collisions and energy exchange, enabling adaptive feature refinement via physics‑informed interactions. Additionally, we reveal that this mechanism induces network parameter condensation during training, where parameters progressively concentrate into a sparse subset of dominant channels. Experiments on scientific computation (PDE operator), image classification (CIFAR‑10/100), and text classification (IMDb/SNLI) show consistent improvements over classic network baselines, with negligible increase of FLOPs.

Abstract:
Goal‑conditioned dynamic manipulation is inherently challenging due to complex system dynamics and stringent task constraints, particularly in deformable object scenarios characterized by high degrees of freedom and underactuation. Prior methods often simplify the problem to low‑speed or 2D settings, limiting their applicability to real‑world 3D tasks. In this work, we explore 3D goal‑conditioned rope manipulation as a representative challenge. To mitigate data scarcity, we introduce a novel simulation framework and benchmark grounded in reduced‑order dynamics, which enables compact state representation and facilitates efficient policy learning. Building on this, we propose Dynamics Informed Diffusion Policy (DIDP), a framework that integrates imitation pretraining with physics‑informed test‑time adaptation. First, we design a diffusion policy that learns inverse dynamics within the reduced‑order space, enabling imitation learning to move beyond naïve data fitting and capture the underlying physical structure. Second, we propose a physics‑informed test‑time adaptation scheme that imposes kinematic boundary conditions and structured dynamics priors on the diffusion process, ensuring consistency and reliability in manipulation execution. Extensive experiments validate the proposed approach, demonstrating strong performance in terms of accuracy and robustness in the learned policy.

Abstract:
Physics‑informed neural networks (PINNs) have proven an effective tool for solving differential equations, in particular when considering non‑standard or ill‑posed settings. When inferring solutions and parameters of the differential equation from data, uncertainty estimates are preferable to point estimates, as they give an idea about the accuracy of the solution. In this work, we consider the inverse problem and employ repulsive ensembles of PINNs (RE‑PINN) for obtaining such estimates. The repulsion is implemented by adding a particular repulsive term to the loss function, which has the property that the ensemble predictions correspond to the true Bayesian posterior in the limit of infinite ensemble members. Where possible, we compare the ensemble predictions to Monte Carlo baselines. Whereas the standard ensemble tends to collapse to maximum‑a‑posteriori solutions, the repulsive ensemble produces significantly more accurate uncertainty estimates and exhibits higher sample diversity.

Abstract:
Inverse problems involving differential equations often require identifying unknown parameters or functions from data. Existing approaches, such as Physics‑Informed Neural Networks (PINNs), Universal Differential Equations (UDEs) and Universal Physics‑Informed Neural Networks (UPINNs), are effective at isolating either parameters or functions but can face challenges when applied simultaneously due to solution non‑uniqueness. In this work, we introduce a framework that addresses these limitations by establishing conditions under which unique solutions can be guaranteed. To illustrate, we apply it to examples from biological systems and ecological dynamics, demonstrating accurate and interpretable results. Our approach significantly enhances the potential of machine learning techniques in modeling complex systems in science and engineering.

Abstract:
This work proposes a higher‑order iterative framework for solving matrix equations, inspired by the structure and functionality of neural networks. A modification of the classical Jacobi iterative method is introduced to compute higher‑order coefficient matrices through matrix‑matrix multiplications. The resulting method, termed the higher order Jacobi method (HOJM), structurally resembles a shallow linear network and allows direct computation of the inverse of the coefficient matrix. Building on this, an iterative scheme is developed that allows efficient resolution of system variations without recomputing the coefficients, once the network parameters are trained for a known system. This iterative process naturally assumes the form of a deep recurrent neural network. The proposed approach goes beyond conventional physics‑informed neural networks (PINNs) by providing an explicit, training‑free definition of network parameters rooted in physical and mathematical formulations. Computational analysis on GPU reveals significant enhancement in the order of complexity, highlighting a compelling and transformative direction for advancing algorithmic efficiency in solving linear systems. This methodology opens avenues for interpretable and scalable solutions to physically motivated problems in computational science.

Abstract:
Reconstructing complex 3D interfaces from indirect measurements remains a grand challenge in scientific computing, particularly for ill‑posed inverse problems like Electrical Impedance Tomography (EIT). Traditional shape optimization struggles with topological changes and regularization tuning, while emerging deep learning approaches often compromise physical fidelity or require prohibitive amounts of paired training data. We present a transformative ``solver‑in‑the‑loop'' framework that bridges this divide by coupling a pre‑trained 3D generative prior with a rigorous boundary integral equation (BIE) solver. Unlike Physics‑Informed Neural Networks (PINNs) that treat physics as soft constraints, our architecture enforces the governing elliptic PDE as a hard constraint at every optimization step, ensuring strict physical consistency. Simultaneously, we navigate a compact latent manifold of plausible geometries learned by a differentiable neural shape representation, effectively regularizing the ill‑posed problem through data‑driven priors rather than heuristic smoothing. By propagating adjoint shape derivatives directly through the neural decoder, we achieve fast, stable convergence with dramatically reduced degrees of freedom. Extensive experiments on 3D high‑contrast EIT demonstrate that this principled hybrid approach yields superior geometric accuracy and data efficiency which is difficult to achieve using traditional methods, establishing a robust new paradigm for physics‑constrained geometric discovery.

Abstract:
Panchromatic (PAN) assisted Dual‑Camera Compressive Hyperspectral Imaging (DCCHI) is a key technology in snapshot hyperspectral imaging. Existing research primarily focuses on exploring spectral information from 2D compressive measurements and spatial information from PAN images in an explicit manner, leading to a bottleneck in HSI reconstruction. Various physical factors, such as temperature, emissivity, and multiple reflections between objects, play a critical role in the process of a sensor acquiring hyperspectral thermal signals. Inspired by this, we attempt to investigate the interrelationships between physical properties to provide deeper theoretical insights for HSI reconstruction. In this paper, we propose a Physics‑Informed Cross‑Modal State Space Model Network (PCMamba) for DCCHI, which incorporates the forward physical imaging process of HSI into the linear complexity of Mamba to facilitate lightweight and high‑quality HSI reconstruction. Specifically, we analyze the imaging process of hyperspectral thermal signals to enable the network to disentangle the three key physical properties‑temperature, emissivity, and texture. By fully exploiting the potential information embedded in 2D measurements and PAN images, the HSIs are reconstructed through a physics‑driven synthesis process. Furthermore, we design a Cross‑Modal Scanning Mamba Block (CSMB) that introduces inter‑modal pixel‑wise interaction with positional inductive bias by cross‑scanning the backbone features and PAN features. Extensive experiments conducted on both real and simulated datasets demonstrate that our method significantly outperforms SOTA methods in both quantitative and qualitative metrics.

Abstract:
We introduce Equivariant Neural Eikonal Solvers, a novel framework that integrates Equivariant Neural Fields (ENFs) with Neural Eikonal Solvers. Our approach employs a single neural field where a unified shared backbone is conditioned on signal‑specific latent variables ‑ represented as point clouds in a Lie group ‑ to model diverse Eikonal solutions. The ENF integration ensures equivariant mapping from these latent representations to the solution field, delivering three key benefits: enhanced representation efficiency through weight‑sharing, robust geometric grounding, and solution steerability. This steerability allows transformations applied to the latent point cloud to induce predictable, geometrically meaningful modifications in the resulting Eikonal solution. By coupling these steerable representations with Physics‑Informed Neural Networks (PINNs), our framework accurately models Eikonal travel‑time solutions while generalizing to arbitrary Riemannian manifolds with regular group actions. This includes homogeneous spaces such as Euclidean, position‑orientation, spherical, and hyperbolic manifolds. We validate our approach through applications in seismic travel‑time modeling of 2D, 3D, and spherical benchmark datasets. Experimental results demonstrate superior performance, scalability, adaptability, and user controllability compared to existing Neural Operator‑based Eikonal solver methods.

Abstract:
Extremum Seeking (ES) is an effective real‑time optimization method for PDE systems in cascade with nonlinear quadratic maps. To address PDEs in the feedback loop, a boundary control law and a re‑design of the additive probing signal are mandatory. The latter, commonly called "trajectory generation" or "motion planning," involves designing perturbation signals that anticipate their propagation through PDEs. Specifically, this requires solving motion planning problems for systems governed by parabolic and hyperbolic PDEs. Physics‑Informed Neural Networks (PINN) is a powerful tool for solving PDEs by embedding physical laws as constraints in the neural network's loss function, enabling efficient solutions for high‑dimensional, nonlinear, and complex problems. This paper proposes a novel construction integrating PINN and ES, automating the motion planning process for specific PDE systems and eliminating the need for case‑by‑case analytical derivations. The proposed strategy efficiently extracts perturbation signals, optimizing the PDE system.

Abstract:
We present the Fourier‑Invertible Neural Encoder (FINE), a compact and interpretable architecture for dimension reduction in translation‑equivariant datasets. FINE integrates reversible filters and monotonic activation functions with a Fourier truncation bottleneck, achieving information‑preserving compression that respects translational symmetry. This design offers a new perspective on symmetry‑aware learning, linking spectral truncation to group‑equivariant representations. The proposed FINE architecture is tested on one‑dimensional nonlinear wave interaction, one‑dimensional Kuramoto‑Sivashinsky turbulence dataset, and a two‑dimensional turbulence dataset. FINE achieves an overall 4.9‑9.1 times lower reconstruction error than convolutional autoencoders while using only 13‑21% of their parameters. The results highlight FINE's effectiveness in representing complex physical systems with minimal dimension in the latent space. The proposed framework provides a principled framework for interpretable, low‑parameter, and symmetry‑preserving dimensional reduction, bridging the gap between Fourier representations and modern neural architectures for scientific and physics‑informed learning.

Abstract:
Reduced‑order modeling (ROM) of time‑dependent and parameterized differential equations aims to accelerate the simulation of complex high‑dimensional systems by learning a compact latent manifold representation that captures the characteristics of the solution fields and their time‑dependent dynamics. Although high‑fidelity numerical solvers generate the training datasets, they have thus far been excluded from the training process, causing the learned latent dynamics to drift away from the discretized governing physics. This mismatch often limits generalization and forecasting capabilities. In this work, we propose Physics‑informed ROM (Φ‑ROM) by incorporating differentiable PDE solvers into the training procedure. Specifically, the latent space dynamics and its dependence on PDE parameters are shaped directly by the governing physics encoded in the solver, ensuring a strong correspondence between the full and reduced systems. Our model outperforms state‑of‑the‑art data‑driven ROMs and other physics‑informed strategies by accurately generalizing to new dynamics arising from unseen parameters, enabling long‑term forecasting beyond the training horizon, maintaining continuity in both time and space, and reducing the data cost. Furthermore, Φ‑ROM learns to recover and forecast the solution fields even when trained or evaluated with sparse and irregular observations of the fields, providing a flexible framework for field reconstruction and data assimilation. We demonstrate the framework's robustness across various PDE solvers and highlight its broad applicability by providing an open‑source JAX implementation that is readily extensible to other PDE systems and differentiable solvers, available at https://phi‑rom.github.io.

Abstract:
Weather forecasting is essential but remains computationally intensive and physically incomplete in traditional numerical weather prediction (NWP) methods. Deep learning (DL) models offer efficiency and accuracy but often ignore physical laws, limiting interpretability and generalization. We propose PhyDL‑NWP, a physics‑guided deep learning framework that integrates physical equations with latent force parameterization into data‑driven models. It predicts weather variables from arbitrary spatiotemporal coordinates, computes physical terms via automatic differentiation, and uses a physics‑informed loss to align predictions with governing dynamics. PhyDL‑NWP enables resolution‑free downscaling by modeling weather as a continuous function and fine‑tunes pre‑trained models with minimal overhead, achieving up to 170x faster inference with only 55K parameters. Experiments show that PhyDL‑NWP improves both forecasting performance and physical consistency.

Abstract:
In this work, we explore the integration of Sequence Encoding for Online Parameter Identification with Physics‑Informed Neural Networks to create a model that, once trained, can be utilized for real time applications with variable parameters, boundary conditions, and initial conditions. Recently, the combination of PINNs with Sparse Regression has emerged as a method for performing dynamical system identification through supervised learning and sparse regression optimization, while also solving the dynamics using PINNs. However, this approach can be limited by variations in parameters or boundary and initial conditions, requiring retraining of the model whenever changes occur. In this work, we introduce an architecture that employs Deep Sets or Sequence Encoders to encode dynamic parameters, boundary conditions, and initial conditions, using these encoded features as inputs for the PINN, enabling the model to adapt to changes in parameters, BCs, and ICs. We apply this approach to three different problems. First, we analyze the Rossler ODE system, demonstrating the robustness of the model with respect to noise and its ability to generalize. Next, we explore the model's capability in a 2D Navier‑Stokes PDE problem involving flow past a cylinder with a parametric sinusoidal inlet velocity function, showing that the model can encode pressure data from a few points to identify the inlet velocity profile and utilize physics to compute velocity and pressure throughout the domain. Finally, we address a 1D heat monitoring problem using real data from the heating of glass fiber and thermoplastic composite plates.

Abstract:
The Gray Radiative Transfer Equations (GRTEs) are high‑dimensional, multiscale problems that pose significant computational challenges for traditional numerical methods. Current deep learning approaches, including Physics‑Informed Neural Networks (PINNs) and Asymptotically Preserving Neural Networks (APNNs), are largely restricted to low‑dimensional or linear GRTEs. To address these challenges, we propose the Radiative Transfer Asymptotically Preserving Neural Network (RT‑APNN), an innovative framework extending APNNs. RT‑APNN integrates multiple neural networks into a cohesive architecture, reducing training time while ensuring high solution accuracy. Advanced techniques such as pre‑training and Markov Chain Monte Carlo (MCMC) adaptive sampling are employed to tackle the complexities of long‑term simulations and intricate boundary conditions. RT‑APNN is the first deep learning method to successfully simulate the Marshak wave problem. Numerical experiments demonstrate its superiority over existing methods, including APNNs and MD‑APNNs, in both accuracy and computational efficiency. Furthermore, RT‑APNN excels at solving high‑dimensional, nonlinear problems, underscoring its potential for diverse applications in science and engineering.

Abstract:
We present a unified convergence theory for gradient‑based training of neural network methods for partial differential equations (PDEs), covering both physics‑informed neural networks (PINNs) and the Deep Ritz method. For linear PDEs, we extend the neural tangent kernel (NTK) framework for PINNs to establish global convergence guarantees for a broad class of linear operators. For nonlinear PDEs, we prove convergence to critical points via the Łojasiewicz inequality under the random feature model, eliminating the need for strong over‑parameterization and encompassing both gradient flow and implicit gradient descent dynamics. Our results further reveal that the random feature model exhibits an implicit regularization effect, preventing parameter divergence to infinity. Theoretical findings are corroborated by numerical experiments, providing new insights into the training dynamics and robustness of neural network PDE solvers.

Abstract:
The data‑driven discovery of long‑time macroscopic dynamics and thermodynamics of dissipative systems with particle fidelity is hampered by significant obstacles. These include the strong time‑scale limitations inherent to particle simulations, the non‑uniqueness of the thermodynamic potentials and operators from given macroscopic dynamics, and the need for efficient uncertainty quantification. This paper introduces Statistical‑Physics Informed Epistemic Diffusion Models (SPIEDiff), a machine learning framework designed to overcome these limitations in the context of purely dissipative systems by leveraging statistical physics, conditional diffusion models, and epinets. We evaluate the proposed framework on stochastic Arrhenius particle processes and demonstrate that SPIEDiff can accurately uncover both thermodynamics and kinetics, while enabling reliable long‑time macroscopic predictions using only short‑time particle simulation data. SPIEDiff can deliver accurate predictions with quantified uncertainty in minutes, drastically reducing the computational demand compared to direct particle simulations, which would take days or years in the examples considered. Overall, SPIEDiff offers a robust and trustworthy pathway for the data‑driven discovery of thermodynamic models.

Abstract:
Physics‑informed machine learning (PIML) is crucial in modern traffic flow modeling because it combines the benefits of both physics‑based and data‑driven approaches. In conventional PIML, physical information is typically incorporated by constructing a hybrid loss function that combines data‑driven loss and physics loss through linear scalarization. The goal is to find a trade‑off between these two objectives to improve the accuracy of model predictions. However, from a mathematical perspective, linear scalarization is limited to identifying only the convex region of the Pareto front, as it treats data‑driven and physics losses as separate objectives. Given that most PIML loss functions are non‑convex, linear scalarization restricts the achievable trade‑off solutions. Moreover, tuning the weighting coefficients for the two loss components can be both time‑consuming and computationally challenging. To address these limitations, this paper introduces a paradigm shift in PIML by reformulating the training process as a multi‑objective optimization problem, treating data‑driven loss and physics loss independently. We apply several multi‑gradient descent algorithms (MGDAs), including traditional multi‑gradient descent (TMGD) and dual cone gradient descent (DCGD), to explore the Pareto front in this multi‑objective setting. These methods are evaluated on both macroscopic and microscopic traffic flow models. In the macroscopic case, MGDAs achieved comparable performance to traditional linear scalarization methods. Notably, in the microscopic case, MGDAs significantly outperformed their scalarization‑based counterparts, demonstrating the advantages of a multi‑objective optimization approach in complex PIML scenarios.

Abstract:
This study investigates the application of Physics‑Informed Neural Networks (PINNs) to inverse problems in acoustic tube analysis, focusing on reconstructing acoustic fields from noisy and limited observation data. Specifically, we address scenarios where the radiation model is unknown, and pressure data is only available at the tube's radiation end. A PINNs framework is proposed to reconstruct the acoustic field, along with the PINN Fine‑Tuning Method (PINN‑FTM) and a traditional optimization method (TOM) for predicting radiation model coefficients. The results demonstrate that PINNs can effectively reconstruct the tube's acoustic field under noisy conditions, even with unknown radiation parameters. PINN‑FTM outperforms TOM by delivering balanced and reliable predictions and exhibiting robust noise‑tolerance capabilities.

Abstract:
Real‑world systems, from aerospace to railway engineering, are modeled with partial differential equations (PDEs) describing the physics of the system. Estimating robust solutions for such problems is essential. Deep learning‑based architectures, such as neural PDE solvers, have recently gained traction as a reliable solution method. The current state of development of these approaches, however, primarily focuses on improving accuracy. The environmental impact of excessive computation, leading to increased carbon emissions, has largely been overlooked. This paper introduces a carbon emission measure for a range of PDE solvers. Our proposed metric, EcoL2, balances model accuracy with emissions across data collection, model training, and deployment. Experiments across both physics‑informed machine learning and operator learning architectures demonstrate that the proposed metric presents a holistic assessment of model performance and emission cost. As such solvers grow in scale and deployment, EcoL2 represents a step toward building performant scientific machine learning systems with lower long‑term environmental impact.

Abstract:
The Saint‑Venant torsion theory is a classical theory for analyzing the torsional behavior of structural components, and it remains critically important in modern computational design workflows. Conventional numerical methods, including the finite element method (FEM), typically rely on mesh‑based approaches to obtain approximate solutions. However, these methods often require complex and computationally intensive techniques to overcome the limitations of approximation, leading to significant increases in computational cost. The objective of this study is to develop a series of novel numerical methods based on physics‑informed neural networks (PINN) for solving the Saint‑Venant torsion equations. Utilizing the expressive power and the automatic differentiation capability of neural networks, the PINN can solve partial differential equations (PDEs) along with boundary conditions without the need for intricate computational techniques. First, a PINN solver was developed to compute the torsional constant for bars with arbitrary cross‑sectional geometries. This was followed by the development of a solver capable of handling cases with sharp geometric transitions; variable‑scaling PINN (VS‑PINN). Finally, a parametric PINN was constructed to address the limitations of conventional single‑instance PINN. The results from all three solvers showed good agreement with reference solutions, demonstrating their accuracy and robustness. Each solver can be selectively utilized depending on the specific requirements of torsional behavior analysis.

Abstract:
Accelerating the solution of nonlinear partial differential equations (PDEs) while maintaining accuracy at coarse spatiotemporal resolution remains a key challenge in scientific computing. Physics‑informed machine learning (ML) methods such as Physics‑Informed Neural Networks (PINNs) introduce prior knowledge through loss functions to ensure physical consistency, but their "soft constraints" are usually not strictly satisfied. Here, we propose LaPON, an operator network inspired by the Lagrange's mean value theorem, which embeds prior knowledge directly into the neural network architecture instead of the loss function, making the neural network naturally satisfy the given constraints. This is a hybrid framework that combines neural operators with traditional numerical methods, where neural operators are used to compensate for the effect of discretization errors on the analytical scale in under‑resolution simulations. As evaluated on turbulence problem modeled by the Navier‑Stokes equations (NSE), the multiple time step extrapolation accuracy and stability of LaPON exceed the direct numerical simulation baseline at 8x coarser grids and 8x larger time steps, while achieving a vorticity correlation of more than 0.98 with the ground truth. It is worth noting that the model can be well generalized to unseen flow states, such as turbulence with different forcing, without retraining. In addition, with the same training data, LaPON's comprehensive metrics on the out‑of‑distribution test set are at least approximately twice as good as two popular ML baseline methods. By combining numerical computing with machine learning, LaPON provides a scalable and reliable solution for high‑fidelity fluid dynamics simulation, showing the potential for wide application in fields such as weather forecasting and engineering design.

Abstract:
Power flow estimation plays a vital role in ensuring the stability and reliability of electrical power systems, particularly in the context of growing network complexities and renewable energy integration. However, existing studies often fail to adequately address the unique characteristics of power systems, such as the sparsity of network connections and the critical importance of the unique Slack node, which poses significant challenges in achieving high‑accuracy estimations. In this paper, we present SenseFlow, a novel physics‑informed and self‑ensembling iterative framework that integrates two main designs, the Physics‑Informed Power Flow Network (FlowNet) and Self‑Ensembling Iterative Estimation (SeIter), to carefully address the unique properties of the power system and thereby enhance the power flow estimation. Specifically, SenseFlow enforces the FlowNet to gradually predict high‑precision voltage magnitudes and phase angles through the iterative SeIter process. On the one hand, FlowNet employs the Virtual Node Attention and Slack‑Gated Feed‑Forward modules to facilitate efficient global‑local communication in the face of network sparsity and amplify the influence of the Slack node on angle predictions, respectively. On the other hand, SeIter maintains an exponential moving average of FlowNet's parameters to create a robust ensemble model that refines power state predictions throughout the iterative fitting process. Experimental results demonstrate that SenseFlow outperforms existing methods, providing a promising solution for high‑accuracy power flow estimation across diverse grid configurations.

Abstract:
Natural gradient methods significantly accelerate the training of Physics‑Informed Neural Networks (PINNs), but are often prohibitively costly. We introduce a suite of techniques to improve the accuracy and efficiency of energy natural gradient descent (ENGD) for PINNs. First, we leverage the Woodbury formula to dramatically reduce the computational complexity of ENGD. Second, we adapt the Subsampled Projected‑Increment Natural Gradient Descent algorithm from the variational Monte Carlo literature to accelerate the convergence. Third, we explore the use of randomized algorithms to further reduce the computational cost in the case of large batch sizes. We find that randomization accelerates progress in the early stages of training for low‑dimensional problems, and we identify key barriers to attaining acceleration in other scenarios. Our numerical experiments demonstrate that our methods outperform previous approaches, achieving the same L^2 error as the original ENGD up to 75× faster.

Abstract:
Recent developments in autonomous driving and robotics underscore the necessity of safety‑critical controllers. Control barrier functions (CBFs) are a popular method for appending safety guarantees to a general control framework, but they are notoriously difficult to generate beyond low dimensions. Existing methods often yield non‑differentiable or inaccurate approximations that lack integrity, and thus fail to ensure safety. In this work, we use physics‑informed neural networks (PINNs) to generate smooth approximations of CBFs by computing Hamilton‑Jacobi (HJ) optimal control solutions. These reachability barrier networks (RBNs) avoid traditional dimensionality constraints and support the tuning of their conservativeness post‑training through a parameterized discount term. To ensure robustness of the discounted solutions, we leverage conformal prediction methods to derive probabilistic safety guarantees for RBNs. We demonstrate that RBNs are highly accurate in low dimensions, and safer than the standard neural CBF approach in high dimensions. Namely, we showcase the RBNs in a 9D multi‑vehicle collision avoidance problem where it empirically proves to be 5.5x safer and 1.9x less conservative than the neural CBFs, offering a promising method to synthesize CBFs for general nonlinear autonomous systems.

Abstract:
Parameter estimation in inverse problems involving partial differential equations (PDEs) underpins modeling across scientific disciplines, especially when parameters vary in space or time. Physics‑informed Machine Learning (PhiML) integrates PDE constraints into deep learning, but prevailing approaches depend on recursive automatic differentiation (autodiff), which produces inaccurate high‑order derivatives, inflates memory usage, and underperforms in noisy settings. We propose Mollifier Layers, a lightweight, architecture‑agnostic module that replaces autodiff with convolutional operations using analytically defined mollifiers. This reframing of derivative computation as smoothing integration enables efficient, noise‑robust estimation of high‑order derivatives directly from network outputs. Mollifier Layers attach at the output layer and require no architectural modifications. We compare them with three distinct architectures and benchmark performance across first‑, second‑, and fourth‑order PDEs ‑‑ including Langevin dynamics, heat diffusion, and reaction‑diffusion systems ‑‑ observing significant improvements in memory efficiency, training time and accuracy for parameter recovery across tasks. To demonstrate practical relevance, we apply Mollifier Layers to infer spatially varying epigenetic reaction rates from super‑resolution chromatin imaging data ‑‑ a real‑world inverse problem with biomedical significance. Our results establish Mollifier Layers as an efficient and scalable tool for physics‑constrained learning.

Abstract:
Natural Gradient Descent (NGD) has emerged as a promising optimization algorithm for training neural network‑based solvers for partial differential equations (PDEs), such as Physics‑Informed Neural Networks (PINNs). However, its practical use is often limited by the high computational cost of solving linear systems involving the Gramian matrix. While matrix‑free NGD methods based on the conjugate gradient (CG) method avoid explicit matrix inversion, the ill‑conditioning of the Gramian significantly slows the convergence of CG. In this work, we extend matrix‑free NGD to broader classes of problems than previously considered and propose the use of Randomized Numerical Linear Algebra (RandNLA) techniques for efficient preconditioning of the inner CG solver. The resulting algorithm demonstrates substantial performance improvements over existing NGD‑based methods and other state‑of‑the‑art optimizers on a range of PDE problems discretized using neural networks.

Abstract:
This research confronts the challenge of substantial physical equation discrepancies encountered in the generation of spatiotemporal physical fields through data‑driven trained models. A spatiotemporal physical field generation model, named HMT‑PF, is developed based on the hybrid Mamba‑Transformer architecture, incorporating unstructured grid information as input. A fine‑tuning block, enhanced with physical information, is introduced to effectively reduce the physical equation discrepancies. The physical equation residuals are computed through a point query mechanism for efficient gradient evaluation, then encoded into latent space for refinement. The fine‑tuning process employs a self‑supervised learning approach to achieve physical consistency while maintaining essential field characteristics. Results show that the hybrid Mamba‑Transformer model achieves good performance in generating spatiotemporal fields, while the physics‑informed fine‑tuning mechanism further reduces significant physical errors effectively. A MSE‑R evaluation method is developed to assess the accuracy and realism of physical field generation.

Abstract:
This study investigates why physics‑informed machine learning (PIML) can fail in macroscopic traffic flow modeling. We define failure as cases where a PIML model underperforms both purely data‑driven and purely physics‑based baselines by a given threshold. Unlike in other fields, physics residuals themselves do not hinder optimization in this setting. Instead, effective updates require both data and physics gradients to form acute angles with the true gradient, a condition difficult to satisfy with low‑resolution loop data. In such cases, neural networks cannot accurately approximate density and speed, and the constructed physics residuals, already degraded by discrete sampling and temporal averaging, lose their ability to capture PDE dynamics, which directly leads to PIML failure. Theoretically, although LWR and ARZ solutions are weak solutions, for piecewise C^k initial data they remain C^k off the shock set under mild conditions, which has Lebesgue measure zero. Thus, almost all detector or collocation points lie in smooth regions where residuals are valid, and the MLP's inability to exactly represent discontinuities is immaterial. Finally, we establish MSE lower bounds of physics residuals: higher‑order models such as ARZ have strictly larger consistency error bounds than LWR under mild conditions. This explains why LWR‑based PIML can outperform ARZ‑based PIML even with high‑resolution data, with the gap shrinking as resolution increases, consistent with prior empirical findings.

Abstract:
Physics Informed Neural Networks (PINNs) often exhibit failure modes in which the PDE residual loss converges while the solution error stays large, a phenomenon traditionally blamed on local optima separated from the true solution by steep loss barriers. We challenge this understanding by demonstrate that the real culprit is insufficient arithmetic precision: with standard FP32, the LBFGS optimizer prematurely satisfies its convergence test, freezing the network in a spurious failure phase. Simply upgrading to FP64 rescues optimization, enabling vanilla PINNs to solve PDEs without any failure modes. These results reframe PINN failure modes as precision induced stalls rather than inescapable local minima and expose a three stage training dynamic unconverged, failure, success whose boundaries shift with numerical precision. Our findings emphasize that rigorous arithmetic precision is the key to dependable PDE solving with neural networks.

Abstract:
This study presents a discrete physics‑informed neural network (dPINN) framework, enhanced with enforced interface constraints (EIC), for modeling physical systems using the domain decomposition method (DDM). Built upon finite element‑style mesh discretization, the dPINN accurately evaluates system energy through Gaussian quadrature‑based element‑wise integration. To ensure physical field continuity across subdomain interfaces, the EIC mechanism enforces interfacial displacement constraints without requiring auxiliary sampling or loss penalties.This formulation supports independent meshing in each subdomain, simplifying preprocessing and improving computational flexibility. Additionally, by eliminating the influence of weak spatial constraints (WSC) commonly observed in traditional PINNs, the EIC‑dPINN delivers more stable and physically consistent predictions.Extensive two‑ and three‑dimensional numerical experiments validate the proposed framework's accuracy and demonstrate the computational efficiency gains achieved through parallel training. The results highlight the framework's scalability, robustness, and potential for solving large‑scale, geometrically complex problems.

Abstract:
Fluid thermodynamics underpins atmospheric dynamics, climate science, industrial applications, and energy systems. However, direct numerical simulations (DNS) of such systems can be computationally prohibitive. To address this, we present a novel physics‑informed spatiotemporal surrogate model for Rayleigh‑Benard convection (RBC), a canonical example of convective fluid flow. Our approach combines convolutional neural networks, for spatial dimension reduction, with an innovative recurrent architecture, inspired by large language models, to model long‑range temporal dynamics. Inference is penalized with respect to the governing partial differential equations to ensure physical interpretability. Since RBC exhibits turbulent behavior, we quantify uncertainty using a conformal prediction framework. This model replicates key physical features of RBC dynamics while significantly reducing computational cost, offering a scalable alternative to DNS for long‑term simulations.

Abstract:
This paper proposes CTP, a novel deep learning framework that integrates convolutional neural network(CNN), Transformer architectures, and physics‑informed neural network(PINN) for ocean front prediction. Ocean fronts, as dynamic interfaces between distinct water masses, play critical roles in marine biogeochemical and physical processes. Existing methods such as LSTM, ConvLSTM, and AttentionConv often struggle to maintain spatial continuity and physical consistency over multi‑step forecasts. CTP addresses these challenges by combining localized spatial encoding, long‑range temporal attention, and physical constraint enforcement. Experimental results across south China sea(SCS) and Kuroshio(KUR) regions from 1993 to 2020 demonstrate that CTP achieves state‑of‑the‑art(SOTA) performance in both single‑step and multi‑step predictions, significantly outperforming baseline models in accuracy, F_1 score, and temporal stability.

Abstract:
We present the first results from the Manticore project, dubbed Manticore‑Local, a suite of Bayesian constrained simulations of the nearby Universe, generated by fitting a physical structure formation model to the 2M++ galaxy catalogue using the BORG algorithm. This field‑level inference yields physically consistent realizations of cosmic structure, leveraging a nonlinear gravitational solver, a refined galaxy bias model, and physics‑informed priors. The Manticore‑Local posterior realizations evolve within a parent cosmological volume statistically consistent with LCDM, demonstrated through extensive posterior predictive tests of power spectra, bispectra, initial condition Gaussianity, and the halo mass function. The inferred local supervolume shows no significant deviation from cosmological expectations; notably, we find no evidence for a large local underdensity. Our model identifies high‑significance counterparts for fourteen prominent galaxy clusters each within one degree of its observed sky position. Across the posterior ensemble, these counterparts are consistently detected with 2‑4 sigma significance, and their reconstructed masses and redshifts agree closely with observational estimates, confirming the inference's spatial and dynamical fidelity. The peculiar velocity field recovered by Manticore‑Local achieves the highest Bayesian evidence across five datasets, surpassing state‑of‑the‑art models. Unlike methods yielding only point estimates or using simplified dynamics, Manticore‑Local provides a full Bayesian posterior over cosmic structure and evolution, enabling rigorous uncertainty quantification. These results establish Manticore‑Local as the most advanced constrained realization suite of the Local Universe to date, offering a robust statistical foundation for future studies of galaxy formation, velocity flows, and environmental dependencies in our cosmic neighbourhood.

Abstract:
We investigate the efficient learning of magnetic phases using artificial neural networks trained on synthetic data, combining computational simplicity with physics‑informed strategies. Focusing on the diluted Ising model, which lacks an exact analytical solution, we explore two complementary approaches: a supervised classification using simple dense neural networks, and an unsupervised detection of phase transitions using convolutional autoencoders trained solely on idealized spin configurations. To enhance model performance, we incorporate two key forms of physics‑informed guidance. First, we exploit architectural biases which preferentially amplify features related to symmetry breaking. Second, we include training configurations that explicitly break \mathbbZ_2 symmetry, reinforcing the network's ability to detect ordered phases. These mechanisms, acting in tandem, increase the network's sensitivity to phase structure even in the absence of explicit labels. We validate the machine learning predictions through comparison with direct numerical estimates of critical temperatures and percolation thresholds. Our results show that synthetic, structured, and computationally efficient training schemes can reveal physically meaningful phase boundaries, even in complex systems. This framework offers a low‑cost and robust alternative to conventional methods, with potential applications in broader condensed matter and statistical physics contexts.

Abstract:
We compare the efficiency and ease‑of‑use of the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm and Sparse Physics‑Informed Discovery of Empirical Relations (SPIDER) framework in recovering the relevant governing equations and boundary conditions from data generated by direct numerical simulations (DNS) of turbulent convective flows. In the former case, a weak‑form implementation pySINDy is used. Time‑dependent data for two‑ (2D) and three‑dimensional (3D) DNS simulation of Rayleigh‑Benard convection and convective plane Couette flow is generated using the Dedalus PDE framework for spectrally solving differential equations. Using pySINDy we are able to recover the governing equations of 2D models of Rayleigh‑Benard convection at Rayleigh numbers, R, from laminar, through transitional to moderately turbulent flow conditions, albeit with increasing difficulty with larger Rayleigh number, especially in recovery of the diffusive terms (with coefficient magnitude proportional to 1/R^0.5). SPIDER requires a much smaller library of terms and we are able to recover more easily the governing equations for a wider range of R in 2D and 3D convection and plane flow models and go on to recover constraints (the incompressibility condition) and boundary conditions, demonstrating the benefits and capabilities of SPIDER to go beyond pySINDy for these fluid problems governed by second‑order PDEs. [We] demonstrat[e] the potential of machine‑learning methods to validate numerical solvers and solutions for such flow problems. We also find that properties of the flow, specifically the correlation time and spatial scales, should inform the initial selection of spatiotemporal subdomain sizes [abbreviated]

Abstract:
Disordered materials such as glasses, unlike crystals, lack long range atomic order and have no periodic unit cells, yielding a high dimensional configuration space with widely varying properties. The complexity not only increases computational costs for atomistic simulations but also makes it difficult for generative AI models to deliver accurate property predictions and realistic structure generation. In this work, we introduce GlassVAE, a hierarchical graph variational autoencoder that uses graph representations to learn compact, rotation, translation, and permutation invariant embeddings of atomic configurations. The resulting structured latent space not only enables efficient generation of novel, physically plausible structures but also supports exploration of the glass energy landscape. To enforce structural realism and physical fidelity, we augment GlassVAE with two physics informed regularizers, a radial distribution function (RDF) loss that captures characteristic short and medium range ordering and an energy regression loss that reflects the broad configurational energetics. Both theoretical analysis and experimental results highlight the critical impact of these regularizers. By encoding high dimensional atomistic data into a compact latent vector and decoding it into structures with accurate energy predictions, GlassVAE provides a fast, physics aware path for modeling and designing disordered materials.

Abstract:
Cancer remains a leading cause of death worldwide, necessitating personalized treatment approaches to improve outcomes. Theranostics, combining molecular‑level imaging with targeted therapy, offers potential for precision oncology but requires optimized, patient‑specific care plans. This paper investigates state‑of‑the‑art data‑driven decision support applications with a reinforcement learning focus in precision oncology. We review current applications, training environments, state‑space representation, performance evaluation criteria, and measurement of risk and reward, highlighting key challenges. We propose a framework integrating data‑driven modeling with reinforcement learning‑based decision support to optimize radiopharmaceutical therapy dosing, addressing identified challenges and setting directions for future research. The framework leverages Neural Ordinary Differential Equations and Physics‑Informed Neural Networks to enhance Physiologically Based Pharmacokinetic models while applying reinforcement learning algorithms to iteratively refine treatment policies based on patient‑specific data.

Abstract:
We present "torchGDM", a numerical framework for nano‑optical simulations based on the Green's Dyadic Method (GDM). This toolkit combines a hybrid approach, allowing for both fully discretized nano‑structures and structures approximated by sets of effective electric and magnetic dipoles. It supports simulations in three dimensions and for infinitely long, two‑dimensional structures. This capability is particularly suited for multi‑scale modeling, enabling accurate near‑field calculations within or around a discretized structure embedded in a complex environment of scatterers represented by effective models. Importantly, torchGDM is entirely implemented in PyTorch, a well‑optimized and GPU‑enabled automatic differentiation framework. This allows for the efficient calculation of exact derivatives of any simulated observable with respect to various inputs, including positions, wavelengths or permittivity, but also intermediate parameters like Green's tensor components, which can be interesting for physics informed deep learning applications. We anticipate that this toolkit will be valuable for applications merging nano‑photonics and machine learning, as well as for solving nano‑photonic optimization and inverse problems, such as the global design and characterization of metasurfaces, where optical interactions between structures are critical.

Abstract:
We present a hybrid quantum‑classical electrostatic Particle‑in‑Cell (PIC) method, where the electrostatic field Poisson solver is implemented on a quantum computer simulator using a hybrid classical‑quantum Neural Network (HNN) using data‑driven and physics‑informed learning approaches. The HNN is trained on classical PIC simulation results and executed via a PennyLane quantum simulator. The remaining computational steps, including particle motion and field interpolation, are performed on a classical system. To evaluate the accuracy and computational cost of this hybrid approach, we test the hybrid quantum‑classical electrostatic PIC against the two‑stream instability, a standard benchmark in plasma physics. Our results show that the quantum Poisson solver achieves comparable accuracy to classical methods. It also provides insights into the feasibility of using quantum computing and HNNs for plasma simulations. We also discuss the computational overhead associated with current quantum computer simulators, showing the challenges and potential advantages of hybrid quantum‑classical numerical methods.

Abstract:
Multi‑technique high resolution X‑ray mapping enhanced by the recent advent of 4th generation synchrotron facilities can produce colossal datasets, challenging traditional analysis methods. Such difficulty is clearly materialized when probing crystal structure of inhomogeneous samples, where the number of diffraction patterns quickly increases with map resolution, making the identification of crystal phases within a vast collection of reflections unfeasibly challenging by direct human inspection. Here we develop a novel analysis approach based on unsupervised clustering algorithms for identifying independent phases within a diffraction spatial map, which allowed us to identify the material distribution across a high‑pressure cerium hydride. By investigating the specific compound, we also contribute to the understanding of synthesis inhomogeneities among the superhydrides, a prominent superconductor class in condensed matter physics whose characterization is highly challenging even for state‑of‑the‑art materials techniques. The analysis framework we present may be readily extended to any correlated set of curves whose features are tied to specific entities, such as structural phases.

Abstract:
Kolmogorov‑Arnold Networks (KANs) have recently shown promise for solving partial differential equations (PDEs). Yet their original formulation is computationally and memory intensive, motivating the introduction of Chebyshev Type‑I‑based KANs (Chebyshev1KANs). Although Chebyshev1KANs have outperformed the vanilla KANs architecture, our rigorous theoretical analysis reveals that they still suffer from rank collapse, ultimately limiting their expressive capacity. To overcome these limitations, we enhance Chebyshev1KANs by integrating wavelet‑activated MLPs with learnable parameters and an internal attention mechanism. We prove that this design preserves a full‑rank Jacobian and is capable of approximating solutions to PDEs of arbitrary order. Furthermore, to alleviate the loss instability and imbalance introduced by the Chebyshev polynomial basis, we externally incorporate a Residual Gradient Attention (RGA) mechanism that dynamically re‑weights individual loss terms according to their gradient norms and residual magnitudes. By jointly leveraging internal and external attention, we present AC‑PKAN, a novel architecture that constitutes an enhancement to weakly supervised Physics‑Informed Neural Networks (PINNs) and extends the expressive power of KANs. Experimental results from nine benchmark tasks across three domains show that AC‑PKAN consistently outperforms or matches state‑of‑the‑art models such as PINNsFormer, establishing it as a highly effective tool for solving complex real‑world engineering problems in zero‑data or data‑sparse regimes. The code will be made publicly available upon acceptance.

Abstract:
Multi‑material decomposition (MMD) enables quantitative reconstruction of tissue compositions in the human body, supporting a wide range of clinical applications. However, traditional MMD typically requires spectral CT scanners and pre‑measured X‑ray energy spectra, significantly limiting clinical applicability. To this end, various methods have been developed to perform MMD using conventional (i.e., single‑energy, SE) CT systems, commonly referred to as SEMMD. Despite promising progress, most SEMMD methods follow a two‑step image decomposition pipeline, which first reconstructs monochromatic CT images using algorithms such as FBP, and then performs decomposition on these images. The initial reconstruction step, however, neglects the energy‑dependent attenuation of human tissues, introducing severe nonlinear beam hardening artifacts and noise into the subsequent decomposition. This paper proposes JSover, a fundamentally reformulated one‑step SEMMD framework that jointly reconstructs multi‑material compositions and estimates the energy spectrum directly from SECT projections. By explicitly incorporating physics‑informed spectral priors into the SEMMD process, JSover accurately simulates a virtual spectral CT system from SE acquisitions, thereby improving the reliability and accuracy of decomposition. Furthermore, we introduce implicit neural representation (INR) as an unsupervised deep learning solver for representing the underlying material maps. The inductive bias of INR toward continuous image patterns constrains the solution space and further enhances estimation quality. Extensive experiments on both simulated and real CT datasets show that JSover outperforms state‑of‑the‑art SEMMD methods in accuracy and computational efficiency.

Abstract:
Accurate and interpretable motion planning is essential for autonomous vehicles (AVs) navigating complex and uncertain environments. While recent end‑to‑end occupancy prediction methods have improved environmental understanding, they typically lack explicit physical constraints, limiting safety and generalization. In this paper, we propose a unified end‑to‑end framework that integrates verifiable physical rules into the occupancy learning process. Specifically, we embed artificial potential fields (APF) as physics‑informed guidance during network training to ensure that predicted occupancy maps are both data‑efficient and physically plausible. Our architecture combines convolutional and recurrent neural networks to capture spatial and temporal dependencies while preserving model flexibility. Experimental results demonstrate that our method improves task completion rate, safety margins, and planning efficiency across diverse driving scenarios, confirming its potential for reliable deployment in real‑world AV systems.

Abstract:
We propose a novel framework for solving nonlinear PDEs using sparse radial basis function (RBF) networks. Sparsity‑promoting regularization is employed to prevent over‑parameterization and reduce redundant features. This work is motivated by longstanding challenges in traditional RBF collocation methods, along with the limitations of physics‑informed neural networks (PINNs) and Gaussian process (GP) approaches, aiming to blend their respective strengths in a unified framework. The theoretical foundation of our approach lies in the function space of Reproducing Kernel Banach Spaces (RKBS) induced by one‑hidden‑layer neural networks of possibly infinite width. We prove a representer theorem showing that the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. The algorithmic framework is based on a three‑phase algorithm to maintain computational efficiency through adaptive feature selection, second‑order optimization, and pruning of inactive neurons. Numerical experiments demonstrate the effectiveness of our method and highlight cases where it offers notable advantages over GP approaches. This work opens new directions for adaptive PDE solvers grounded in rigorous analysis with efficient, learning‑inspired implementation.

Abstract:
Complexity science offers a wide range of measures for quantifying unpredictability, structure, and information. Yet, a systematic conceptual organization of these measures is still missing. We present a unified framework that locates statistical, algorithmic, and dynamical measures along three axes (regularity, randomness, and complexity) and situates them in a common conceptual space. We map statistical, algorithmic, and dynamical measures into this conceptual space, discussing their computational accessibility and approximability. This taxonomy reveals the deep challenges posed by uncomputability and highlights the emergence of modern data‑driven methods (including autoencoders, latent dynamical models, symbolic regression, and physics‑informed neural networks) as pragmatic approximations to classical complexity ideals. Latent spaces emerge as operational arenas where regularity extraction, noise management, and structured compression converge, bridging theoretical foundations with practical modeling in high‑dimensional systems. We close by outlining implications for physics‑informed AI and AI‑guided discovery in complex physical systems, arguing that classical questions of complexity remain central to next‑generation scientific modeling.

Abstract:
Finite element (FE) modeling is essential for structural analysis but remains computationally intensive, especially under dynamic loading. While operator learning models have shown promise in replicating static structural responses at FEM level accuracy, modeling dynamic behavior remains more challenging. This work presents a Multiple Input Operator Network (MIONet) that incorporates a second trunk network to explicitly encode temporal dynamics, enabling accurate prediction of structural responses under moving loads. Traditional DeepONet architectures using recurrent neural networks (RNNs) are limited by fixed time discretization and struggle to capture continuous dynamics. In contrast, MIONet predicts responses continuously over both space and time, removing the need for step wise modeling. It maps scalar inputs including load type, velocity, spatial mesh, and time steps to full field structural responses. To improve efficiency and enforce physical consistency, we introduce a physics informed loss based on dynamic equilibrium using precomputed mass, damping, and stiffness matrices, without solving the governing PDEs directly. Further, a Schur complement formulation reduces the training domain, significantly cutting computational costs while preserving global accuracy. The model is validated on both a simple beam and the KW‑51 bridge, achieving FEM level accuracy within seconds. Compared to GRU based DeepONet, our model offers comparable accuracy with improved temporal continuity and over 100 times faster inference, making it well suited for real‑time structural monitoring and digital twin applications.

Abstract:
To mitigate the barren plateau problem, effective parameter initialization is crucial for optimizing the Quantum Approximate Optimization Algorithm (QAOA) in the near‑term Noisy Intermediate‑Scale Quantum (NISQ) era. Prior physics‑driven approaches leveraged the optimal parameter concentration phenomenon, utilizing medium values of previously optimized QAOA parameters stored in databases as initialization for new graphs. However, this medium‑value‑based strategy lacks generalization capability. Conversely, prior computer‑science‑based approaches employed graph neural networks (GNNs) trained on previously optimized QAOA parameters to predict initialization values for new graphs. However, these approaches neglect key physics‑informed QAOA principles, such as parameter concentration, symmetry, and adiabatic evolution, resulting in suboptimal parameter predictions and limited performance improvements. Furthermore, no existing GNN‑based methods support parameter initialization for QAOA circuits with variable depths or for solving weighted Max‑Cut problems. This paper introduces QSeer, a quantum‑inspired GNN designed for accurate QAOA parameter prediction. Compared to prior physics‑ and computer‑science‑driven methods, QSeer improves the initial approximation ratio and convergence speed of QAOA circuits across diverse graphs by 6%‑68% and 5x‑10x, respectively.

Abstract:
Accurately capturing the nonlinear dynamic behavior of structures remains a significant challenge in mechanics and engineering. Traditional physics‑based models and data‑driven approaches often struggle to simultaneously ensure model interpretability, noise robustness, and estimation optimality. To address this issue, this paper proposes an Adaptive Physics‑Informed System Modeling with Control (APSMC) framework. By integrating Kalman filter‑based state estimation with physics‑constrained proximal gradient optimization, the framework adaptively updates time‑varying state‑space model parameters while processing real‑time input‑output data under white noise disturbances. Theoretically, this process is equivalent to real‑time tracking of the Jacobian matrix of a nonlinear dynamical system. Within this framework, we leverage the theoretical foundation of stochastic subspace identification to demonstrate that, as observational data accumulates, the APSMC algorithm yields state‑space model estimates that converge to the theoretically optimal solution. The effectiveness of the proposed framework is validated through numerical simulations of a Duffing oscillator and the seismic response of a frame structure, as well as experimental tests on a scaled bridge model. Experimental results show that, under noisy conditions, APSMC successfully predicts 19 consecutive 10‑second time series using only a single initial 10‑second segment for model updating, achieving a minimum normalized mean square error (NMSE) of 0.398%. These findings demonstrate that the APSMC framework not only offers superior online identification and denoising performance but also provides a reliable foundation for downstream applications such as structural health monitoring, real‑time control, adaptive filtering, and system identification.

Abstract:
Physics‑Informed Neural Networks (PINNs) have been widely used to obtain solutions to various physical phenomena modeled as Differential Equations. As PINNs are not naturally equipped with mechanisms for Uncertainty Quantification, some work has been done to quantify the different uncertainties that arise when dealing with PINNs. In this paper, we use a two‑step procedure to train Bayesian Neural Networks that provide uncertainties over the solutions to differential equation systems provided by PINNs. We use available error bounds over PINNs to formulate a heteroscedastic variance that improves the uncertainty estimation. Furthermore, we solve forward problems and utilize the obtained uncertainties when doing parameter estimation in inverse problems in cosmology.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical laws directly into the loss function. However, as a fundamental optimization issue, internal covariate shift (ICS) hinders the stable and effective training of PINNs by disrupting feature distributions and limiting model expressiveness. Unlike standard deep learning tasks, conventional remedies for ICS ‑‑ such as Batch Normalization and Layer Normalization ‑‑ are not directly applicable to PINNs, as they distort the physical consistency required for reliable PDE solutions. To address this issue, we propose Mask‑PINNs, a novel architecture that introduces a learnable mask function to regulate feature distributions while preserving the underlying physical constraints of PINNs. We provide a theoretical analysis showing that the mask suppresses the expansion of feature representations through a carefully designed modulation mechanism. Empirically, we validate the method on multiple PDE benchmarks ‑‑ including convection, wave propagation, and Helmholtz equations ‑‑ across diverse activation functions. Our results show consistent improvements in prediction accuracy, convergence stability, and robustness. Furthermore, we demonstrate that Mask‑PINNs enable the effective use of wider networks, overcoming a key limitation in existing PINN frameworks.

Abstract:
Wave‑like images‑‑from attosecond streaking spectrograms to optical spectra, audio mel‑spectrograms and periodic video frames‑‑encode critical harmonic structures that elude conventional feature extractors. We propose a unified, matrix‑equivalent framework that reinterprets convolution and attention as linear transforms on flattened inputs, revealing filter weights as basis vectors spanning latent feature subspaces. To infuse spectral priors we apply elementwise \(\sin(\cdot)\) mappings to each weight matrix. Embedding these transforms into CNN, ViT and Capsule architectures yields Sin‑Basis Networks with heightened sensitivity to periodic motifs and built‑in invariance to spatial shifts. Experiments on a diverse collection of wave‑like image datasets‑‑including 80,000 synthetic attosecond streaking spectrograms, thousands of Raman, photoluminescence and FTIR spectra, mel‑spectrograms from AudioSet and cycle‑pattern frames from Kinetics‑‑demonstrate substantial gains in reconstruction accuracy, translational robustness and zero‑shot cross‑domain transfer. Theoretical analysis via matrix isomorphism and Mercer‑kernel truncation quantifies how sinusoidal reparametrization enriches expressivity while preserving stability in data‑scarce regimes. Sin‑Basis Networks thus offer a lightweight, physics‑informed approach to deep learning across all wave‑form imaging modalities.

Abstract:
The motion planning problem involves finding a collision‑free path from a robot's starting to its target configuration. Recently, self‑supervised learning methods have emerged to tackle motion planning problems without requiring expensive expert demonstrations. They solve the Eikonal equation for training neural networks and lead to efficient solutions. However, these methods struggle in complex environments because they fail to maintain key properties of the Eikonal equation, such as optimal value functions and geodesic distances. To overcome these limitations, we propose a novel self‑supervised temporal difference metric learning approach that solves the Eikonal equation more accurately and enhances performance in solving complex and unseen planning tasks. Our method enforces Bellman's principle of optimality over finite regions, using temporal difference learning to avoid spurious local minima while incorporating metric learning to preserve the Eikonal equation's essential geodesic properties. We demonstrate that our approach significantly outperforms existing self‑supervised learning methods in handling complex environments and generalizing to unseen environments, with robot configurations ranging from 2 to 12 degrees of freedom (DOF).

Abstract:
Estimating rate coefficients from complex chemical reactions is essential for advancing detailed chemistry. However, the stiffness inherent in real‑world atmospheric chemistry systems poses severe challenges, leading to training instability and poor convergence, which hinder effective rate coefficient estimation using learning‑based approaches. To address this, we propose a Stiff Physics‑Informed Neural ODE framework (SPIN‑ODE) for chemical reaction modelling. Our method introduces a three‑stage optimisation process: first, a black‑box neural ODE is trained to fit concentration trajectories; second, a Chemical Reaction Neural Network (CRNN) is pre‑trained to learn the mapping between concentrations and their time derivatives; and third, the rate coefficients are fine‑tuned by integrating with the pre‑trained CRNN. Extensive experiments on both synthetic and newly proposed real‑world datasets validate the effectiveness and robustness of our approach. As the first work addressing stiff neural ODE for chemical rate coefficient discovery, our study opens promising directions for integrating neural networks with detailed chemistry.

Abstract:
Simulating seismic first‑arrival traveltime plays a crucial role in seismic tomography. First‑arrival traveltime simulation relies on solving the eikonal equation. The accuracy of conventional numerical solvers is limited to a finite‑difference approximation. In recent years, physics‑informed neural networks (PINNs) have been applied to achieve this task. However, traditional PINNs encounter challenges in accurately solving the eikonal equation, especially in cases where the model exhibits directional scaling differences. These challenges result in substantial traveltime prediction errors when the traveling distance is long. To improve the accuracy of PINN in traveltime prediction, we incorporate the reciprocity principle as a constraint into the PINN training framework. Based on the reciprocity principle, which states that the traveltime between two points remains invariant when their roles as source and receiver are exchanged, we propose to apply this principle to multiple source‑receiver pairs in PINN‑based traveltime prediction. Furthermore, a dynamic weighting mechanism is proposed to balance the contributions of the eikonal equation loss and the reciprocity‑constrained loss during the training process. This adaptive weighting evolves dynamically with the training epochs, enhancing the convergency of the training process. Experiments conducted on a simple lens velocity model, the Overthrust velocity model, and a 3D velocity model demonstrate that the introduction of the reciprocity‑constrained PINN significantly improves the accuracy of traveltime predictions.

Abstract:
One use case of ``physics‑informed neural networks'' (PINNs) is solution reconstruction, which aims to estimate the full‑field state of a physical system from sparse measurements. Parameterized governing equations of the system are used in tandem with the measurements to regularize the regression problem. However, in real‑world solution reconstruction problems, the parameterized governing equation may be inconsistent with the physical phenomena that give rise to the measurement data. We show that due to assuming consistency between the true and parameterized physics, PINNs‑based approaches may fail to satisfy three basic criteria of interpretability, robustness, and data consistency. As we argue, these criteria ensure that (i) the quality of the reconstruction can be assessed, (ii) the reconstruction does not depend strongly on the choice of physics loss, and (iii) that in certain situations, the physics parameters can be uniquely recovered. In the context of elasticity and heat transfer, we demonstrate how standard formulations of the physics loss and techniques for constraining the solution to respect the measurement data lead to different ``constraint forces" ‑‑ which we define as additional source terms arising from the constraints ‑‑ and that these constraint forces can significantly influence the reconstructed solution. To avoid the potentially substantial influence of the choice of physics loss and method of constraint enforcement on the reconstructed solution, we propose the ``explicit constraint force method'' (ECFM) to gain control of the source term introduced by the constraint. We then show that by satisfying the criteria of interpretability, robustness, and data consistency, this approach leads to more predictable and customizable reconstructions from noisy measurement data, even when the parameterization of the missing physics is inconsistent with the measured system.

Abstract:
Current, realistic numerical simulations of the solar atmosphere reproduce observations in a statistical sense; they do not replicate observations such as a movie of solar granulation. Inversions on the other hand reproduce observations by design, but the resulting models are often not physically self‑consistent. Physics‑informed neural networks (PINNs) offer a new approach to solving the time‑dependent radiative hydrodynamics equations and matching observations as boundary conditions. PINNs approximate the solution of the integro‑differential equations with a deep neural network. The parameters of this network are determined by minimizing the residuals with respect to the physics equations and the observations. The resulting models are continuous in all dimensions, can zoom into local areas of interest in space and time, and provide information on physical parameters that are not necessarily directly observed such as horizontal velocities. Here we present the first proof of concept of this novel approach, explain the underlying methodology in detail, and provide an outlook to the many applications that PINNs enable.

Abstract:
We systematically investigated a reinforcement learning (RL)‑based closed‑loop active flow control strategy to enhance the lift‑to‑drag ratio of a wing section with an NLF(1)‑0115 airfoil at an angle of attack 5 degree. The effects of key control parameters, including actuation location, observed state, reward function, and control update interval, are evaluated at a chord‑based Reynolds number of Re=20,000. Results show that all parameters significantly influence control performance, with the update interval playing a particularly critical role. Properly chosen update intervals introduce a broader spectrum of actuation frequencies, enabling more effective interactions with a wider range of flow structures and contributing to improved control effectiveness. The optimally trained RL controller is further evaluated in a three‑dimensional numerical setup at the same Reynolds number. Actuation is applied using both spanwise‑uniform and spanwise‑varying control profiles. The results demonstrate that the pretrained controller, combined with a physics‑informed spanwise distribution, achieves substantial performance gains. These findings extend the feasibility and scalability of a pretrained RL‑based control strategy to more complex airfoil flows.

Abstract:
The rise of deep learning challenges the longstanding scientific ideal of insight ‑ the human capacity to understand phenomena by uncovering underlying mechanisms. In many modern applications, accurate predictions no longer require interpretable models, prompting debate about whether explainability is a realistic or even meaningful goal. From our perspective in physics, we examine this tension through a concrete case study: a physics‑informed neural network (PINN) trained on a rarefied gas dynamics problem governed by the Boltzmann equation. Despite the system's clear structure and well‑understood governing laws, the trained network's weights resemble Gaussian‑distributed random matrices, with no evident trace of the physical principles involved. This suggests that deep learning and traditional simulation may follow distinct cognitive paths to the same outcome ‑ one grounded in mechanistic insight, the other in statistical interpolation. Our findings raise critical questions about the limits of explainable AI and whether interpretability can ‑ or should‑remain a universal standard in artificial reasoning.

Abstract:
We present a systematic framework for shortening and automating molecular dynamics equilibration through improved position initialization methods and uncertainty quantification analysis, using the Yukawa one‑component plasma as an exemplar system. Our comprehensive evaluation of seven initialization approaches (uniform random, uniform random with rejection, Halton and Sobol sequences, perfect and perturbed lattices, and a Monte Carlo pair distribution method) demonstrates that initialization significantly impacts equilibration efficiency, with microfield distribution analysis providing diagnostic insights into thermal behaviors. Our results establish that initialization method selection is relatively inconsequential at low coupling strengths, while physics‑informed methods demonstrate superior performance at high coupling strengths, reducing equilibration time. We establish direct relationships between temperature stability and uncertainties in transport properties (diffusion coefficient and viscosity), comparing thermostating protocols including ON‑OFF versus OFF‑ON duty cycles, Berendsen versus Langevin thermostats, and thermostat coupling strengths. Our findings demonstrate that weaker thermostat coupling generally requires fewer equilibration cycles, and OFF‑ON thermostating sequences outperform ON‑OFF approaches for most initialization methods. The methodology implements temperature forecasting as a quantitative metric for system thermalization, enabling users to determine equilibration adequacy based on specified uncertainty tolerances in desired output properties, thus transforming equilibration from a heuristic process to a rigorously quantifiable procedure with clear termination criteria.

Abstract:
We develop a novel physics informed deep learning approach for solving nonlinear drift‑diffusion equations on metric graphs. These models represent an important model class with a large number of applications in areas ranging from transport in biological cells to the motion of human crowds. While traditional numerical schemes require a large amount of tailoring, especially in the case of model design or parameter identification problems, physics informed deep operator networks (DeepONet) have emerged as a versatile tool for the solution of partial differential equations with the particular advantage that they easily incorporate parameter identification questions. We here present an approach where we first learn three DeepONet models for representative inflow, inner and outflow edges, resp., and then subsequently couple these models for the solution of the drift‑diffusion metric graph problem by relying on an edge‑based domain decomposition approach. We illustrate that our framework is applicable for the accurate evaluation of graph‑coupled physics models and is well suited for solving optimization or inverse problems on these coupled networks.

Abstract:
Modal identification is crucial for structural health monitoring and structural control, providing critical insights into structural dynamics and performance. This study presents a novel deep learning framework that integrates graph neural networks (GNNs), transformers, and a physics‑informed loss function to achieve modal decomposition and identification across a population of structures. The transformer module decomposes multi‑degrees‑of‑freedom (MDOF) structural dynamic measurements into single‑degree‑of‑freedom (SDOF) modal responses, facilitating the identification of natural frequencies and damping ratios. Concurrently, the GNN captures the structural configurations and identifies mode shapes corresponding to the decomposed SDOF modal responses. The proposed model is trained in a purely physics‑informed and unsupervised manner, leveraging modal decomposition theory and the independence of structural modes to guide learning without the need for labeled data. Validation through numerical simulations and laboratory experiments demonstrates its effectiveness in accurately decomposing dynamic responses and identifying modal properties from sparse structural dynamic measurements, regardless of variations in external loads or structural configurations. Comparative analyses against established modal identification techniques and model variations further underscore its superior performance, positioning it as a favorable approach for population‑based structural health monitoring.

Abstract:
Tissue viscoelasticity has been recognized as a crucial biomechanical indicator for disease diagnosis and therapeutic monitoring. Conventional shear wave elastography techniques depend on dispersion analysis and face fundamental limitations in clinical scenarios. Particularly, limited wave propagation data with low signal‑to‑noise ratios, along with challenges in discriminating between dual dispersion sources stemming from viscoelasticity and finite tissue dimensions, pose great difficulties for extracting dispersion relation. In this study, we introduce SWVE‑Net, a framework for shear wave viscoelasticity imaging based on a physics‑informed neural network (PINN). SWVE‑Net circumvents dispersion analysis by directly incorporating the viscoelasticity wave motion equation into the loss functions of the PINN. Finite element simulations reveal that SWVE‑Net quantifies viscosity parameters within a wide range (0.15‑1.5 Pas), even for samples just a few millimeters in size, where substantial wave reflections and dispersion occur. Ex vivo experiments demonstrate its applicability across various organs, including brain, liver, kidney, and spleen, each with distinct viscoelasticity. In in vivo human trials on breast and skeletal muscle tissues, SWVE‑Net reliably assesses viscoelastic properties with standard deviation‑to‑mean ratios below 15%, highlighting robustness under real‑world constraints. SWVE‑Net overcomes the core limitations of conventional elastography and enables reliable viscoelastic characterization where traditional methods fall short. It holds promise for applications such as grading hepatic lipid accumulation, detecting myocardial infarction boundaries, and distinguishing malignant from benign tumors.

Abstract:
This article introduces Perception‑Informed Neural Networks (PrINNs), a framework designed to incorporate perception‑based information into neural networks, addressing both systems with known and unknown physics laws or differential equations. Moreover, PrINNs extend the concept of Physics‑Informed Neural Networks (PINNs) and their variants, offering a platform for the integration of diverse forms of perception precisiation, including singular, probability distribution, possibility distribution, interval, and fuzzy graph. In fact, PrINNs allow neural networks to model dynamical systems by integrating expert knowledge and perception‑based information through loss functions, enabling the creation of modern data‑driven models. Some of the key contributions include Mixture of Experts Informed Neural Networks (MOEINNs), which combine heterogeneous expert knowledge into the network, and Transformed‑Knowledge Informed Neural Networks (TKINNs), which facilitate the incorporation of meta‑information for enhanced model performance. Additionally, Fuzzy‑Informed Neural Networks (FINNs) as a modern class of fuzzy deep neural networks leverage fuzzy logic constraints within a deep learning architecture, allowing online training without pre‑training and eliminating the need for defuzzification. PrINNs represent a significant step forward in bridging the gap between traditional physics‑based modeling and modern data‑driven approaches, enabling neural networks to learn from both structured physics laws and flexible perception‑based rules. This approach empowers neural networks to operate in uncertain environments, model complex systems, and discover new forms of differential equations, making PrINNs a powerful tool for advancing computational science and engineering.

Abstract:
Accurate modeling of closure terms is a critical challenge in engineering and scientific research, particularly when data is sparse (scarse or incomplete), making widely applicable models difficult to develop. This study proposes a novel approach for constructing closure models in such challenging scenarios. We introduce a Series‑Parallel Multi‑Network Architecture that integrates Physics‑Informed Neural Networks (PINNs) to incorporate physical constraints and heterogeneous data from multiple initial and boundary conditions, while employing dedicated subnetworks to independently model unknown closure terms, enhancing generalizability across diverse problems. These closure models are integrated into an accurate Partial Differential Equation (PDE) solver, enabling robust solutions to complex predictive simulations in engineering applications.

Abstract:
Magnetic resonance spectroscopy (MRS) is a non‑invasive technique to measure the metabolic composition of tissues, offering valuable insights into neurological disorders, tumor detection, and other metabolic dysfunctions. However, accurate metabolite quantification is hindered by challenges such as spectral overlap, low signal‑to‑noise ratio, and various artifacts. Traditional methods like linear‑combination modeling are susceptible to ambiguities and commonly only provide a theoretical lower bound on estimation accuracy in the form of the Cramér‑Rao bound. This work introduces a Bayesian inference framework using Sylvester normalizing flows (SNFs) to approximate posterior distributions over metabolite concentrations, enhancing quantification reliability. A physics‑based decoder incorporates prior knowledge of MRS signal formation, ensuring realistic distribution representations. We validate the method on simulated 7T proton MRS data, demonstrating accurate metabolite quantification, well‑calibrated uncertainties, and insights into parameter correlations and multi‑modal distributions.

Abstract:
Active stress models in cardiac biomechanics account for the mechanical deformation caused by muscle activity, thus providing a link between the electrophysiological and mechanical properties of the tissue. The accurate assessment of active stress parameters is fundamental for a precise understanding of myocardial function but remains difficult to achieve in a clinical setting, especially when only displacement and strain data from medical imaging modalities are available. This work investigates, through an in‑silico study, the application of physics‑informed neural networks (PINNs) for inferring active contractility parameters in time‑dependent cardiac biomechanical models from these types of imaging data. In particular, by parametrising the sought state and parameter field with two neural networks, respectively, and formulating an energy minimisation problem to search for the optimal network parameters, we are able to reconstruct in various settings active stress fields in the presence of noise and with a high spatial resolution. To this end, we also advance the vanilla PINN learning algorithm with the use of adaptive weighting schemes, ad‑hoc regularisation strategies, Fourier features, and suitable network architectures. In addition, we thoroughly analyse the influence of the loss weights in the reconstruction of active stress parameters. Finally, we apply the method to the characterisation of tissue inhomogeneities and detection of fibrotic scars in myocardial tissue. This approach opens a new pathway to significantly improve the diagnosis, treatment planning, and management of heart conditions associated with cardiac fibrosis.

Abstract:
The fusion of artificial intelligence (AI) with physics‑guided frameworks has opened transformative avenues for advancing the design and optimization of electromagnetic and nanophotonic systems. Innovations in deep neural networks (DNNs) and physics‑informed neural networks (PINNs) now provide robust tools to tackle longstanding challenges in light scattering engineering, meta‑optics, and nonlinear photonics. This review outlines recent progress in leveraging these computational methodologies to enhance device performance across domains such as dynamic light modulation, antenna design, and nonlinear optical phenomena. We systematically survey advancements in AI‑driven forward and inverse design strategies, which bypass conventional trial‑and‑error approaches by embedding physical laws directly into optimization workflows. Furthermore, the integration of AI accelerates electromagnetic simulations and enables precise modelling of complex optical effects, including topological photonic states and nonlinear interactions. A comparative evaluation of algorithmic frameworks highlights their strengths in balancing computational efficiency, multi‑objective optimization, and fabrication feasibility. Challenges such as limited interpretability of AI models and data scarcity for unconventional optical modes are critically addressed. Finally, we emphasize future opportunities in scalable multi‑physics modelling, adaptive architectures, and practical deployment of AI‑optimized photonic devices. This work underscores the pivotal role of AI in transcending traditional design limitations, thereby propelling the development of next‑generation photonic technologies with unprecedented functionality and efficiency.

Abstract:
Quantum machine learning for spin and molecular systems faces critical challenges of scarce labeled data and computationally expensive simulations. To address these limitations, we introduce Hamiltonian‑Masked Autoencoding (HMAE), a novel self‑supervised framework that pre‑trains transformers on unlabeled quantum Hamiltonians, enabling efficient few‑shot transfer learning. Unlike random masking approaches, HMAE employs a physics‑informed strategy based on quantum information theory to selectively mask Hamiltonian terms based on their physical significance. Experiments on 12,500 quantum Hamiltonians (60% real‑world, 40% synthetic) demonstrate that HMAE achieves 85.3% \pm 1.5% accuracy in phase classification and 0.15 \pm 0.02 eV MAE in ground state energy prediction with merely 10 labeled examples ‑ a statistically significant improvement (p < 0.01) over classical graph neural networks (78.1% \pm 2.1%) and quantum neural networks (76.8% \pm 2.3%). Our method's primary advantage is exceptional sample efficiency ‑ reducing required labeled examples by 3‑5x compared to baseline methods ‑ though we emphasize that ground truth values for fine‑tuning and evaluation still require exact diagonalization or tensor networks. We explicitly acknowledge that our current approach is limited to small quantum systems (specifically limited to 12 qubits during training, with limited extension to 16‑20 qubits in testing) and that, while promising within this regime, this size restriction prevents immediate application to larger systems of practical interest in materials science and quantum chemistry.

Abstract:
Dielectric response (DR) of insulating materials is key input information for designing electrical insulation systems and defining safe operating conditions of various HV devices. In dielectric materials, different polarization and conduction processes occur at different time scales, making it challenging to physically interpret raw measured data. To analyze DR measurement results, equivalent circuit models (ECMs) are commonly used, reducing the complexity of the physical system to a number of circuit elements that capture the dominant response. This paper examines the use of physics‑informed neural networks (PINNs) for inverse modeling of DR in time domain using parallel RC circuits. To assess their performance, we test PINNs on synthetic data generated from analytical solutions of corresponding ECMs, incorporating Gaussian noise to simulate measurement errors. Our results show that PINNs are highly effective at solving well‑conditioned inverse problems, accurately estimating up to five unknown RC parameters with minimal requirements on neural network size, training duration, and hyperparameter tuning. Furthermore, we extend the ECMs to incorporate temperature dependence and demonstrate that PINNs can accurately recover embedded, nonlinear temperature functions from noisy DR data sampled at different temperatures. This case study in modeling DR in time domain presents a solution with wide‑ranging potential applications in disciplines relying on ECMs, utilizing the latest technology in machine learning for scientific computation.

Abstract:
Deep learning has emerged as a powerful tool in scientific modeling, particularly for complex dynamical systems; however, accurately capturing age‑structured population dynamics under policy‑driven fertility changes remains a significant challenge due to the lack of effective integration between domain knowledge and long‑term temporal dependencies. To address this issue, we propose two physics‑informed deep learning frameworks‑‑PINN and LSTM‑PINN‑‑that incorporate policy‑aware fertility functions into a transport‑reaction partial differential equation to simulate population evolution from 2024 to 2054. The standard PINN model enforces the governing equation and boundary conditions via collocation‑based training, enabling accurate learning of underlying population dynamics and ensuring stable convergence. Building on this, the LSTM‑PINN framework integrates sequential memory mechanisms to effectively capture long‑range dependencies in the age‑time domain, achieving robust training performance across multiple loss components. Simulation results under three distinct fertility policy scenarios‑the Three‑child policy, the Universal two‑child policy, and the Separate two‑child policy‑‑demonstrate the models' ability to reflect policy‑sensitive demographic shifts and highlight the effectiveness of integrating domain knowledge into data‑driven forecasting. This study provides a novel and extensible framework for modeling age‑structured population dynamics under policy interventions, offering valuable insights for data‑informed demographic forecasting and long‑term policy planning in the face of emerging population challenges.

Abstract:
Passivity‑based control ensures system stability by leveraging dissipative properties and is widely applied in electrical and mechanical systems. Port‑Hamiltonian systems (PHS), in particular, are well‑suited for interconnection and damping assignment passivity‑based control (IDA‑PBC) due to their structured, energy‑centric modeling approach. However, current IDA‑PBC faces two key challenges: (i) it requires precise system knowledge, which is often unavailable due to model uncertainties, and (ii) it is typically limited to set‑point control. To address these limitations, we propose a data‑driven tracking control approach based on a physics‑informed model, namely Gaussian process Port‑Hamiltonian systems, along with the modified matching equation. By leveraging the Bayesian nature of the model, we establish probabilistic stability and passivity guarantees. A simulation demonstrates the effectiveness of our approach.

Abstract:
Material stress analysis is a critical aspect of material design and performance optimization. Under dynamic loading, the global stress evolution in materials exhibits complex spatiotemporal characteristics, especially in two‑phase random materials (TRMs). Such kind of material failure is often associated with stress concentration, and the phase boundaries are key locations where stress concentration occurs. In practical engineering applications, the spatiotemporal resolution of acquired microstructural data and its dynamic stress evolution is often limited. This poses challenges for deep learning methods in generating high‑resolution spatiotemporal stress fields, particularly for accurately capturing stress concentration regions. In this study, we propose a framework for global stress generation and spatiotemporal super‑resolution in TRMs under dynamic loading. First, we introduce a diffusion model‑based approach, named as Spatiotemporal Stress Diffusion (STS‑diffusion), for generating global spatiotemporal stress data. This framework incorporates Space‑Time U‑Net (STU‑net), and we systematically investigate the impact of different attention positions on model accuracy. Next, we develop a physics‑informed network for spatiotemporal super‑resolution, termed as Spatiotemporal Super‑Resolution Physics‑Informed Operator (ST‑SRPINN). The proposed ST‑SRPINN is an unsupervised learning method. The influence of data‑driven and physics‑informed loss function weights on model accuracy is explored in detail. Benefiting from physics‑based constraints, ST‑SRPINN requires only low‑resolution stress field data during training and can upscale the spatiotemporal resolution of stress fields to arbitrary magnifications.

Abstract:
Metal additive manufacturing enables unprecedented design freedom and the production of customized, complex components. However, the rapid melting and solidification dynamics inherent to metal AM processes generate heterogeneous, non‑equilibrium microstructures that significantly impact mechanical properties and subsequent functionality. Predicting microstructure and its evolution across spatial and temporal scales remains a central challenge for process optimization and defect mitigation. While conventional experimental techniques and physics‑based simulations provide a physical foundation and valuable insights, they face critical limitations. In contrast, data‑driven machine learning offers an alternative prediction approach and powerful pattern recognition but often operate as black‑box, lacking generalizability and physical consistency. To overcome these limitations, physics‑informed machine learning, including physics‑informed neural networks, has emerged as a promising paradigm by embedding governing physical laws into neural network architectures, thereby enhancing accuracy, transparency, data efficiency, and extrapolation capabilities. This work presents a comprehensive evaluation of modeling strategies for microstructure prediction in metal AM. The strengths and limitations of experimental, computational, and data‑driven methods are analyzed in depth, and highlight recent advances in hybrid PIML frameworks that integrate physical knowledge with ML. Key challenges, such as data scarcity, multi‑scale coupling, and uncertainty quantification, are discussed alongside future directions. Ultimately, this assessment underscores the importance of PIML‑based hybrid approaches in enabling predictive, scalable, and physically consistent microstructure modeling for site‑specific, microstructure‑aware process control and the reliable production of high‑performance AM components.

Abstract:
In this article, our goal is to solve two‑parameter singular perturbation problems (SPPs) in one‑ and two‑dimensions using an adapted Physics‑Informed Neural Networks (PINNs) approach. Such problems are of major importance in engineering and sciences as it appears in control theory, fluid and gas dynamics, financial modelling and so on. Solutions of such problems exhibit boundary and/or interior layers, which make them difficult to handle. It has been validated in the literature that standard PINNs have low accuracy and can't handle such problems efficiently. Recently Cao et. al \citecao2023physics proposed a new parameter asymptotic PINNs (PA‑PINNs) to solve one‑parameter singularly perturbed convection‑dominated problems. It was observed that PA‑PINNs works better than standard PINNs and gPINNs in terms of accuracy, convergence and stability. In this article, for the first time robustness of PA‑PINNs will be validated for solving two‑parameter SPPs.

Abstract:
Backward stochastic differential equation (BSDE)‑based deep learning methods provide an alternative to Physics‑Informed Neural Networks (PINNs) for solving high‑dimensional partial differential equations (PDEs), offering potential algorithmic advantages in settings such as stochastic optimal control, where the PDEs of interest are tied to an underlying dynamical system. However, standard BSDE‑based solvers have empirically been shown to underperform relative to PINNs in the literature. In this paper, we identify the root cause of this performance gap as a discretization bias introduced by the standard Euler‑Maruyama (EM) integration scheme applied to one‑step self‑consistency BSDE losses, which shifts the optimization landscape off target. We find that this bias cannot be satisfactorily addressed through finer step‑sizes or multi‑step self‑consistency losses. To properly handle this issue, we propose a Stratonovich‑based BSDE formulation, which we implement with stochastic Heun integration. We show that our proposed approach completely eliminates the bias issues faced by EM integration. Furthermore, our empirical results show that our Heun‑based BSDE method consistently outperforms EM‑based variants and achieves competitive results with PINNs across multiple high‑dimensional benchmarks. Our findings highlight the critical role of integration schemes in BSDE‑based PDE solvers, an algorithmic detail that has received little attention thus far in the literature.

Abstract:
Physics‑Informed Machine Learning (PIML) offers a powerful paradigm of integrating data with physical laws to address important scientific problems, such as parameter estimation, inferring hidden physics, equation discovery, and state prediction, etc. However, PIML still faces many serious optimization challenges that significantly restrict its applications. In this study, we propose a comprehensive framework that transforms PIML to convex optimization to overcome all these limitations, referred to as Convex‑PIML. The linear combination of B‑splines is utilized to approximate the data, promoting the convexity of the loss function. By replacing the non‑convex components of the loss function with convex approximations, the problem is further converted into a sequence of successively refined approximated convex optimization problems. This conversion allows the use of well‑established convex optimization algorithms, obtaining solutions effectively and efficiently. Furthermore, an adaptive knot optimization method based on error estimate is introduced to mitigate the spectral bias issue of PIML, further improving the performance. The proposed theoretically guaranteed framework is tested in scenarios with distinct types of physical prior. The results indicate that optimization problems are effectively solved in these scenarios, highlighting the potential of the framework for broad applications.

Abstract:
Gravitational‑wave astronomy shows great promise in determining nuclear physics in a regime not accessible to terrestrial experiments. We introduce physics‑informed priors constrained by nuclear theory and perturbative Quantum Chromodynamics calculations, as well as astrophysical measurements of neutron‑star masses and radii. When these priors are used in gravitational‑wave astrophysical inference, we show a significant improvement on nuclear equation of state constraints. Applying these to the first observed gravitational‑wave binary neutron‑star merger GW170817, the constraints on the radius of a 1.4\,M_\odot neutron star improve from R_1.4 =12.54^+1.05_‑1.54 \, \rm km to R_1.4 = 12.11^+0.91_‑1.11 \,\rm km and those on the tidal deformability from \tildeΛ_1.186 < 720 to \tildeΛ_1.186 = 384^+306_‑158 (90% confidence intervals) at the events measured chirp mass \mathcalM=1.186\,M_\odot. We also show these priors can be used to perform model selection between binary neutron star and neutron star‑black hole mergers; in the case of GW190425, the results provide only marginal evidence with a Bayes factor \mathcalBF=1.33 in favour of the binary neutron star merger hypothesis. Given their ability to improve the astrophysical inference of binary mergers involving neutron stars, we advocate for these physics‑informed priors to be used as standard in the literature and provide open‑source code for reproducibility and adaptation of the method.

Abstract:
In this paper, we develop a new optimization framework for the least squares learning problem via fully connected neural networks or physics‑informed neural networks. The gradient descent sometimes behaves inefficiently in deep learning because of the high non‑convexity of loss functions and the vanishing gradient issue. Our idea is to introduce auxiliary variables to separate the layers of the deep neural networks and reformulate the loss functions for ease of optimization. We design the self‑adaptive weights to preserve the consistency between the reformulated loss and the original mean squared loss, which guarantees that optimizing the new loss helps optimize the original problem. Numerical experiments are presented to verify the consistency and show the effectiveness and robustness of our models over gradient descent.

Abstract:
Recently, a novel linear model predictive control algorithm based on a physics‑informed Gaussian Process has been introduced, whose realizations strictly follow a system of underlying linear ordinary differential equations with constant coefficients. The control task is formulated as an inference problem by conditioning the Gaussian process prior on the setpoints and incorporating pointwise soft‑constraints as further virtual setpoints. We apply this method to systems of nonlinear differential equations, obtaining a local approximation through the linearization around an equilibrium point. In the case of an asymptotically stable equilibrium point convergence is given through the Bayesian inference schema of the Gaussian Process. Results for this are demonstrated in a numerical example.

Abstract:
Physics‑Informed Neural Networks have emerged as a promising methodology for solving PDEs, gaining significant attention in computer science and various physics‑related fields. Despite being demonstrated the ability to incorporate the physics of laws for versatile applications, PINNs still struggle with the challenging problems which are stiff to be solved and/or have high‑frequency components in the solutions, resulting in accuracy and convergence issues. It may not only increase computational costs, but also lead to accuracy loss or solution divergence. In this study, an alternative approach is proposed to mitigate the above‑mentioned problems. Inspired by the multi‑grid method in CFD community, the underlying idea of the current approach is to efficiently remove different frequency errors via training with different levels of training samples, resulting in a simpler way to improve the training accuracy without spending time in fine‑tuning of neural network structures, loss weights as well as hyperparameters. To demonstrate the efficacy of current approach, we first investigate canonical 1D ODE with high‑frequency component and 2D convection‑diffusion equation with V‑cycle training strategy. Finally, the current method is employed for the classical benchmark problem of steady Lid‑driven cavity flows at different Reynolds numbers, to investigate the applicability and efficacy for the problem involved multiple modes of high and low frequency. By virtue of various training sequence modes, improvement through predictions lead to 30% to 60% accuracy improvement. We also investigate the synergies between current method and transfer learning techniques for more challenging problems (i.e., higher Re). From the present results, it also revealed that the current framework can produce good predictions even for the case of Re=5000, demonstrating the ability to solve complex high‑frequency PDEs.

Abstract:
Our contributions are motivated by fusion reactors that rely on maintaining magnetohydrodynamic (MHD) equilibrium, where the balance between plasma pressure and confining magnetic fields is required for stable operation. In axisymmetric tokamak reactors in particular, and under the assumption of toroidal symmetry, this equilibrium can be mathematically modelled using the Grad‑Shafranov Equation (GSE). Recent works have demonstrated the potential of using Physics‑Informed Neural Networks (PINNs) to model the GSE. Existing studies did not examine realistic scenarios in which a single network generalizes to a variety of boundary conditions. Addressing that limitation, we evaluate a PINN architecture that incorporates boundary points as network inputs. Additionally, we compare PINN model accuracy and inference speeds with a Fourier Neural Operator (FNO) model. Finding the PINN model to be the most performant, and accurate in our setting, we use the network verification tool Marabou to perform a range of verification tasks. Although we find some discrepancies between evaluations of the networks natively in PyTorch, compared to via Marabou, we are able to demonstrate useful and practical verification workflows. Our study is the first investigation of verification of such networks.

Abstract:
Climate science is the multidisciplinary field that studies the Earth's climate and its evolution. At the very core of climate science are indispensable climate models that predict future climate scenarios, inform policy decisions, and dictate how a country's economy should change in light of the changing climate. Climate models capture a wide range of interacting dynamic processes via extremely complex ordinary and partial differential equations. To model these large‑scale complex processes, climate science leverages supercomputers, advanced simulations, and statistical methods to predict future climate. An area of engineering that is rarely studied in climate science is control engineering. Given that climate systems are inherently dynamic, it is intuitive to analyze them within the framework of dynamic system science. This perspective has been underexplored in the literature. In this manuscript, we provide a tutorial that: (i) introduces the control engineering community to climate dynamics and modeling, including spatiotemporal scales and challenges in climate modeling; (ii) offers a fresh perspective on climate models from a control systems viewpoint; and (iii) explores the relevance and applicability of various advanced graph and network control‑based approaches in building a physics‑informed framework for learning, control and estimation in climate systems. We also present simple and then more complex climate models, depicting fundamental ideas and processes that are instrumental in building climate change projections. This tutorial also builds parallels and observes connections between various contemporary problems at the forefront of climate science and their control theoretic counterparts. We specifically observe that an abundance of climate science problems can be linguistically reworded and mathematically framed as control theoretic ones.

Abstract:
Detecting ship presence via wake signatures in SAR imagery is attracting considerable research interest, but limited annotated data availability poses significant challenges for supervised learning. Physics‑based simulations are commonly used to address this data scarcity, although they are slow and constrain end‑to‑end learning. In this work, we explore a new direction for more efficient and end‑to‑end SAR ship wake simulation using a diffusion model trained on data generated by a physics‑based simulator. The training dataset is built by pairing images produced by the simulator with text prompts derived from simulation parameters. Experimental result show that the model generates realistic Kelvin wake patterns and achieves significantly faster inference than the physics‑based simulator. These results highlight the potential of diffusion models for fast and controllable wake image generation, opening new possibilities for end‑to‑end downstream tasks in maritime SAR analysis.

Abstract:
Physics‑informed neural networks (PINNs) integrate physical laws with data‑driven models to improve generalization and sample efficiency. This work introduces an open‑source implementation of the Physics‑Informed Neural Network with Control (PINC) framework, designed to model the dynamics of an underwater vehicle. Using initial states, control actions, and time inputs, PINC extends PINNs to enable physically consistent transitions beyond the training domain. Various PINC configurations are tested, including differing loss functions, gradient‑weighting schemes, and hyperparameters. Validation on a simulated underwater vehicle demonstrates more accurate long‑horizon predictions compared to a non‑physics‑informed baseline

Abstract:
Digital twin (DT) techniques have been proposed for the autonomous operation and lifecycle management of next‑generation optical networks. To fully utilize potential capacity and accommodate dynamic services, the DT must dynamically update in sync with deployed optical networks throughout their lifecycle, ensuring low‑margin operation. This paper proposes a dynamic‑updating DT for the lifecycle management of optical networks, employing a hybrid approach that integrates data‑driven and physics‑informed techniques for fiber channel modeling. This integration ensures both rapid calculation speed and high physics consistency in optical performance prediction while enabling the dynamic updating of critical physical parameters for DT. The lifecycle management of optical networks, covering accurate performance prediction at the network deployment and dynamic updating during network operation, is demonstrated through simulation in a large‑scale network. Up to 100 times speedup in prediction is observed compared to classical numerical methods. In addition, the fiber Raman gain strength, amplifier frequency‑dependent gain profile, and connector loss between fiber and amplifier on C and L bands can be simultaneously updated. Moreover, the dynamic‑updating DT is verified on a field‑trial C+L‑band transmission link, achieving a maximum accuracy improvement of 1.4 dB for performance estimation post‑device replacement. Overall, the dynamic‑updating DT holds promise for driving the next‑generation optical networks towards lifecycle autonomous management.

Abstract:
The discovery of constitutive models for hyperelastic materials is essential yet challenging due to their nonlinear behavior and the limited availability of experimental data. Traditional methods typically require extensive stress‑strain or full‑field measurements, which are often difficult to obtain in practical settings. To overcome these challenges, we propose a physics‑informed neural network (PINN)‑based framework that enables the discovery of constitutive models using only sparse measurement data ‑ such as displacement and reaction force ‑ that can be acquired from a single material test. By integrating PINNs with finite element discretization, the framework reconstructs full‑field displacement and identifies the underlying strain energy density from predefined candidates, while ensuring consistency with physical laws. A two‑stage training process is employed: the Adam optimizer jointly updates neural network parameters and model coefficients to obtain an initial solution, followed by L‑BFGS refinement and sparse regression with l_p regularization to extract a parsimonious constitutive model. Validation on benchmark hyperelastic models demonstrates that the proposed method can accurately recover constitutive laws and displacement fields, even when the input data are limited and noisy. These findings highlight the applicability of the proposed framework to experimental scenarios where measurement data are both scarce and noisy.

Abstract:
Marine remote sensing enhances maritime surveillance, environmental monitoring, and naval operations. Vessel length estimation, a key component of this technology, supports effective maritime surveillance by empowering features such as vessel classification. Departing from traditional methods relying on two‑dimensional hydrodynamic wakes or computationally intensive satellite imagery, this paper introduces an innovative approach for vessel length estimation that leverages the subtle magnetic wake signatures of vessels, captured through a low‑complexity one‑dimensional profile from a single airborne magnetic sensor scan. The proposed method centers around our characterized nonlinear integral equations that connect the magnetic wake to the vessel length within a realistic finite‑depth marine environment. To solve the derived equations, we initially leverage a deep residual neural network (DRNN). The proposed DRNN‑based solution framework is shown to be unable to exactly learn the intricate relationships between parameters when constrained by a limited training‑dataset. To overcome this issue, we introduce an innovative approach leveraging a physics‑informed residual neural network (PIRNN). This model integrates physical formulations directly into the loss function, leading to improved performance in terms of both accuracy and convergence speed. Considering a sensor scan angle of less than 15^\circ, which maintains a reasonable margin below Kelvin's limit angle of 19.5^\circ, we explore the impact of various parameters on the accuracy of the vessel length estimation, including sensor scan angle, vessel speed, and sea depth. Numerical simulations demonstrate the superiority of the proposed PIRNN method, achieving mean length estimation errors consistently below 5% for vessels longer than 100m. For shorter vessels, the errors generally remain under 10%.

Abstract:
Physics‑Informed Neural Networks (PINNs) are a novel computational approach for solving partial differential equations (PDEs) with noisy and sparse initial and boundary data. Although, efficient quantification of epistemic and aleatoric uncertainties in big multi‑scale problems remains challenging. We propose \PINN a novel method of computing global uncertainty in PDEs using a Bayesian framework, by combining local Bayesian Physics‑Informed Neural Networks (BPINN) with domain decomposition. The solution continuity across subdomains is obtained by imposing the flux continuity across the interface of neighboring subdomains. To demonstrate the effectiveness of \PINN, we conduct a series of computational experiments on PDEs in 1D and 2D spatial domains. Although we have adopted conservative PINNs (cPINNs), the method can be seamlessly extended to other domain decomposition techniques. The results infer that the proposed method recovers the global uncertainty by computing the local uncertainty exactly more efficiently as the uncertainty in each subdomain can be computed concurrently. The robustness of \PINN is verified by adding uncorrelated random noise to the training data up to 15% and testing for different domain sizes.

Abstract:
Stress analysis is an important part of material design. For materials with complex microstructures, such as two‑phase random materials (TRMs), material failure is often accompanied by stress concentration. Phase interfaces in two‑phase materials are critical for stress concentration. Therefore, the prediction error of stress at phase boundaries is crucial. In practical engineering, the pixels of the obtained material microstructure images are limited, which limits the resolution of stress images generated by deep learning methods, making it difficult to observe stress concentration regions. Existing Image Super‑Resolution (ISR) technologies are all based on data‑driven supervised learning. However, stress images have natural physical constraints, which provide new ideas for new ISR technologies. In this study, we constructed a stress prediction framework for TRMs. First, the framework uses a proposed Multiple Compositions U‑net (MC U‑net) to predict stress in low‑resolution material microstructures. By considering the phase interface information of the microstructure, the MC U‑net effectively reduces the problem of excessive prediction errors at phase boundaries. Secondly, a Mixed Physics‑Informed Neural Network (MPINN) based method for stress ISR (SRPINN) was proposed. By introducing the constraints of physical information, the new method does not require paired stress images for training and can increase the resolution of stress images to any multiple. This enables a multiscale analysis of the stress concentration regions at phase boundaries. Finally, we performed stress analysis on TRMs with different phase volume fractions and loading states through transfer learning. The results show the proposed stress prediction framework has satisfactory accuracy and generalization ability.

Abstract:
Liquid electrolytes are critical components of next‑generation energy storage systems, enabling fast ion transport, minimizing interfacial resistance, and ensuring electrochemical stability for long‑term battery performance. However, measuring electrolyte properties and designing formulations remain experimentally and computationally expensive. In this work, we present a unified framework for designing liquid electrolyte formulation, integrating a forward predictive model with an inverse generative approach. Leveraging both computational and experimental data collected from literature and extensive molecular simulations, we train a predictive model capable of accurately estimating electrolyte properties from ionic conductivity to solvation structure. Our physics‑informed architecture preserves permutation invariance and incorporates empirical dependencies on temperature and salt concentration, making it broadly applicable to property prediction tasks across molecular mixtures. Furthermore, we introduce ‑‑ to the best of our knowledge ‑‑ the first generative machine learning framework for molecular mixture design, demonstrated on electrolyte systems. This framework supports multi‑condition‑constrained generation, addressing the inherently multi‑objective nature of materials design. As a proof of concept, we experimentally identified three liquid electrolytes with both high ionic conductivity and anion‑concentrated solvation structure. This unified framework advances data‑driven electrolyte design and can be readily extended to other complex chemical systems beyond electrolytes.

Abstract:
Physics‑informed neural networks have attracted significant attention in scientific machine learning for their capability to solve forward and inverse problems governed by partial differential equations. However, the accuracy of PINN solutions is often limited by the treatment of boundary conditions. Conventional penalty‑based methods, which incorporate boundary conditions as penalty terms in the loss function, cannot guarantee exact satisfaction of the given boundary conditions and are highly sensitive to the choice of penalty parameters. This paper demonstrates that distance functions, specifically R‑functions, can be leveraged to enforce boundary conditions, overcoming these limitations. R‑functions provide normalized distance fields, enabling flexible representation of boundary geometries, including non‑convex domains, and facilitating various types of boundary conditions. Nevertheless, distance functions alone are insufficient for accurate inverse analysis in PINNs. To address this, we propose an integrated framework that combines the normalized distance field with bias‑corrected adaptive weight tuning to improve both accuracy and efficiency. Numerical results show that the proposed method provides more accurate and efficient solutions to various inverse problems than penalty‑based approaches, even in the presence of non‑convex geometries with complex boundary conditions. This approach offers a reliable and efficient framework for inverse analysis using PINNs, with potential applications across a wide range of engineering problems.

Abstract:
This paper presents a digital‑twin platform for active safety analysis in mixed traffic environments. The platform is built using a multi‑modal data‑enabled traffic environment constructed from drone‑based aerial LiDAR, OpenStreetMap, and vehicle sensor data (e.g., GPS and inclinometer readings). High‑resolution 3D road geometries are generated through AI‑powered semantic segmentation and georeferencing of aerial LiDAR data. To simulate real‑world driving scenarios, the platform integrates the CAR Learning to Act (CARLA) simulator, Simulation of Urban MObility (SUMO) traffic model, and NVIDIA PhysX vehicle dynamics engine. CARLA provides detailed micro‑level sensor and perception data, while SUMO manages macro‑level traffic flow. NVIDIA PhysX enables accurate modeling of vehicle behaviors under diverse conditions, accounting for mass distribution, tire friction, and center of mass. This integrated system supports high‑fidelity simulations that capture the complex interactions between autonomous and conventional vehicles. Experimental results demonstrate the platform's ability to reproduce realistic vehicle dynamics and traffic scenarios, enhancing the analysis of active safety measures. Overall, the proposed framework advances traffic safety research by enabling in‑depth, physics‑informed evaluation of vehicle behavior in dynamic and heterogeneous traffic environments.

Abstract:
The ability to predict trajectories of surrounding agents and obstacles is a crucial component in many robotic applications. Data‑driven approaches are commonly adopted for state prediction in scenarios where the underlying dynamics are unknown. However, the performance, reliability, and uncertainty of data‑driven predictors become compromised when encountering out‑of‑distribution observations relative to the training data. In this paper, we introduce a Plug‑and‑Play Physics‑Informed Machine Learning (PnP‑PIML) framework to address this challenge. Our method employs conformal prediction to identify outlier dynamics and, in that case, switches from a nominal predictor to a physics‑consistent model, namely distributed Port‑Hamiltonian systems (dPHS). We leverage Gaussian processes to model the energy function of the dPHS, enabling not only the learning of system dynamics but also the quantification of predictive uncertainty through its Bayesian nature. In this way, the proposed framework produces reliable physics‑informed predictions even for the out‑of‑distribution scenarios.

Abstract:
Accurate myocardial image registration is essential for cardiac strain analysis and disease diagnosis. However, spectral bias in neural networks impedes modeling high‑frequency deformations, producing inaccurate, biomechanically implausible results, particularly in pathological data. This paper addresses spectral bias in physics‑informed neural networks (PINNs) by integrating Fourier Feature mappings and introducing modulation strategies into a PINN framework. Experiments on two distinct datasets demonstrate that the proposed methods enhance the PINN's ability to capture complex, high‑frequency deformations in cardiomyopathies, achieving superior registration accuracy while maintaining biomechanical plausibility ‑ thus providing a foundation for scalable cardiac image registration and generalization across multiple patients and pathologies.

Abstract:
In this study, fluid flow and concentration distribution inside a 3D T‑shaped micromixer with various fin shapes and configurations are investigated using a Flexible Physics‑Informed Neural Network (FlexPINN), which includes modifications over the vanilla PINN architecture. Three types of fins (rectangular, elliptical, and triangular) are considered to evaluate the influence of fin geometry, along with four different fin configurations inside the 3D channel to examine the effect of placement. The simulations are conducted at four Reynolds numbers: 5, 20, 40, and 80, in both single‑unit (four fins) and double‑unit (eight fins) configurations. The goal is to assess pressure drop coefficient, mixing index, and mixing efficiency using the FlexPINN method. Given the challenges in simulating 3D problems with standard PINN, several improvements are introduced. The governing equations are injected into the network as first‑order, dimensionless derivatives to enhance accuracy. Transfer learning is used to reduce computational cost, and adaptive loss weighting is applied to improve convergence compared to the vanilla PINN approach. These modifications enable a consistent and flexible architecture that can be used across numerous tested cases. Using the proposed FlexPINN method, the pressure drop coefficient and mixing index are predicted with maximum errors of 3.25% and 2.86%, respectively, compared to Computational Fluid Dynamics (CFD) results. Among all the tested cases, the rectangular fin with configuration C in the double‑unit setup at Reynolds number 40 shows the highest mixing efficiency, reaching a value of 1.63. The FlexPINN framework demonstrates strong capabilities in simulating fluid flow and species transport in complex 3D geometries.

Abstract:
We present a physics‑embedded Bayesian neural network (PE‑BNN) framework that integrates fission product yields (FPYs) with prior nuclear physics knowledge to predict energy‑dependent FPY data with fine structure. By incorporating an energy‑independent phenomenological shell factor as a single input feature, the PE‑BNN captures both fine structures and global energy trends. The combination of this physics‑informed input with hyperparameter optimization via the Watanabe‑Akaike Information Criterion (WAIC) significantly enhances predictive performance. Our results demonstrate that the PE‑BNN framework is well‑suited for target observables with systematic features that can be embedded as model inputs, achieving close agreement with known shell effects and prompt neutron multiplicities.

Abstract:
Many data‑driven modules in smart grid rely on access to high‑quality power flow data; however, real‑world data are often limited due to privacy and operational constraints. This paper presents a physics‑informed generative framework based on Denoising Diffusion Probabilistic Models (DDPMs) for synthesizing feasible power flow data. By incorporating auxiliary training and physics‑informed loss functions, the proposed method ensures that the generated data exhibit both statistical fidelity and adherence to power system feasibility. We evaluate the approach on the IEEE 14‑bus and 30‑bus benchmark systems, demonstrating its ability to capture key distributional properties and generalize to out‑of‑distribution scenarios. Comparative results show that the proposed model outperforms three baseline models in terms of feasibility, diversity, and accuracy of statistical features. This work highlights the potential of integrating generative modelling into data‑driven power system applications.

Abstract:
Supervised machine learning involves approximating an unknown functional relationship from a limited dataset of features and corresponding labels. The classical approach to feature‑based machine learning typically relies on applying linear regression to standardized features, without considering their physical meaning. This may limit model explainability, particularly in scientific applications. This study proposes a physics‑informed approach to feature‑based machine learning that constructs non‑linear feature maps informed by physical laws and dimensional analysis. These maps enhance model interpretability and, when physical laws are unknown, allow for the identification of relevant mechanisms through feature ranking. The method aims to improve both predictive performance in regression tasks and classification skill scores by integrating domain knowledge into the learning process, while also enabling the potential discovery of new physical equations within the context of explainable machine learning.

Abstract:
While non‑prehensile manipulation (e.g., controlled pushing/poking) constitutes a foundational robotic skill, its learning remains challenging due to the high sensitivity to complex physical interactions involving friction and restitution. To achieve robust policy learning and generalization, we opt to learn a world model of the 3D rigid body dynamics involved in non‑prehensile manipulations and use it for model‑based reinforcement learning. We propose PIN‑WM, a Physics‑INformed World Model that enables efficient end‑to‑end identification of a 3D rigid body dynamical system from visual observations. Adopting differentiable physics simulation, PIN‑WM can be learned with only few‑shot and task‑agnostic physical interaction trajectories. Further, PIN‑WM is learned with observational loss induced by Gaussian Splatting without needing state estimation. To bridge Sim2Real gaps, we turn the learned PIN‑WM into a group of Digital Cousins via physics‑aware randomizations which perturb physics and rendering parameters to generate diverse and meaningful variations of the PIN‑WM. Extensive evaluations on both simulation and real‑world tests demonstrate that PIN‑WM, enhanced with physics‑aware digital cousins, facilitates learning robust non‑prehensile manipulation skills with Sim2Real transfer, surpassing the Real2Sim2Real state‑of‑the‑arts.

Abstract:
Physics‑Informed Neural Networks (PINNs) have shown promise in solving partial differential equations (PDEs), including the frequency‑domain Helmholtz equation. However, standard training of PINNs using gradient descent (GD) suffers from slow convergence and instability, particularly for high‑frequency wavefields. For scattered acoustic wavefield simulation based on Helmholtz equation, we derive a hybrid optimization framework that accelerates training convergence by embedding a least‑squares (LS) solver directly into the GD loss function. This formulation enables optimal updates for the linear output layer. Our method is applicable with or without perfectly matched layers (PML), and we provide practical tensor‑based implementations for both scenarios. Numerical experiments on benchmark velocity models demonstrate that our approach achieves faster convergence, higher accuracy, and improved stability compared to conventional PINN training. In particular, our results show that the LS‑enhanced method converges rapidly even in cases where standard GD‑based training fails. The LS solver operates on a small normal matrix, ensuring minimal computational overhead and making the method scalable for large‑scale wavefield simulations.

Abstract:
Severe accidents (SAs) in nuclear power plants have been analyzed using thermal‑hydraulic (TH) system codes such as MELCOR and MAAP. These codes efficiently simulate the progression of SAs, while they still have inherent limitations due to their inconsistent finite difference schemes. The use of empirical schemes incorporating both implicit and explicit formulations inherently induces unidirectional coupling in multi‑physics analyses. The objective of this study is to develop a novel numerical method for TH system codes using physics‑informed neural network (PINN). They have shown strength in solving multi‑physics due to the innate feature of neural networks‑automatic differentiation. We propose a node‑assigned PINN (NA‑PINN) that is suitable for the control volume approach‑based system codes. NA‑PINN addresses the issue of spatial governing equation variation by assigning an individual network to each nodalization of the system code, such that spatial information is excluded from both the input and output domains, and each subnetwork learns to approximate a purely temporal solution. In this phase, we evaluated the accuracy of the PINN methods for the hydrodynamic module. In the 6 water tank simulation, PINN and NA‑PINN showed maximum absolute errors of 1.678 and 0.007, respectively. It should be noted that only NA‑PINN demonstrated acceptable accuracy. To the best of the authors' knowledge, this is the first study to successfully implement a system code using PINN. Our future work involves extending NA‑PINN to a multi‑physics solver and developing it in a surrogate manner.

Abstract:
Measuring stress fields in fluids and soft materials is crucial in various fields such as mechanical engineering, medicine, and bioengineering. However, conventional methods that calculate stress fields from velocity fields struggle to measure complex fluids where the stress constitutive equation is unknown. To address this, we propose a novel approach that combines photoelastic measurements ‑‑ which can non‑invasively visualize internal stresses ‑‑ with machine learning to measure stress fields. The machine learning model, which we named physics‑informed convolutional encoder‑decoder (PICED), integrates a convolutional neural network (CNN)‑based encoder‑decoder model with a physics‑informed neural network (PINN). Using this approach, three‑dimensional stress fields can be predicted with high accuracy for multiple interpolated data points in a rectangular channel flow.

Abstract:
Kolmogorov‑Arnold Networks (KANs) have emerged as a promising alternative to Multi‑layer Perceptrons (MLPs) due to their superior function‑fitting abilities in data‑driven modeling. In this paper, we propose a novel framework, DAE‑KAN, for solving high‑index differential‑algebraic equations (DAEs) by integrating KANs with Physics‑Informed Neural Networks (PINNs). This framework not only preserves the ability of traditional PINNs to model complex systems governed by physical laws but also enhances their performance by leveraging the function‑fitting strengths of KANs. Numerical experiments demonstrate that for DAE systems ranging from index‑1 to index‑3, DAE‑KAN reduces the absolute errors of both differential and algebraic variables by 1 to 2 orders of magnitude compared to traditional PINNs. To assess the effectiveness of this approach, we analyze the drift‑off error and find that both PINNs and DAE‑KAN outperform classical numerical methods in controlling this phenomenon. Our results highlight the potential of neural network methods, particularly DAE‑KAN, in solving high‑index DAEs with substantial computational accuracy and generalization, offering a promising solution for challenging partial differential‑algebraic equations.

Abstract:
In this paper, we propose a novel physics‑informed generative learning approach, named RadioDiff‑k^2, for accurate and efficient multipath‑aware radio map (RM) construction. As future wireless communication evolves towards environment‑aware paradigms, the accurate construction of RMs becomes crucial yet highly challenging. Conventional electromagnetic (EM)‑based methods, such as full‑wave solvers and ray‑tracing approaches, exhibit substantial computational overhead and limited adaptability to dynamic scenarios. Although existing neural network (NN) approaches have efficient inferencing speed, they lack sufficient consideration of the underlying physics of EM wave propagation, limiting their effectiveness in accurately modeling critical EM singularities induced by complex multipath environments. To address these fundamental limitations, we propose a novel physics‑inspired RM construction method guided explicitly by the Helmholtz equation, which inherently governs EM wave propagation. Specifically, based on the analysis of partial differential equations (PDEs), we theoretically establish a direct correspondence between EM singularities, which correspond to the critical spatial features influencing wireless propagation, and regions defined by negative wave numbers in the Helmholtz equation. We then design an innovative dual diffusion model (DM)‑based large artificial intelligence framework comprising one DM dedicated to accurately inferring EM singularities and another DM responsible for reconstructing the complete RM using these singularities along with environmental contextual information. Experimental results demonstrate that the proposed RadioDiff‑k^2 framework achieves state‑of‑the‑art (SOTA) performance in both image‑level RM construction and localization tasks, while maintaining inference latency within a few hundred milliseconds.

Abstract:
Due to its ability to work in non‑line‑of‑sight and low‑light environments, radio frequency (RF) imaging technology is expected to bring new possibilities for embodied intelligence and multimodal sensing. However, widely used RF devices (such as Wi‑Fi) often struggle to provide high‑precision electromagnetic measurements and large‑scale datasets, hindering the application of RF imaging technology. In this paper, we combine the ideas of PINN to design the RINN network, using physical constraints instead of true value comparison constraints and adapting it with the characteristics of ubiquitous RF signals, allowing the RINN network to achieve RF imaging using only one sample without phase and with amplitude noise. Our numerical evaluation results show that compared with 5 classic algorithms based on phase data for imaging results, RINN's imaging results based on phaseless data are good, with indicators such as RRMSE (0.11) performing similarly well. RINN provides new possibilities for the universal development of radio frequency imaging technology.

Abstract:
Metasurfaces, sub‑wavelength artificial structures, can control light's amplitude, phase, and polar ization, enabling applications in efficient imaging, holograms, and sensing. Recent years, AI has witnessed remarkable progress and spurred scientific discovery. In metasurface design, optical inverse design has recently emerged as a revolutionary approach. It uses deep learning to create a nonlinear mapping between optical structures and functions, bypassing time‑consuming traditional design and attaining higher accuracy. Yet, current deep‑learning models for optical design face limitations. They often work only for fixed wavelengths and polarizations, and lack universality as input‑output vector size changes may require retraining. There's also a lack of compatibility across different application scenarios. This paper introduces MetasurfaceViT, a revolutionary generic AI model. It leverages a large amount of data using Jones matrices and physics‑informed data augmentation. By pre‑training through masking wavelengths and polarization channels, it can reconstruct full‑wavelength Jones matrices, which will be utilized by fine‑tuning model to enable inverse design. Finally, a tandem workflow appended by a forward prediction network is introduced to evaluate performance. The versatility of MetasurfaceViT with high prediction accuracy will open a new paradigm for optical inverse design.

Abstract:
This position paper argues that the next generation of artificial intelligence in meteorological and climate sciences must transition from fragmented hybrid heuristics toward a unified paradigm of physics‑guided multimodal transformers. While purely data‑driven models have achieved significant gains in predictive accuracy, they often treat atmospheric processes as mere visual patterns, frequently producing results that lack scientific consistency or violate fundamental physical laws. We contend that current ``hybrid'' attempts to bridge this gap remain ad‑hoc and struggle to scale across the heterogeneous nature of meteorological data ranging from satellite imagery to sparse sensor measurements. We argue that the transformer architecture, through its inherent capacity for cross‑modal alignment, provides the only viable foundation for a systematic integration of domain knowledge via physical constraint embedding and physics‑informed loss functions. By advocating for this unified architectural shift, we aim to steer the community away from ``black‑box'' curve fitting and toward AI systems that are inherently falsifiable, scientifically grounded, and robust enough to address the existential challenges of extreme weather and climate change.

Abstract:
We present a parameter‑decoupled superresolution framework for estimating sub‑wavelength separations of passive two‑point sources without requiring prior knowledge or control of the source. Our theoretical foundation circumvents the need to estimate multiple challenging parameters such as partial coherence, brightness imbalance, random relative phase, and photon statistics. A physics‑informed machine learning (ML) model (trained with a standard desktop workstation), synergistically integrating this theory, further addresses practical imperfections including background noise, photon loss, and centroid/orientation misalignment. The integrated parameter‑decoupling superresolution method achieves resolution 14 and more times below the diffraction limit (corresponding to ~ 13.5 nm in optical microscopy) on experimentally generated realistic images with >82% fidelity, performance rivaling state‑of‑the‑art techniques for actively controllable sources. Critically, our method's robustness against source parameter variability and source‑independent noises enables potential applications in realistic scenarios where source control is infeasible, such as astrophysical imaging, live‑cell microscopy, and quantum metrology. This work bridges a critical gap between theoretical superresolution limits and practical implementations for passive systems.

Abstract:
This paper presents a physics‑informed training framework for projection‑based Reduced Order Models (ROMs). We extend the PROM‑ANN architecture by complementing snapshot‑based training with a FEM‑based, discrete physics‑informed residual loss, bridging the gap between traditional projection‑based ROMs and physics‑informed neural networks (PINNs). Unlike conventional PINNs that rely on analytical PDEs, our approach leverages FEM residuals to guide the learning of the ROM approximation manifold. Key contributions include: (1) a parameter‑agnostic, discrete residual loss applicable to non‑linear problems, (2) an architectural modification to PROM‑ANN improving accuracy for fast‑decaying singular values, and (3) an empirical study on the proposed physics informed training process for ROMs. The method is demonstrated on a non‑linear hyperelasticity problem, simulating a rubber cantilever under multi‑axial loads. The main accomplishment in regards to the proposed residual‑based loss is its applicability on non‑linear problems by interfacing with FEM software while maintaining reasonable training times. The modified PROM‑ANN outperforms POD by orders of magnitude in snapshot reconstruction accuracy, while the original formulation is not able to learn a proper mapping for this use‑case. Finally, the application of physics informed training in ANN‑PROM modestly narrows the gap between data reconstruction and ROM accuracy, however it highlights the untapped potential of the proposed residual‑driven optimization for future ROM development. This work underscores the critical role of FEM residuals in ROM construction and calls for further exploration on architectures beyond PROM‑ANN.

Abstract:
Predicting the remaining useful life (RUL) of rotating machinery is critical for industrial safety and maintenance, but existing methods struggle with scarce target‑domain data and unclear degradation dynamics. We propose a Meta‑Learning and Knowledge Discovery‑based Physics‑Informed Neural Network (MKDPINN) to address these challenges. The method first maps noisy sensor data to a low‑dimensional hidden state space via a Hidden State Mapper (HSM). A Physics‑Guided Regulator (PGR) then learns unknown nonlinear PDEs governing degradation evolution, embedding these physical constraints into the PINN framework. This integrates data‑driven and physics‑based approaches. The framework uses meta‑learning, optimizing across source‑domain meta‑tasks to enable few‑shot adaptation to new target tasks. Experiments on industrial data and the C‑MAPSS benchmark show MKDPINN outperforms baselines in generalization and accuracy, proving its effectiveness for RUL prediction under data scarcity

Abstract:
Accurate real‑time modeling of multi‑body dynamical systems is essential for enabling digital twin applications across industries. While many data‑driven approaches aim to learn system dynamics, jointly predicting internal loads and system trajectories remains a key challenge. This dual prediction is especially important for fault detection and predictive maintenance, where internal loads‑such as contact forces‑act as early indicators of faults, reflecting wear or misalignment before affecting motion. These forces also serve as inputs to degradation models (e.g., crack growth), enabling damage prediction and remaining useful life estimation. We propose Equi‑Euler GraphNet, a physics‑informed graph neural network (GNN) that simultaneously predicts internal forces and global trajectories in multi‑body systems. In this mesh‑free framework, nodes represent system components and edges encode interactions. Equi‑Euler GraphNet introduces two inductive biases: (1) an equivariant message‑passing scheme, interpreting edge messages as interaction forces consistent under Euclidean transformations; and (2) a temporal‑aware iterative node update mechanism, based on Euler integration, to capture influence of distant interactions over time. Tailored for cylindrical roller bearings, it decouples ring dynamics from constrained motion of rolling elements. Trained on high‑fidelity multiphysics simulations, Equi‑Euler GraphNet generalizes beyond the training distribution, accurately predicting loads and trajectories under unseen speeds, loads, and configurations. It outperforms state‑of‑the‑art GNNs focused on trajectory prediction, delivering stable rollouts over thousands of time steps with minimal error accumulation. Achieving up to a 200x speedup over conventional solvers while maintaining comparable accuracy, it serves as an efficient reduced‑order model for digital twins, design, and maintenance.

Abstract:
We propose a distinct approach to solving linear and nonlinear differential equations (DEs) on quantum computers by encoding the problem into ground states of effective Hamiltonian operators. Our algorithm relies on constructing such operators in the Chebyshev space, where an effective Hamiltonian is a sum of global differential and data constraints. Once the effective Hamiltonian is formed, solutions of differential equations can be obtained using the ground state preparation techniques (e.g. imaginary‑time evolution and quantum singular value transformation), bypassing variational search. Unlike approaches based on discrete grids, the algorithm enables evaluation of solutions beyond fixed grid points and implements constraints in the physics‑informed way. Our proposal inherits the best traits from quantum machine learning‑based DE solving (compact basis representation, automatic differentiation, nonlinearity) and quantum linear algebra‑based approaches (fine‑grid encoding, provable speed‑up for state preparation), offering a robust strategy for quantum scientific computing in the early fault‑tolerant era.

Abstract:
This tutorial paper focuses on safe physics‑informed machine learning in the context of dynamics and control, providing a comprehensive overview of how to integrate physical models and safety guarantees. As machine learning techniques enhance the modeling and control of complex dynamical systems, ensuring safety and stability remains a critical challenge, especially in safety‑critical applications like autonomous vehicles, robotics, medical decision‑making, and energy systems. We explore various approaches for embedding and ensuring safety constraints, including structural priors, Lyapunov and Control Barrier Functions, predictive control, projections, and robust optimization techniques. Additionally, we delve into methods for uncertainty quantification and safety verification, including reachability analysis and neural network verification tools, which help validate that control policies remain within safe operating bounds even in uncertain environments. The paper includes illustrative examples demonstrating the implementation aspects of safe learning frameworks that combine the strengths of data‑driven approaches with the rigor of physical principles, offering a path toward the safe control of complex dynamical systems.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs). However, their performance heavily relies on the strategy used to select training points. Conventional adaptive sampling methods, such as residual‑based refinement, often require multi‑round sampling and repeated retraining of PINNs, leading to computational inefficiency due to redundant points and costly gradient computations‑particularly in high‑dimensional or high‑order derivative scenarios. To address these limitations, we propose RL‑PINNs, a reinforcement learning(RL)‑driven adaptive sampling framework that enables efficient training with only a single round of sampling. Our approach formulates adaptive sampling as a Markov decision process, where an RL agent dynamically selects optimal training points by maximizing a long‑term utility metric. Critically, we replace gradient‑dependent residual metrics with a computationally efficient function variation as the reward signal, eliminating the overhead of derivative calculations. Furthermore, we employ a delayed reward mechanism to prioritize long‑term training stability over short‑term gains. Extensive experiments across diverse PDE benchmarks, including low‑regular, nonlinear, high‑dimensional, and high‑order problems, demonstrate that RL‑PINNs significantly outperforms existing residual‑driven adaptive methods in accuracy. Notably, RL‑PINNs achieve this with negligible sampling overhead, making them scalable to high‑dimensional and high‑order problems.

Abstract:
Accurately modeling friction in robotics remains a core challenge, as robotics simulators like MuJoCo and PyBullet use simplified friction models or heuristics to balance computational efficiency with accuracy, where these simplifications and approximations can lead to substantial differences between simulated and physical performance. In this paper, we present a physics‑informed friction estimation framework that enables the integration of well‑established friction models with learnable components, requiring only minimal, generic measurement data. Our approach enforces physical consistency yet retains the flexibility to capture complex friction phenomena. We demonstrate, on an underactuated and nonlinear system, that the learned friction models, trained solely on small and noisy datasets, accurately reproduce dynamic friction properties with significantly higher fidelity than the simplified models commonly used in robotics simulators. Crucially, we show that our approach enables the learned models to be transferable to systems they are not trained on. This ability to generalize across multiple systems streamlines friction modeling for complex, underactuated tasks, offering a scalable and interpretable path toward improving friction model accuracy in robotics and control.

Abstract:
The study of the gravitational wave signatures of neutron star oscillations may provide important information of their interior structure and Equation of State (EoS) at high densities. We present a novel technique based on physically informed neural networks (PINNs) to solve the eigenvalue problem associated with normal oscillation modes of neutron stars. The procedure is tested in a simplified scenario, with an analytical solution, that can be used to test the performance and the accuracy of the method. We show that it is possible to get accurate results of both the eigenfrequencies and the eigenfunctions with this scheme. The flexibility of the method and its capability of adapting to complex scenarios may serve in the future as a path to include more physics into these systems.

Abstract:
Low‑light conditions pose significant challenges for both human and machine annotation. This in turn has led to a lack of research into machine understanding for low‑light images and (in particular) videos. A common approach is to apply annotations obtained from high quality datasets to synthetically created low light versions. In addition, these approaches are often limited through the use of unrealistic noise models. In this paper, we propose a new Degradation Estimation Network (DEN), which synthetically generates realistic standard RGB (sRGB) noise without the requirement for camera metadata. This is achieved by estimating the parameters of physics‑informed noise distributions, trained in a self‑supervised manner. This zero‑shot approach allows our method to generate synthetic noisy content with a diverse range of realistic noise characteristics, unlike other methods which focus on recreating the noise characteristics of the training data. We evaluate our proposed synthetic pipeline using various methods trained on its synthetic data for typical low‑light tasks including synthetic noise replication, video enhancement, and object detection, showing improvements of up to 24% KLD, 21% LPIPS, and 62% AP_50‑95, respectively.

Abstract:
The Deep Material Network (DMN) has emerged as a powerful framework for multiscale materials modeling, enabling efficient and accurate prediction of material behavior across different length scales. Unlike conventional data‑driven approaches, the trainable parameters in DMN possess clear physical interpretations‑they encode the geometric characteristics of representative volume elements (RVEs) rather than serving as purely statistical fitting parameters . By employing a hierarchical tree structure, DMN learns the homogenization behavior associated with microstructural geometry. Consequently, it can be trained exclusively on linear elastic datasets while effectively extrapolating to nonlinear responses during online prediction, making it a highly efficient and scalable approach for multiscale simulations. From a broader perspective, DMN can be viewed as a physics‑informed reduced‑order model that captures the essential micromechanical features governing macroscopic behavior. Its hierarchical formulation provides a compact yet interpretable representation of the RVE response, significantly reducing computational costs compared to direct numerical simulations. This review elaborates on the theoretical foundation, training methodology, and recent extensions of DMN, emphasizing its role as a unifying framework that connects data‑driven learning with physically interpretable multiscale modeling.

Abstract:
Physics‑Informed Neural Networks (PINNs) have recently emerged as a promising alternative for solving partial differential equations, offering a mesh‑free framework that incorporates physical laws directly into the learning process. In this study, we explore the application of PINNs for solving unsteady Maxwell's equations and compare their performance with two established numerical methods: the Finite‑Difference Time‑Domain (FDTD) method and a compact Pade scheme with filtering. Three benchmark problems are considered, ranging from 1D free‑space wave propagation to 2D Gaussian pulses in periodic and dielectric media. We assess the effectiveness of convergence‑enhancing strategies for PINNs, including random Fourier features, spatio‑temporal periodicity, and temporal causality training. An ablation study highlights that architectural choices must align with the underlying physics. Additionally, we employ a Neural Tangent Kernel framework to examine the spatio‑temporal convergence behavior of PINNs. Results show that convergence rates correlate with error over time but not in space, revealing a limitation in how training dynamics allocate learning effort. Overall, this study demonstrates that PINNs, when properly configured, can match or surpass traditional solvers in accuracy and flexibility, though challenges remain in addressing spatial inhomogeneity and adapting training to localized complexity.

Abstract:
This study proposes a novel imaging and reconstruction framework for dual‑energy cone‑beam CT (DECBCT) using only two orthogonal X‑ray projections at different energy levels (2V‑DECBCT). The goal is to enable fast and low‑dose DE volumetric imaging with high spectral fidelity and structural accuracy, suitable for DECBCT‑guided radiation therapy. We introduce a framework for 2V‑DECBCT based on physics‑informed dual‑domain diffusion models. A cycle‑domain training strategy is employed to enforce consistency between projection and volume reconstructions through a differentiable physics‑informed module. Furthermore, a spectral‑consistency loss is introduced to preserve inter‑energy contrast during the generative process. The model is trained and evaluated using 4D XCAT phantom data under realistic anatomical motion. The method produces high‑fidelity DECBCT volumes from only two views, accurately preserving anatomical boundaries and suppressing artifacts. Subtraction maps computed from the reconstructed energy volumes show strong visual and numerical agreement with ground truth. This work presents the first diffusion model‑based framework for 2V‑DECBCT reconstruction, demonstrating accurate structural and spectral recovery from extremely sparse inputs.

Abstract:
Image decomposition offers deep insights into the imaging factors of visual data and significantly enhances various advanced computer vision tasks. In this work, we introduce a novel approach to low‑light image enhancement based on decomposed physics‑informed priors. Existing methods that directly map low‑light to normal‑light images in the sRGB color space suffer from inconsistent color predictions and high sensitivity to spectral power distribution (SPD) variations, resulting in unstable performance under diverse lighting conditions. To address these challenges, we introduce a Physics‑informed Color‑aware Transform (PiCat), a learning‑based framework that converts low‑light images from the sRGB color space into deep illumination‑invariant descriptors via our proposed Color‑aware Transform (CAT). This transformation enables robust handling of complex lighting and SPD variations. Complementing this, we propose the Content‑Noise Decomposition Network (CNDN), which refines the descriptor distributions to better align with well‑lit conditions by mitigating noise and other distortions, thereby effectively restoring content representations to low‑light images. The CAT and the CNDN collectively act as a physical prior, guiding the transformation process from low‑light to normal‑light domains. Our proposed PiCat framework demonstrates superior performance compared to state‑of‑the‑art methods across five benchmark datasets.

Abstract:
Power flow (PF) calculations are fundamental to power system analysis to ensure stable and reliable grid operation. The Newton‑Raphson (NR) method is commonly used for PF analysis due to its rapid convergence when initialized properly. However, as power grids operate closer to their capacity limits, ill‑conditioned cases and convergence issues pose significant challenges. This work, therefore, addresses these challenges by proposing strategies to improve NR initialization, hence minimizing iterations and avoiding divergence. We explore three approaches: (i) an analytical method that estimates the basin of attraction using mathematical bounds on voltages, (ii) Two data‑driven models leveraging supervised learning or physics‑informed neural networks (PINNs) to predict optimal initial guesses, and (iii) a reinforcement learning (RL) approach that incrementally adjusts voltages to accelerate convergence. These methods are tested on benchmark systems. This research is particularly relevant for modern power systems, where high penetration of renewables and decentralized generation require robust and scalable PF solutions. In experiments, all three proposed methods demonstrate a strong ability to provide an initial guess for Newton‑Raphson method to converge with fewer steps. The findings provide a pathway for more efficient real‑time grid operations, which, in turn, support the transition toward smarter and more resilient electricity networks.

Abstract:
Numerical solvers for PDEs often struggle to balance computational cost with accuracy, especially in multiscale and time‑dependent systems. Neural operators offer a promising way to accelerate simulations, but their practical deployment is hindered by several challenges: they typically require large volumes of training data generated from high‑fidelity solvers, tend to accumulate errors over time in dynamical settings, and often exhibit poor generalization in multiphysics scenarios. This work introduces a novel hybrid framework that integrates physics‑informed deep operator network with FEM through domain decomposition and leverages numerical analysis for time marching. Our innovation lies in efficient coupling FE and DeepONet subdomains via a Schwarz method, expecting to solve complex and nonlinear regions by a pretrained DeepONet, while the remainder is handled by conventional FE. To address the challenges of dynamic systems, we embed a time stepping scheme directly into the DeepONet, substantially reducing long‑term error propagation. Furthermore, an adaptive subdomain evolution strategy enables the ML‑resolved region to expand dynamically, capturing fine‑scale features without remeshing. Our framework shows accelerated convergence rates (up to 20% improvement in convergence rates compared to conventional FE coupling approaches) while preserving solution fidelity with error margins consistently below 3%. Our study shows that our proposed hybrid solver: (1) reduces computational costs by eliminating fine mesh requirements, (2) mitigates error accumulation in time‑dependent simulations, and (3) enables automatic adaptation to evolving physical phenomena. This work establishes a new paradigm for coupling state of the art physics based and machine learning solvers in a unified framework, offering a robust, reliable, and scalable pathway for high fidelity multiscale simulations.

Abstract:
Solving partial differential equations (PDEs) has been indispensable in scientific and engineering applications. Recently, deep learning methods have been widely used to solve high‑dimensional problems, one of which is the physics‑informed neural network (PINN). Typically, a deep learning method has three main components: a neural network, a loss function, and an optimizer. While the construction of the loss function is rooted in the definition of solution space, how to choose a optimal neural network is somewhat ad hoc, leaving much room for improvement. In the framework of PINN, we propose an unsupervised network architecture search method for solving PDEs, termed PINN‑DARTS, which applies the differentiable architecture search (DARTS) to find the optimal network architecture structure in a given set of neural networks. In this set, the number of layers and the number of neurons in each layer can change. In the searching phase, both network and architecture parameters are updated simultaneously, so the running time is close to that of PINN with a pre‑determined network structure. Unlike available works, our approach is unsupervised and purely based on the PDE residual without any prior usage of solutions. PINN‑DARTS outputs the optimal network structure as well as the associated numerical solution. The performance of PINN‑DARTS is verified on several benchmark PDEs, including elliptic, parabolic, wave, and Burgers' equations. Compared to traditional architecture search methods, PINN‑DARTS achieves significantly higher architectural accuracy. Another interesting observation is that both the solution complexity and the PDE type have a prominent impact on the optimal network architecture. Our study suggests that architectures with uneven widths from layer to layer may have superior performance across different solution complexities and different PDE types.

Abstract:
As autonomous systems become increasingly prevalent in daily life, ensuring their safety is paramount. Control Barrier Functions (CBFs) have emerged as an effective tool for guaranteeing safety; however, manually designing them for specific applications remains a significant challenge. With the advent of deep learning techniques, recent research has explored synthesizing CBFs using neural networks‑commonly referred to as neural CBFs. This paper introduces a novel class of neural CBFs that leverages a physics‑inspired neural network framework by incorporating Zubov's Partial Differential Equation (PDE) within the context of safety. This approach provides a scalable methodology for synthesizing neural CBFs applicable to high‑dimensional systems. Furthermore, by utilizing reciprocal CBFs instead of zeroing CBFs, the proposed framework allows for the specification of flexible, user‑defined safe regions. To validate the effectiveness of the approach, we present case studies on three different systems: an inverted pendulum, autonomous ground navigation, and aerial navigation in obstacle‑laden environments.

Abstract:
This paper presents an improved approach for preserving sharp interfaces in multiphase Lattice Boltzmann Method (LBM) simulations using Physics‑Informed Neural Networks (PINNs). Interface diffusion is a common challenge in multiphase LBM, leading to reduced accuracy in simulating phenomena where interfacial dynamics are critical. We propose a coupled PINN‑LBM framework that maintains interface sharpness while preserving the physical accuracy of the simulation. Our approach is validated through droplet simulations, with quantitative metrics measuring interface width, maximum gradient, phase separation, effective interface width, and interface energy. The enhanced visualization techniques employed in this work clearly demonstrate the superior performance of PINN‑LBM over standard LBM for multiphase simulations, particularly in maintaining well‑defined interfaces throughout the simulation. We provide a comprehensive analysis of the results, showcasing how the neural network integration effectively counteracts numerical diffusion, while maintaining physical consistency with the underlying fluid dynamics.

Abstract:
Estimating forest height from Synthetic Aperture Radar (SAR) images often relies on traditional physical models, which, while interpretable and data‑efficient, can struggle with generalization. In contrast, Deep Learning (DL) approaches lack physical insight. To address this, we propose CoHNet ‑ an end‑to‑end framework that combines the best of both worlds: DL optimized with physics‑informed constraints. We leverage a pre‑trained neural surrogate model to enforce physical plausibility through a unique training loss. Our experiments show that this approach not only improves forest height estimation accuracy but also produces meaningful features that enhance the reliability of predictions.

Abstract:
Physics‑informed neural networks (PINNs) is becoming a popular alternative method for solving partial differential equations (PDEs). However, they require dedicated manual modifications to the hyperparameters of the network, the sampling methods and loss function weights for different PDEs, which reduces the efficiency of the solvers. In this paper, we pro‑ pose a general multi‑stage framework, i.e. BO‑SA‑PINNs to alleviate this issue. In the first stage, Bayesian optimization (BO) is used to select hyperparameters for the training process, and based on the results of the pre‑training, the network architecture, learning rate, sampling points distribution and loss function weights suitable for the PDEs are automatically determined. The proposed hyperparameters search space based on experimental results can enhance the efficiency of BO in identifying optimal hyperparameters. After selecting the appropriate hyperparameters, we incorporate a global self‑adaptive (SA) mechanism the second stage. Using the pre‑trained model and loss information in the second‑stage training, the exponential moving average (EMA) method is employed to optimize the loss function weights, and residual‑based adaptive refinement with distribution (RAR‑D) is used to optimize the sampling points distribution. In the third stage, L‑BFGS is used for stable training. In addition, we introduce a new activation function that enables BO‑SA‑PINNs to achieve higher accuracy. In numerical experiments, we conduct comparative and ablation experiments to verify the performance of the model on Helmholtz, Maxwell, Burgers and high‑dimensional Poisson equations. The comparative experiment results show that our model can achieve higher accuracy and fewer iterations in test cases, and the ablation experiments demonstrate the positive impact of every improvement.

Abstract:
Video imaging is often affected by complex degradations such as blur, noise, and compression artifacts. Traditional restoration methods follow a "single‑task single‑model" paradigm, resulting in poor generalization and high computational cost, limiting their applicability in real‑world scenarios with diverse degradation types. We propose UniFlowRestore, a general video restoration framework that models restoration as a time‑continuous evolution under a prompt‑guided and physics‑informed vector field. A physics‑aware backbone PhysicsUNet encodes degradation priors as potential energy, while PromptGenerator produces task‑relevant prompts as momentum. These components define a Hamiltonian system whose vector field integrates inertial dynamics, decaying physical gradients, and prompt‑based guidance. The system is optimized via a fixed‑step ODE solver to achieve efficient and unified restoration across tasks. Experiments show that UniFlowRestore delivers stateof‑the‑art performance with strong generalization and efficiency. Quantitative results demonstrate that UniFlowRestore achieves state‑of‑the‑art performance, attaining the highest PSNR (33.89 dB) and SSIM (0.97) on the video denoising task, while maintaining top or second‑best scores across all evaluated tasks.

Abstract:
Physics‑Informed Neural Networks (PINNs) have gained widespread popularity for solving inverse and forward problems across a range of scientific and engineering domains. However, most existing PINN frameworks are limited to the Eulerian domain, where physical quantities are described at fixed spatial locations. In this work, we propose a novel PINN‑based framework that couples Eulerian and Lagrangian perspectives by using particle trajectory data to reconstruct Eulerian velocity and pressure fields. We evaluate the performance of our method across three distinct fluid flow scenarios: two‑dimensional external flow past a cylinder, two‑dimensional internal flow in a confined geometry, and three‑dimensional internal flow inside an airplane cabin. In all three cases, we successfully reconstruct the velocity field from Lagrangian particle data. Moreover, for the 2D external and internal flows, we recover the pressure field solely through the physics‑informed learning process, without using any direct pressure measurements. We also conduct a sensitivity analysis to understand the effects of temporal resolution and particle count on the reconstruction accuracy. Our results show that smaller time‑step sizes significantly improve the predictions, while the total number of particles has a comparatively smaller influence. These findings establish the potential of our coupled Eulerian‑Lagrangian PINN framework as a powerful tool for enhancing experimental methods such as Particle Tracking Velocimetry (PTV). Looking ahead, this approach may be extended to infer hidden quantities such as pressure in three‑dimensional flows or material properties like viscosity, opening new avenues for data‑driven fluid dynamics in complex geometries.

Abstract:
We present a deep learning approach for computing multi‑phase solutions to the semiclassical limit of the Schrödinger equation. Traditional methods require deriving a multi‑phase ansatz to close the moment system of the Liouville equation, a process that is often computationally intensive and impractical. Our method offers an efficient alternative by introducing a novel two‑stage neural network framework to close the 2N× 2N moment system, where N represents the number of phases in the solution ansatz. In the first stage, we train neural networks to learn the mapping between higher‑order moments and lower‑order moments (along with their derivatives). The second stage incorporates physics‑informed neural networks (PINNs), where we substitute the learned higher‑order moments to systematically close the system. We provide theoretical guarantees for the convergence of both the loss functions and the neural network approximations. Numerical experiments demonstrate the effectiveness of our method for one‑ and two‑dimensional problems with various phase numbers N in the multi‑phase solutions. The results confirm the accuracy and computational efficiency of the proposed approach compared to conventional techniques.

Abstract:
Physics‑informed neural operators offer a powerful framework for learning solution operators of partial differential equations (PDEs) by combining data and physics losses. However, these physics losses rely on derivatives. Computing these derivatives remains challenging, with spectral and finite difference methods introducing approximation errors due to finite resolution. Here, we propose the mollified graph neural operator (mGNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries. This enhancement enables efficient training on irregular grids and varying geometries while allowing seamless evaluation of physics losses at randomly sampled points for improved generalization. For a PDE example on regular grids, mGNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences, although training was slower. It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough. On these unstructured point clouds, mGNO leads to errors that are consistently 2 orders of magnitude lower than machine learning baselines (Meta‑PDE, which accelerates PINNs) for comparable runtimes, and also delivers speedups from 1 to 3 orders of magnitude compared to the numerical solver for similar accuracy. mGNOs can also be used to solve inverse design and shape optimization problems on complex geometries.

Abstract:
In this article we develop a Physics Informed Neural Network (PINN) approach to simulate ice sheet dynamics governed by the Shallow Ice Approximation. This problem takes the form of a time‑dependent parabolic obstacle problem. Prior work has used this approach to address the stationary obstacle problem and here we extend it to the time dependent problem. Through comprehensive 1D and 2D simulations, we validate the model's effectiveness in capturing complex free‑boundary conditions. By merging traditional mathematical modeling with cutting‑edge deep learning methods, this approach provides a scalable and robust solution for predicting temporal variations in ice thickness. To illustrate this approach in a real world setting, we simulate the dynamics of the Devon Ice Cap, incorporating aerogeophysical data from 2000 and 2018.

Abstract:
The flight time of multirotor unmanned aerial vehicles (UAVs) is typically constrained by their high power consumption. Tethered power systems present a viable solution to extend flight times while maintaining the advantages of multirotor UAVs, such as hover capability and agility. This paper addresses the critical aspect of cable selection for tether‑powered multirotor UAVs, considering both hover and forward flight. Existing research often overlooks the trade‑offs between cable mass, power losses, and system constraints. We propose a novel methodology to optimize cable selection, accounting for thrust requirements and power efficiency across various flight conditions. The approach combines physics‑informed modeling with system identification to combine hover and forward flight dynamics, incorporating factors such as motor efficiency, tether resistance, and aerodynamic drag. This work provides an intuitive and practical framework for optimizing tethered UAV designs, ensuring efficient power transmission and flight performance. Thus allowing for better, safer, and more efficient tethered drones.

Abstract:
Accurate separation of P‑ and S‑waves is essential for multi‑component seismic data processing, as it helps eliminate interference between wave modes during imaging or inversion, which leads to high‑accuracy results. Traditional methods for separating P‑ and S‑waves rely on the Christoffel equation to compute the polarization direction of the waves in the wavenumber domain, which is computationally expensive. Although machine learning has been employed to improve the computational efficiency of the separation process, most methods still require supervised learning with labeled data, which is often unavailable for field data. To address this limitation, we propose a wavefield separation technique based on the physics‑informed neural network (PINN). This unsupervised machine learning approach is applicable to unlabeled data. Furthermore, the trained PINN model provides a mesh‑free numerical solution that effectively captures wavefield features at multiple scales. Numerical tests demonstrate that the proposed PINN‑based separation method can accurately separate P‑ and S‑waves in both homogeneous and heterogeneous media.

Abstract:
Land surface temperature (LST) retrieval from remote sensing data is pivotal for analyzing climate processes and surface energy budgets. However, LST retrieval is an ill‑posed inverse problem, which becomes particularly severe when only a single band is available. In this paper, we propose a deeply coupled framework integrating mechanistic modeling and machine learning to enhance the accuracy and generalizability of single‑channel LST retrieval. Training samples are generated using a physically‑based radiative transfer model and a global collection of 5810 atmospheric profiles. A physics‑informed machine learning framework is proposed to systematically incorporate the first principles from classical physical inversion models into the learning workflow, with optimization constrained by radiative transfer equations. Global validation demonstrated a 30% reduction in root‑mean‑square error versus standalone methods. Under extreme humidity, the mean absolute error decreased from 4.87 K to 2.29 K (53% improvement). Continental‑scale tests across five continents confirmed the superior generalizability of this model.

Abstract:
Physics‑Informed Kolmogorov‑Arnold Networks (PIKANs) are gaining attention as an effective counterpart to the original multilayer perceptron‑based Physics‑Informed Neural Networks (PINNs). Both representation models can address inverse problems and facilitate gray‑box system identification. However, a comprehensive understanding of their performance in terms of accuracy and speed remains underexplored. In particular, we introduce a modified PIKAN architecture, tanh‑cPIKAN, which is based on Chebyshev polynomials for parametrization of the univariate functions with an extra nonlinearity for enhanced performance. We then present a systematic investigation of how choices of the optimizer, representation, and training configuration influence the performance of PINNs and PIKANs in the context of systems pharmacology modeling. We benchmark a wide range of first‑order, second‑order, and hybrid optimizers, including various learning rate schedulers. We use the new Optax library to identify the most effective combinations for learning gray‑boxes under ill‑posed, non‑unique, and data‑sparse conditions. We examine the influence of model architecture (MLP vs. KAN), numerical precision (single vs. double), the need for warm‑up phases for second‑order methods, and sensitivity to the initial learning rate. We also assess the optimizer scalability for larger models and analyze the trade‑offs introduced by JAX in terms of computational efficiency and numerical accuracy. Using two representative systems pharmacology case studies ‑ a pharmacokinetics model and a chemotherapy drug‑response model ‑ we offer practical guidance on selecting optimizers and representation models/architectures for robust and efficient gray‑box discovery. Our findings provide actionable insights for improving the training of physics‑informed networks in biomedical applications and beyond.

Abstract:
In this work, we propose the Residual‑Weighted Physics‑Informed Neural Network (RW‑PINN), a new method designed to enhance the accuracy of Physics‑Informed Neural Network (PINN) based algorithms. We construct a deep learning framework with two residual‑weighting schemes to solve reaction diffusion equations and evaluate its performance on both forward and inverse problems. The approach computes weights proportional to the PDE residuals, rescales them, and incorporates these scaled residuals into the loss function, leading to more stable training. Furthermore, we establish generalized error bounds that account for training and quadrature errors, and we analyze the convergence and stability of the method. The proposed algorithms are validated through numerical experiments on nonlinear equations, supported by statistical error analysis. To further demonstrate the effectiveness of our methodology, we implemented PINN‑based forward and inverse frameworks for the nonlinear equations and conducted a comparative analysis with the proposed RW‑PINN approach.

Abstract:
Central idea: To obtain the interaction potential using the inverse scattering method, we have employed the Physics‑Informed Machine Learning (PIML) approach. In this framework, the machine learning algorithm is guided by the underlying physical laws, enabling the accurate extraction of the inverse scattering potential from the elastic scattering data. Methodology: As a reference potential, a combination of three smoothly joined Morse functions has been utilized, characterized by ten model parameters. These parameters are optimized in an iterative fashion using a Genetic Algorithm to ensure the best fit to the phase shifts extracted from the experimental scattering data. The process of optimization is guided by the computed scattering phase shifts by solving the phase equation using 5th order RK‑method for the reference potential in each iteration Results: Our approach yields inverse potentials for both single and multi channel scattering. Using the Scattering Phase Shifts obtained from these inverse potentials, we calculate the partial cross‑section to determine the resonance energies and decay width. The obtain values of resonance energies and decay width for 3D1, 3D2 and 3D3 states of alpha‑deuteron are in correspondence with the experimental results. Conclusion: It can be concluded that our machine learning‑based approach for constructing the inverse potential offers a novel and complementary technique to existing direct methods.

Abstract:
Kolmogorov‑Arnold Networks (KANs) have gained attention as an alternative to traditional multilayer perceptrons (MLPs) for deep learning applications in computational physics, particularly for solving inverse problems with sparse data, as exemplified by the physics‑informed Kolmogorov‑Arnold network (PIKAN). However, the capability of KANs to simultaneously solve inverse problems over multiple irregular geometries within a single training run remains unexplored. To address this gap, we introduce the physics‑informed Kolmogorov‑Arnold PointNet (PI‑KAN‑PointNet), in which shared KANs are integrated into the PointNet architecture to capture the geometric features of computational domains. The loss function comprises the squared residuals of the governing equations, computed via automatic differentiation, along with sparse observations and partially known boundary conditions. We construct shared KANs using Jacobi polynomials and investigate their performance by considering Jacobi polynomials of different degrees and types in terms of both computational cost and prediction accuracy. As a benchmark test case, we consider natural convection in a square enclosure with a cylinder, where the cylinder's shape varies across a dataset of 135 geometries. PI‑KAN‑PointNet offers two main advantages. First, it overcomes the limitation of current PIKANs, which are restricted to solving only a single computational domain per training run, thereby reducing computational costs. Second, when comparing the performance of PI‑KAN‑PointNet with that of the physics‑informed PointNet using MLPs, we observe that, with approximately the same number of trainable parameters and comparable computational cost in terms of the number of epochs, training time per epoch, and memory usage, PI‑KAN‑PointNet yields more accurate predictions, particularly for values on unknown boundary conditions involving nonsmooth geometries.

Abstract:
In robotics, control barrier function (CBF)‑based safety filters are commonly used to enforce state constraints. A critical challenge arises when the relative degree of the CBF varies across the state space. This variability can create regions within the safe set where the control input becomes unconstrained. When implemented as a safety filter, this may result in chattering near the safety boundary and ultimately compromise system safety. To address this issue, we propose a novel approach for CBF synthesis by formulating it as solving a set of boundary value problems. The solutions to the boundary value problems are determined using physics‑informed neural networks (PINNs). Our approach ensures that the synthesized CBFs maintain a constant relative degree across the set of admissible states, thereby preventing unconstrained control scenarios. We illustrate the approach in simulation and further verify it through real‑world quadrotor experiments, demonstrating its effectiveness in preserving desired system safety properties.

Abstract:
Accurately predicting fluid dynamics and evolution has been a long‑standing challenge in physical sciences. Conventional deep learning methods often rely on the nonlinear modeling capabilities of neural networks to establish mappings between past and future states, overlooking the fluid dynamics, or only modeling the velocity field, neglecting the coupling of multiple physical quantities. In this paper, we propose a new physics‑informed learning approach that incorporates coupled physical quantities into the prediction process to assist with forecasting. Central to our method lies in the discretization of physical equations, which are directly integrated into the model architecture and loss function. This integration enables the model to provide robust, long‑term future predictions. By incorporating physical equations, our model demonstrates temporal extrapolation and spatial generalization capabilities. Experimental results show that our approach achieves the state‑of‑the‑art performance in spatiotemporal prediction across both numerical simulations and real‑world extreme‑precipitation nowcasting benchmarks.

Abstract:
A physics‑constrained neural network is presented for predicting the optical response of metasurfaces. Our approach incorporates physical laws directly into the neural network architecture and loss function, addressing critical challenges in the modeling of metasurfaces. Unlike methods that require specialized weighting strategies or separate architectural branches to handle different data regimes and phase wrapping discontinuities, this unified approach effectively addresses phase discontinuities, energy conservation constraints, and complex gap‑dependent behavior. We implement sine‑cosine phase representation with Euclidean normalization as a non‑trainable layer within the network, enabling the model to account for the periodic nature of phase while enforcing the mathematical constraint \sin^2 ϕ+ \cos^2 ϕ= 1. A Euclidean distance‑based loss function in the sine‑cosine space ensures a physically meaningful error metric while preventing discontinuity issues. The model achieves good, consistent performance (e.g., coefficient of determinations above 0.9) with small, imbalanced datasets of 580 and 1075 data points, compared to several thousand typically required by alternative approaches. This physics‑informed approach preserves physical interpretability while reducing reliance on large datasets and could be extended to systems involving periodic or wrapped quantities.

Abstract:
Physics‑informed machine learning (PIML) provides a promising solution for building energy modeling and can serve as a virtual environment to enable reinforcement learning (RL) agents to interact and learn. However, challenges remain in efficiently integrating physics priors, evaluating the effectiveness of physics constraints, balancing model accuracy and physics consistency, and enabling real‑world implementation. To address these gaps, this study introduces a Physics‑Informed Modularized Neural Network (PI‑ModNN), which incorporates physics priors through a physics‑informed model structure, loss functions, and hard constraints. A new evaluation metric called "temperature response violation" is developed to quantify the physical consistency of data‑driven building dynamic models under varying control inputs and training data sizes. Additionally, a physics prior evaluation framework based on rule importance is proposed to assess the contribution of each individual physics prior, offering guidance on selecting appropriate PIML techniques. Results indicate that incorporating physical priors does not always improve model performance; inappropriate priors may decrease model accuracy and consistency. However, hard constraints are effective in enforcing model consistency. Furthermore, we present a general workflow for developing control‑oriented PIML models and integrating them with deep reinforcement learning (DRL). Following this framework, a case study implementing DRL in an office space over three months demonstrates potential energy savings of 31.4%. Finally, we provide a general guideline for integrating data‑driven models with advanced building control through a four‑step evaluation framework, paving the way for reliable and scalable deployment of advanced building controls.

Abstract:
We implement physics‑informed neural networks (PINNs) to solve the time‑independent Schrödinger equation for three canonical one‑dimensional quantum potentials: an infinite square well, a finite square well, and a finite barrier. The PINN models incorporate trial wavefunctions that exactly satisfy boundary conditions (Dirichlet zeros at domain boundaries), and they optimize a loss functional combining the PDE residual with a normalization constraint. For the infinite well, the ground‑state energy is known (E = pi^2 in dimensionless units) and held fixed in training, whereas for the finite well and barrier, the eigenenergy is treated as a trainable parameter. We use fully‑connected neural networks with smooth activation functions to represent the wavefunction and demonstrate that PINNs can learn the ground‑state eigenfunctions and eigenvalues for these quantum systems. The results show that the PINN‑predicted wavefunctions closely match analytical solutions or expected behaviors, and the learned eigenenergies converge to known values. We present training logs and convergence of the energy parameter, as well as figures comparing the PINN solutions to exact results. The discussion addresses the performance of PINNs relative to traditional numerical methods, highlighting challenges such as convergence to the correct eigenvalue, sensitivity to initialization, and the difficulty of modeling discontinuous potentials. We also discuss the importance of the normalization term to resolve the scaling ambiguity of the wavefunction. Finally, we conclude that PINNs are a viable approach for quantum eigenvalue problems, and we outline future directions including extensions to higher‑dimensional and time‑dependent Schrödinger equations.

Abstract:
Heat exchangers (HEXs) play a central role in process industries for thermal energy transfer. Fouling, the gradual accumulation of solids on heat transfer surfaces, causes a time‑varying decrease in the overall heat transfer coefficient (U(t)), significantly impacting the efficiency of heat transfer. Good estimation and modeling of fouling (the heat transfer coefficient) will lead to better fouling mitigation strategies. This study investigates the identifiability of the time‑varying U(t) in HEXs from closed‑loop operational data, without external excitation of reference signals or knowledge of the controller parameters. We establish that while the complete system model cannot be identified under these given constraints, the time‑varying heat transfer coefficient U(t) remains identifiable. Further, we propose a neural network based architecture, called (Per‑PINN), for estimation and modeling the heat transfer coefficient from the closed‑loop system data. This Per‑PINN model is shown to perform better than the existing Physics‑Informed Neural Networks (PINN) based models for inverse parameter learning as it inherently fixes the underlying physical equations and learns only the time‑varying parameter U(t).

Abstract:
Parameter estimation for differential equations from measured data is an inverse problem prevalent across quantitative sciences. Physics‑Informed Neural Networks (PINNs) have emerged as effective tools for solving such problems, especially with sparse measurements and incomplete system information. However, PINNs face convergence issues, stability problems, overfitting, and complex loss function design. Here we introduce PINNverse, a training paradigm that addresses these limitations by reformulating the learning process as a constrained differential optimization problem. This approach achieves a dynamic balance between data loss and differential equation residual loss during training while preventing overfitting. PINNverse combines the advantages of PINNs with the Modified Differential Method of Multipliers to enable convergence on any point on the Pareto front. We demonstrate robust and accurate parameter estimation from noisy data in four classical ODE and PDE models from physics and biology. Our method enables accurate parameter inference also when the forward problem is expensive to solve.

Abstract:
Accurate epidemic forecasting is crucial for effective disease control and prevention. Traditional compartmental models often struggle to estimate temporally and spatially varying epidemiological parameters, while deep learning models typically overlook disease transmission dynamics and lack interpretability in the epidemiological context. To address these limitations, we propose a novel Causal Spatiotemporal Graph Neural Network (CSTGNN), a hybrid framework that integrates a Spatio‑Contact SIR model with Graph Neural Networks (GNNs) to capture the spatiotemporal propagation of epidemics. Inter‑regional human mobility exhibits continuous and smooth spatiotemporal patterns, leading to adjacent graph structures that share underlying mobility dynamics. To model these dynamics, we employ an adaptive static connectivity graph to represent the stable components of human mobility and utilize a temporal dynamics model to capture fluctuations within these patterns. By integrating the adaptive static connectivity graph with the temporal dynamics graph, we construct a dynamic graph that encapsulates the comprehensive properties of human mobility networks. Additionally, to capture temporal trends and variations in infectious disease spread, we introduce a temporal decomposition model to handle temporal dependence. This model is then integrated with a dynamic graph convolutional network for epidemic forecasting. We validate our model using real‑world datasets at the provincial level in China and the state level in Germany. Extensive studies demonstrate that our method effectively models the spatiotemporal dynamics of infectious diseases, providing a valuable tool for forecasting and intervention strategies. Furthermore, analysis of the learned parameters offers insights into disease transmission mechanisms, enhancing the interpretability and practical applicability of our model.

Abstract:
Data centers (DCs) as mission‑critical infrastructures are pivotal in powering the growth of artificial intelligence (AI) and the digital economy. The evolution from Internet DC to AI DC has introduced new challenges in operating and managing data centers for improved business resilience and reduced total cost of ownership. As a result, new paradigms, beyond the traditional approaches based on best practices, must be in order for future data centers. In this research, we propose and develop a novel Physical AI (PhyAI) framework for advancing DC operations and management. Our system leverages the emerging capabilities of state‑of‑the‑art industrial products and our in‑house research and development. Specifically, it presents three core modules, namely: 1) an industry‑grade in‑house simulation engine to simulate DC operations in a highly accurate manner, 2) an AI engine built upon NVIDIA PhysicsNemo for the training and evaluation of physics‑informed machine learning (PIML) models, and 3) a digital twin platform built upon NVIDIA Omniverse for our proposed 5‑tier digital twin framework. This system presents a scalable and adaptable solution to digitalize, optimize, and automate future data center operations and management, by enabling real‑time digital twins for future data centers. To illustrate its effectiveness, we present a compelling case study on building a surrogate model for predicting the thermal and airflow profiles of a large‑scale DC in a real‑time manner. Our results demonstrate its superior performance over traditional time‑consuming Computational Fluid Dynamics/Heat Transfer (CFD/HT) simulation, with a median absolute temperature prediction error of 0.18 °C. This emerging approach would open doors to several potential research directions for advancing Physical AI in future DC operations.

Abstract:
Scientific machine learning is an emerging field that broadly describes the combination of scientific computing and machine learning to address challenges in science and engineering. Within the context of differential equations, this has produced highly influential methods, such as neural ordinary differential equations (NODEs). Recent works extend this line of research to consider neural differential‑algebraic systems of equations (DAEs), where some unknown relationships within the DAE are learned from data. Training neural DAEs, similarly to neural ODEs, is computationally expensive, as it requires the solution of a DAE for every parameter update. Further, the rigorous consideration of algebraic constraints is difficult within common deep learning training algorithms such as stochastic gradient descent. In this work, we apply the simultaneous approach to neural DAE problems, resulting in a fully discretized nonlinear optimization problem, which is solved to local optimality and simultaneously obtains the neural network parameters and the solution to the corresponding DAE. We extend recent work demonstrating the simultaneous approach for neural ODEs, by presenting a general framework to solve neural DAEs, with explicit consideration of hybrid models, where some components of the DAE are known, e.g. physics‑informed constraints. Furthermore, we present a general strategy for improving the performance and convergence of the nonlinear programming solver, based on solving an auxiliary problem for initialization and approximating Hessian terms. We achieve promising results in terms of accuracy, model generalizability and computational cost, across different problem settings such as sparse data, unobserved states and multiple trajectories. Lastly, we provide several promising future directions to improve the scalability and robustness of our approach.

Abstract:
Foundation models have demonstrated strong reasoning and generalization capabilities in driving‑related tasks, including scene understanding, planning, and control. However, they still face challenges in hallucinations, uncertainty, and long inference latency. While existing foundation models have general knowledge of avoiding collisions, they often lack transportation‑specific safety knowledge. To overcome these limitations, we introduce LetsPi, a physics‑informed, dual‑phase, knowledge‑driven framework for safe, human‑like trajectory planning. To prevent hallucinations and minimize uncertainty, this hybrid framework integrates Large Language Model (LLM) reasoning with physics‑informed social force dynamics. LetsPi leverages the LLM to analyze driving scenes and historical information, providing appropriate parameters and target destinations (goals) for the social force model, which then generates the future trajectory. Moreover, the dual‑phase architecture balances reasoning and computational efficiency through its Memory Collection phase and Fast Inference phase. The Memory Collection phase leverages the physics‑informed LLM to process and refine planning results through reasoning, reflection, and memory modules, storing safe, high‑quality driving experiences in a memory bank. Surrogate safety measures and physics‑informed prompt techniques are introduced to enhance the LLM's knowledge of transportation safety and physical force, respectively. The Fast Inference phase extracts similar driving experiences as few‑shot examples for new scenarios, while simplifying input‑output requirements to enable rapid trajectory planning without compromising safety. Extensive experiments using the HighD dataset demonstrate that LetsPi outperforms baseline models across five safety metrics.See PDF for project Github link.

Abstract:
Recently, data‑driven simulators based on graph neural networks have gained attention in modeling physical systems on unstructured meshes. However, they struggle with long‑range dependencies in fluid flows, particularly in refined mesh regions. This challenge, known as the 'over‑squashing' problem, hinders information propagation. While existing graph rewiring methods address this issue to some extent, they only consider graph topology, overlooking the underlying physical phenomena. We propose Physics‑Informed Ollivier‑Ricci Flow (PIORF), a novel rewiring method that combines physical correlations with graph topology. PIORF uses Ollivier‑Ricci curvature (ORC) to identify bottleneck regions and connects these areas with nodes in high‑velocity gradient nodes, enabling long‑range interactions and mitigating over‑squashing. Our approach is computationally efficient in rewiring edges and can scale to larger simulations. Experimental results on 3 fluid dynamics benchmark datasets show that PIORF consistently outperforms baseline models and existing rewiring methods, achieving up to 26.2 improvement.

Abstract:
We propose an adaptive randomized truncation estimator for Krylov subspace methods that optimizes the trade‑off between the solution variance and the computational cost, while remaining unbiased. The estimator solves a constrained optimization problem to compute the truncation probabilities on the fly, with minimal computational overhead. The problem has a closed‑form solution when the improvement of the deterministic algorithm satisfies a diminishing returns property. We prove that obtaining the optimal adaptive truncation distribution is impossible in the general case. Without the diminishing return condition, our estimator provides a suboptimal but still unbiased solution. We present experimental results in GP hyperparameter training and competitive physics‑informed neural networks problem to demonstrate the effectiveness of our approach.

Abstract:
The degree of polymerization (DP) is one of the methods for estimating the aging of the polymer based insulation systems, such as cellulose insulation in power components. The main degradation mechanisms in polymers are hydrolysis, pyrolysis, and oxidation. These mechanisms combined cause a reduction of the DP. However, the data availability for these types of problems is usually scarce. This study analyzes insulation aging using cellulose degradation data from power transformers. The aging problem for the cellulose immersed in mineral oil inside power transformers is modeled with ordinary differential equations (ODEs). We recover the governing equations of the degradation system using Physics‑Informed Neural Networks (PINNs) and symbolic regression. We apply PINNs to discover the Arrhenius equation's unknown parameters in the Ekenstam ODE describing cellulose contamination content and the material aging process related to temperature for synthetic data and real DP values. A modification of the Ekenstam ODE is given by Emsley's system of ODEs, where the rate constant expressed by the Arrhenius equation decreases in time with the new formulation. We use PINNs and symbolic regression to recover the functional form of one of the ODEs of the system and to identify an unknown parameter.

Abstract:
Recent works on the application of Physics‑Informed Neural Networks to traffic density estimation have shown to be promising for future developments due to their robustness to model errors and noisy data. In this paper, we introduce a methodology for online approximation of the traffic density using measurements from probe vehicles in two settings: one using the Greenshield model and the other considering a high‑fidelity traffic simulation. The proposed method continuously estimates the real‑time traffic density in space and performs model identification with each new set of measurements. The density estimate is updated in almost real‑time using gradient descent and adaptive weights. In the case of full model knowledge, the resulting algorithm has similar performance to the classical open‑loop one. However, in the case of model mismatch, the iterative solution behaves as a closed‑loop observer and outperforms the baseline method. Similarly, in the high‑fidelity setting, the proposed algorithm correctly reproduces the traffic characteristics.

Abstract:
The unprecedented X‑ray flux density provided by modern X‑ray sources offers new spatiotemporal possibilities for X‑ray imaging of fast dynamic processes. Approaches to exploit such possibilities often result in either i) a limited number of projections or spatial information due to limited scanning speed, as in time‑resolved tomography, or ii) a limited number of time points, as in stroboscopic imaging, making the reconstruction problem ill‑posed and unlikely to be solved by classical reconstruction approaches. 4D reconstruction from such data requires sample priors, which can be included via deep learning (DL). State‑of‑the‑art 4D reconstruction methods for X‑ray imaging combine the power of AI and the physics of X‑ray propagation to tackle the challenge of sparse views. However, most approaches do not constrain the physics of the studied process, i.e., a full physical model. Here we present 4D physics‑informed optimized neural implicit X‑ray imaging (4D‑PIONIX), a novel physics‑informed 4D X‑ray image reconstruction method combining the full physical model and a state‑of‑the‑art DL‑based reconstruction method for 4D X‑ray imaging from sparse views. We demonstrate and evaluate the potential of our approach by retrieving 4D information from ultra‑sparse spatiotemporal acquisitions of simulated binary droplet collisions, a relevant fluid dynamic process. We envision that this work will open new spatiotemporal possibilities for various 4D X‑ray imaging modalities, such as time‑resolved X‑ray tomography and more novel sparse acquisition approaches like X‑ray multi‑projection imaging, which will pave the way for investigations of various rapid 4D dynamics, such as fluid dynamics and composite testing.

Abstract:
We solve the anharmonic oscillator with physics‑informed renormalisation group (PIRG) flows, with an emphasis on the weak coupling regime with its instanton‑dominated tunnelling processes. We show that the instanton physics behind the exponential decay of the energy gap is already covered in the first order of the derivative expansion of the PIRG. The crucial new ingredients in the present analysis are the use of the ground state expansion within PIRG flows, as well as precision numerics based on Galerkin methods. Our result a_\mathrminst = 1.910(2) for the decay constant is in quantitative agreement with the analytic one, a_\mathrminst = 1.886 with a deviation of 1%. This illustrates very impressively the capacity of the PIRG for fully capturing non‑perturbative physics already in relatively simple approximations.

Abstract:
In this paper, we propose the Adaptive Movement Sampling Physics‑Informed Residual Network (AM‑PIRN) to address challenges in solving nonlinear option pricing PDE models, where solutions often exhibit significant curvature or shock waves over time. The AM‑PIRN architecture is designed to concurrently minimize PDE residuals and achieve high‑fidelity option price approximations by dynamically redistributing training points based on evolving PDE residuals, while maintaining a fixed total number of points. To enhance stability and training efficiency, we integrate a ResNet backbone, replacing conventional fully connected neural networks used in Physics‑Informed Neural Networks (PINNs). Numerical experiments across nonlinear option pricing models demonstrate that AM‑PIRN outperforms PINN, RAM‑PINN, and WAM‑PINN in both resolving PDE constraints and accurately estimating option prices. The method's advantages are particularly pronounced in complex or multi‑dimensional models, where its adaptive sampling and robust architecture effectively mitigate challenges posed by sharp gradients and high nonlinearity.

Abstract:
Neural network‑based methods are effective for solving equilibria in Mean‑Field Games (MFGs), particularly in high‑dimensional settings. However, solving the coupled partial differential equations (PDEs) in MFGs limits their applicability since solving coupled PDEs is computationally expensive. Additionally, modifying boundary conditions, such as the initial state distribution or terminal value function, necessitates extensive retraining, reducing scalability. To address these challenges, we propose a generalized framework, PIONM (Physics‑Informed Neural Operator NF‑MKV Net), which leverages physics‑informed neural operators to solve MFGs equations. PIONM utilizes neural operators to compute MFGs equilibria for arbitrary boundary conditions. The method encodes boundary conditions as input features and trains the model to align them with density evolution, modeled using discrete‑time normalizing flows. Once trained, the algorithm efficiently computes the density distribution at any time step for modified boundary condition, ensuring efficient adaptation to different boundary conditions in MFGs equilibria. Unlike traditional MFGs methods constrained by fixed coefficients, PIONM efficiently computes equilibria under varying boundary conditions, including obstacles, diffusion coefficients, initial densities, and terminal functions. PIONM can adapt to modified conditions while preserving density distribution constraints, demonstrating superior scalability and generalization capabilities compared to existing methods.

Abstract:
A physics‑constrained deep learning surrogate that predicts the exponential ``avalanche'' growth rate of runaway electrons (REs) for a plasma containing partially ionized impurities is developed. Specifically, a physics‑informed neural network (PINN) that learns the adjoint of the relativistic Fokker‑Planck equation in steady‑state is derived, enabling a rapid surrogate of the RE avalanche for a broad range of plasma parameters, motivating a path towards an ML‑accelerated integrated description of a tokamak disruption. A steady‑state power balance equation together with atomic physics data is embedded directly into the PINN, thus limiting the PINN to train across physically consistent temperatures and charge state distributions. This restricted training domain enables accurate predictions of the PINN while drastically reducing the computational cost of training the model. In addition, a novel closure for the relativistic electron population used when evaluating the secondary source of REs is developed that enables improved accuracy compared to a Rosenbluth‑Putvinski source. The avalanche surrogate is verified against Monte Carlo simulations, where it is shown to accurately predict the RE avalanche growth rate across a broad range of plasma parameters encompassing distinct tokamak disruption scenarios.

Abstract:
The rapid advancement of foundation models has revolutionized visual representation learning in a self‑supervised manner. However, their application in remote sensing (RS) remains constrained by a fundamental gap: existing models predominantly handle single or limited modalities, overlooking the inherently multi‑modal nature of RS observations. Optical, synthetic aperture radar (SAR), and multi‑spectral data offer complementary insights that significantly reduce the inherent ambiguity and uncertainty in single‑source analysis. To bridge this gap, we introduce RingMoE, a unified multi‑modal RS foundation model with 14.7 billion parameters, pre‑trained on 400 million multi‑modal RS images from nine satellites. RingMoE incorporates three key innovations: (1) A hierarchical Mixture‑of‑Experts (MoE) architecture comprising modal‑specialized, collaborative, and shared experts, effectively modeling intra‑modal knowledge while capturing cross‑modal dependencies to mitigate conflicts between modal representations; (2) Physics‑informed self‑supervised learning, explicitly embedding sensor‑specific radiometric characteristics into the pre‑training objectives; (3) Dynamic expert pruning, enabling adaptive model compression from 14.7B to 1B parameters while maintaining performance, facilitating efficient deployment in Earth observation applications. Evaluated across 23 benchmarks spanning six key RS tasks (i.e., classification, detection, segmentation, tracking, change detection, and depth estimation), RingMoE outperforms existing foundation models and sets new SOTAs, demonstrating remarkable adaptability from single‑modal to multi‑modal scenarios. Beyond theoretical progress, it has been deployed and trialed in multiple sectors, including emergency response, land management, marine sciences, and urban planning.

Abstract:
Optical nanoscopy is crucial in life and materials sciences, revealing subtle cellular processes and nanomaterial properties. Scattering‑type Scanning Near‑field Optical Microscopy (s‑SNOM) provides nanoscale resolution, relying on the interactions taking place between a laser beam, a sharp tip and the sample. The Atomic Force Microscope (AFM) is a fundamental part of an s‑SNOM system, providing the necessary probe‑sample feedback mechanisms for data acquisition. In this Letter, we demonstrate that s‑SNOM data can be partially inferred from AFM images. We first show that a generative artificial intelligence (AI) model (pix2pix) can generate synthetic s‑SNOM data from experimental AFM images. Second, we demonstrate that virtual s‑SNOM data can be extrapolated from knowledge of the tip position and, consequently, from AFM signals. To this end, we introduce an analytical model that explains the mechanisms underlying AFM‑to‑s‑SNOM image translation. These insights have the potential to be integrated into future physics‑informed explainable AI models. The two proposed approaches generate pseudo s‑SNOM data without direct optical measurements, significantly expanding access to optical nanoscopy through widely available AFM systems. This advancement holds great promise for reducing both time and costs associated with nanoscale imaging.

Abstract:
Recent advances in image and video generation raise hopes that these models possess world modeling capabilities, the ability to generate realistic, physically plausible videos. This could revolutionize applications in robotics, autonomous driving, and scientific simulation. However, before treating these models as world models, we must ask: Do they adhere to physical conservation laws? To answer this, we introduce Morpheus, a benchmark for evaluating video generation models on physical reasoning. It features 80 real‑world videos capturing physical phenomena, guided by conservation laws. Since artificial generations lack ground truth, we assess physical plausibility using physics‑informed metrics evaluated with respect to infallible conservation laws known per physical setting, leveraging advances in physics‑informed neural networks and vision‑language foundation models. Our findings reveal that even with advanced prompting and video conditioning, current models struggle to encode physical principles despite generating aesthetically pleasing videos. All data, leaderboard, and code are open‑sourced at our project page.

Abstract:
Accurately predicting nonlinear transient thermal fields in two‑dimensional domains is a significant challenge in various engineering fields, where conventional analytical and numerical methods struggle to balance physical fidelity with computational efficiency when dealing with strong material nonlinearities and evolving multiphysics boundary conditions. To address this challenge, we propose a novel cross‑disciplinary approach integrating Green's function formulations with adaptive neural operators, enabling a new paradigm for multiphysics thermal analysis. Our methodology combines rigorous analytical derivations with a physics‑informed neural architecture consisting of five adaptive hidden layers (64 neurons per layer) that incorporates solutions as physical constraints, optimizing learning rates to balance convergence stability and computational speed. Extensive validation demonstrates superior performance in handling rapid thermal transients and strongly coupled nonlinear responses, which significantly improves computational efficiency while maintaining high agreement with analytical benchmarks across a range of material configurations and boundary conditions.

Abstract:
The Earth's main geomagnetic field arises from the constant motion of the fluid outer core. By assuming that the field changes are advection‑dominated, the fluid motion at the core surface can be related to the secular variation of the geomagnetic field. The majority of existing core flow models are global, showing features such as an eccentric planetary gyre, with some evidence of rapid regional changes. By construction, the flow defined at any location by such a model depends on all magnetic field variations across the entire core‑mantle boundary making it challenging to interpret local structures in the flow as due to specific local changes in magnetic field. Here we present an alternative strategy in which we construct regional flow models that rely only on local secular changes. We use a novel technique based on machine learning termed Physics‑Informed Neural Networks (PINNs), in which we seek a regional flow model that simultaneously fits both the local magnetic field variation and dynamical conditions assumed satisfied by the flow. Although we present results using the Tangentially Geostrophic flow constraint, we set out a modelling framework for which the physics constraint can be easily changed by altering a single line of code. After validating the PINN‑based method on synthetic flows, we apply our method to the CHAOS‑8.1 geomagnetic field model, itself based on data from Swarm. Constructing a global mosaic of regional flows, we reproduce the planetary gyre, providing independent evidence that the strong secular changes at high latitude and in equatorial regions are part of the same global feature. Our models also corroborate regional changes in core flows over the last decade. Furthermore, our models endorse the existence of a dynamic high latitude jet, which began accelerating around 2005 but has been weakening since 2017.

Abstract:
This paper presents a method for generating probabilistic descent trajectories in simulations of real‑world airspace. A dataset of 116,066 trajectories harvested from Mode S radar returns in UK airspace was used to train and test the model. Thirteen aircraft types with varying performance characteristics were investigated. It was found that the error in the mean prediction of time to reach the bottom of descent for the proposed method was less than that of the the Base of Aircraft Data (BADA) model by a factor of 10. Furthermore, the method was capable of generating a range of trajectories that were similar to the held out test dataset when analysed in distribution. The proposed method is hybrid, with aircraft drag and calibrated airspeed functions generated probabilistically to parameterise the BADA equations, ensuring the physical plausibility of generated trajectories.

Abstract:
In this work, we introduce implicit Finite Operator Learning (iFOL) for the continuous and parametric solution of partial differential equations (PDEs) on arbitrary geometries. We propose a physics‑informed encoder‑decoder network to establish the mapping between continuous parameter and solution spaces. The decoder constructs the parametric solution field by leveraging an implicit neural field network conditioned on a latent or feature code. Instance‑specific codes are derived through a PDE encoding process based on the second‑order meta‑learning technique. In training and inference, a physics‑informed loss function is minimized during the PDE encoding and decoding. iFOL expresses the loss function in an energy or weighted residual form and evaluates it using discrete residuals derived from standard numerical PDE methods. This approach results in the backpropagation of discrete residuals during both training and inference. iFOL features several key properties: (1) its unique loss formulation eliminates the need for the conventional encode‑process‑decode pipeline previously used in operator learning with conditional neural fields for PDEs; (2) it not only provides accurate parametric and continuous fields but also delivers solution‑to‑parameter gradients without requiring additional loss terms or sensitivity analysis; (3) it can effectively capture sharp discontinuities in the solution; and (4) it removes constraints on the geometry and mesh, making it applicable to arbitrary geometries and spatial sampling (zero‑shot super‑resolution capability). We critically assess these features and analyze the network's ability to generalize to unseen samples across both stationary and transient PDEs. The overall performance of the proposed method is promising, demonstrating its applicability to a range of challenging problems in computational mechanics.

Abstract:
In this paper, we demonstrate the feasibility of performing automatic Technology Computer Aided Design (TCAD) parameter calibration and extraction using machine learning, with the machine trained solely by TCAD simulation data. The methodology is validated using experimental data. Schottky Barrier Diodes (SBDs) with different effective anode workfunction (WF) are fabricated with emerging ultra‑wide bandgap material, Gallium Oxide (Ga2O3), and are measured at various temperatures (T). Their current voltage curves are used for automatic Ga2O3 Philips Unified Mobility (PhuMob) model parameters calibration. Five critical PhuMob parameters were calibrated. The machine consists of an autoencoder and a neural network and is trained solely by TCAD simulation data with variations in WF, T, and the five PhuMob parameters (seven variations in total). Then, Ga2O3 PhuMob parameters are extracted from the noisy experimental curves. Subsequent TCAD simulation using the extracted parameters shows that the quality of the parameters is as good as an expert's calibration at the pre‑turned on regime, but not in the on state regime. By using a simple physics‑informed neural network, the machine performs as well as the human expert in all regimes.

Abstract:
Accurately modeling open quantum system dynamics is crucial for advancing quantum technologies, yet traditional methods struggle to balance accuracy and efficiency. Machine learning (ML) provides a promising alternative, particularly through recursive models that predict system evolution based on past history. While these models have shown success in predicting single observables, their effectiveness in more complex tasks, such as forecasting the full reduced density matrix (RDM), remains unclear. We extend history‑based recursive ML approaches to complex quantum systems, comparing four physics‑informed neural network (PINN) architectures: (i) single‑RDM‑predicting PINN (SR‑PINN), (ii) SR‑PINN with simulation parameters (PSR‑PINN), (iii) multi‑RDMs‑predicting PINN (MR‑PINN), and (iv) MR‑PINN with simulation parameters (PMR‑PINN). These models are applied to the spin‑boson (SB) model and the Fenna‑Matthews‑Olson (FMO) complex. Our results show that SR‑PINN and PSR‑PINN, constrained by a narrow history window, fail to capture complex quantum evolution, leading to unstable long‑term predictions, especially in nonlinear and highly correlated dynamics. In contrast, MR‑PINN and PMR‑PINN improve accuracy by extending the forecast horizon, incorporating long‑range correlations, and reducing error propagation. Surprisingly, explicitly including simulation parameters such as temperature and reorganization energy in PSR‑PINN and PMR‑PINN does not consistently enhance accuracy and can even reduce performance, suggesting that these effects are already encoded in the RDM evolution. These findings highlight the limitations of short‑sighted recursive forecasting and demonstrate the superior stability and accuracy of far‑sighted approaches for long‑term predictions.

Abstract:
Satellites are used for a multitude of applications, including communications, Earth observation, and space science. Neural networks and deep learning‑based approaches now represent the state‑of‑the‑art to enhance the performance and efficiency of these tasks. Given that satellites are susceptible to various faults, one critical application of Artificial Intelligence (AI) is fault detection. However, despite the advantages of neural networks, these systems are vulnerable to radiation errors, which can significantly impact their reliability. Ensuring the dependability of these solutions requires extensive testing and validation, particularly using fault injection methods. This study analyses a physics‑informed (PI) real‑valued non‑volume preserving (Real NVP) normalizing flow model for fault detection in space systems, with a focus on resilience to Single‑Event Upsets (SEUs). We present a customized fault injection framework in TensorFlow to assess neural network resilience. Fault injections are applied through two primary methods: Layer State injection, targeting internal network components such as weights and biases, and Layer Output injection, which modifies layer outputs across various activations. Fault types include zeros, random values, and bit‑flip operations, applied at varying levels and across different network layers. Our findings reveal several critical insights, such as the significance of bit‑flip errors in critical bits, that can lead to substantial performance degradation or even system failure. With this work, we aim to exhaustively study the resilience of Real NVP models against errors due to radiation, providing a means to guide the implementation of fault tolerance measures.

Abstract:
Accurately reconstructing continuous flow fields from sparse or indirect measurements remains an open challenge, as existing techniques often suffer from oversmoothing artifacts, reliance on heterogeneous architectures, and the computational burden of enforcing physics‑informed losses in implicit neural representations (INRs). In this paper, we introduce a novel flow field reconstruction framework based on divergence‑free kernels (DFKs), which inherently enforce incompressibility while capturing fine structures without relying on hierarchical or heterogeneous representations. Through qualitative analysis and quantitative ablation studies, we identify the matrix‑valued radial basis functions derived from Wendland's \mathcalC^4 polynomial (DFKs‑Wen4) as the optimal form of analytically divergence‑free approximation for velocity fields, owing to their favorable numerical properties, including compact support, positive definiteness, and second‑order differentiablility. Experiments across various reconstruction tasks, spanning data compression, inpainting, super‑resolution, and time‑continuous flow inference, has demonstrated that DFKs‑Wen4 outperform INRs and other divergence‑free representations in both reconstruction accuracy and computational efficiency while requiring the fewest trainable parameters.

Abstract:
Understanding and solving fluid dynamics equations efficiently remains a fundamental challenge in computational physics. Traditional numerical solvers and physics‑informed neural networks struggle to capture the full range of frequency components in partial differential equation solutions, limiting their accuracy and efficiency. Here, we propose the Multi‑stream Physics Hybrid Network, a novel neural architecture that integrates quantum and classical layers in parallel to improve the accuracy of solving fluid dynamics equations, namely ''Kovasznay flow'' problem. This approach decomposes the solution into separate frequency components, each predicted by independent Parallel Hybrid Networks, simplifying the training process and enhancing performance. We evaluated the proposed model against a comparable classical neural network, the Multi‑stream Physics Classical Network, in both data‑driven and physics‑driven scenarios. Our results show that the Multi‑stream Physics Hybrid Network achieves a reduction in root mean square error by 36% for velocity components and 41% for pressure prediction compared to the classical model, while using 24% fewer trainable parameters. These findings highlight the potential of hybrid quantum‑classical architectures for advancing computational fluid dynamics.

Abstract:
Machine learning methods have shown promise in learning chaotic dynamical systems, enabling model‑free short‑term prediction and attractor reconstruction. However, when applied to large‑scale, spatiotemporally chaotic systems, purely data‑driven machine learning methods often suffer from inefficiencies, as they require a large learning model size and a massive amount of training data to achieve acceptable performance. To address this challenge, we incorporate the spatial coupling structure of the target system as an inductive bias in the network design. Specifically, we introduce physics‑guided clustered echo state networks, leveraging the efficiency of the echo state networks as a base model. Experimental results on benchmark chaotic systems demonstrate that our physics‑informed method outperforms existing echo state network models in learning the target chaotic systems. Additionally, we numerically demonstrate that leveraging coupling knowledge into ESN models can enhance their robustness to variations of training and target system conditions. We further show that our proposed model remains effective even when the coupling knowledge is imperfect or extracted directly from time series data. We believe this approach has the potential to enhance other machine‑learning methods.

Abstract:
This paper presents a novel approach for accelerating n‑body simulations by integrating a physics‑informed graph neural networks (GNN) with traditional numerical methods. Our method implements a leapfrog‑based simulation engine to generate datasets from diverse astrophysical scenarios which are then transformed into graph representations. A custom‑designed GNN is trained to predict particle accelerations with high precision. Experiments, conducted on 60 training and 6 testing simulations spanning from 3 to 500 bodies over 1000 time steps, demonstrate that the proposed model achieves extremely low prediction errors‑loss values while maintaining robust long‑term stability, with accumulated errors in position, velocity, and acceleration remaining insignificant. Furthermore, our method yields a modest speedup of approximately 17% over conventional simulation techniques. These results indicate that the integration of deep learning with traditional physical simulation methods offers a promising pathway to significantly enhance computational efficiency without compromising accuracy.

Abstract:
We present a novel approach to hard‑constrain Neumann boundary conditions in physics‑informed neural networks (PINNs) using Fourier feature embeddings. Neumann boundary conditions are used to described critical processes in various application, yet they are more challenging to hard‑constrain in PINNs than Dirichlet conditions. Our method employs specific Fourier feature embeddings to directly incorporate Neumann boundary conditions into the neural network's architecture instead of learning them. The embedding can be naturally extended by high frequency modes to better capture high frequency phenomena. We demonstrate the efficacy of our approach through experiments on a diffusion problem, for which our method outperforms existing hard‑constraining methods and classical PINNs, particularly in multiscale and high frequency scenarios.

Abstract:
Building performance simulation (BPS) is critical for understanding building dynamics and behavior, analyzing performance of the built environment, optimizing energy efficiency, improving demand flexibility, and enhancing building resilience. However, conducting BPS is not trivial. Traditional BPS relies on an accurate building energy model, mostly physics‑based, which depends heavily on detailed building information, expert knowledge, and case‑by‑case model calibrations, thereby significantly limiting their scalability. With the development of sensing technology and increased data availability, there is a growing attention and interest in data‑driven BPS. However, purely data‑driven models often suffer from limited generalization ability and a lack of physical consistency, resulting in poor performance in real‑world applications. To address these limitations, recent studies have started to incorporate physics priors into data‑driven models, a methodology called physics‑informed machine learning (PIML). PIML is an emerging field with the definitions, methodologies, evaluation criteria, application scenarios, and future directions that remain open. To bridge those gaps, this study systematically reviews the state‑of‑art PIML for BPS, offering a comprehensive definition of PIML, and comparing it to traditional BPS approaches regarding data requirements, modeling effort, performance and computation cost. We also summarize the commonly used methodologies, validation approaches, application domains, available data sources, open‑source packages and testbeds. In addition, this study provides a general guideline for selecting appropriate PIML models based on BPS applications. Finally, this study identifies key challenges and outlines future research directions, providing a solid foundation and valuable insights to advance R&D of PIML in BPS.

Abstract:
Despite considerable scientific advances in numerical simulation, efficiently solving PDEs remains a complex and often expensive problem. Physics‑informed Neural Networks (PINN) have emerged as an efficient way to learn surrogate solvers by embedding the PDE in the loss function and minimizing its residuals using automatic differentiation at so‑called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements for PINNs. In this paper, leveraging a new quadrature method for approximating definite integrals, we introduce a provably accurate sampling method for collocation points based on the Hessian of the PDE residuals. Comparative experiments conducted on a set of 1D and 2D PDEs demonstrate the benefits of our method.

Abstract:
Data‑enabled predictive control (DeePC) has emerged as a powerful technique to control complex systems without the need for extensive modeling efforts. However, relying solely on offline collected data trajectories to represent the system dynamics introduces certain drawbacks. Therefore, we present a novel semi‑data‑driven model predictive control (SD‑MPC) framework that combines (limited) model information with DeePC to address a range of these drawbacks, including sensitivity to noisy data and a lack of robustness. In this work, we focus on the performance of DeePC in operating regimes not captured by the offline collected data trajectories and demonstrate how incorporating an underlying parametric model can counteract this issue. SD‑MPC exhibits equivalent closed‑loop performance as DeePC for deterministic linear time‑invariant systems. Simulations demonstrate the general control performance of the proposed SD‑MPC for both a linear time‑invariant system and a nonlinear system modeled as a linear parameter‑varying system. These results provide numerical evidence of the enhanced robustness of SD‑MPC over classical DeePC.

Abstract:
In this paper, we introduce a formulation of Physics‑Informed Neural Networks (PINNs), based on learning the form of the Fourier decomposition, and a training methodology based on a spread of randomly chosen boundary conditions. By training in this way we produce a PINN that generalises; after training it can be used to correctly predict the solution for an arbitrary set of boundary conditions and interpolate this solution between the samples that spanned the training domain. We demonstrate for a toy system of two coupled oscillators that this gives the PINN formulation genuine predictive capability owing to an effective reduction of the training to evaluation times ratio due to this decoupling of the solution from specific boundary conditions.

Abstract:
Advances in ultra‑intense laser technology have increased repetition rates and average power for chirped‑pulse laser systems, which offers a promising solution for many applications including energetic proton sources. An important challenge is the need to optimize and control the proton source by varying some of the many degrees of freedom inherent to the laser‑plasma interactions. Machine learning can play an important role in this task, as our work examines. Building on our earlier work in Desai et al. 2024, we generate a large ～1.5 million data point synthetic data set for proton acceleration using a physics‑informed analytic model that we improved to include pre‑pulse physics. Then, we train different machine learning methods on this data set to determine which methods perform efficiently and accurately. Generally, we find that quasi‑real‑time training of neural network models using single shot data from a kHz repetition rate ultra‑intense laser system should typically be feasible on a single GPU. We also find that a less sophisticated model like a polynomial regression can be trained even faster and that the accuracy of these models is still good enough to be useful. We provide our source code and example synthetic data for others to test new machine learning methods and approaches to automated learning in this regime.

Abstract:
Simultaneously detecting hidden solid boundaries and reconstructing flow fields from sparse observations poses a significant inverse challenge in fluid mechanics. This study presents a physics‑informed neural network (PINN) framework designed to infer the presence, shape, and motion of static or moving solid boundaries within a flow field. By integrating a body fraction parameter into the governing equations, the model enforces no‑slip/no‑penetration boundary conditions in solid regions while preserving conservation laws of fluid dynamics. Using partial flow field data, the method simultaneously reconstructs the unknown flow field and infers the body fraction distribution, thereby revealing solid boundaries. The framework is validated across diverse scenarios, including incompressible Navier‑Stokes and compressible Euler flows, such as steady flow past a fixed cylinder, an inline oscillating cylinder, and subsonic flow over an airfoil. The results demonstrate accurate detection of hidden boundaries, reconstruction of missing flow data, and estimation of trajectories and velocities of a moving body. Further analysis examines the effects of data sparsity, velocity‑only measurements, and noise on inference accuracy. The proposed method exhibits robustness and versatility, highlighting its potential for applications when only limited experimental or numerical data are available.

Abstract:
We present a method to infer temperature fields from stereo particle‑image velocimetry (PIV) data in turbulent Rayleigh‑Bénard convection (RBC) using Physics‑informed neural networks (PINNs). The physical setup is a cubic RBC cell with Rayleigh number \textRa=10^7 and Prandtl number \textPr=0.7. With data only available in a vertical plane A:x=x_0, the residuals of the governing partial differential equations are minimised in an enclosing 3D domain around A with thickness δ_x. Dynamic collocation point sampling strategies are used to overcome the lack of 3D labelled information and to optimize the overall convergence of the PINN. In particular, in the out‑of‑plane direction x, the collocation points are distributed according to a normal distribution, in order to emphasize the region where data is provided. Along the vertical direction, we leverage meshing information and sample points from a distribution designed based on the grid of a direct numerical simulation (DNS). This approach points greater attention to critical regions, particularly the areas with high temperature gradients within the thermal boundary layers. Using planar three‑component velocity data from a DNS, we successfully validate the reconstruction of the temperature fields in the PIV plane. We evaluate the robustness of our method with respect to characteristics of the labelled data used for training: the data time span, the sampling frequency, some noisy data and boundary data omission, aiming to better accommodate the challenges associated with experimental data. Developing PINNs on controlled simulation data is a crucial step toward their effective deployment on experimental data. The key is to systematically introduce noise, gaps, and uncertainties in simulated data to mimic real‑world conditions and ensure robust generalization.

Abstract:
Evolution equations, including both ordinary differential equations (ODEs) and partial differential equations (PDEs), play a pivotal role in modeling dynamic systems. However, achieving accurate long‑time integration for these equations remains a significant challenge. While physics‑informed neural networks (PINNs) provide a mesh‑free framework for solving PDEs, they often suffer from temporal error accumulation, which limits their effectiveness in capturing long‑time behaviors. To alleviate this issue, we propose integral regularization PINNs (IR‑PINNs), a novel approach that enhances temporal accuracy by incorporating an integral‑based residual term into the loss function. This method divides the entire time interval into smaller sub‑intervals and enforces constraints over these sub‑intervals, thereby improving the resolution and correlation of temporal dynamics. Furthermore, IR‑PINNs leverage adaptive sampling to dynamically refine the distribution of collocation points based on the evolving solution, ensuring higher accuracy in regions with sharp gradients or rapid variations. Numerical experiments on benchmark problems demonstrate that IR‑PINNs outperform original PINNs and other state‑of‑the‑art methods in capturing long‑time behaviors, offering a robust and accurate solution for evolution equations.

Abstract:
Koopman operator has been recognized as an ongoing data‑driven modeling method for vehicle dynamics which lifts the original state space into a high‑dimensional linear state space. The deep neural networks (DNNs) are verified to be useful for the approximation of Koopman operator. To further improve the accuracy of Koopman operator approximation, this paper introduces a physical loss function term from the concept of physics‑informed neural networks (PINNs), i.e., the acceleration loss between neural network output and sensor measurements, to improve the efficiency of network learning and its interpretability. Moreover, we utilize the sliding window least squares (SWLS) to update the system matrix and input matrix online in the lifted space, therefore enabling the deep Koopman operator to adapt to the rapid dynamics of autonomous vehicles in real events. The data collection and validation are conducted on CarSim/Simlink co‑simulation platform. With comparison to other physics‑based and data‑driven approaches on various scenarios, the results reveal that the acceleration loss‑informed network refines the accuracy of Koopman operator approximation and renders it with inherent generalization, and the SWLS enforces the deep Koopman operator's capability to cope with changes in vehicle parameters, road conditions, and rapid maneuvers. This indicates the proposed physics‑informed adaptive deep Koopman operator is a performant and efficient data‑driven modeling tool.

Abstract:
We humans rely on a wide range of commonsense knowledge to interact with an extensive number and categories of objects in the physical world. Likewise, such commonsense knowledge is also crucial for robots to successfully develop generalized object manipulation skills. While recent advancements in Multi‑modal Large Language Models (MLLMs) have showcased their impressive capabilities in acquiring commonsense knowledge and conducting commonsense reasoning, effectively grounding this semantic‑level knowledge produced by MLLMs to the physical world to thoroughly guide robots in generalized articulated object manipulation remains a challenge that has not been sufficiently addressed. To this end, we introduce analytic concepts, procedurally defined upon mathematical symbolism that can be directly computed and simulated by machines. By leveraging the analytic concepts as a bridge between the semantic‑level knowledge inferred by MLLMs and the physical world where real robots operate, we can figure out the knowledge of object structure and functionality with physics‑informed representations, and then use the physically grounded knowledge to instruct robot control policies for generalized and accurate articulated object manipulation. Extensive experiments in both real world and simulation demonstrate the superiority of our approach.

Abstract:
Neural networks have emerged as powerful tools for modeling complex physical systems, yet balancing high accuracy with computational efficiency remains a critical challenge in their convergence behavior. In this work, we propose the Hybrid Parallel Kolmogorov‑Arnold Network (KAN) and Multi‑Layer Perceptron (MLP) Physics‑Informed Neural Network (HPKM‑PINN), a novel architecture that synergistically integrates parallelized KAN and MLP branches within a unified PINN framework. The HPKM‑PINN introduces a scaling factor ξ, to optimally balance the complementary strengths of KAN's interpretable function approximation and MLP's nonlinear feature learning, thereby enhancing predictive performance through a weighted fusion of their outputs. Through systematic numerical evaluations, we elucidate the impact of the scaling factor ξ on the model's performance in both function approximation and partial differential equation (PDE) solving tasks. Benchmark experiments across canonical PDEs, such as the Poisson and Advection equations, demonstrate that HPKM‑PINN achieves a marked decrease in loss values (reducing relative error by two orders of magnitude) compared to standalone KAN or MLP models. Furthermore, the framework exhibits numerical stability and robustness when applied to various physical systems. These findings highlight the HPKM‑PINN's ability to leverage KAN's interpretability and MLP's expressivity, positioning it as a versatile and scalable tool for solving complex PDE‑driven problems in computational science and engineering.

Abstract:
We use the physics‑informed renormalisation group (PIRG) for the construction of gauge invariant renormalisation group flows. The respective effective action is a sum of a gauge invariant quantum part and the classical gauge fixing part which arranges for invertibility of the gauge field two‑point function. Thus, the BRST transformations simply accommodate the gauge consistency of the gauge fixing sector, while the quantum part of the effective action is gauge and BRST invariant. We apply this physics‑informed approach to Yang‑Mills theory and gravity and show how the flowing gauge fields arrange for full gauge invariance. We also embed the background field approximation to the functional renormalisation group (fRG) in an exact gauge invariant PIRG flow. This allows us to discuss the dynamics of the correction terms, and the non‑trivial ultraviolet or infrared relevant terms are elucidated within a one‑loop approximation. The background field approximation of the latter is known for violating one‑loop universality for specific regulators and we show how the present setup reinstates universality in a constructive way. Finally, we discuss the landscape of fRG flows in gauge theories through the lens of the novel PIRG approach as well as potential applications.

Abstract:
We introduce MixFunn, a novel neural network architecture designed to solve differential equations with enhanced precision, interpretability, and generalization capability. The architecture comprises two key components: the mixed‑function neuron, which integrates multiple parameterized nonlinear functions to improve representational flexibility, and the second‑order neuron, which combines a linear transformation of its inputs with a quadratic term to capture cross‑combinations of input variables. These features significantly enhance the expressive power of the network, enabling it to achieve comparable or superior results with drastically fewer parameters and a reduction of up to four orders of magnitude compared to conventional approaches. We applied MixFunn in a physics‑informed setting to solve differential equations in classical mechanics, quantum mechanics, and fluid dynamics, demonstrating its effectiveness in achieving higher accuracy and improved generalization to regions outside the training domain relative to standard machine learning models. Furthermore, the architecture facilitates the extraction of interpretable analytical expressions, offering valuable insights into the underlying solutions.

Abstract:
This paper presents a novel physical parameter estimation framework for on‑site model characterization, using a two‑phase modelling strategy with Physics‑Informed Neural Networks (PINNs) and transfer learning (TL). In the first phase, a PINN is trained using only the physical principles of the single particle model (SPM) equations. In the second phase, the majority of the PINN parameters are frozen, while critical electrochemical parameters are set as trainable and adjusted using real‑world voltage profile data. The proposed approach significantly reduces computational costs, making it suitable for real‑time implementation on Battery Management Systems (BMS). Additionally, as the initial phase does not require field data, the model is easy to deploy with minimal setup requirements. With the proposed methodology, we have been able to effectively estimate relevant electrochemical parameters with operating data. This has been proved estimating diffusivities and active material volume fractions with charge data in different degradation conditions. The methodology is experimentally validated in a Raspberry Pi device using data from a standard charge profile with a 3.89% relative accuracy estimating the active material volume fractions of a NMC cell with 82.09% of its nominal capacity.

Abstract:
The study of parametric differential equations plays a crucial role in weather forecasting and epidemiological modeling. These phenomena are better represented using fractional derivatives due to their inherent memory or hereditary effects. This paper introduces a novel scientific machine learning approach for solving parametric time‑fractional differential equations by combining traditional spectral methods with neural networks. Instead of relying on automatic differentiation techniques, commonly used in traditional Physics‑Informed Neural Networks (PINNs), we propose a more efficient global discretization method based on Legendre polynomials. This approach eliminates the need to simulate the parametric fractional differential equations across multiple parameter values. By applying the Legendre‑Galerkin weak formulation to the differential equation, we construct a loss function for training the neural network. The trial solutions are represented as linear combinations of Legendre polynomials, with the coefficients learned by the neural network. The convergence of this method is theoretically established, and the theoretical results are validated through numerical experiments on several well‑known differential equations.

Abstract:
This study introduces a physics‑informed machine learning framework to accelerate the computation of the microkinetic model of Fischer‑Tropsch synthesis. A neural network, trained within the NVIDIA Modulus framework, approximates the fraction of vacant catalytic sites with high accuracy. The combination of implicit differentiation and the Newton‑Raphson method enhances derivative calculations, ensuring physical consistency. Computational efficiency improves significantly, with speedups up to 10^4 times on a GPU. This versatile methodology generalizes across catalysts and reactors, offering a robust tool for chemical engineering applications, including model approximation and catalyst parameter fitting from experimental data.

Abstract:
Grid‑forming converters (GFCs) are crucial for frequency and voltage stability in modern power systems. However, their performance under overload conditions remains a challenge. This paper highlights the limitations of existing approaches in managing DC source saturation and AC current limits, emphasizing the need for improved control strategies to ensure system stability. This paper proposes a control strategy based on a physics‑informed neural network (PINN) to improve GFC performance under overloaded conditions, effectively preventing switch failures and mitigating DC source saturation. This approach outperforms conventional methods by maintaining stable voltage and frequency, even under significant load increase where traditional droop control alone proves inadequate. The post‑disturbance operating point of GFCs remains unchanged using PINN‑based control with an improvement of 0.245 Hz in frequency and 0.03 p.u. in active power when compared to an already existing current limitation strategy. Additionally, it reduces peak voltage deviations during transients by 24.14%, lowers the rate of change of frequency (ROCOF) from 0.02 Hz/s to 0.005 Hz/s, and improves the rate of change of voltage (ROCOV), keeping both within acceptable limits. These improvements significantly enhance system resilience, especially in inertia‑less power networks.

Abstract:
Structural uncertainties and unresolved features in fault zones hinder the assessment of leakage risks in subsurface CO2 storage. Understanding multi‑scale uncertainties in fracture network conductivity is crucial for mitigating risks and reliably modelling upscaled fault leakage rates. Conventional models, such as the Cubic Law, which is based on mechanical aperture measurements, often neglect fracture roughness, leading to model misspecifications and inaccurate conductivity estimates. Here, we develop a physics‑informed, AI‑driven correction of these model misspecifications by automatically integrating roughness effects and small‑scale structural uncertainties. Using Bayesian inference combined with data‑driven and geometric corrections, we reconstruct local hydraulic aperture fields that reliably estimate fracture conductivities. By leveraging interactions across scales, we improve upon traditional empirical corrections and provide a framework for propagating uncertainties from individual fractures to network scales. Our approach thereby supports robust calibration of conductivity ranges for fault leakage sensitivity analyses, offering a scalable solution for subsurface risk assessment.

Abstract:
The complexity of condensed matter arises from emergent behaviors that cannot be understood by analyzing individual constituents in isolation. While traditional condensed‑matter approaches‑developed primarily for ideal crystalline solids‑have provided deep insights into symmetry, order, and electronic structure, they fall short in describing the rich, multiscale organization of hierarchical and soft materials. These systems exhibit structural correlations across multiple length and time scales, often governed by nonlinear interactions that span from molecular to macroscopic domains. This review explores how the convergence of emerging experimental and computational strategies are redefining our ability to characterize and model such systems. We examine how multimodal techniques‑combining scattering, imaging, and spectroscopy‑can map structural order and dynamics across scales, with methods like small‑angle scattering tensor tomography, dark‑field imaging, and ultrafast spectroscopies providing unprecedented spatiotemporal resolution. On the computational front, machine learning approaches such as graph neural networks, neural operators, and physics‑informed models offer powerful tools to connect disparate scales and uncover hidden correlations in high‑dimensional data. These advancements have the potential to close the gap between structure and function in complex materials, thereby addressing one of the grand challenges of contemporary material science: understanding and engineering multiscale architectures, whose emergent properties underpin the behavior of next‑generation functional materials, biological systems, and adaptive technologies.

Abstract:
Physics‑informed neural networks offered an alternate way to solve several differential equations that govern complicated physics. However, their success in predicting the acoustic field is limited by the vanishing‑gradient problem that occurs when solving the Helmholtz equation. In this paper, a formulation is presented that addresses this difficulty. The problem of solving the two‑dimensional Helmholtz equation with the prescribed boundary conditions is posed as an unconstrained optimization problem using trial solution method. According to this method, a trial neural network that satisfies the given boundary conditions prior to the training process is constructed using the technique of transfinite interpolation and the theory of R‑functions. This ansatz is initially applied to the rectangular domain and later extended to the circular and elliptical domains. The acoustic field predicted from the proposed formulation is compared with that obtained from the two‑dimensional finite element methods. Good agreement is observed in all three domains considered. Minor limitations associated with the proposed formulation and their remedies are also discussed.

Abstract:
A significant advancement in Neural Network (NN) research is the integration of domain‑specific knowledge through custom loss functions. This approach addresses a crucial challenge: how can models utilize physics or mathematical principles to enhance predictions when dealing with sparse, noisy, or incomplete data? Physics‑Informed Neural Networks (PINNs) put this idea into practice by incorporating physical equations, such as Partial Differential Equations (PDEs), as soft constraints. This guidance helps the networks find solutions that align with established laws. Recently, researchers have expanded this framework to include Bayesian NNs (BNNs), which allow for uncertainty quantification while still adhering to physical principles. But what happens when the governing equations of a system are not known? In this work, we introduce methods to automatically extract PDEs from historical data. We then integrate these learned equations into three different modeling approaches: PINNs, Bayesian‑PINNs (B‑PINNs), and Bayesian Linear Regression (BLR). To assess these frameworks, we evaluate them on a real‑world Multivariate Time Series (MTS) dataset. We compare their effectiveness in forecasting future states under different scenarios: with and without PDE constraints and accuracy considerations. This research aims to bridge the gap between data‑driven discovery and physics‑guided learning, providing valuable insights for practical applications.

Abstract:
Physics‑informed neural networks (PINNs) have demonstrated promise as a framework for solving forward and inverse problems involving partial differential equations. Despite recent progress in the field, it remains challenging to quantify uncertainty in these networks. While techniques such as Bayesian PINNs (B‑PINNs) provide a principled approach to capturing epistemic uncertainty through Bayesian inference, they can be computationally expensive for large‑scale applications. In this work, we propose Epistemic Physics‑Informed Neural Networks (E‑PINNs), a framework that uses a small network, the epinet, to efficiently quantify epistemic uncertainty in PINNs. The proposed approach works as an add‑on to existing, pre‑trained PINNs with a small computational overhead. We demonstrate the applicability of the proposed framework in various test cases and compare the results with B‑PINNs using Hamiltonian Monte Carlo (HMC) posterior estimation and dropout‑equipped PINNs (Dropout‑PINNs). In our experiments, E‑PINNs achieve calibrated coverage with competitive sharpness at substantially lower cost. We demonstrate that when B‑PINNs produce narrower bands, they under‑cover in our tests. E‑PINNs also show better calibration than Dropout‑PINNs in these examples, indicating a favorable accuracy‑efficiency trade‑off.

Abstract:
The reconstruction of deep ocean currents is a major challenge in data assimilation due to the scarcity of interior data. In this work, we present a proof of concept for deep ocean flow reconstruction using a Physics‑Informed Neural Network (PINN), a machine learning approach that offers an alternative to traditional data assimilation methods. We introduce an efficient algorithm called StrAssPINN (for Stratified Assimilation PINNs), which assigns a separate network to each layer of the ocean model while allowing them to interact during training. The neural network takes spatiotemporal coordinates as input and predicts the velocity field at those points. Using a SIREN architecture (a multilayer perceptron with sine activation functions), which has proven effective in various contexts, the network is trained using both available observational data and dynamical priors enforced at several collocation points. We apply this method to pseudo‑observed ocean data generated from a 3‑layer quasi‑geostrophic model, where the pseudo‑observations include surface‑level data akin to SWOT observations of sea surface height, interior data similar to ARGO floats, and a limited number of deep ARGO‑like measurements in the lower layers. Our approach successfully reconstructs ocean flows in both the interior and surface layers, demonstrating a strong ability to resolve key ocean mesoscale features, including vortex rings, eastward jets associated with potential vorticity fronts, and smoother Rossby waves. This work serves as a prelude to applying StrAssPINN to real‑world observational data.

Abstract:
Type 1 Diabetes (T1D) management is a complex task due to many variability factors. Artificial Pancreas (AP) systems have alleviated patient burden by automating insulin delivery through advanced control algorithms. However, the effectiveness of these systems depends on accurate modeling of glucose‑insulin dynamics, which traditional mathematical models often fail to capture due to their inability to adapt to patient‑specific variations. This study introduces a Biological‑Informed Recurrent Neural Network (BIRNN) framework to address these limitations. The BIRNN leverages a Gated Recurrent Units (GRU) architecture augmented with physics‑informed loss functions that embed physiological constraints, ensuring a balance between predictive accuracy and consistency with biological principles. The framework is validated using the commercial UVA/Padova simulator, outperforming traditional linear models in glucose prediction accuracy and reconstruction of unmeasured states, even under circadian variations in insulin sensitivity. The results demonstrate the potential of BIRNN for personalized glucose regulation and future adaptive control strategies in AP systems.

Abstract:
This paper investigates the application of Physics‑Informed Neural Networks (PINNs) for solving the inverse advection‑diffusion problem to localize pollution sources. The study focuses on optimizing neural network architectures to accurately model pollutant dispersion dynamics under diverse conditions, including scenarios with weak and strong winds and multiple pollution sources. Various PINN configurations are evaluated, showing the strong dependence of solution accuracy on hyperparameter selection. Recommendations for efficient PINN configurations are provided based on these comparisons. The approach is tested across multiple scenarios and validated using real‑world data that accounts for atmospheric variability. The results demonstrate that the proposed methodology achieves high accuracy in source localization, showcasing the stability and potential of PINNs for addressing environmental monitoring and pollution management challenges under complex weather conditions.

Abstract:
Reinforcement learning (RL) can be used to tune data‑driven (economic) nonlinear model predictive controllers ((e)NMPCs) for optimal performance in a specific control task by optimizing the dynamic model or parameters in the policy's objective function or constraints, such as state bounds. However, the sample efficiency of RL is crucial, and to improve it, we combine a model‑based RL algorithm with our published method that turns Koopman (e)NMPCs into automatically differentiable policies. We apply our approach to an eNMPC case study of a continuous stirred‑tank reactor (CSTR) model from the literature. The approach outperforms benchmark methods, i.e., data‑driven eNMPCs using models based on system identification without further RL tuning of the resulting policy, and neural network controllers trained with model‑based RL, by achieving superior control performance and higher sample efficiency. Furthermore, utilizing partial prior knowledge about the system dynamics via physics‑informed learning further increases sample efficiency.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data‑scarce scenarios, due to their unsupervised training capability. However, limitations related to convergence and the need for re‑optimization with each change in PDE parameters hinder their widespread adoption across scientific and engineering applications. This survey reviews existing research that addresses these limitations through transfer learning and meta‑learning. The covered methods improve the training efficiency, allowing faster adaptation to new PDEs with fewer data and computational resources. While traditional numerical methods solve systems of differential equations directly, neural networks learn solutions implicitly by adjusting their parameters. One notable advantage of neural networks is their ability to abstract away from specific problem domains, allowing them to retain, discard, or adapt learned representations to efficiently address similar problems. By exploring the application of these techniques to PINNs, this survey identifies promising directions for future research to facilitate the broader adoption of PINNs in a wide range of scientific and engineering applications.

Abstract:
Physics‑informed deep learning approaches have been developed to solve forward and inverse stochastic differential equation (SDE) problems with high‑dimensional stochastic space. However, the existing deep learning models have difficulties solving SDEs with high‑dimensional spatial space. In the present study, we propose a scalable physics‑informed deep generative model (sPI‑GeM), which is capable of solving SDE problems with both high‑dimensional stochastic and spatial space. The sPI‑GeM consists of two deep learning models, i.e., (1) physics‑informed basis networks (PI‑BasisNet), which are used to learn the basis functions as well as the coefficients given data on a certain stochastic process or random field, and (2) physics‑informed deep generative model (PI‑GeM), which learns the distribution over the coefficients obtained from the PI‑BasisNet. The new samples for the learned stochastic process can then be obtained using the inner product between the output of the generator and the basis functions from the trained PI‑BasisNet. The sPI‑GeM addresses the scalability in the spatial space in a similar way as in the widely used dimensionality reduction technique, i.e., principal component analysis (PCA). A series of numerical experiments, including approximation of Gaussian and non‑Gaussian stochastic processes, forward and inverse SDE problems, are performed to demonstrate the accuracy of the proposed model. Furthermore, we also show the scalability of the sPI‑GeM in both the stochastic and spatial space using an example of a forward SDE problem with 38‑ and 20‑dimension stochastic and spatial space, respectively.

Abstract:
Human activity recognition (HAR) with deep learning models relies on large amounts of labeled data, often challenging to obtain due to associated cost, time, and labor. Self‑supervised learning (SSL) has emerged as an effective approach to leverage unlabeled data through pretext tasks, such as masked reconstruction and multitask learning with signal processing‑based data augmentations, to pre‑train encoder models. However, such methods are often derived from computer vision approaches that disregard physical mechanisms and constraints that govern wearable sensor data and the phenomena they reflect. In this paper, we propose a physics‑informed multi‑task pre‑training (PIM) framework for IMU‑based HAR. PIM generates pre‑text tasks based on the understanding of basic physical aspects of human motion: including movement speed, angles of movement, and symmetry between sensor placements. Given a sensor signal, we calculate corresponding features using physics‑based equations and use them as pretext tasks for SSL. This enables the model to capture fundamental physical characteristics of human activities, which is especially relevant for multi‑sensor systems. Experimental evaluations on four HAR benchmark datasets demonstrate that the proposed method outperforms existing state‑of‑the‑art methods, including data augmentation and masked reconstruction, in terms of accuracy and F1 score. We have observed gains of almost 10% in macro f1 score and accuracy with only 2 to 8 labeled examples per class and up to 3% when there is no reduction in the amount of training data.

Abstract:
Physics‑informed neural networks (PINNs) demonstrate promising potential in parameterized engineering turbulence optimization problems but face challenges, such as high data requirements and low computational accuracy when applied to engineering turbulence problems. This study proposes a framework that enhances the ability of PINNs to solve parametric turbulence problems without training datasets from experiments or CFD‑Parametric Turbulence PINNs (PT‑PINNs)). Two key methods are introduced to improve the accuracy and robustness of this framework. The first is a soft constraint method for turbulent viscosity calculation. The second is a pre‑training method based on the conservation of flow rate in the flow field. The effectiveness of PT‑PINNs is validated using a three‑dimensional backward‑facing step (BFS) turbulence problem with two varying parameters (Re = 3000‑200000, ER = 1.1‑1.5). PT‑PINNs produce predictions that closely match experimental data and computational fluid dynamics (CFD) results across various conditions. Moreover, PT‑PINNs offer a computational efficiency advantage over traditional CFD methods. The total time required to construct the parametric BFS turbulence model is 39 hours, one‑sixteenth of the time required by traditional numerical methods. The inference time for a single‑condition prediction is just 40 seconds‑only 0.5% of a single CFD computation. These findings highlight the potential of PT‑PINNs for future applications in engineering turbulence optimization problems.

Abstract:
Machine learning simulations of open quantum dynamics often rely on recursive predictors that accumulate error. We develop a non‑recursive convolutional neural networks (CNNs) that maps system parameters and a redundant time encoding directly to excitation‑energy‑transfer populations in the Fenna‑Matthews‑Olson complex. The encoding‑modified logistic plus \tanh functions‑normalizes time and resolves fast, transitional, and quasi‑steady regimes, while physics‑informed labels enforce population conservation and inter‑site consistency. Trained only on 0～ 7 ps reference trajectories generated with a Lindblad model in QuTiP, the network accurately predicts 0～100 ps dynamics across a range of reorganization energies, bath rates, and temperatures. Beyond 20 ps, the absolute relative error remains below 0.05, demonstrating stable long‑time extrapolation. By avoiding step‑by‑step recursion, the method suppresses error accumulation and generalizes across timescales. These results show that redundant time encoding enables data‑efficient inference of long‑time quantum dissipative dynamics in realistic pigment‑protein complexes, and may aid the data‑driven design of light‑harvesting materials.

Abstract:
Due to the limited accuracy of 4D Magnetic Resonance Imaging (MRI) in identifying hemodynamics in cardiovascular diseases, the challenges in obtaining patient‑specific flow boundary conditions, and the computationally demanding and time‑consuming nature of Computational Fluid Dynamics (CFD) simulations, it is crucial to explore new data assimilation algorithms that offer possible alternatives to these limitations. In the present work, we study Physics‑Informed Neural Networks (PINNs), Deep Operator Networks (DeepONets), and their Physics‑Informed extensions (PI‑DeepONets) in predicting vascular flow simulations in the context of a 3D Abdominal Aortic Aneurysm (AAA) idealized model. PINN is a technique that combines deep neural networks with the fundamental principles of physics, incorporating the physics laws, which are given as partial differential equations, directly into loss functions used during the training process. On the other hand, DeepONet is designed to learn nonlinear operators from data and is particularly useful in studying parametric partial differential equations (PDEs), e.g., families of PDEs with different source terms, boundary conditions, or initial conditions. Here, we adapt the approaches to address the particular use case of AAA by integrating the 3D Navier‑Stokes equations (NSE) as the physical laws governing fluid dynamics. In addition, we follow best practices to enhance the capabilities of the models by effectively capturing the underlying physics of the problem under study. The advantages and limitations of each approach are highlighted through a series of relevant application cases. We validate our results by comparing them with CFD simulations for benchmark datasets, demonstrating good agreements and emphasizing those cases where improvements in computational efficiency are observed.

Abstract:
In contrast to the assumptions of most existing Electromigration (EM) analysis tools, the evolution of EM‑induced stress is inherently non‑deterministic, influenced by factors such as input current fluctuations and manufacturing non‑idealities. Traditional approaches for estimating stress variations typically involve computationally expensive and inefficient Monte Carlo simulations with industrial solvers, which quantify variations using mean and variance metrics. In this work, we introduce a novel machine learning‑based framework, termed BPINN‑EM‑ Post, for efficient stochastic analysis of EM‑induced post‑voiding aging processes. For the first time, our new approach integrates closed‑form analytical solutions with a Bayesian Physics‑ Informed Neural Network (BPINN) framework to accelerate the analysis. The closed‑form solutions enforce physical laws at the individual wire segment level, while the BPINN ensures that physics constraints at inter‑segment junctions are satisfied and stochastic behaviors are accurately modeled. By reducing the number of variables in the loss functions through utilizing analytical solutions, our method significantly improves training efficiency without accuracy loss and naturally incorporates variational effects. Additionally, the analytical solutions effectively address the challenge of incorporating initial stress distributions in interconnect structures during post‑void stress calculations. Numerical results demonstrate that BPINN‑EM‑Post achieves over 240x and more than 67x speedup compared to Monte Carlo simulations using the FEM‑based COMSOL solver and FDM‑based EMSpice, respectively, with marginal accuracy loss.

Abstract:
Solving partial differential equations (PDEs) with discontinuous solutions , such as shock waves in multiphase viscous flow in porous media , is critical for a wide range of scientific and engineering applications, as they represent sudden changes in physical quantities. Physics‑Informed Neural Networks (PINNs), an approach proposed for solving PDEs, encounter significant challenges when applied to such systems. Accurately solving PDEs with discontinuities using PINNs requires specialized techniques to ensure effective solution accuracy and numerical stability. A benchmarking study was conducted on two multiphase flow problems in porous media: the classic Buckley‑Leverett (BL) problem and a fully coupled system of equations involving shock waves but with varying levels of solution complexity. The findings show that PM and LM approaches can provide accurate solutions for the BL problem by effectively addressing the infinite gradients associated with shock occurrences. In contrast, AM methods failed to effectively resolve the shock waves. When applied to fully coupled PDEs (with more complex loss landscape), the generalization error in the solutions quickly increased, highlighting the need for ongoing innovation. This study provides a comprehensive review of existing techniques for managing PDE discontinuities using PINNs, offering information on their strengths and limitations. The results underscore the necessity for further research to improve PINNs ability to handle complex discontinuities, particularly in more challenging problems with complex loss landscapes. This includes problems involving higher dimensions or multiphysics systems, where current methods often struggle to maintain accuracy and efficiency.

Abstract:
Evolution‑based protein structure prediction models have achieved breakthrough success in recent years. However, they struggle to generalize beyond evolutionary priors and on sequences lacking rich homologous data. Here we present a novel, out‑of‑domain benchmark based on sactipeptides, a rare class of ribosomally synthesized and post‑translationally modified peptides (RiPPs) characterized by sulfur‑to‑α‑carbon thioether bridges creating cross‑links between cysteine residues and backbone. We evaluate recent models on predicting conformations compatible with these cross‑links bridges for the 10 known sactipeptides with elucidated post‑translational modifications. Crucially, the structures of 5 of them have not yet been experimentally resolved. This makes the task a challenging problem for evolution‑based models, which we find exhibit limited performance (0.0% to 19.2% GDT‑TS on sulfur‑to‑α‑carbon distance). Our results point at the need for physics‑informed models to sustain progress in biomolecular structure prediction.

Abstract:
Vortex flows are ubiquitous in both natural processes and engineering applications, including phenomena such as typhoons, water currents, and aerospace fluid dynamics. The vortex particle method, a computational approach grounded in vortex dynamics, has been extensively applied in aerodynamics, oceanography, turbulence, and aeroacoustics. With the recent introduction of machine learning into computational fluid dynamics, a hybrid framework known as the differentiable vortex particle method (DVPM) has been proposed, which integrates the vortex particle method with deep learning to enable efficient learning and prediction. However, the original formulation of DVPM is limited to ideal vortex flow conditions, such as inviscid flows without non‑conservative body forces, which significantly restricts its practical applicability. In this study, we extend the differentiable vortex particle method beyond idealized flow scenarios to encompass more realistic, non‑ideal conditions, including viscous flow and flow subjected to non‑conservative body forces. We establish the Lamb‑Oseen vortex as a benchmark case, representing a fundamental viscous vortex flow in fluid mechanics. This selection offers significant analytical advantages, as the Lamb‑Oseen vortex possesses an exact analytical solution derived from the Navier‑Stokes (NS) equations, thereby providing definitive ground truth data for training and validation purposes. Through rigorous evaluation across a spectrum of Reynolds numbers, we demonstrate that DVPM achieves superior accuracy in modeling the Lamb‑Oseen vortex compared to conventional convolutional neural networks (CNNs) and physics‑informed neural networks (PINNs). Our results substantiate DVPM's robust capabilities in modeling non‑ideal vortex flows, establishing its distinct advantages over contemporary deep learning methodologies in fluid dynamics applications.

Abstract:
This work investigates the feasibility of using Physics‑Informed Neural Networks (PINNs) as surrogate models for river stage prediction, aiming to reduce computational cost while maintaining predictive accuracy. Our primary contribution demonstrates that PINNs can successfully approximate HEC‑RAS numerical solutions when trained on a single river, achieving strong predictive accuracy with generally low relative errors, though some river segments exhibit higher deviations. By integrating the governing Saint‑Venant equations into the learning process, the proposed PINN‑based surrogate model enforces physical consistency and significantly improves computational efficiency compared to HEC‑RAS. We evaluate the model's performance in terms of accuracy and computational speed, demonstrating that it closely approximates HEC‑RAS predictions while enabling real‑time inference. These results highlight the potential of PINNs as effective surrogate models for single‑river hydrodynamics, offering a promising alternative for computationally efficient river stage forecasting. Future work will explore techniques to enhance PINN training stability and robustness across a more generalized multi‑river model.

Abstract:
Physics‑informed machine learning offers a promising framework for solving complex partial differential equations (PDEs) by integrating observational data with governing physical laws. However, learning PDEs with varying parameters and changing initial conditions and boundary conditions (ICBCs) with theoretical guarantees remains an open challenge. In this paper, we propose physics‑informed deep B‑spline networks, a novel technique that approximates a family of PDEs with different parameters and ICBCs by learning B‑spline control points through neural networks. The proposed B‑spline representation reduces the learning task from predicting solution values over the entire domain to learning a compact set of control points, enforces strict compliance to initial and Dirichlet boundary conditions by construction, and enables analytical computation of derivatives for incorporating PDE residual losses. While existing approximation and generalization theories are not applicable in this setting ‑ where solutions of parametrized PDE families are represented via B‑spline bases ‑ we fill this gap by showing that B‑spline networks are universal approximators for such families under mild conditions. We also derive generalization error bounds for physics‑informed learning in both elliptic and parabolic PDE settings, establishing new theoretical guarantees. Finally, we demonstrate in experiments that the proposed technique has improved efficiency‑accuracy tradeoffs compared to existing techniques in a dynamical system problem with discontinuous ICBCs and can handle nonhomogeneous ICBCs and non‑rectangular domains.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large number of parameters to achieve reasonable accuracy, particularly for complex PDEs. In this paper, we present a quantum‑classical physics‑informed neural network (QCPINN) that combines quantum and classical components, allowing us to solve PDEs with significantly fewer parameters while maintaining comparable accuracy and convergence to classical PINNs. We systematically evaluated two quantum circuit architectures across various configurations on five benchmark PDEs to identify optimal QCPINN designs. Our results demonstrate that the QCPINN achieves stable convergence and comparable accuracy while using only 10‑30% of the trainable parameters required by classical PINNs. This approach also results in a significant reduction in the relative L_2 error for Helmholtz, Klein‑Gordon, and Convection‑diffusion equations, with a reduction ranging from 4% to 64% across various fields. These findings demonstrate the potential of parameter efficiency and solution accuracy in physics‑informed machine learning, allowing for a substantial decrease in model complexity without compromising solution quality.QCPINN presents a promising pathway to address the computational challenges associated with solving PDEs.

Abstract:
Quantitative estimation of human joint motion in daily living spaces is essential for early detection and rehabilitation tracking of neuromusculoskeletal disorders (e.g., Parkinson's) and mitigating trip and fall risks for older adults. Existing approaches involve monitoring devices such as cameras, wearables, and pressure mats, but have operational constraints such as direct line‑of‑sight, carrying devices, and dense deployment. To overcome these limitations, we leverage gait‑induced floor vibration to estimate lower‑limb joint motion (e.g., ankle, knee, and hip flexion angles), allowing non‑intrusive and contactless gait health monitoring in people's living spaces. To overcome the high uncertainty in lower‑limb movement given the limited information provided by the gait‑induced floor vibrations, we formulate a physics‑informed graph to integrate domain knowledge of gait biomechanics and structural dynamics into the model. Specifically, different types of nodes represent heterogeneous information from joint motions and floor vibrations; Their connecting edges represent the physiological relationships between joints and forces governed by gait biomechanics, as well as the relationships between forces and floor responses governed by the structural dynamics. As a result, our model poses physical constraints to reduce uncertainty while allowing information sharing between the body and the floor to make more accurate predictions. We evaluate our approach with 20 participants through a real‑world walking experiment. We achieved an average of 3.7 degrees of mean absolute error in estimating 12 joint flexion angles (38% error reduction from baseline), which is comparable to the performance of cameras and wearables in current medical practices.

Abstract:
NeuralFoil is an open‑source Python‑based tool for rapid aerodynamics analysis of airfoils, similar in purpose to XFoil. Speedups ranging from 8x to 1,000x over XFoil are demonstrated, after controlling for equivalent accuracy. NeuralFoil computes both global and local quantities (lift, drag, velocity distribution, etc.) over a broad input space, including: an 18‑dimensional space of airfoil shapes, possibly including control deflections; a 360 degree range of angles of attack; Reynolds numbers from 10^2 to 10^10; subsonic flows up to the transonic drag rise; and with varying turbulence parameters. Results match those of XFoil closely: the mean relative error of drag is 0.37% on simple cases, and remains as low as 2.0% on a test dataset with numerous post‑stall and transitional cases. NeuralFoil facilitates gradient‑based design optimization, due to its C^\infty‑continuous solutions, automatic‑differentiation‑compatibility, and bounded computational cost without non‑convergence issues. NeuralFoil is a hybrid of physics‑informed machine learning techniques and analytical models. Here, physics information includes symmetries that are structurally embedded into the model architecture, feature engineering using domain knowledge, and guaranteed extrapolation to known limit cases. This work also introduces a new approach for surrogate model uncertainty quantification that enables robust design optimization. This work discusses the methodology and performance of NeuralFoil with several case studies, including a practical airfoil design optimization study including both aerodynamic and non‑aerodynamic constraints. Here, NeuralFoil optimization is able to produce airfoils nearly identical in performance and shape to expert‑designed airfoils within seconds; these computationally‑optimized airfoils provide a useful starting point for further expert refinement.

Abstract:
This paper presents a rigorous evaluation of Real‑to‑Sim parameter estimation approaches for fabric manipulation in robotics. The study systematically assesses three state‑of‑the‑art approaches, namely two differential pipelines and a data‑driven approach. We also devise a novel physics‑informed neural network approach for physics parameter estimation. These approaches are interfaced with two simulations across multiple Real‑to‑Sim scenarios (lifting, wind blowing, and stretching) for five different fabric types and evaluated on three unseen scenarios (folding, fling, and shaking). We found that the simulation engines and the choice of Real‑to‑Sim approaches significantly impact fabric manipulation performance in our evaluation scenarios. Moreover, PINN observes superior performance in quasi‑static tasks but shows limitations in dynamic scenarios.

Abstract:
This article sheds light on the late‑time influence of initial conditions in Boussinesq Rayleigh‑Taylor turbulence using an approach combining direct numerical simulations, machine learning and theory. The initial conditions are characterized by four non‑dimensional numbers describing the statistical properties of random‑phase multi‑mode perturbations of an initial diffuse interface. Based on high‑fidelity data, a surrogate physics‑informed neural network is used to extrapolate the dynamics to very late times and unseen initial conditions, beyond the reach of simulations. This enables uncertainty and global sensitivity analyses to be carried out, revealing the influence of initial conditions in the late‑time regime. While the results support the idea of a universal self‑similar growth rate, the virtual time origin is found to be strongly sensitive to the initial Reynolds, perturbation steepness and bandwidth numbers. An analytical model based on the phenomenology of Rayleigh‑Taylor mixing layers explains most of this dependency, and provide accurate predictions for the virtual time origin. It turns out that when the initial perturbation reaches nonlinear saturation earlier, the mixing layer also re‑accelerates earlier, while the virtual time origin is larger.

Abstract:
For a data‑driven and physics combined modelling of unsteady flow systems with moving immersed boundaries, Sundar \it et al. introduced an immersed boundary‑aware (IBA) framework, combining Physics‑Informed Neural Networks (PINNs) and the immersed boundary method (IBM). This approach was beneficial because it avoided case‑specific transformations to a body‑attached reference frame. Building on this, we now address the challenges of long time integration in velocity reconstruction and pressure recovery by extending this IBA framework with sequential learning strategies. Key difficulties for PINNs in long time integration include temporal sparsity, long temporal domains and rich spectral content. To tackle these, a moving boundary‑enabled PINN is developed, proposing two sequential learning strategies: ‑ a time marching with gradual increase in time domain size, however, this approach struggles with error accumulation over long time domains; and ‑ a time decomposition which divides the temporal domain into smaller segments, combined with transfer learning it effectively reduces error propagation and computational complexity. The key findings for modelling of incompressible unsteady flows past a flapping airfoil include: ‑ for quasi‑periodic flows, the time decomposition approach with preferential spatio‑temporal sampling improves accuracy and efficiency for pressure recovery and aerodynamic load reconstruction, and, ‑ for long time domains, decomposing it into smaller temporal segments and employing multiple sub‑networks, simplifies the problem ensuring stability and reduced network sizes. This study highlights the limitations of traditional PINNs for long time integration of flow‑structure interaction problems and demonstrates the benefits of decomposition‑based strategies for addressing error accumulation, computational cost, and complex dynamics.

Abstract:
We introduce a simultaneous and meshfree topology optimization (TO) framework based on physics‑informed Gaussian processes (GPs). Our framework endows all design and state variables via GP priors which have a shared, multi‑output mean function that is parametrized via a customized deep neural network (DNN). The parameters of this mean function are estimated by minimizing a multi‑component loss function that depends on the performance metric, design constraints, and the residuals on the state equations. Our TO approach yields well‑defined material interfaces and has a built‑in continuation nature that promotes global optimality. Other unique features of our approach include (1) its customized DNN which, unlike fully connected feed‑forward DNNs, has a localized learning capacity that enables capturing intricate topologies and reducing residuals in high gradient fields, (2) its loss function that leverages localized weights to promote solution accuracy around interfaces, and (3) its use of curriculum training to avoid local optimality.To demonstrate the power of our framework, we validate it against commercial TO package COMSOL on three problems involving dissipated power minimization in Stokes flow.

Abstract:
World Models help Artificial Intelligence (AI) predict outcomes, reason about its environment, and guide decision‑making. While widely used in reinforcement learning, they lack the structured, adaptive representations that even young children intuitively develop. Advancing beyond pattern recognition requires dynamic, interpretable frameworks inspired by Piaget's cognitive development theory. We highlight six key research areas ‑‑ physics‑informed learning, neurosymbolic learning, continual learning, causal inference, human‑in‑the‑loop AI, and responsible AI ‑‑ as essential for enabling true reasoning in AI. By integrating statistical learning with advances in these areas, AI can evolve from pattern recognition to genuine understanding, adaptation and reasoning capabilities.

Abstract:
In this paper, we propose a hybrid method that combines finite element method (FEM) and physics‑informed neural network (PINN) for solving linear elliptic problems. This method contains three steps: (1) train a PINN and obtain an approximate solution u_θ; (2) enrich the finite element space with u_θ; (3) obtain the final solution by FEM in the enriched space. In the second step, the enriched space is constructed by addition v + u_θ or multiplication v \cdot u_θ, where v belongs to the standard finite element space. We conduct the convergence analysis for the proposed method. Compared to the standard FEM, the same convergence order is obtained and higher accuracy can be achieved when solution derivatives are well approximated in PINN. Numerical examples from one dimension to three dimensions verify these theoretical results. For some examples, the accuracy of the proposed method can be reduced by a couple of orders of magnitude compared to the standard FEM.

Abstract:
We introduce a novel approach for end‑to‑end black‑box optimization of high energy physics (HEP) detectors using local deep learning (DL) surrogates. These surrogates approximate a scalar objective function that encapsulates the complex interplay of particle‑matter interactions and physics analysis goals. In addition to a standard reconstruction‑based metric commonly used in the field, we investigate the information‑theoretic metric of mutual information. Unlike traditional methods, mutual information is inherently task‑agnostic, offering a broader optimization paradigm that is less constrained by predefined targets. We demonstrate the effectiveness of our method in a realistic physics analysis scenario: optimizing the thicknesses of calorimeter detector layers based on simulated particle interactions. The surrogate model learns to approximate objective gradients, enabling efficient optimization with respect to energy resolution. Our findings reveal three key insights: (1) end‑to‑end black‑box optimization using local surrogates is a practical and compelling approach for detector design, providing direct optimization of detector parameters in alignment with physics analysis goals; (2) mutual information‑based optimization yields design choices that closely match those from state‑of‑the‑art physics‑informed methods, indicating that these approaches operate near optimality and reinforcing their reliability in HEP detector design; and (3) information‑theoretic methods provide a powerful, generalizable framework for optimizing scientific instruments. By reframing the optimization process through an information‑theoretic lens rather than domain‑specific heuristics, mutual information enables the exploration of new avenues for discovery beyond conventional approaches.

Abstract:
In a more connected world, modeling multi‑agent systems with hyperbolic partial differential equations (PDEs) offers a compact, physics‑consistent description of collective dynamics. However, classical control tools need adaptation for these complex systems. Physics‑informed neural networks (PINNs) provide a powerful framework to fix this issue by inferring solutions to PDEs by embedding governing equations into the neural network. A major limitation of original PINNs is their inability to capture steep gradients and discontinuities in hyperbolic PDEs. To tackle this problem, we propose a stacked residual PINN method enhanced with a vanishing viscosity mechanism. Initially, a basic PINN with a small viscosity coefficient provides a stable, low‑fidelity solution. Residual correction blocks with learnable scaling parameters then iteratively refine this solution, progressively decreasing the viscosity coefficient to transition from parabolic to hyperbolic PDEs. Applying this method to traffic state reconstruction improved results by an order of magnitude in relative \mathcalL^2 error, demonstrating its potential to accurately estimate solutions where original PINNs struggle with instability and low fidelity.

Abstract:
Simulating the complex interactions between soft tissues and rigid anatomy is critical for applications in surgical training, planning, and robotic‑assisted interventions. Traditional Finite Element Method (FEM)‑based simulations, while accurate, are computationally expensive and impractical for real‑time scenarios. Learning‑based approaches have shown promise in accelerating predictions but have fallen short in modeling soft‑rigid interactions effectively. We introduce MIXPINN, a physics‑informed Graph Neural Network (GNN) framework for mixed‑material simulations, explicitly capturing soft‑rigid interactions using graph‑based augmentations. Our approach integrates Virtual Nodes (VNs) and Virtual Edges (VEs) to enhance rigid body constraint satisfaction while preserving computational efficiency. By leveraging a graph‑based representation of biomechanical structures, MIXPINN learns high‑fidelity deformations from FEM‑generated data and achieves real‑time inference with sub‑millimeter accuracy. We validate our method in a realistic clinical scenario, demonstrating superior performance compared to baseline GNN models and traditional FEM methods. Our results show that MIXPINN reduces computational cost by an order of magnitude while maintaining high physical accuracy, making it a viable solution for real‑time surgical simulation and robotic‑assisted procedures.

Abstract:
Quantum Physics‑Informed Neural Networks (QPINNs) integrate quantum computing and machine learning to impose physical biases on the output of a quantum neural network, aiming to either solve or discover differential equations. The approach has recently been implemented on both the gate model and continuous variable quantum computing architecture, where it has been demonstrated capable of solving ordinary differential equations. Here, we aim to extend the method to effectively address a wider range of equations, such as the Poisson equation and the heat equation. To achieve this goal, we introduce an architecture specifically designed to compute second‑order (and higher‑order) derivatives without relying on nested automatic differentiation methods. This approach mitigates the unwanted side effects associated with nested gradients in simulations, paving the way for more efficient and accurate implementations. By leveraging such an approach, the quantum circuit addresses partial differential equations, a challenge not yet tackled using this approach on continuous‑variable quantum computers. As a proof‑of‑concept, we solve a one‑dimensional instance of the heat equation, demonstrating its effectiveness in handling PDEs, both in an ideal and a noisy regime. We report our experiment on a photonic hardware to address a realistic noise scenario for our simulations. Such a framework paves the way for further developments in continuous‑variable quantum computing and underscores its potential contributions to advancing quantum machine learning.

Abstract:
While magnetic micro‑robots have demonstrated significant potential across various applications, including drug delivery and microsurgery, the open issue of precise navigation and control in complex fluid environments is crucial for in vivo implementation. This paper introduces a novel flow‑aware navigation and control strategy for magnetic micro‑robots that explicitly accounts for the impact of fluid flow on their movement. First, the proposed method employs a Physics‑Informed U‑Net (PI‑UNet) to refine the numerically predicted fluid velocity using local observations. Then, the predicted velocity is incorporated in a flow‑aware A path planning algorithm, ensuring efficient navigation while mitigating flow‑induced disturbances. Finally, a control scheme is developed to compensate for the predicted fluid velocity, thereby optimizing the micro‑robot's performance. A series of simulation studies and real‑world experiments are conducted to validate the efficacy of the proposed approach. This method enhances both planning accuracy and control precision, expanding the potential applications of magnetic micro‑robots in fluid‑affected environments typical of many medical scenarios.

Abstract:
Nonlinear plasma physics problems are usually simulated through comprehensive modeling of phase space. The extreme computational cost of such simulations has motivated the development of multi‑moment fluid models. However, a major challenge has been finding a suitable fluid closure for these fluid models. Recent developments in physics‑informed machine learning have led to a renewed interest in constructing accurate fluid closure terms. In this study, we take an approach that integrates kinetic physics from the first‑principles Vlasov simulations into a fluid model (through the heat flux closure term) using the Fourier neural operator ‑ a neural network architecture. Without resolving the phase space dynamics, this new fluid model is capable of capturing the nonlinear evolution of the Landau damping process that exactly matches the Vlasov simulation results. This machine learning‑assisted new approach provides a computationally affordable framework that surpasses previous fluid models in accurately modeling the kinetic evolution of complex plasma systems.

Abstract:
Spectral bias is a significant phenomenon in neural network training and can be explained by neural tangent kernel (NTK) theory. In this work, we develop the NTK theory for deep neural networks with physics‑informed loss, providing insights into the convergence of NTK during initialization and training, and revealing its explicit structure. We find that, in most cases, the differential operators in the loss function do not induce a faster eigenvalue decay rate and stronger spectral bias. Some experimental results are also presented to verify the theory.

Abstract:
Accurate forecasting of viral disease outbreaks is crucial for guiding public health responses and preventing widespread loss of life. In recent years, Physics‑Informed Neural Networks (PINNs) have emerged as a promising framework that can capture the intricate dynamics of viral infection and reliably predict its future progression. However, despite notable advances, the application of PINNs in disease modeling remains limited. Standard PINNs are effective in simulating disease dynamics through forward modeling but often face challenges in estimating key biological parameters from sparse or noisy experimental data when applied in an inverse framework. To overcome these limitations, a recent extension known as Disease Informed Neural Networks (DINNs) has emerged, offering a more robust approach to parameter estimation tasks. In this work, we apply this DINNs technique on a recently proposed hepatitis B virus (HBV) infection dynamics model to predict infection transmission within the liver. This model consists of four compartments: uninfected and infected hepatocytes, rcDNA‑containing capsids, and free viruses. Leveraging the power of DINNs, we study the impacts of (i) variations in parameter range, (ii) experimental noise in data, (iii) sample sizes, (iv) network architecture and (v) learning rate. We employ this methodology in experimental data collected from nine HBV‑infected chimpanzees and observe that it reliably estimates the model parameters. DINNs can capture infection dynamics and predict their future progression even when data of some compartments of the system are missing. Additionally, it identifies the influential model parameters that determine whether the HBV infection is cleared or persists within the host.

Abstract:
Deep learning has shown strong potential in modeling complex spatiotemporal dynamics. However, most existing methods depend on densely and uniformly sampled data, which is often unavailable in practice due to sensor and cost limitations. In many real‑world settings, such as mobile sensing and physical experiments, data are burst‑sampled with short high‑frequency segments followed by long gaps, making it difficult to learn accurate dynamics from sparse observations. To address this issue, we propose Physics‑Informed Multi‑Scale Recurrent Learning (PIMRL), a novel framework specifically designed for burst‑sampled spatiotemporal data. PIMRL combines macro‑scale latent dynamics inference with micro‑scale adaptive refinement guided by incomplete prior information from partial differential equations (PDEs). It further introduces a temporal message‑passing mechanism to effectively propagate information across burst intervals. This multi‑scale architecture enables PIMRL to model complex systems accurately even under severe data scarcity. We evaluate our approach on five benchmark datasets involving 1D to 3D multi‑scale PDEs. The results show that PIMRL consistently outperforms state‑of‑the‑art baselines, achieving substantial improvements and reducing errors by up to 80% in the most challenging settings, which demonstrates the clear advantage of our model. Our work demonstrates the effectiveness of physics‑informed recurrent learning for accurate and efficient modeling of sparse spatiotemporal systems.

Abstract:
Extreme learning machines (ELMs), which preset hidden layer parameters and solve for last layer coefficients via a least squares method, can typically solve partial differential equations faster and more accurately than Physics Informed Neural Networks. However, they remain computationally expensive when high accuracy requires large least squares problems to be solved. Domain decomposition methods (DDMs) for ELMs have allowed parallel computation to reduce training times of large systems. This paper constructs a coarse space for ELMs, which enables further acceleration of their training. By partitioning interface variables into coarse and non‑coarse variables, selective elimination introduces a Schur complement system on the non‑coarse variables with the coarse problem embedded. Key to the performance of the proposed method is a Neumann‑Neumann acceleration that utilizes the coarse space. Numerical experiments demonstrate significant speedup compared to a previous DDM method for ELMs.

Abstract:
Knowledge of the force time history of a structure is essential to assess its behaviour, ensure safety and maintain reliability. However, direct measurement of external forces is often challenging due to sensor limitations, unknown force characteristics, or inaccessible load points. This paper presents an efficient dynamic load reconstruction method using physics‑informed Gaussian processes (GP) based on frequency‑sparse Fourier basis functions. The GP's covariance matrices are built using the description of the system dynamics, and the model is trained using structural response measurements. This provides support and interpretability to the machine learning model, in contrast to purely data‑driven methods. In addition, the model filters out irrelevant components in the Fourier basis function by leveraging the sparsity of structural responses in the frequency domain, thereby reducing computational complexity during optimization. The trained model for structural responses is then integrated with the differential equation for a harmonic oscillator, creating a probabilistic dynamic load model that predicts load patterns without requiring force data during training. The model's effectiveness is validated through two case studies: a numerical model of a wind‑excited 76‑story building and an experiment using a physical scale model of the Lillebælt Bridge in Denmark, excited by a servo motor. For both cases, validation of the reconstructed forces is provided using comparison metrics for several signal properties. The developed model holds potential for applications in structural health monitoring, damage prognosis, and load model validation.

Abstract:
Outdoor thermal comfort is a critical determinant of urban livability, particularly in hot desert climates where extreme heat poses challenges to public health, energy consumption, and urban planning. Mean Radiant Temperature (T_mrt) is a key parameter for evaluating outdoor thermal comfort, especially in urban environments where radiation dynamics significantly impact human thermal exposure. Traditional methods of estimating T_mrt rely on field measurements and computational simulations, both of which are resource intensive. This study introduces a Physics‑Informed Neural Network (PINN) approach that integrates shortwave and longwave radiation modeling with deep learning techniques. By leveraging a multimodal dataset that includes meteorological data, built environment characteristics, and fisheye image‑derived shading information, our model enhances predictive accuracy while maintaining physical consistency. Our experimental results demonstrate that the proposed PINN framework outperforms conventional deep learning models, with the best‑performing configurations achieving an RMSE of 3.50 and an R^2 of 0.88. This approach highlights the potential of physics‑informed machine learning in bridging the gap between computational modeling and real‑world applications, offering a scalable and interpretable solution for urban thermal comfort assessments.

Abstract:
Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers ‑‑ Bayesian statistical models mostly based on Gaussian process (GP) priors which seamlessly combine empirical measurements and mechanistic knowledge. As such, they quantify uncertainties arising from e.g. noisy or missing data, unknown PDE parameters or discretization error by design. Prior work has established connections to classical PDE solvers and provided solid theoretical guarantees. However, scaling such methods to large‑scale problems remains a fundamental challenge primarily due to dense covariance matrices. Our approach addresses the scalability issues by leveraging the Markov property of many commonly used GP priors. It has been shown that such priors are solutions to stochastic PDEs (SPDEs) which when discretized allow for highly efficient GP regression through sparse linear algebra. In this work, we show how to leverage this prior class to make probabilistic PDE solvers practical, even for large‑scale nonlinear PDEs, through greatly accelerated inference mechanisms. Additionally, our approach also allows for flexible and physically meaningful priors beyond what can be modeled with covariance functions. Experiments confirm substantial speedups and accelerated convergence of our physics‑informed priors in nonlinear settings.

Abstract:
Accurate free energy representations are crucial for understanding phase dynamics in materials. We employ a scale‑bridging approach to incorporate atomistic information into our free energy model by training a neural network on DFT‑informed Monte Carlo data. To optimize sampling in the high‑dimensional Monte Carlo space, we present an Active Learning framework that integrates space‑filling sampling, uncertainty‑based sampling, and physics‑informed sampling. Additionally, our approach includes methods such as hyperparameter tuning, dynamic sampling, and novelty enforcement. These strategies can be combined to reduce MSE,either globally or in targeted regions of interest,while minimizing the number of required data points. The framework introduced here is broadly applicable to Monte Carlo sampling of a range of materials systems.

Abstract:
The precision, stability, and performance of lightweight high‑strength steel structures in heavy machinery is affected by their highly nonlinear dynamics. This, in turn, makes control more difficult, simulation more computationally intensive, and achieving real‑time autonomy, using standard approaches, impossible. Machine learning through data‑driven, physics‑informed and physics‑inspired networks, however, promises more computationally efficient and accurate solutions to nonlinear dynamic problems. This study proposes a novel framework that has been developed to estimate real‑time structural deflection in hydraulically actuated three‑dimensional systems. It is based on SLIDE, a machine‑learning‑based method to estimate dynamic responses of mechanical systems subjected to forced excitations.~Further, an algorithm is introduced for the data acquisition from a hydraulically actuated system using randomized initial configurations and hydraulic pressures.~The new framework was tested on a hydraulically actuated flexible boom with various sensor combinations and lifting various payloads. The neural network was successfully trained in less time using standard parameters from PyTorch, ADAM optimizer, the various sensor inputs, and minimal output data. The SLIDE‑trained neural network accelerated deflection estimation solutions by a factor of 10^7 in reference to flexible multibody simulation batches and provided reasonable accuracy. These results support the studies goal of providing robust, real‑time solutions for control, robotic manipulators, structural health monitoring, and automation problems.

Abstract:
In many science and engineering settings, system dynamics are characterized by governing PDEs, and a major challenge is to solve inverse problems (IPs) where unknown PDE parameters are inferred based on observational data gathered under limited budget. Due to the high costs of setting up and running experiments, experimental design (ED) is often done with the help of PDE simulations to optimize for the most informative design parameters to solve such IPs, prior to actual data collection. This process of optimizing design parameters is especially critical when the budget and other practical constraints make it infeasible to adjust the design parameters between trials during the experiments. However, existing experimental design (ED) methods tend to require sequential and frequent design parameter adjustments between trials. Furthermore, they also have significant computational bottlenecks due to the need for complex numerical simulations for PDEs, and do not exploit the advantages provided by physics informed neural networks (PINNs), such as its meshless solutions, differentiability, and amortized training. This work presents PIED, the first ED framework that makes use of PINNs in a fully differentiable architecture to perform continuous optimization of design parameters for IPs for one‑shot deployments. PIED overcomes existing methods' computational bottlenecks through parallelized computation and meta‑learning of PINN parameter initialization, and proposes novel methods to effectively take into account PINN training dynamics in optimizing the ED parameters. Through experiments based on noisy simulated data and even real world experimental data, we empirically show that given limited observation budget, PIED significantly outperforms existing ED methods in solving IPs, including challenging settings where the inverse parameters are unknown functions rather than just finite‑dimensional.

Abstract:
Amorphous multi‑element materials offer unprecedented tunability in composition and properties, yet their rational design remains challenging due to the lack of predictive structure‑property relationships and the vast configurational space. Traditional modeling struggles to capture the intricate short‑range order that dictates their stability and functionality. We here introduce ApolloX, a pioneering predictive framework for amorphous multi‑element materials, establishing a new paradigm by integrating physics‑informed generative modeling with particle swarm optimization, using chemical short‑range order as an explicit constraint. By systematically navigating the disordered energy landscape, ApolloX enables the targeted design of thermodynamically stable amorphous configurations. It accurately predicts atomic‑scale arrangements, including composition‑driven metal clustering and amorphization trends, which are well‑validated by experiments, while also guiding synthesis by leveraging sluggish diffusion to control elemental distribution and disorder. The resulting structural evolution, governed by composition, directly impacts catalytic performance, leading to improved activity and stability with increasing amorphization. This predictive‑experimental synergy transforms the discovery of amorphous materials, unlocking new frontiers in catalysis, energy storage, and functional disordered systems.

Abstract:
This study introduces a unified control framework that addresses the challenge of precise quadruped locomotion with unknown payloads, named as online payload identification‑based physics‑informed neural network predictive control (OPI‑PINNPC). By integrating online payload identification with physics‑informed neural networks (PINNs), our approach embeds identified mass parameters directly into the neural network's loss function, ensuring physical consistency while adapting to changing load conditions. The physics‑constrained neural representation serves as an efficient surrogate model within our nonlinear model predictive controller, enabling real‑time optimization despite the complex dynamics of legged locomotion. Experimental validation on our quadruped robot platform demonstrates 35% improvement in position and orientation tracking accuracy across diverse payload conditions (25‑100 kg), with substantially faster convergence compared to previous adaptive control methods. Our framework provides a adaptive solution for maintaining locomotion performance under variable payload conditions without sacrificing computational efficiency.

Abstract:
General‑sum differential games can approximate values solved by Hamilton‑Jacobi‑Isaacs (HJI) equations for efficient inference when information is incomplete. However, solving such games through conventional methods encounters the curse of dimensionality (CoD). Physics‑informed neural networks (PINNs) offer a scalable approach to alleviate the CoD and approximate values, but there exist convergence issues for value approximations through vanilla PINNs when state constraints lead to values with large Lipschitz constants, particularly in safety‑critical applications. In addition to addressing CoD, it is necessary to learn a generalizable value across a parametric space of games, rather than training multiple ones for each specific player‑type configuration. To overcome these challenges, we propose a Hybrid Neural Operator (HNO), which is an operator that can map parameter functions for games to value functions. HNO leverages informative supervised data and samples PDE‑driven data across entire spatial‑temporal space for model refinement. We evaluate HNO on 9D and 13D scenarios with nonlinear dynamics and state constraints, comparing it against a Supervised Neural Operator (a variant of DeepONet). Under the same computational budget and training data, HNO outperforms SNO for safety performance. This work provides a step toward scalable and generalizable value function approximation, enabling real‑time inference for complex human‑robot or multi‑agent interactions.

Abstract:
Accurate tropical cyclone (TC) intensity prediction is crucial for mitigating storm hazards, yet its complex dynamics pose challenges to traditional methods. Here, we introduce a Physics‑Informed Residual Neural Ordinary Differential Equation (PIR‑NODE) model to precisely forecast TC intensity evolution. This model leverages the powerful non‑linear fitting capabilities of deep learning, integrates residual connections to enhance model depth and training stability, and explicitly models the continuous temporal evolution of TC intensity using Neural ODEs. Experimental results in the SHIPS dataset demonstrate that the PIR‑NODE model achieves a significant improvement in 24‑hour intensity prediction accuracy compared to traditional statistical models and benchmark deep learning methods, with a 25. 2% reduction in the root mean square error (RMSE) and a 19.5% increase in R‑square (R2) relative to a baseline of neural network. Crucially, the residual structure effectively preserves initial state information, and the model exhibits robust generalization capabilities. This study details the PIR‑NODE model architecture, physics‑informed integration strategies, and comprehensive experimental validation, revealing the substantial potential of deep learning techniques in predicting complex geophysical systems and laying the foundation for future refined TC forecasting research.

Abstract:
Recently, physics‑informed neural networks (PINNs) and their variants have gained significant popularity as a scientific computing method for solving partial differential equations (PDEs), whereas accuracy is still its main shortcoming. Despite numerous development efforts, there is no literature demonstrating that these methods surpass classic numerical algorithms in solving the forward issue. In this paper, by analyzing the disparities between PINNs and traditional numerical methods based on mesh discretization, we investigate the underlying causes for the in adequate precision of PINNs and introduce a novel approach named global physics‑informed neural networks (GPINNs). Inspired by the crucial concept of global nodal association in conventional numerical algorithms, GPINNs leverages the prior field distribution information from pre‑trained PINNs to estimate the association weights between arbitrary nodes in space. GPINNs can not only be regarded as a meshless approach but also be demonstrated, both theoretically and in practical circumstances, to have the ability of second‑order convergence when trained with equidistant nodes. Overall, GPINNs may be seen as an ideal approach to inheriting the merits of scientific machine learning (SciML) and conventional numerical computing, which also represent the first SciML algorithm to surpass standard numerical methods in terms of accuracy.

Abstract:
This paper presents two novel, physics‑informed extreme learning machine (PIELM)‑based algorithms for solving steady and unsteady nonlinear partial differential equations (PDEs) related to fluid flow. Although single‑hidden‑layer PIELMs outperform deep physics‑informed neural networks (PINNs) in speed and accuracy for linear and quasilinear PDEs, their extension to nonlinear problems remains challenging. To address this, we introduce a curriculum learning strategy that reformulates nonlinear PDEs as a sequence of increasingly complex quasilinear PDEs. Additionally, our approach enables a physically interpretable initialization of network parameters by leveraging Radial Basis Functions (RBFs). The performance of the proposed algorithms is validated on two benchmark incompressible flow problems: the viscous Burgers equation and lid‑driven cavity flow. To the best of our knowledge, this is the first work to extend PIELM to solving Burgers' shock solution as well as lid‑driven cavity flow up to a Reynolds number of 100. As a practical application, we employ PIELM to predict blood flow in a stenotic vessel. The results confirm that PIELM efficiently handles nonlinear PDEs, positioning it as a promising alternative to PINNs for both linear and nonlinear PDEs.

Abstract:
We explore the capability of physics‑informed neural networks (PINNs) to discover multiple solutions. Many real‑world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty is giving appropriate initial conditions or initial guesses, to which the widely used time‑marching schemes and Newton's iteration method are very sensitive in finding solutions for complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first reveal that PINNs, when combined with random initialization and deep ensemble method ‑‑ originally developed for uncertainty quantification ‑‑ can effectively uncover multiple solutions to nonlinear ordinary and partial differential equations (ODEs/PDEs). Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often‑overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN‑generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen‑Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear differential equations.

Abstract:
As técnicas de aprendizado de máquina emergiram no contexto científico e se desenvolveram como ferramentas poderosas para enfrentar uma ampla gama de desafios na sociedade. A integração dessas técnicas com a física tem conduzido a abordagens inovadoras na compreensão, controle e simulação de fenômenos físicos. Este artigo visa proporcionar uma introdução prática às redes neurais e seus conceitos fundamentais, destacando perspectivas recentes dos avanços na interseção entre modelos de aprendizado de máquina e sistemas físicos. Além disso, apresentamos um material prático para orientar o leitor em seus primeiros passos na aplicação de redes neurais para resolver problemas físicos. Como exemplo ilustrativo, fornecemos quatro aplicações de complexidades crescentes para o problema de um pêndulo simples, a saber: fit de parâmetros da Equação Diferencial Ordinária (EDO) do pêndulo para aproximação de ângulo pequeno; Physics Informed Neural Networks (PINNs) para encontrar soluções da EDO do pêndulo em ângulo pequeno; Autoencoders em conjunto de dados de imagens do pêndulo para estimação de dimensionalidade do espaço de parâmetros do problema físico; uso de arquiteturas Sparse Identification of Non‑Linear Dynamics (SINDy) para descoberta de modelos e expressões analíticas para o problema do pêndulo não linear (ângulos grandes).

Abstract:
Physics‑Informed Neural Networks (PINNs) have gained significant attention for their simplicity and flexibility in engineering and scientific computing. In this study, we introduce a normalized PINN (NPINN) framework to solve a class of wave propagation equations in non‑unitized domains over extended time ranges. This is achieved through a normalization technique that involves either spatial or temporal variable normalization. To enhance the capability of NPINN in solving wave equations, we integrate a Fourier‑induced deep neural network as the solver, leading to a novel architecture termed NFPINN. Furthermore, we explore different normalization strategies for spatial and temporal variables and identify the optimal normalization approach for our method. To assess the effectiveness and robustness of the proposed NFPINN, we present numerical experiments in both two‑dimensional and three‑dimensional Euclidean spaces, considering regular and irregular domains. The results confirm the accuracy and stability of our approach.

Abstract:
Electromyography (EMG)‑‑based computational musculoskeletal modeling is a non‑invasive method for studying musculotendon function, human movement, and neuromuscular control, providing estimates of internal variables like muscle forces and joint torques. However, EMG signals from deeper muscles are often challenging to measure by placing the surface EMG electrodes and unfeasible to measure directly using invasive methods. The restriction to the access of EMG data from deeper muscles poses a considerable obstacle to the broad adoption of EMG‑driven modeling techniques. A strategic alternative is to use an estimation algorithm to approximate the missing EMG signals from deeper muscle. A similar strategy is used in physics‑informed deep learning, where the features of physical systems are learned without labeled data. In this work, we propose a hybrid deep learning algorithm, namely the neural musculoskeletal model (NMM), that integrates physics‑informed and data‑driven deep learning to approximate the EMG signals from the deeper muscles. While data‑driven modeling is used to predict the missing EMG signals, physics‑based modeling engraves the subject‑specific information into the predictions. Experimental verifications on five test subjects are carried out to investigate the performance of the proposed hybrid framework. The proposed NMM is validated against the joint torque computed from 'OpenSim' software. The predicted deep EMG signals are also compared against the state‑of‑the‑art muscle synergy extrapolation (MSE) approach, where the proposed NMM completely outperforms the existing MSE framework by a significant margin.

Abstract:
Seismic inversion‑including post‑stack, pre‑stack, and full waveform inversion is compute and memory‑intensive. Recently, several approaches, including physics‑informed machine learning, have been developed to address some of these limitations. Motivated by the potential of quantum computing, we report on our attempt to map one such classical physics‑informed algorithm to a quantum framework. The primary goal is to investigate the technical challenges of this mapping, given that quantum algorithms rely on computing principles fundamentally different from those in classical computing. Quantum computers operate using qubits, which exploit superposition and entanglement, offering the potential to solve classically intractable problems. While current quantum hardware is limited, hybrid quantum‑classical algorithms‑particularly in quantum machine learning (QML)‑demonstrate potential for near‑term applications and can be readily simulated. We apply QML to subsurface imaging through the development of a hybrid quantum physics‑informed neural network (HQ‑PINN) for post‑stack and pre‑stack seismic inversion. The HQ‑PINN architecture adopts an encoder‑decoder structure: a hybrid quantum neural network encoder estimates P‑ and S‑impedances from seismic data, while the decoder reconstructs seismic responses using geophysical relationships. Training is guided by minimizing the misfit between the input and reconstructed seismic traces. We systematically assess the impact of quantum layer design, differentiation strategies, and simulator backends on inversion performance. We demonstrate the efficacy of our approach through the inversion of both synthetic and the Sleipner field datasets. The HQ‑PINN framework consistently yields accurate results, showcasing quantum computing's promise for geosciences and paving the way for future quantum‑enhanced geophysical workflows.

Abstract:
First formulated by Sir Isaac Newton in his work "Philosophiae Naturalis Principia Mathematica", the concept of the Three‑Body Problem was put forth as a study of the motion of the three celestial bodies within the Earth‑Sun‑Moon system. In a generalized definition, it seeks to predict the motion for an isolated system composed of three point masses freely interacting under Newton's law of universal attraction. This proves to be analogous to a multitude of interactions between celestial bodies, and thus, the problem finds applicability within the studies of celestial mechanics. Despite numerous attempts by renowned physicists to solve it throughout the last three centuries, no general closed‑form solutions have been reached due to its inherently chaotic nature for most initial conditions. Current state‑of‑the‑art solutions are based on two approaches, either numerical high‑precision integration or machine learning‑based. Notwithstanding the breakthroughs of neural networks, these present a significant limitation, which is their ignorance of any prior knowledge of the chaotic systems presented. Thus, in this work, we propose a novel method that utilizes Physics‑Informed Neural Networks (PINNs). These deep neural networks are able to incorporate any prior system knowledge expressible as an Ordinary Differential Equation (ODE) into their learning processes as a regularizing agent. Our findings showcase that PINNs surpass current state‑of‑the‑art machine learning methods with comparable prediction quality. Despite a better prediction quality, the usability of numerical integrators suffers due to their prohibitively high computational cost. These findings confirm that PINNs are both effective and time‑efficient open‑form solvers of the Three‑Body Problem that capitalize on the extensive knowledge we hold of classical mechanics.

Abstract:
Geodesic distances play a fundamental role in robotics, as they efficiently encode global geometric information of the domain. Recent methods use neural networks to approximate geodesic distances by solving the Eikonal equation through physics‑informed approaches. While effective, these approaches often suffer from unstable convergence during training in complex environments. We propose a framework to learn geodesic distances in irregular domains by using the Soner boundary condition, and systematically evaluate the impact of data losses on training stability and solution accuracy. Our experiments demonstrate that incorporating data losses significantly improves convergence robustness, reducing training instabilities and sensitivity to initialization. These findings suggest that hybrid data‑physics approaches can effectively enhance the reliability of learning‑based geodesic distance solvers with sparse data.

Abstract:
We propose GAGrasp, a novel framework for dexterous grasp generation that leverages geometric algebra representations to enforce equivariance to SE(3) transformations. By encoding the SE(3) symmetry constraint directly into the architecture, our method improves data and parameter efficiency while enabling robust grasp generation across diverse object poses. Additionally, we incorporate a differentiable physics‑informed refinement layer, which ensures that generated grasps are physically plausible and stable. Extensive experiments demonstrate the model's superior performance in generalization, stability, and adaptability compared to existing methods. Additional details at https://gagrasp.github.io/

Abstract:
Physics‑informed neural network (PINN) is a powerful emerging method for studying forward‑inverse problems of partial differential equations (PDEs), even from limited sample data. Variable coefficient PDEs, which model real‑world phenomena, are of considerable physical significance and research value. This study proposes a gradient‑enhanced PINN with residual unit (R‑gPINN) method to solve the data‑driven solution and function discovery for variable coefficient PDEs. On the one hand, the proposed method incorporates residual units into the neural networks to mitigate gradient vanishing and network degradation, unify linear and nonlinear coefficient problem. We present two types of residual unit structures in this work to offer more flexible solutions in problem‑solving. On the other hand, by including gradient terms of variable coefficients, the method penalizes collocation points that fail to satisfy physical properties. This enhancement improves the network's adherence to physical constraints and aligns the prediction function more closely with the objective function. Numerical experiments including solve the forward‑inverse problems of variable coefficient Burgers equation, variable coefficient KdV equation, variable coefficient Sine‑Gordon equation, and high‑dimensional variable coefficient Kadomtsev‑Petviashvili equation. The results show that using R‑gPINN method can greatly improve the accuracy of predict solution and predict variable coefficient in solving variable coefficient equations.

Abstract:
Physics‑Informed Neural Networks (PINNs) offer a promising approach to simulating physical systems. Still, their application is limited by optimization challenges, mainly due to the lack of activation functions that generalize well across several physical systems. Existing activation functions often lack such flexibility and generalization power. To address this issue, we introduce Rational Exponential Activation (REAct), a generalized form of tanh consisting of four learnable shape parameters. Experiments show that REAct outperforms many standard and benchmark activations, achieving an MSE three orders of magnitude lower than tanh on heat problems and generalizing well to finer grids and points beyond the training domain. It also excels at function approximation tasks and improves noise rejection in inverse problems, leading to more accurate parameter estimates across varying noise levels.

Abstract:
This study benchmarks hybrid quantum physics‑informed neural network (HQPINN) to model high‑speed flows, compared against classical physics‑informed neural networks (PINNs) and fully quantum neural networks (QNNs). The HQPINN architecture integrates a parameterized quantum circuit (PQC) with a classical neural network in parallel, trained via a physics‑informed loss. Across harmonic, non‑harmonic, and transonic benchmarks, HQPINNs demonstrate balanced performance, offering competitive accuracy and stability with reduced parameter cost. Quantum PINNs are highly efficient for harmonic problems achieving the lowest loss with minimal parameters due to their Fourier structure, but struggle to generalize in non‑harmonic settings involving shocks and discontinuities. HQPINNs mitigate such artifacts, and with sufficient parameterization, can match the performance of classical models in more complex regimes. Although constrained by current quantum emulation costs and scalability, HQPINNs show promise as general‑purpose solvers, offering parameter efficiency with robust fallback behavior, particularly suited for problems where the nature of the solution is not known a‑priori.

Abstract:
Reducing radiation doses benefits patients, however, the resultant low‑dose computed tomography (LDCT) images often suffer from clinically unacceptable noise and artifacts. While deep learning (DL) shows promise in LDCT reconstruction, it requires large‑scale data collection from multiple clients, raising privacy concerns. Federated learning (FL) has been introduced to address these privacy concerns; however, current methods are typically tailored to specific scanning protocols, which limits their generalizability and makes them less effective for unseen protocols. To address these issues, we propose SCAN‑PhysFed, a novel SCanning‑ and ANatomy‑level personalized Physics‑Driven Federated learning paradigm for LDCT reconstruction. Since the noise distribution in LDCT data is closely tied to scanning protocols and anatomical structures being scanned, we design a dual‑level physics‑informed way to address these challenges. Specifically, we incorporate physical and anatomical prompts into our physics‑informed hypernetworks to capture scanning‑ and anatomy‑specific information, enabling dual‑level physics‑driven personalization of imaging features. These prompts are derived from the scanning protocol and the radiology report generated by a medical large language model (MLLM), respectively. Subsequently, client‑specific decoders project these dual‑level personalized imaging features back into the image domain. Besides, to tackle the challenge of unseen data, we introduce a novel protocol vector‑quantization strategy (PVQS), which ensures consistent performance across new clients by quantifying the unseen scanning code as one of the codes in the scanning codebook. Extensive experimental results demonstrate the superior performance of SCAN‑PhysFed on public datasets.

Abstract:
In numerical simulation, structured mesh generation often requires a lot of time and manpower investment. The general scheme for structured quad mesh generation is to find a mapping between the computational domain and the physical domain. This mapping can be obtained by solving partial differential equations. However, existing structured mesh generation methods are difficult to ensure both efficiency and mesh quality. In this paper, we propose a structured mesh generation method based on physics‑informed neural network, PINN‑MG. It takes boundary curves as input and then utilizes an attention network to capture the potential mapping between computational and physical domains, generating structured meshes for the input physical domain. PINN‑MG introduces the Navier‑Lamé equation in linear elastic as a partial differential equation term in the loss function, ensuring that the neural network conforms to the law of elastic body deformation when optimizing the loss value. The training process of PINN‑MG is completely unsupervised and does not require any prior knowledge or datasets, which greatly reduces the previous workload of producing structured mesh datasets. Experimental results show that PINN‑MG can generate higher quality structured quad meshes than other methods, and has the advantages of traditional algebraic methods and differential methods.

Abstract:
Advanced slot and winding designs are imperative to create future high performance electrical machines (EM). As a result, the development of methods to design and improve slot filling factor (SFF) has attracted considerable research. Recent developments in manufacturing processes, such as additive manufacturing and alternative materials, has also highlighted a need for novel high‑fidelity design techniques to develop high performance complex geometries and topologies. This study therefore introduces a novel physics‑informed machine learning (PIML) design optimization process for improving SFF in traction electrical machines used in electric vehicles. A maximum entropy sampling algorithm (MESA) is used to seed a physics‑informed Bayesian optimization (PIBO) algorithm, where the target function and its approximations are produced by Gaussian processes (GP)s. The proposed PIBO‑MESA is coupled with a 2D finite element model (FEM) to perform a GP‑based surrogate and provide the first demonstration of the optimal combination of complex design variables for an electrical machine. Significant computational gains were achieved using the new PIBO‑MESA approach, which is 45% faster than existing stochastic methods, such as the non‑dominated sorting genetic algorithm II (NSGA‑II). The FEM results confirm that the new design optimization process and keystone shaped wires lead to a higher SFF (i.e. by 20%) and electromagnetic improvements (e.g. maximum torque by 12%) with similar resistivity. The newly developed PIBO‑MESA design optimization process therefore presents significant benefits in the design of high‑performance electric machines, with reduced development time and costs.

Abstract:
The advancement of smart grid technologies necessitates the integration of cutting‑edge computational methods to enhance predictive energy optimization. This study proposes a multi‑faceted approach by incorporating (1) Deep Reinforcement Learning (DRL) agents trained using data from Digital Twins (DTs) to optimize energy consumption in real time, (2) Physics‑Informed Neural Networks (PINNs) to seamlessly embed physical laws within the optimization process, ensuring model accuracy and interpretability, and (3) Blockchain (BC) technology to facilitate secure and transparent communication across the smart grid infrastructure. The model was trained and validated using comprehensive datasets, including smart meter energy consumption data, renewable energy outputs, dynamic pricing, and user preferences collected from IoT devices. The proposed framework achieved superior predictive performance with a Mean Absolute Error (MAE) of 0.237 kWh, Root Mean Square Error (RMSE) of 0.298 kWh, and an R‑squared (R2) value of 0.978, indicating a 97.8% explanation of data variance. Classification metrics further demonstrated the model's robustness, achieving 97.7% accuracy, 97.8% precision, 97.6% recall, and an F1 Score of 97.7%. Comparative analysis with traditional models like Linear Regression, Random Forest, SVM, LSTM, and XGBoost revealed the superior accuracy and real‑time adaptability of the proposed method. In addition to enhancing energy efficiency, the model reduced energy costs by 35%, maintained a 96% user comfort index, and increased renewable energy utilization to 40%. This study demonstrates the transformative potential of integrating PINNs, DT, and Blockchain technologies to optimize energy consumption in smart grids, paving the way for sustainable, secure, and efficient energy management systems.

Abstract:
Deep operator networks (DeepONets) represent a powerful class of data‑driven methods for operator learning, demonstrating strong approximation capabilities for a wide range of linear and nonlinear operators. They have shown promising performance in learning operators that govern partial differential equations (PDEs), including diffusion‑reaction systems and Burgers' equations. However, the accuracy of DeepONets is often constrained by computational limitations and optimization challenges inherent in training deep neural networks. Furthermore, the computational cost associated with training these networks is typically very high. To address these challenges, we leverage randomized neural networks (RaNNs), in which the parameters of the hidden layers remain fixed following random initialization. RaNNs compute the output layer parameters using the least‑squares method, significantly reducing training time and mitigating optimization errors. In this work, we integrate DeepONets with RaNNs to propose RaNN‑DeepONets, a hybrid architecture designed to balance accuracy and efficiency. Furthermore, to mitigate the need for extensive data preparation, we introduce the concept of physics‑informed RaNN‑DeepONets. Instead of relying on data generated through other time‑consuming numerical methods, we incorporate PDE information directly into the training process. We evaluate the proposed model on three benchmark PDE problems: diffusion‑reaction dynamics, Burgers' equation, and the Darcy flow problem. Through these tests, we assess its ability to learn nonlinear operators with varying input types. When compared to the standard DeepONet framework, RaNN‑DeepONets achieves comparable accuracy while reducing computational costs by orders of magnitude. These results highlight the potential of RaNN‑DeepONets as an efficient alternative for operator learning in PDE‑based systems.

Abstract:
Many inverse problems require reconstructing physical fields from limited and noisy data while incorporating known governing equations. A growing body of work within probabilistic numerics formalizes such tasks via Bayesian inference in function spaces by assigning a physically meaningful prior to the latent field. In this work, we demonstrate that Brownian bridge Gaussian processes can be viewed as a softly‑enforced physics‑constrained prior for the Poisson equation. We first show equivalence between the variational problem associated with the Poisson equation and a kernel ridge regression objective. Then, through the connection between Gaussian process regression and kernel methods, we identify a Gaussian process for which the posterior mean function and the minimizer to the variational problem agree, thereby placing this PDE‑based regularization within a fully Bayesian framework. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We then develop a finite‑dimensional representation in function space and prove convergence of the projected prior and resulting posterior in Wasserstein distance. Finally, we connect the method to the important problem of identifying model‑form error in applications, providing a diagnostic for model misspecification.

Abstract:
We present a seeded topology optimization methodology for integrated photonic devices fabricated on foundry platforms that yields improved performance compared to traditional topology optimization. We employ blurring filters and a design rule check correction algorithm to more readily meet fabrication constraints, resulting in devices with fewer artifacts and improved correlation between simulation and measurements. A statistical study is performed on a 2D modal multiplexer, revealing that 87% of devices optimized using this strategy conform to foundry constraints, compared to 13% of devices optimized using traditional TO. We apply seeded topology optimization to an ultra‑compact TE modal multiplexer, a TE mode converter, a polarization rotator, and a high‑contrast grating reflector. Using this optimization strategy, the measured insertion loss of the TE mode converter was reduced from < 1.50 dB to < 0.64 dB, and the measured TE1 insertion loss of the TE modal multiplexer was reduced from < 3.95 dB to < 1.38 dB over C‑band. This approach enables a two‑step inverse design process, merging of physics‑informed design strategies with inverse design, and ensures strict compliance with foundry constraints throughout optimization.

Abstract:
In this work, we integrate the predictive capabilities of compartmental disease dynamics models with machine learning ability to analyze complex, high‑dimensional data and uncover patterns that conventional models may overlook. Specifically, we present a proof of concept demonstrating the application of data‑driven methods and deep neural networks to a recently introduced Susceptible‑Infected‑Recovered type model with social features, including a saturated incidence rate, to improve epidemic prediction and forecasting. Our results show that a robust data augmentation strategy trough suitable data‑driven models can improve the reliability of Feed‑Forward Neural Networks and Nonlinear Autoregressive Networks, providing a complementary strategy to Physics‑Informed Neural Networks, particularly in settings where data augmentation from mechanistic models can enhance learning. This approach enhances the ability to handle nonlinear dynamics and offers scalable, data‑driven solutions for epidemic forecasting, prioritizing predictive accuracy over the constraints of physics‑based models. Numerical simulations of the lockdown and post‑lockdown phase of the COVID‑19 epidemic in Italy and Spain validate our methodology.

Abstract:
Imitating how humans move their gaze in a visual scene is a vital research problem for both visual understanding and psychology, kindling crucial applications such as building alive virtual characters. Previous studies aim to predict gaze trajectories when humans are free‑viewing an image, searching for required targets, or looking for clues to answer questions in an image. While these tasks focus on visual‑centric scenarios, humans move their gaze also along with audio signal inputs in more common scenarios. To fill this gap, we introduce a new task that predicts human gaze trajectories in a visual scene with synchronized audio inputs and provide a new dataset containing 20k gaze points from 8 subjects. To effectively integrate audio information and simulate the dynamic process of human gaze motion, we propose a novel learning framework called EyEar (Eye moving while Ear listening) based on physics‑informed dynamics, which considers three key factors to predict gazes: eye inherent motion tendency, vision salient attraction, and audio semantic attraction. We also propose a probability density score to overcome the high individual variability of gaze trajectories, thereby improving the stabilization of optimization and the reliability of the evaluation. Experimental results show that EyEar outperforms all the baselines in the context of all evaluation metrics, thanks to the proposed components in the learning model.

Abstract:
Accurate state estimation is fundamental to intelligent vehicles. Wheel load, one of the most important chassis states, serves as an essential input for advanced driver assistance systems (ADAS) and exerts a direct influence on vehicle stability and safety. However, wheel load estimation remains challenging due to the complexity of chassis modeling and the susceptibility of nonlinear systems to noise. To address these issues, this paper first introduces a refined suspension linkage‑level modeling approach that constructs a nonlinear instantaneous dynamic model by explicitly considering the complex geometric structure of the suspension. Building upon this, we propose a damper characteristics‑based Bayesian physics‑informed neural network (Damper‑B‑PINN) framework to estimate dynamic wheel load, which leverages the suspension dynamics as physical guidance of PINN while employing Bayesian inference to mitigate the effects of system noise and uncertainty. Moreover, a damper‑characteristic physics conditioning (DPC) module is designed for embedding physical prior. The proposed Damper‑B‑PINN is evaluated using both high‑fidelity simulation datasets generated by CarSim software and real‑world datasets collected from a Formula Student race car. Experimental results demonstrate that our Damper‑B‑PINN consistently outperforms existing methods across various test conditions, particularly extreme ones. These findings highlight the potential of the proposed Damper‑B‑PINN framework to enhance the accuracy and robustness of dynamic wheel load estimation, thereby improving the reliability and safety of ADAS applications.

Abstract:
This work focuses on understanding the minimum eradication time for the controlled Susceptible‑Infectious‑Recovered (SIR) model in the time‑homogeneous setting, where the infection and recovery rates are constant. The eradication time is defined as the earliest time the infectious population drops below a given threshold and remains below it. For time‑homogeneous models, the eradication time is well‑defined due to the predictable dynamics of the infectious population, and optimal control strategies can be systematically studied. We utilize Physics‑Informed Neural Networks (PINNs) to solve the partial differential equation (PDE) governing the eradication time and derive the corresponding optimal vaccination control. The PINN framework enables a mesh‑free solution to the PDE by embedding the dynamics directly into the loss function of a deep neural network. We use a variable scaling method to ensure stable training of PINN and mathematically analyze that this method is effective in our setting. This approach provides an efficient computational alternative to traditional numerical methods, allowing for an approximation of the eradication time and the optimal control strategy. Through numerical experiments, we validate the effectiveness of the proposed method in computing the minimum eradication time and achieving optimal control. This work offers a novel application of PINNs to epidemic modeling, bridging mathematical theory and computational practice for time‑homogeneous SIR models.

Abstract:
I will demonstrate the effectiveness of Physics‑Informed Neural Networks (PINNs) in solving partial differential equations (PDEs) when training data are scarce or noisy. The training data can be located either at the boundaries or within the domain. Additionally, PINNs can be used as an inverse method to determine unknown coefficients in the equations. This study will highlight the application of PINNs in modeling magnetohydrodynamic processes relevant to strongly magnetized plasmas, such as those found in the solar corona.

Abstract:
Modelling complex multiphysics systems governed by nonlinear and strongly coupled partial differential equations (PDEs) is a cornerstone in computational science and engineering. However, it remains a formidable challenge for traditional numerical solvers due to high computational cost, making them impractical for large‑scale applications. Neural operators' reliance on data‑driven training limits their applicability in real‑world scenarios, as data is often scarce or expensive to obtain. Here, we propose a novel paradigm, physics‑informed parallel neural operator (PIPNO), a scalable and unsupervised learning framework that enables data‑free PDE modelling by leveraging only governing physical laws. The parallel kernel integration design, incorporating ensemble learning, significantly enhances both compatibility and computational efficiency, enabling scalable operator learning for nonlinear and strongly coupled PDEs. PIPNO efficiently captures nonlinear operator mappings across diverse physics, including geotechnical engineering, material science, electromagnetism, quantum mechanics, and fluid dynamics. The proposed method achieves high‑fidelity and rapid predictions, outperforming existing operator learning approaches in modelling nonlinear and strongly coupled multiphysics systems. Therefore, PIPNO offers a powerful alternative to conventional solvers, broadening the applicability of neural operators for multiphysics modelling while ensuring efficiency, robustness, and scalability.

Abstract:
Fatigue modeling is essential for motion synthesis tasks to model human motions under fatigued conditions and biomechanical engineering applications, such as investigating the variations in movement patterns and posture due to fatigue, defining injury risk mitigation and prevention strategies, formulating fatigue minimization schemes, and creating improved ergonomic designs. Nevertheless, employing datadriven methods for synthesizing the impact of fatigue on motion, receives little to no attention in the literature. In this work, we present Fatigue‑PINN, a deep learning framework based on Physics‑Informed Neural Networks, for modeling fatigued human movements, while providing joint‑specific fatigue configurations for adaptation and mitigation of motion artifacts on a joint level, resulting in more smooth, hence physicallyplausible animations. To account for muscle fatigue, we simulate the fatigue‑induced fluctuations in the maximum exerted joint torques by leveraging a PINN adaptation of the Three‑Compartment Controller model to exploit physics‑domain knowledge for improving accuracy. This model also introduces parametric motion alignment with respect to joint‑specific fatigue, hence avoiding sharp frame transitions. Our results indicate that Fatigue‑PINN accurately simulates the effects of externally perceived fatigue on open‑type human movements being consistent with findings from real‑world experimental fatigue studies. Since fatigue is incorporated in torque space, Fatigue‑PINN provides an end‑to‑end encoder‑decoder‑like architecture, to ensure transforming joint angles to joint torques and vice‑versa, thus, being compatible with motion synthesis frameworks operating on joint angles.

Abstract:
The ground state of second‑quantized quantum chemistry Hamiltonians is key to determining molecular properties. Neural quantum states (NQS) offer flexible and expressive wavefunction ansatze for this task but face two main challenges: highly peaked ground‑state wavefunctions hinder efficient sampling, and local energy evaluations scale quartically with system size, incurring significant computational costs. In this work, we overcome these challenges by introducing a suite of algorithmic enhancements, which includes efficient periodic compact subspace construction, truncated local energy evaluations, improved stochastic sampling, and physics‑informed modifications. Applying these techniques to the neural network backflow (NNBF) ansatz, we demonstrate significant gains in both accuracy and scalability. Our enhanced method surpasses traditional quantum chemistry methods like CCSD and CCSD(T), outperforms other NQS approaches, and achieves competitive energies with state‑of‑the‑art ab initio techniques such as HCI, ASCI, FCIQMC, and DMRG. A series of ablation and comparative studies quantifies the contribution of each enhancement to the observed improvements in accuracy and efficiency. Furthermore, we investigate the representational capacity of the ansatz, finding that its performance correlates with the inverse participation ratio (IPR), with more delocalized states being more challenging to approximate.

Abstract:
We parametrically solve the Boltzmann equations governing freeze‑in dark matter (DM) in alternative cosmologies with Physics‑Informed Neural Networks (PINNs), a mesh‑free method. Through inverse PINNs, using a single DM experimental point ‑‑ observed relic density ‑‑ we determine the physical attributes of the theory, namely power‑law cosmologies, inspired by braneworld scenarios, and particle interaction cross sections. The expansion of the Universe in such alternative cosmologies has been parameterized through a switch‑like function reproducing the Hubble law at later times. Without loss of generality, we model more realistically this transition with a smooth function. We predict a distinct pair‑wise relationship between power‑law exponent and particle interactions: for a given cosmology with negative (positive) exponent, smaller (larger) cross sections are required to reproduce the data. Lastly, via Bayesian methods, we quantify the epistemic uncertainty of theoretical parameters found in inverse problems.

Abstract:
This review explores the integration of deep learning (DL) with full‑waveform inversion (FWI) for enhanced seismic imaging and subsurface characterization. It covers FWI and DL fundamentals, geophysical applications (velocity estimation, deconvolution, tomography), and challenges (model complexity, data quality). The review also outlines future research directions, including hybrid, generative, and physics‑informed models for improved accuracy, efficiency, and reliability in subsurface property estimation. The synergy between DL and FWI has the potential to transform geophysics, providing new insights into Earth's subsurface.

Abstract:
Purpose: T2 quantification from gradient echo magnetic resonance imaging is particularly affected by subject motion due to the high sensitivity to magnetic field inhomogeneities, which are influenced by motion and might cause signal loss. Thus, motion correction is crucial to obtain high‑quality T2 maps. Methods: We extend our previously introduced learning‑based physics‑informed motion correction method, PHIMO, by utilizing acquisition knowledge to enhance the reconstruction performance for challenging motion patterns and increase PHIMO's robustness to varying strengths of magnetic field inhomogeneities across the brain. We perform comprehensive evaluations regarding motion detection accuracy and image quality for data with simulated and real motion. Results: Our extended version of PHIMO outperforms the learning‑based baseline methods both qualitatively and quantitatively with respect to line detection and image quality. Moreover, PHIMO performs on‑par with a conventional state‑of‑the‑art motion correction method for T2 quantification from gradient echo MRI, which relies on redundant data acquisition. Conclusion: PHIMO's competitive motion correction performance, combined with a reduction in acquisition time by over 40% compared to the state‑of‑the‑art method, make it a promising solution for motion‑robust T2 quantification in research settings and clinical routine.

Abstract:
We study entanglement percolation in qubit‑based planar quantum network models of arbitrary topology, where neighboring nodes are initially connected by pure states with quenched disorder in their entanglement. To address this, we develop a physics‑informed heuristic algorithm designed to find a sequence of entanglement swapping and distillation operations to connect any pair of distant nodes. The algorithm combines locally optimal percolation strategies between nodes at a maximum distance of one swapping operation. If this fails to produce a maximally entangled state, it looks for alternative paths surrounding intermediate states within the process. We analytically find and numerically verify thresholds in quantum percolation, which depend on the initial network configuration and entanglement, and are associated with specific percolation strategies. We classify these strategies based on the connectivity, a quantity that relates the entanglement in the final state and the level of integrity of the network at the end of the process. We find distinct regimes of quantum percolation, which are clearly separated by the percolation thresholds of the employed strategies and vastly vary according to the network topology.

Abstract:
Physics‑Informed Neural Networks (PINNs) have gained increasing attention for solving partial differential equations, including the Helmholtz equation, due to their flexibility and mesh‑free formulation. However, their low‑frequency bias limits their accuracy and convergence speed for high‑frequency wavefield simulations. To alleviate these problems, we propose a simplified PINN framework that incorporates Gabor functions, designed to capture the oscillatory and localized nature of wavefields more effectively. Unlike previous attempts that rely on auxiliary networks to learn Gabor parameters, we redefine the network's task to map input coordinates to a custom Gabor coordinate system, simplifying the training process without increasing the number of trainable parameters compared to a simple PINN. We validate the proposed method across multiple velocity models, including the complex Marmousi and Overthrust models, and demonstrate its superior accuracy, faster convergence, and better robustness features compared to both traditional PINNs and earlier Gabor‑based PINNs. Additionally, we propose an efficient integration of a Perfectly Matched Layer (PML) to enhance wavefield behavior near the boundaries. These results suggest that our approach offers an efficient and accurate alternative for scattered wavefield modeling and lays the groundwork for future improvements in PINN‑based seismic applications.

Abstract:
Energy landscapes play a crucial role in shaping dynamics of many real‑world complex systems. System evolution is often modeled as particles moving on a landscape under the combined effect of energy‑driven drift and noise‑induced diffusion, where the energy governs the long‑term motion of the particles. Estimating the energy landscape of a system has been a longstanding interdisciplinary challenge, hindered by the high operational costs or the difficulty of obtaining supervisory signals. Therefore, the question of how to infer the energy landscape in the absence of true energy values is critical. In this paper, we propose a physics‑informed self‑supervised learning method to learn the energy landscape from the evolution trajectories of the system. It first maps the system state from the observation space to a discrete landscape space by an adaptive codebook, and then explicitly integrates energy into the graph neural Fokker‑Planck equation, enabling the joint learning of energy estimation and evolution prediction. Experimental results across interdisciplinary systems demonstrate that our estimated energy has a correlation coefficient above 0.9 with the ground truth, and evolution prediction accuracy exceeds the baseline by an average of 17.65%. The code is available at github.com/tsinghua‑fib‑lab/PESLA.

Abstract:
The discovery of novel high‑temperature superconductor materials holds transformative potential for a wide array of technological applications. However, the combinatorially vast chemical and configurational search space poses a significant bottleneck for both experimental and theoretical investigations. In this study, we employ the design of high‑temperature ternary superhydride superconductors as a representative case to demonstrate how this challenge can be well addressed through a deep‑learning‑driven theoretical framework. This framework integrates high‑throughput crystal structure exploration, physics‑informed screening, and accurate prediction of superconducting critical temperatures. Our approach enabled the exploration of approximately 36 million ternary hydride structures across a chemical space of 29 elements, leading to the identification of 144 potential high‑Tc superconductors with predicted Tc > 200 K and superior thermodynamic stability at 200 GPa. Among these, 129 compounds spanning 27 novel structural prototypes are reported for the first time, representing a significant expansion of the known structural landscape for hydride superconductors. This work not only greatly expands the known repertoire of high‑Tc hydride superconductors but also establishes a scalable and efficient methodology for navigating the complex landscape of multinary hydrides.

Abstract:
Data assimilation (DA) enables hydrologic models to update their internal states using near‑real‑time observations for more accurate forecasts. With deep neural networks like long short‑term memory (LSTM), using either lagged observations as inputs (called "data integration") or variational DA has shown success in improving forecasts. However, it is unclear which methods are performant or optimal for physics‑informed machine learning ("differentiable") models, which represent only a small amount of physically‑meaningful states while using deep networks to supply parameters or missing processes. Here we developed variational DA methods for differentiable models, including optimizing adjusters for just precipitation data, just model internal hydrological states, or both. Our results demonstrated that differentiable streamflow models using the CAMELS dataset can benefit strongly and equivalently from variational DA as LSTM, with one‑day lead time median Nash‑Sutcliffe efficiency (NSE) elevated from 0.75 to 0.82. The resulting forecast matched or outperformed LSTM with DA in the eastern, northwestern, and central Great Plains regions of the conterminous United States. Both precipitation and state adjusters were needed to achieve these results, with the latter being substantially more effective on its own, and the former adding moderate benefits for high flows. Our DA framework does not need systematic training data and could serve as a practical DA scheme for whole river networks.

Abstract:
The optimal power flow (OPF) problem can be rapidly and reliably solved by employing responsive online solvers based on neural networks. The dynamic nature of renewable energy generation and the variability of power grid conditions necessitate frequent neural network updates with new data instances. To address this need and reduce the time required for data preparation time, we propose a semi‑supervised learning framework aided by data augmentation. In this context, ridge regression replaces the traditional solver, facilitating swift prediction of optimal solutions for the given input load demands. Additionally, to accelerate the backpropagation during training, we develop novel batch‑mean gradient estimation approaches along with a reduced branch set to alleviate the complexity of gradient computation. Numerical simulations demonstrate that our neural network, equipped with the proposed gradient estimators, consistently achieves feasible and near‑optimal solutions. These results underline the effectiveness of our approach for practical implementation in real‑time OPF applications.

Abstract:
The wavelength and state of polarization (SOP) are fundamental properties of an optical field which are essential for applications in optical communications, imaging and other fields. However, it is challenging for existing spectrometers and polarimeters to measure these parameters simultaneously, resulting in reduced spatial and temporal efficiency. To overcome this limitation, we propose and demonstrate a compact multimode fiber (MMF)‑based high‑dimensional light analyzer capable of simultaneously performing high‑precision measurements of both wavelength and SOP. Core‑offset launching is introduced in the MMF to reshuffle the mode coupling. A neural network named WP‑Net has been designed dedicated to wavelength and SOP synchronization measurements. Physics‑informed loss function based on optical prior knowledge is used to optimize the learning process. These advancements have enhanced the sensitivity, achieving a wavelength resolution of 0.045 pm and an SOP resolution of 0.0088.

Abstract:
Data‑driven machine learning models often require extensive datasets, which can be costly or inaccessible, and their predictions may fail to comply with established physical laws. Current approaches for incorporating physical priors mitigate these issues by penalizing deviations from known physical laws, as in physics‑informed neural networks, or by designing architectures that automatically satisfy specific invariants. However, penalization approaches do not guarantee compliance with physical constraints for unseen inputs, and invariant‑based methods lack flexibility and generality. We propose a novel physics‑consistent machine learning method that directly enforces compliance with physical principles by projecting model outputs onto the manifold defined by these laws. This procedure ensures that predictions inherently adhere to the chosen physical constraints, improving reliability and interpretability. Our method is demonstrated on two systems: a spring‑mass system and a low‑temperature reactive plasma. Compared to purely data‑driven models, our approach significantly reduces errors in physical law compliance, enhances predictive accuracy of physical quantities, and outperforms alternatives when working with simpler models or limited datasets. The proposed projection‑based technique is versatile and can function independently or in conjunction with existing physics‑informed neural networks, offering a powerful, general, and scalable solution for developing fast and reliable surrogate models of complex physical systems, particularly in resource‑constrained scenarios.

Abstract:
As the key component that facilitates long‑haul transmission in optical fiber communications by increasing capacity and reducing costs, accurate characterization and gain settings of erbium‑doped fiber amplifiers (EDFAs) are essential for quality of transmission estimation and system configuration optimization. However, it is difficult to construct accurate and reliable EDFA models due to complex physical mechanisms and dynamic loading conditions. Although some mathematical and data‑driven models have been proposed, their practical applications will face limitations of intricate parameter measurements and high data requirements, respectively. To overcome limitations of both methods, a physics‑informed machine learning (PIML) method for parameter identification and gain estimation of EDFA is proposed, which greatly reduces the data requirements by embedding physical prior knowledge in the neural network. In this approach, the gain of EDFA can be accurately estimated by a physics‑informed neural network (PINN)‑based forward model when parameters including absorption, gain, saturation, and background loss are known. For practical scenarios where parameters are unknown, PINN‑based inverse models are established first to identify actual values of parameters from only several sets of input‑output data pairs, and PINN‑based forward models are accordingly established for gain estimation with identified values. Moreover, an experimental system is constructed to verify the feasibility and performance of proposed method in practical scenarios. Results show that PIML‑based method can effectively identify physical parameters from measured data, and better gain estimation results are achieved with mean absolute error of 0.127 dB and standard deviation of 0.065 dB using identified values than typical values of parameters.

Abstract:
Machine learning (ML) has emerged as a promising tool for simulating quantum dissipative dynamics. However, existing methods often struggle to enforce key physical constraints, such as trace conservation, when modeling reduced density matrices (RDMs). While Physics‑Informed Neural Networks (PINN) aim to address these challenges, they frequently fail to achieve full physical consistency. In this work, we introduce a novel approach that leverages the \mathfraksu(n) Lie algebra to represent RDMs as a combination of an identity matrix and n^2 ‑ 1 Hermitian, traceless, and orthogonal basis operators,where n is the system's dimension. By learning only the coefficients associated with this basis, our framework inherently ensures exact trace conservation, as the traceless nature of the basis restricts the trace contribution solely to the identity matrix. This eliminates the need for explicit trace‑preserving penalty terms in the loss function, simplifying optimization and improving learning efficiency. We validate our approach on two benchmark quantum systems: the spin‑boson model and the Fenna‑Matthews‑Olson complex. By comparing the performance of four neural network (NN) architectures ‑‑ Purely Data‑driven Physics‑Uninformed Neural Networks (PUNN), \mathfraksu(n) Lie algebra‑based PUNN (\mathfraksu(n)‑PUNN), traditional PINN, and \mathfraksu(n) Lie algebra‑based PINN (\mathfraksu(n)‑PINN) ‑‑ we highlight the limitations of conventional methods and demonstrate the superior accuracy, robustness, and efficiency of our approach in learning quantum dissipative dynamics.

Abstract:
What if our clothes could capture our body motion accurately? This paper introduces Flexible Inertial Poser (FIP), a novel motion‑capturing system using daily garments with two elbow‑attached flex sensors and four Inertial Measurement Units (IMUs). To address the inevitable sensor displacements in loose wearables which degrade joint tracking accuracy significantly, we identify the distinct characteristics of the flex and inertial sensor displacements and develop a Displacement Latent Diffusion Model and a Physics‑informed Calibrator to compensate for sensor displacements based on such observations, resulting in a substantial improvement in motion capture accuracy. We also introduce a Pose Fusion Predictor to enhance multimodal sensor fusion. Extensive experiments demonstrate that our method achieves robust performance across varying body shapes and motions, significantly outperforming SOTA IMU approaches with a 19.5% improvement in angular error, a 26.4% improvement in elbow angular error, and a 30.1% improvement in positional error. FIP opens up opportunities for ubiquitous human‑computer interactions and diverse interactive applications such as Metaverse, rehabilitation, and fitness analysis.

Abstract:
Hamiltonian neural networks (HNNs) represent a promising class of physics‑informed deep learning methods that utilize Hamiltonian theory as foundational knowledge within neural networks. However, their direct application to engineering systems is often challenged by practical issues, including the presence of external inputs, dissipation, and noisy measurements. This paper introduces a novel framework that enhances the capabilities of HNNs to address these real‑life factors. We integrate port‑Hamiltonian theory into the neural network structure, allowing for the inclusion of external inputs and dissipation, while mitigating the impact of measurement noise through an output‑error (OE) model structure. The resulting output error port‑Hamiltonian neural networks (OE‑pHNNs) can be adapted to tackle modeling complex engineering systems with noisy measurements. Furthermore, we propose the identification of OE‑pHNNs based on the subspace encoder approach (SUBNET), which efficiently approximates the complete simulation loss using subsections of the data and uses an encoder function to predict initial states. By integrating SUBNET with OE‑pHNNs, we achieve consistent models of complex engineering systems under noisy measurements. In addition, we perform a consistency analysis to ensure the reliability of the proposed data‑driven model learning method. We demonstrate the effectiveness of our approach on system identification benchmarks, showing its potential as a powerful tool for modeling dynamic systems in real‑world applications.

Abstract:
Spectropolarimetric inversions of solar observations are fundamental for the estimation of the magnetic field in the solar atmosphere. However, instrumental noise, computational requirements, and varying levels of physical realism make it challenging to derive reliable solar magnetic field estimates. In this study, we present a novel approach for spectropolarimetric inversions based on Physics Informed Neural Networks (PINNs) to infer the photospheric magnetic field under the Milne‑Eddington approximation (PINN ME). Our model acts as a representation of the parameter space, mapping input coordinates (t, x, y) to the respective spectropolarimetric parameters, which are used to synthesize the corresponding stokes profiles. By iteratively sampling coordinate points, synthesizing profiles, and minimizing the deviation from the observed stokes profiles, our method can find the set of Milne‑Eddington parameters that best fit the observations. In addition, we directly include the point‑spread‑function to account for instrumental effects. We use a predefined parameter space as well as synthetic profiles from a radiative MHD simulation to evaluate the performance of our method and to estimate the impact of instrumental noise. Our results demonstrate that PINN ME achieves an intrinsic spatio‑temporal coupling, which can largely mitigate observational noise and provides a memory‑efficient inversion even for extended fields‑of‑view. Finally, we apply our method to observations and show that our method provides a high spatial coherence and can resolve small‑scale features both in strong‑ and weak‑field regions.

Abstract:
Inverse problems arise almost everywhere in science and engineering where we need to infer on a quantity from indirect observation. The cases of medical, biomedical, and industrial imaging systems are the typical examples. A very high overview of classification of the inverse problems method can be: i) Analytical, ii) Regularization, and iii) Bayesian inference methods. Even if there are straight links between them, we can say that the Bayesian inference based methods are the most powerful, as they give the possibility of accounting for prior knowledge and can account for errors and uncertainties in general. One of the main limitations stay in computational costs in particular for high dimensional imaging systems. Neural Networks (NN), and in particular Deep NNs (DNN), have been considered as a way to push farther this limit. Physics Informed Neural Networks (PINN) concept integrates physical laws with deep learning techniques to enhance the speed, accuracy and efficiency of the above mentioned problems. In this work, a new Bayesian framework for the concept of PINN (BPINN) is presented and discussed which includes the deterministic one if we use the Maximum A Posteriori (MAP) estimation framework. We consider two cases of supervised and unsupervised for training step, obtain the expressions of the posterior probability of the unknown variables, and deduce the posterior laws of the NN's parameters. We also discuss about the challenges of implementation of these methods in real applications.

Abstract:
The acceptance rate in Woodcock tracking algorithm is generalized to an arbitrary position‑dependent variable q(x). A neural network is used to optimize q(x), and the FOM value is used as the loss function. This idea comes from physics informed neural network(PINN), where a neural network is used to represent the solution of differential equations. Here the neural network q(x) should solve the functional equations that optimize FOM. For a 1d transmission problem with Gaussian absorption cross section, we observe a significant improvement of the FOM value compared to the constant q case and the original Woodcock method. Generalizations of the neural network Woodcock(NNW) method to 3d voxel models are waiting to be explored.

Abstract:
In an increasingly interconnected world, Cyber‑Physical Systems (CPS) are essential to critical industries like healthcare, transportation, and manufacturing, merging physical processes with computational intelligence. However, the security of these systems is a major concern. Anomalies, whether from sensor malfunctions or cyberattacks, can lead to catastrophic failures, making effective detection vital for preventing harm and service disruptions. This paper provides a comprehensive review of anomaly detection techniques in CPS. We categorize and compare various methods, including data‑driven approaches (machine learning, deep learning, machine learning‑deep learning ensemble), model‑driven approaches (mathematical, invariant‑based), hybrid datamodel approaches (Physics‑Informed Neural Networks), and system‑oriented approaches. Our analysis highlights the strengths and weaknesses of each technique, offering a practical guide for creating safer and more reliable systems. By identifying current research gaps, we aim to inspire future work that will enhance the security and adaptability of CPS in our automated world.

Abstract:
Physics intelligence and digital twins often require rapid and repeated performance evaluation of various engineering systems (e.g. robots, autonomous vehicles, semiconductor chips) to enable (almost) real‑time actions or decision making. This has motivated the development of accelerated partial differential equation (PDE) solvers, in resource‑constrained scenarios if the PDE solvers are to be deployed on the edge. Physics‑informed neural networks (PINNs) have shown promise in solving high‑dimensional PDEs, but the training time on state‑of‑the‑art digital hardware (e.g., GPUs) is still orders‑of‑magnitude longer than the latency required for enabling real‑time decision making. Photonic computing offers a potential solution to address this huge latency gap because of its ultra‑high operation speed. However, the lack of photonic memory and the large device sizes prevent training real‑size PINNs on photonic chips. This paper proposes a completely back‑propagation‑free (BP‑free) and highly salable framework for training real‑size PINNs on silicon photonic platforms. Our approach involves three key innovations: (1) a sparse‑grid Stein derivative estimator to avoid the BP in the loss evaluation of a PINN, (2) a dimension‑reduced zeroth‑order optimization via tensor‑train decomposition to achieve better scalability and convergence in BP‑free training, and (3) a scalable on‑chip photonic PINN training accelerator design using photonic tensor cores. We validate our numerical methods on both low‑ and high‑dimensional PDE benchmarks. Through pre‑silicon simulation based on real device parameters, we further demonstrate the significant performance benefit (e.g., real‑time training, huge chip area reduction) of our photonic accelerator.

Abstract:
Solving differential equations is a critical challenge across a host of domains. While many software packages efficiently solve these equations using classical numerical approaches, there has been less effort in developing a library for researchers interested in solving such systems using neural networks. With PyTorch as its backend, NeuroDiffEq is a software library that exploits neural networks to solve differential equations. In this paper, we highlight the latest features of the NeuroDiffEq library since its debut. We show that NeuroDiffEq can solve complex boundary value problems in arbitrary dimensions, tackle boundary conditions at infinity, and maintain flexibility for dynamic injection at runtime.

Abstract:
Water distribution systems (WDSs) are an important part of critical infrastructure becoming increasingly significant in the face of climate change and urban population growth. We propose a robust and scalable surrogate deep learning (DL) model to enable efficient planning, expansion, and rehabilitation of WDSs. Our approach incorporates an improved graph neural network architecture, an adapted physics‑informed algorithm, an innovative training scheme, and a physics‑preserving data normalization method. Evaluation results on a number of WDSs demonstrate that our model outperforms the current state‑of‑the‑art DL model. Moreover, our method allows us to scale the model to bigger and more realistic WDSs. Furthermore, our approach makes the model more robust to out‑of‑distribution input features (demands, pipe diameters). Hence, our proposed method constitutes a significant step towards bridging the simulation‑to‑real gap in the use of artificial intelligence for WDSs.

Abstract:
Earthquake forecasting remains a significant scientific challenge, with current methods falling short of achieving the performance necessary for meaningful societal benefits. Traditional models, primarily based on past seismicity and geomechanical data, struggle to capture the complexity of seismic patterns and often overlook valuable non‑seismic precursors such as geophysical, geochemical, and atmospheric anomalies. The integration of such diverse data sources into forecasting models, combined with advancements in AI technologies, offers a promising path forward. AI methods, particularly deep learning, excel at processing complex, large‑scale datasets, identifying subtle patterns, and handling multidimensional relationships, making them well‑suited for overcoming the limitations of conventional approaches. This review highlights the importance of combining AI with geophysical knowledge to create robust, physics‑informed forecasting models. It explores current AI methods, input data types, loss functions, and practical considerations for model development, offering guidance to both geophysicists and AI researchers. While many AI‑based studies oversimplify earthquake prediction, neglecting critical features such as data imbalance and spatio‑temporal clustering, the integration of specialized geophysical insights into AI models can address these shortcomings. We emphasize the importance of interdisciplinary collaboration, urging geophysicists to experiment with AI architectures thoughtfully and encouraging AI experts to deepen their understanding of seismology. By bridging these disciplines, we can develop more accurate, reliable, and societally impactful earthquake forecasting tools.

Abstract:
Weight change estimation is crucial in various applications, particularly for detecting pick‑up and put‑back actions when people interact with the shelf while shopping in autonomous stores. Moreover, accurate weight change estimation allows autonomous stores to automatically identify items being picked up or put back, ensuring precise cost estimation. However, the conventional approach of estimating weight changes requires specialized weight‑sensing shelves, which are densely deployed weight scales, incurring intensive sensor consumption and high costs. Prior works explored the vibration‑based weight sensing method, but they failed when the location of weight change varies. In response to these limitations, we made the following contributions: (1) We propose WeVibe, a first item weight change estimation system through active shelf vibration sensing. The main intuition of the system is that the weight placed on the shelf influences the dynamic vibration response of the shelf, thus altering the shelf vibration patterns. (2) We model a physics‑informed relationship between the shelf vibration response and item weight across multiple locations on the shelf based on structural dynamics theory. This relationship is linear and allows easy training of a weight estimation model at a new location without heavy data collection. (3) We evaluate our system on a gondola shelf organized as the real‑store settings. WeVibe achieved a mean absolute error down to 38.07g and a standard deviation of 31.2g with one sensor and 10% samples from three weight classes on estimating weight change from 0g to 450g, which can be leveraged for differentiating items with more than 100g differences.

Abstract:
Physics‑informed neural networks have shown significant potential in solving partial differential equations (PDEs) across diverse scientific fields. However, their performance often deteriorates when addressing PDEs with intricate and strongly coupled solutions. In this work, we present a novel Sharp‑PINN framework to tackle complex phase field corrosion problems. Instead of minimizing all governing PDE residuals simultaneously, the Sharp‑PINNs introduce a staggered training scheme that alternately minimizes the residuals of Allen‑Cahn and Cahn‑Hilliard equations, which govern the corrosion system. To further enhance its efficiency and accuracy, we design an advanced neural network architecture that integrates random Fourier features as coordinate embeddings, employs a modified multi‑layer perceptron as the primary backbone, and enforces hard constraints in the output layer. This framework is benchmarked through simulations of corrosion problems with multiple pits, where the staggered training scheme and network architecture significantly improve both the efficiency and accuracy of PINNs. Moreover, in three‑dimensional cases, our approach is 5‑10 times faster than traditional finite element methods while maintaining competitive accuracy, demonstrating its potential for real‑world engineering applications in corrosion prediction.

Abstract:
Simulation and optimization are crucial for advancing the engineering design of complex systems and processes. Traditional optimization methods require substantial computational time and effort due to their reliance on resource‑intensive simulations, such as finite element analysis, and the complexity of rigorous optimization algorithms. Data‑agnostic AI‑based surrogate models, such as Physics‑Informed Neural Operators (PINOs), offer a promising alternative to these conventional simulations, providing drastically reduced inference time, unparalleled data efficiency, and zero‑shot super‑resolution capability. However, the predictive accuracy of these models is often constrained to small, low‑dimensional design spaces or systems with relatively simple dynamics. To address this, we introduce a novel Physics‑Informed DeepONet (PIDON) architecture, which extends the capabilities of conventional neural operators to effectively model the nonlinear behavior of complex engineering systems across high‑dimensional design spaces and a wide range of dynamic design configurations. This new architecture outperforms existing SOTA models, enabling better predictions across broader design spaces. Leveraging PIDON's differentiability, we integrate a gradient‑based optimization approach using the Adam optimizer to efficiently determine optimal design variables. This forms an end‑to‑end gradient‑based optimization framework that accelerates the design process while enhancing scalability and efficiency. We demonstrate the effectiveness of this framework in the optimization of aerospace‑grade composites curing processes achieving a 3x speedup in obtaining optimal design variables compared to gradient‑free methods. Beyond composites processing, the proposed model has the potential to be used as a scalable and efficient optimization tool for broader applications in advanced engineering and digital twin systems.

Abstract:
Accurate blur estimation is essential for high‑performance imaging across various applications. Blur is typically represented by the point spread function (PSF). In this paper, we propose a physics‑informed PSF learning framework for imaging systems, consisting of a simple calibration followed by a learning process. Our framework could achieve both high accuracy and universal applicability. Inspired by the Seidel PSF model for representing spatially varying PSF, we identify its limitations in optimization and introduce a novel wavefront‑based PSF model accompanied by an optimization strategy, both reducing optimization complexity and improving estimation accuracy. Moreover, our wavefront‑based PSF model is independent of lens parameters, eliminate the need for prior knowledge of the lens. To validate our approach, we compare it with recent PSF estimation methods (Degradation Transfer and Fast Two‑step) through a deblurring task, where all the estimated PSFs are used to train state‑of‑the‑art deblurring algorithms. Our approach demonstrates improvements in image quality in simulation and also showcases noticeable visual quality improvements on real captured images.

Abstract:
Alloy design can be framed as a constraint‑satisfaction problem. Building on previous methodologies, we propose equipping Gaussian Process Classifiers (GPCs) with physics‑informed prior mean functions to model the boundaries of feasible design spaces. Through three case studies, we highlight the utility of informative priors for handling constraints on continuous and categorical properties. (1) Phase Stability: By incorporating CALPHAD predictions as priors for solid‑solution phase stability, we enhance model validation using a publicly available XRD dataset. (2) Phase Stability Prediction Refinement: We demonstrate an in silico active learning approach to efficiently correct phase diagrams. (3) Continuous Property Thresholds: By embedding priors into continuous property models, we accelerate the discovery of alloys meeting specific property thresholds via active learning. In each case, integrating physics‑based insights into the classification framework substantially improved model performance, demonstrating an efficient strategy for constraint‑aware alloy design.

Abstract:
Selecting optimal kernels for regression in physical systems remains a challenge, often relying on trial‑and‑error with standard functions. In this work, we establish a mathematical correspondence between support vector machine kernels and quantum propagators, demonstrating that kernel efficacy is determined by its spectral alignment with the system's Green's function. Based on this isomorphism, we propose a unified, physics‑informed framework for kernel selection and design. For systems with known propagator forms, we derive analytical selection rules that map standard kernels to physical operators. For complex systems where the Green's function is analytically intractable, we introduce a constructive numerical method using the Kernel Polynomial Method with Jackson smoothing to generate custom, physics‑aligned kernels. Numerical experiments spanning electrical conductivity, electronic band structure, anharmonic oscillators, and photonic crystals demonstrate that this framework consistently performs well as long as there is an alignment with a Green's function.

Abstract:
As autonomous systems become more ubiquitous in daily life, ensuring high performance with guaranteed safety is crucial. However, safety and performance could be competing objectives, which makes their co‑optimization difficult. Learning‑based methods, such as Constrained Reinforcement Learning (CRL), achieve strong performance but lack formal safety guarantees due to safety being enforced as soft constraints, limiting their use in safety‑critical settings. Conversely, formal methods such as Hamilton‑Jacobi (HJ) Reachability Analysis and Control Barrier Functions (CBFs) provide rigorous safety assurances but often neglect performance, resulting in overly conservative controllers. To bridge this gap, we formulate the co‑optimization of safety and performance as a state‑constrained optimal control problem, where performance objectives are encoded via a cost function and safety requirements are imposed as state constraints. We demonstrate that the resultant value function satisfies a Hamilton‑Jacobi‑Bellman (HJB) equation, which we approximate efficiently using a novel physics‑informed machine learning framework. In addition, we introduce a conformal prediction‑based verification strategy to quantify the learning errors, recovering a high‑confidence safety value function, along with a probabilistic error bound on performance degradation. Through several case studies, we demonstrate the efficacy of the proposed framework in enabling scalable learning of safe and performant controllers for complex, high‑dimensional autonomous systems.

Abstract:
We present a method leveraging extreme learning machine (ELM) type randomized neural networks (NNs) for learning the exact time integration algorithm for initial value problems (IVPs). The exact time integration algorithm for non‑autonomous systems can be represented by an algorithmic function in higher dimensions, which satisfies an associated system of partial differential equations with corresponding boundary conditions. Our method learns the algorithmic function by solving this associated system using ELM with a physics informed approach. The trained ELM network serves as the learned algorithm and can be used to solve the IVP with arbitrary initial data or step sizes from some domain. When the right hand side of the non‑autonomous system exhibits a periodicity with respect to any of its arguments, while the solution itself to the problem is not periodic, we show that the algorithmic function is either periodic, or when it is not, satisfies a well‑defined relation for different periods. This property can greatly simplify the algorithm learning in many problems. We consider explicit and implicit NN formulations, leading to explicit or implicit time integration algorithms, and discuss how to train the ELM network by the nonlinear least squares method. Extensive numerical experiments with benchmark problems, including non‑stiff, stiff and chaotic systems, show that the learned NN algorithm produces highly accurate solutions in long‑time simulations, with its time‑marching errors decreasing nearly exponentially with increasing degrees of freedom in the neural network. We compare extensively the computational performance (accuracy vs.~cost) between the current NN algorithm and the leading traditional time integration algorithms. The learned NN algorithm is computationally competitive, markedly outperforming the traditional algorithms in many problems.

Abstract:
We employ deep learning within holographic duality to investigate T‑linear resistivity, a hallmark of strange metals. Utilizing Physics‑Informed Neural Networks, we incorporate boundary data for T‑linear resistivity and bulk differential equations into a loss function. This approach allows us to derive dilaton potentials in Einstein‑Maxwell‑Dilaton‑Axion theories, capturing essential features of strange metals, such as T‑linear resistivity and linear specific heat scaling. We also explore the impact of the resistivity slope on dilaton potentials. Regardless of slope, dilaton potentials exhibit universal exponential growth at low temperatures, driving T‑linear resistivity and matching infrared geometric analyses. At a specific slope, our method rediscovers the Gubser‑Rocha model, a well‑known holographic model of strange metals. Additionally, the robustness of T‑linear resistivity at higher temperatures correlates with the asymptotic AdS behavior of the dilaton coupling to the Maxwell term. Our findings suggest that deep learning could help uncover mechanisms in holographic condensed matter systems and advance our understanding of strange metals.

Abstract:
Unstructured grid data are essential for modelling complex geometries and dynamics in computational physics. Yet, their inherent irregularity presents significant challenges for conventional machine learning (ML) techniques. This paper provides a comprehensive review of advanced ML methodologies designed to handle unstructured grid data in high‑dimensional dynamical systems. Key approaches discussed include graph neural networks, transformer models with spatial attention mechanisms, interpolation‑integrated ML methods, and meshless techniques such as physics‑informed neural networks. These methodologies have proven effective across diverse fields, including fluid dynamics and environmental simulations. This review is intended as a guidebook for computational scientists seeking to apply ML approaches to unstructured grid data in their domains, as well as for ML researchers looking to address challenges in computational physics. It places special focus on how ML methods can overcome the inherent limitations of traditional numerical techniques and, conversely, how insights from computational physics can inform ML development. To support benchmarking, this review also provides a summary of open‑access datasets of unstructured grid data in computational physics. Finally, emerging directions such as generative models with unstructured data, reinforcement learning for mesh generation, and hybrid physics‑data‑driven paradigms are discussed to inspire future advancements in this evolving field.

Abstract:
This study introduces a physics‑based machine learning framework for modeling both brittle and ductile fractures. Unlike physics‑informed neural networks, which solve partial differential equations by embedding physical laws as soft constraints in loss functions and enforcing boundary conditions via collocation points, our framework integrates physical principles, such as the governing equations and constraints, directly into the neural network architecture. This approach eliminates the dependency on problem‑specific retraining for new boundary value problems, ensuring adaptability and consistency. By embedding constitutive behavior into the network's foundational design, our method represents a significant step toward unifying material modeling with machine learning for computational fracture mechanics. Specifically, a feedforward neural network is designed to embed physical laws within its architecture, ensuring thermodynamic consistency. Building on this foundation, synthetic datasets generated from finite element‑based phase‑field simulations are employed to train the proposed framework, focusing on capturing the homogeneous responses of brittle and ductile fractures. Detailed analyses are performed on the stored elastic energy and the dissipated work due to plasticity and fracture, demonstrating the capability of the framework to predict essential fracture features. The proposed physics‑based machine learning framework overcomes the shortcomings of classical machine learning models, which rely heavily on large datasets and lack guarantees of physical principles. By leveraging its physics‑integrated design, the physics‑based machine learning framework demonstrates exceptional performance in predicting key properties of brittle and ductile fractures with limited training data.

Abstract:
In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network‑based methods, such as physics‑informed neural networks, depend on automatic differentiation and the sampling of collocation points, which can result in a lack of interpretability and lower accuracy compared to traditional numerical methods. To address this issue, we propose two approaches for learning discontinuous Galerkin solutions to PDEs using small linear convolutional neural networks. Our first approach is supervised and depends on labeled data, while our second approach is unsupervised and does not rely on any training data. In both cases, our methods use substantially fewer parameters than similar numerics‑based neural networks while also demonstrating comparable accuracy to the true and DG solutions for elliptic problems.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical laws into neural network training. However, traditional PINN models are typically designed for single PDEs, limiting their generalizability across different physical systems. In this work, we explore the potential of a foundation PINN model capable of solving multiple PDEs within a unified architecture. We investigate the efficacy of a single PINN framework trained on four distinct PDEs‑the Simple Harmonic Oscillator (SHO), the 1D Heat Equation, the 1D Wave Equation, and the 2D Laplace Equation, demonstrating its ability to learn diverse physical dynamics. To enhance sample efficiency, we incorporate Active Learning (AL) using Monte Carlo (MC) Dropout‑based uncertainty estimation, selecting the most informative training samples iteratively. We evaluate different active learning strategies, comparing models trained on 10%, 20%, 30%, 40%, and 50% of the full dataset, and analyze their impact on solution accuracy. Our results indicate that targeted uncertainty sampling significantly improves performance with fewer training samples, leading to efficient learning across multiple PDEs. This work highlights the feasibility of a generalizable PINN‑based foundation model, capable of adapting to different physics‑based problems without redesigning network architectures. Our findings suggest that multi‑PDE PINNs with active learning can serve as an effective approach for reducing computational costs while maintaining high accuracy in physics‑based deep learning applications.

Abstract:
This paper proposes a Curriculum‑Transfer‑Learning based physics‑informed neural network (CTL‑PINN) for long‑term simulation of physical and mechanical behaviors. The main innovation of CTL‑PINN lies in decomposing long‑term problems into a sequence of short‑term subproblems. Initially, the standard PINN is employed to solve the first sub‑problem. As the simulation progresses, subsequent time‑domain problems are addressed using a curriculum learning approach that integrates information from previous steps. Furthermore, transfer learning techniques are incorporated, allowing the model to effectively utilize prior training data and solve sequential time domain transfer problems. CTL‑PINN combines the strengths of curriculum learning and transfer learning, overcoming the limitations of standard PINNs, such as local optimization issues, and addressing the inaccuracies over extended time domains encountered in CL‑PINN and the low computational efficiency of TL‑PINN. The efficacy and robustness of CTL‑PINN are demonstrated through applications to nonlinear wave propagation, Kirchhoff plate dynamic response, and the hydrodynamic model of the Three Gorges Reservoir Area, showcasing its superior capability in addressing long‑term computational challenges.

Abstract:
Using machine learning (ML) to construct interatomic interactions and thus potential energy surface (PES) has become a common strategy for materials design and simulations. However, those current models of machine learning interatomic potential (MLIP) provide no relevant physical constrains, and thus may owe intrinsic out‑of‑domain difficulty which underlies the challenges of model generalizability and physical scalability. Here, by incorporating physics‑informed Universal‑Scaling law and nonlinearity‑embedded interaction function, we develop a Super‑linear MLIP with both Ultra‑Small parameterization and greatly expanded expressive capability, named SUS2‑MLIP. Due to the global scaling rooting in universal equation of state (UEOS), SUS2‑MLIP not only has significantly‑reduced parameters by decoupling the element space from coordinate space, but also naturally outcomes the out‑of‑domain difficulty and endows the potentials with inherent generalizability and scalability even with relatively small training dataset. The nonlinearity‑enbeding transformation for interaction function expands the expressive capability and make the potentials super‑linear. The SUS2‑MLIP outperforms the state‑of‑the‑art MLIP models with its exceptional computational efficiency especially for multiple‑element materials and physical scalability in property prediction. This work not only presents a highly‑efficient universal MLIP model but also sheds light on incorporating physical constraints into artificial‑intelligence‑aided materials simulation.

Abstract:
As a part of the integrated energy system (IES), gas pipeline networks can provide additional flexibility to power systems through coordinated optimal dispatch. An accurate pipeline network model is critical for the optimal operation and control of IESs. However, inaccuracies or unavailability of accurate pipeline parameters often introduce errors in the state‑space models of such networks. This paper proposes a physics‑informed recurrent network (PIRN) to identify the state‑space model of gas pipelines. It fuses sparse measurement data with fluid‑dynamic behavior expressed by partial differential equations. By embedding the physical state‑space model within the recurrent network, parameter identification becomes an end‑to‑end PIRN training task. The model can be realized in PyTorch through modifications to a standard RNN backbone. Case studies demonstrate that our proposed PIRN can accurately estimate gas pipeline models from sparse terminal node measurements, providing robust performance and significantly higher parameter efficiency. Furthermore, the identified state‑space model of the pipeline network can be seamlessly integrated into optimization frameworks.

Abstract:
Physics‑Informed Neural Networks (PINNs) seek to solve partial differential equations (PDEs) with deep learning. Mainstream approaches that deploy fully‑connected multi‑layer deep learning architectures require prolonged training to achieve even moderate accuracy, while recent work on feature engineering allows higher accuracy and faster convergence. This paper introduces SAFE‑NET, a Single‑layered Adaptive Feature Engineering NETwork that achieves orders‑of‑magnitude lower errors with far fewer parameters than baseline feature engineering methods. SAFE‑NET returns to basic ideas in machine learning, using Fourier features, a simplified single hidden layer network architecture, and an effective optimizer that improves the conditioning of the PINN optimization problem. Numerical results show that SAFE‑NET converges faster and typically outperforms deeper networks and more complex architectures. It consistently uses fewer parameters ‑‑ on average, 65% fewer than the competing feature engineering methods ‑‑ while achieving comparable accuracy in less than 30% of the training epochs. Moreover, each SAFE‑NET epoch is 95% faster than those of competing feature engineering approaches. These findings challenge the prevailing belief that modern PINNs effectively learn features in these scientific applications and highlight the efficiency gains possible through feature engineering.

Abstract:
In the field of systems biology, differential equations are commonly used to model biological systems, but solving them for large‑scale and complex systems can be computationally expensive. Recently, the integration of machine learning and mathematical modeling has offered new opportunities for scientific discoveries in biology and health. The emerging physics‑informed neural network (PINN) has been proposed as a solution to this problem. However, PINN can be computationally expensive and unreliable for complex biological systems. To address these issues, we propose the Fourier‑enhanced Neural Networks for systems biology (SB‑FNN). SB‑FNN uses an embedded Fourier neural network with an adaptive activation function and a cyclic penalty function to optimize the prediction of biological dynamics, particularly for biological systems that exhibit oscillatory patterns. Experimental results demonstrate that SB‑FNN achieves better performance and is more efficient than PINN for handling complex biological models. Experimental results on cellular and population models demonstrate that SB‑FNN outperforms PINN in both accuracy and efficiency, making it a promising alternative approach for handling complex biological models. The proposed method achieved better performance on six biological models and is expected to replace PINN as the most advanced method in systems biology.

Abstract:
Establishing Lipschitz stability estimates is crucial for ensuring the mathematical robustness of neural network (NN) approximations in machine learning (ML)‑based parameter estimation, particularly in physics‑informed settings. In this work, we derive such estimates for the inverse of a nonlinear map defined on a manifold that captures both unknown parameters and the nonlinear physical processes they influence. Our analysis is based on finite‑dimensional, learnable representations of the manifold and provides Lipschitz stability estimates on the manifold‑based subspaces, for a class of inverse maps associated with parameter dependent linear compact operators. Such operators model scattered and far‑field data that can be used to detect structures such as cracks. We apply our theoretical ML manifold framework to inverse Helmholtz problems in unbounded regions exterior to cracks, addressing the scattered‑field data‑driven inverse problem while ensuring injectivity conditions on the manifold, a requirement for the Lipschitz stability. Our method accurately recovers crack‑defining parameters without requiring prior knowledge of inputs such as incident wave types or external forces on the crack. Numerical experiments using NN approximations confirm the accuracy, efficiency, and robustness of the proposed approach.

Abstract:
Direct numerical simulation of turbulence at realistic Reynolds numbers is still beyond current computational capability, necessitating models that reduce the number of resolved spatial scales. Motivated by phenomenology and recent data‑driven works based on universality of the smallest scales in fully developed turbulence, the statistical dynamics of the velocity gradient tensor (VGT) at the Kolmogorov scale become of critical importance in advancing turbulence models. Physics‑informed machine learning has found considerable success in exploiting large datasets taken from direct numerical simulation of Navier‑Stokes to improve models for the evolution of the VGT. In this work, we follow the long line of blending physical insight with data analysis to simultaneously advance both the modeling and understanding of the phenomenology of the VGT. Using the intimate connection between VGT evolution and fluid deformation, we develop the Lagrangian attention tensor network approach that significantly improves over current physics‑informed machine learning methods. We demonstrate state‑of‑the‑art performance in both a‑priori and a‑posteriori metrics, before interpreting the trained attention mechanisms to discover a surprising connection between the history of the strain‑rate‑tensor and the pressure Hessian.

Abstract:
This paper puts forward the vision of creating a library of neural‑network‑based models for power system simulations. Traditional numerical solvers struggle with the growing complexity of modern power systems, necessitating faster and more scalable alternatives. Physics‑Informed Neural Networks (PINNs) offer promise to solve fast the ordinary differential equations (ODEs) governing power system dynamics. This is vital for the reliability, cost optimization, and real‑time decision‑making in the electricity grid. Despite their potential, standardized frameworks to train PINNs remain scarce. This poses a barrier for the broader adoption and reproducibility of PINNs; it also does not allow the streamlined creation of a PINN‑based model library. This paper addresses these gaps. It introduces a Python‑based toolbox for developing PINNs tailored to power system components, available on GitHub https://github. com/radiakos/PowerPINN. Using this framework, we capture the dynamic characteristics of a 9th‑order system, which is probably the most complex power system component trained with a PINN to date, demonstrating the toolbox capabilities, limitations, and potential improvements. The toolbox is open and free to use by anyone interested in creating PINN‑based models for power system components.

Abstract:
We present a five‑module pedagogical framework for teaching physics‑informed machine learning (ML) through two progressively complex physical systems: a driven, damped nonlinear pendulum and a one‑dimensional quantum anharmonic oscillator. Five model architectures are implemented and compared: a standard artificial neural network (ANN), a one‑dimensional convolutional neural network (CNN), a long short‑term memory (LSTM) network, and two physics‑informed neural networks (PINNs) ‑‑ one per physical system. All models are implemented in PyTorch~2.9 and executed on an NVIDIA RTX~5090 GPU, making the framework directly applicable to modern deep learning laboratory courses. Quantitative benchmarks show that data‑driven models achieve mean absolute errors of 1.3×10^‑2~rad (pendulum ANN) and 4.4×10^‑5~a.u.\ (quantum CNN), while the curriculum‑trained pendulum PINN reaches an MAE of 3.1×10^‑2~rad using only collocation points. A systematic CPU‑vs‑GPU benchmark reveals speedups ranging from 1.2× (small ANN) to 24.6× (LSTM), providing a concrete pedagogical demonstration of when GPU acceleration is ‑‑ and is not ‑‑ warranted. The framework is packaged as self‑contained Jupyter notebooks designed for a graduate‑level \emphDeep Neural Networks for Physical Systems course, with embedded reflection questions that guide students from data‑driven thinking toward physics‑constrained formulations.

Abstract:
Physics‑informed Neural Networks (PINNs) show that embedding physical laws directly into the learning objective can significantly enhance the efficiency and physical consistency of neural network solutions. Similar to optimizing loss functions in machine learning, evolutionary algorithms iteratively optimize objective functions by simulating natural selection processes. Inspired by this principle, we ask a natural question: can physical information be similarly embedded into the fitness function of evolutionary algorithms? In this work, we propose Physics‑informed Evolution (PIE), a novel framework that incorporates physical information derived from governing physical laws into the evolutionary fitness landscape, thereby extending Physics‑informed artificial intelligence methods from machine learning to the broader domain of evolutionary computation. As a concrete instantiation, we apply PIE to quantum control problems governed by the Schrödinger equation, where the goal is to find optimal control fields that drive quantum systems from initial states to desired target states. We validate PIE on three representative quantum control benchmarks: state preparation in V‑type three‑level systems, entangled state generation in superconducting quantum circuits, and two‑atom cavity QED systems. Within the PIE framework, we systematically compare the performance of ten single‑objective and five multi‑objective evolutionary algorithms. Experimental results demonstrate that by embedding physical information into the fitness function, PIE effectively guides evolutionary search, yielding control fields with high fidelity, low state deviation, and robust performance across different scenarios. Our findings further suggest that the Physics‑informed principle extends naturally beyond neural network training to the broader domain of evolutionary computation.

Abstract:
This study introduces a hybrid machine learning‑based scale‑bridging framework for predicting the permeability of fibrous textile structures. By addressing the computational challenges inherent to multiscale modeling, the proposed approach evaluates the efficiency and accuracy of different scale‑bridging methodologies combining traditional surrogate models and even integrating physics‑informed neural networks (PINNs) with numerical solvers, enabling accurate permeability predictions across micro‑ and mesoscales. Four methodologies were evaluated: Single Scale Method (SSM), Simple Upscaling Method (SUM), Scale‑Bridging Method (SBM), and Fully Resolved Model (FRM). SSM, the simplest method, neglects microscale permeability and exhibited permeability values deviating by up to 150% of the FRM model, which was taken as ground truth at an equivalent lower fiber volume content. SUM improved predictions by considering uniform microscale permeability, yielding closer values under similar conditions, but still lacked structural variability. The SBM method, incorporating segment‑based microscale permeability assignments, showed significant enhancements, achieving almost equivalent values while maintaining computational efficiency and modeling runtimes of ~45 minutes per simulation. In contrast, FRM, which provides the highest fidelity by fully resolving microscale and mesoscale geometries, required up to 270 times more computational time than SSM, with model files exceeding 300 GB. Additionally, a hybrid dual‑scale solver incorporating PINNs has been developed and shows the potential to overcome generalization errors and the problem of data scarcity of the data‑driven surrogate approaches. The hybrid framework advances permeability modelling by balancing computational cost and prediction reliability, laying the foundation for further applications in fibrous composite manufacturing.

Abstract:
In this work, we present a study combining two approaches in the context of solving PDEs: the continuous finite element method (FEM) and more recent techniques based on neural networks. In recent years, physics‑informed neural networks (PINNs) have become particularly interesting for rapidly solving PDEs, especially in high dimensions. However, their lack of accuracy can be a significant drawback in this context, hence the interest in combining them with FEM, for which error estimates are already known. The complete pipeline proposed here consists in modifying the classical FEM approximation spaces by taking information from a prior, chosen as the prediction of a neural network. On the one hand, this combination improves and certifies the prediction of neural networks, to obtain a fast and accurate solution. On the other hand, error estimates are proven, showing that such strategies outperform classical ones by a factor that depends only on the quality of the prior. We validate our approach with numerical results performed on parametric problems with 1D, 2D and 3D geometries. These experiments demonstrate that to achieve a given accuracy, a coarser mesh can be used with our enriched FEM compared to the standard FEM, leading to reduced computational time, particularly for parametric problems.

Abstract:
We propose compleX‑PINN, a novel physics‑informed neural network (PINN) architecture incorporating a learnable activation function inspired by the Cauchy integral theorem. By optimizing the activation parameters, compleX‑PINN achieves high accuracy with just a single hidden layer. Empirically, we demonstrate that compleX‑PINN solves high‑dimensional problems that pose significant challenges for PINNs. Our results show that compleX‑PINN consistently achieves substantially greater precision, often improving accuracy by an order of magnitude, on these complex tasks.

Abstract:
Finding an ε‑stationary point of a nonconvex function with a Lipschitz continuous Hessian is a central problem in optimization. Regularized Newton methods are a classical tool and have been studied extensively, yet they still face a trade‑off between global and local convergence. Whether a parameter‑free algorithm of this type can simultaneously achieve optimal global complexity and quadratic local convergence remains an open question. To bridge this long‑standing gap, we propose a new class of regularizers constructed from the current and previous gradients, and leverage the conjugate gradient approach with a negative curvature monitor to solve the regularized Newton equation. The proposed algorithm is adaptive, requiring no prior knowledge of the Hessian Lipschitz constant, and achieves a global complexity of O(ε^‑3/2) in terms of the second‑order oracle calls, and \tildeO(ε^‑7/4) for Hessian‑vector products, respectively. When the iterates converge to a point where the Hessian is positive definite, the method exhibits quadratic local convergence. Preliminary numerical results, including training the physics‑informed neural networks, illustrate the competitiveness of our algorithm.

Abstract:
Deep optics has emerged as a promising approach by co‑designing optical elements with deep learning algorithms. However, current research typically overlooks the analysis and optimization of manufacturing and assembly tolerances. This oversight creates a significant performance gap between designed and fabricated optical systems. To address this challenge, we present the first end‑to‑end tolerance‑aware optimization framework that incorporates multiple tolerance types into the deep optics design pipeline. Our method combines physics‑informed modelling with data‑driven training to enhance optical design by accounting for and compensating for structural deviations in manufacturing and assembly. We validate our approach through computational imaging applications, demonstrating results in both simulations and real‑world experiments. We further examine how our proposed solution improves the robustness of optical systems and vision algorithms against tolerances through qualitative and quantitative analyses. Code and additional visual results are available at openimaginglab.github.io/LensTolerance.

Abstract:
Simulating complex physical systems is crucial for understanding and predicting phenomena across diverse fields, such as fluid dynamics and heat transfer, as well as plasma physics and structural mechanics. Traditional approaches rely on solving partial differential equations (PDEs) using numerical methods, which are computationally expensive and often prohibitively slow for real‑time applications or large‑scale simulations. Neural PDEs have emerged as efficient alternatives to these costly numerical solvers, offering significant computational speed‑ups. However, their lack of robust uncertainty quantification (UQ) limits deployment in critical applications. We introduce a model‑agnostic, physics‑informed conformal prediction (CP) framework that provides guaranteed uncertainty estimates without requiring labelled data. By utilising a physics‑based approach, we can quantify and calibrate the model's inconsistencies with the physics rather than the uncertainty arising from the data. Our approach utilises convolutional layers as finite‑difference stencils and leverages physics residual errors as nonconformity scores, enabling data‑free UQ with marginal and joint coverage guarantees across prediction domains for a range of complex PDEs. We further validate the efficacy of our method on neural PDE models for plasma modelling and shot design in fusion reactors.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs) by incorporating physical constraints into deep learning models. However, standard PINNs often require a large number of training samples to achieve high accuracy, leading to increased computational costs. To address this issue, we propose Active Learning‑Driven PINNs (AL‑PINN), which integrates Uncertainty Quantification (UQ) and Active Learning (AL) strategies to optimize sample selection dynamically. AL‑PINN utilizes Monte Carlo Dropout to estimate epistemic uncertainty in the model predictions, enabling the adaptive selection of high‑uncertainty regions for additional training. This approach significantly enhances learning efficiency by focusing computational resources on the most informative data points. We evaluate AL‑PINN on benchmark PDE problems with known analytical solutions and real‑world WeatherBench climate data. Our results demonstrate that AL‑PINN achieves comparable or superior accuracy compared to traditional PINNs while reducing the number of required training samples. The proposed framework is particularly beneficial for scientific and engineering applications where data collection is expensive or limited, such as climate modeling, medical simulations, and material science. Our findings highlight the potential of active learning in accelerating PINN‑based PDE solvers while maintaining high accuracy and computational efficiency.

Abstract:
Stochastic partial differential equations (SPDEs) describe the evolution of random processes over space and time, but their solutions are often analytically intractable and computationally expensive to estimate. In this paper, we propose the Learned Expectation Collapser (LEC), a physics‑informed neural framework designed to approximate the expected value of linear SPDE solutions without requiring domain discretization. By leveraging randomized sampling of both space‑time coordinates and noise realizations during training, LEC trains standard feedforward neural networks to minimize residual loss across multiple stochastic samples. We hypothesize and empirically confirm that this training regime drives the network to converge toward the expected value of the solution of the SPDE. Using the stochastic heat equation as a testbed, we evaluate performance across a diverse set of 144 experimental configurations that span multiple spatial dimensions, noise models, and forcing functions. The results show that the model consistently learns accurate approximations of the expected value of the solution in lower dimensions and a predictable decrease in accuracy with increased spatial dimensions, with improved stability and robustness under increased Monte Carlo sampling. Our findings offer new insight into how neural networks implicitly learn statistical structure from stochastic differential operators and suggest a pathway toward scalable, simulator‑free SPDE solvers.

Abstract:
Time‑resolved scanning probe microscopy methods, like time‑resolved electrostatic force microscopy (trEFM), enable imaging of dynamic processes ranging from ion motion in batteries to electronic dynamics in microstructured thin film semiconductors for solar cells. Reconstructing the underlying physical dynamics from these techniques can be challenging due to the interplay of cantilever physics with the actual transient kinetics of interest in the resulting signal. Previously, quantitative trEFM used empirical calibration of the cantilever or feed‑forward neural networks trained on simulated data to extract the physical dynamics of interest. Both these approaches are limited by interpreting the underlying signal as a single exponential function, which serves as an approximation but does not adequately reflect many realistic systems. Here, we present a multi‑branched, multi‑output convolutional neural network (CNN) that uses the trEFM signal in addition to the physical cantilever parameters as input. The trained CNN accurately extracts parameters describing both single‑exponential and bi‑exponential underlying functions, and more accurately reconstructs real experimental data in the presence of noise. This work demonstrates an application of physics‑informed machine learning to complex signal processing tasks, enabling more efficient and accurate analysis of trEFM.

Abstract:
Dynamic interacting system modeling is important for understanding and simulating real world systems. The system is typically described as a graph, where multiple objects dynamically interact with each other and evolve over time. In recent years, graph Ordinary Differential Equations (ODE) receive increasing research attentions. While achieving encouraging results, existing solutions prioritize the traditional Euclidean space, and neglect the intrinsic geometry of the system and physics laws, e.g., the principle of entropy increasing. The limitations above motivate us to rethink the system dynamics from a fresh perspective of Riemannian geometry, and pose a more realistic problem of physics‑informed dynamic system modeling, considering the underlying geometry and physics law for the first time. In this paper, we present a novel physics‑informed Riemannian graph ODE for a wide range of entropy‑increasing dynamic systems (termed as Pioneer). In particular, we formulate a differential system on the Riemannian manifold, where a manifold‑valued graph ODE is governed by the proposed constrained Ricci flow, and a manifold preserving Gyro‑transform aware of system geometry. Theoretically, we report the provable entropy non‑decreasing of our formulation, obeying the physics laws. Empirical results show the superiority of Pioneer on real datasets.

Abstract:
Neural operators have shown great potential in surrogate modeling. However, training a well‑performing neural operator typically requires a substantial amount of data, which can pose a major challenge in complex applications. In such scenarios, detailed physical knowledge can be unavailable or difficult to obtain, and collecting extensive data is often prohibitively expensive. To mitigate this challenge, we propose the Pseudo Physics‑Informed Neural Operator (PPI‑NO) framework. PPI‑NO constructs a surrogate physics system for the target system using partial differential equations (PDEs) derived from simple, rudimentary physics principles, such as basic differential operators. This surrogate system is coupled with a neural operator model, using an alternating update and learning process to iteratively enhance the model's predictive power. While the physics derived via PPI‑NO may not mirror the ground‑truth underlying physical laws ‑‑ hence the term ``pseudo physics'' ‑‑ this approach significantly improves the accuracy of standard operator learning models in data‑scarce scenarios, which is evidenced by extensive evaluations across five benchmark tasks and a fatigue modeling application.

Abstract:
This paper advances the use of physics‑informed neural networks (PINNs) architectures to address moving interface problems via the level set method. Originally developed for other PDE‑based problems, we particularly leverage PirateNet's features, including causal training, sequence‑to‑sequence learning, random weight factorization, and Fourier feature embeddings, and tailor them to handle problems with complex interface dynamics. Numerical experiments validate this framework on benchmark problems such as Zalesak's disk rotation and time‑reversed vortex flow. We demonstrate that PINNs can efficiently solve level set problems exhibiting significant interface deformation without the need for upwind numerical stabilization, as generally required by classic discretization methods, or additional mass conservation schemes. However, incorporating an Eikonal regularization term in the loss function with an appropriate weight can further enhance results in specific scenarios. Our results indicate that PINNs with the PirateNet architecture surpass conventional PINNs in accuracy, achieving state‑of‑the‑art error rates of L^2=0.14% for Zalesak's disk and L^2=0.85 % for the time‑reversed vortex flow problem, as compared to reference solutions. Additionally, for a complex two‑phase flow dam break problem coupling the level set with the Navier‑Stokes equations, we propose a geometric reinitialization method embedded within the sequence‑to‑sequence training scheme to ensure long‑term stability and accurate inference of the level set field.

Abstract:
Soft robots can revolutionize several applications with high demands on dexterity and safety. When operating these systems, real‑time estimation and control require fast and accurate models. However, prediction with first‑principles (FP) models is slow, and learned black‑box models have poor generalizability. Physics‑informed machine learning offers excellent advantages here, but it is currently limited to simple, often simulated systems without considering changes after training. We propose physics‑informed neural networks (PINNs) for articulated soft robots (ASRs) with a focus on data efficiency. The amount of expensive real‑world training data is reduced to a minimum ‑‑ one dataset in one system domain. Two hours of data in different domains are used for a comparison against two gold‑standard approaches: In contrast to a recurrent neural network, the PINN provides a high generalizability. The prediction speed of an accurate FP model is exceeded with the PINN by up to a factor of 467 at slightly reduced accuracy. This enables nonlinear model predictive control (MPC) of a pneumatic ASR. Accurate position tracking with the MPC running at 47 Hz is achieved in six dynamic experiments.

Abstract:
Modeling plays a critical role in additive manufacturing (AM), enabling a deeper understanding of underlying processes. Parametric solutions for such models are of great importance, enabling the optimization of production processes and considerable cost reductions. However, the complexity of the problem and diversity of spatio‑temporal scales involved in the process pose significant challenges for traditional numerical methods. Surrogate models offer a powerful alternative by accelerating simulations and facilitating real‑time monitoring and control. The present study presents an operator learning approach that relies on the deep operator network (DeepONet) and physics‑informed neural networks (PINN) to predict the three‑dimensional temperature distribution during melting and consolidation in laser powder bed fusion (LPBF). Parametric solutions for both single‑track and multi‑track scenarios with respect to tool path are obtained. To address the challenges in obtaining parametric solutions for multi‑track scenarios using DeepONet architecture, a sequential PINN approach is proposed to efficiently manage the increased training complexity inherent in those scenarios. The accuracy and consistency of the model are verified against finite‑difference computations. The developed surrogate allows us to efficiently analyze the effect of scanning paths and laser parameters on the thermal history.

Abstract:
Physics‑informed neural networks (PINNs) face significant challenges in modeling multi‑frequency wavefields in complex velocity models due to their slow convergence, difficulty in representing high‑frequency details, and lack of generalization to varying frequencies and velocity scenarios. To address these issues, we propose Meta‑LRPINN, a novel framework that combines low‑rank parameterization using singular value decomposition (SVD) with meta‑learning and frequency embedding. Specifically, we decompose the weights of PINN's hidden layers using SVD and introduce an innovative frequency embedding hypernetwork (FEH) that links input frequencies with the singular values, enabling efficient and frequency‑adaptive wavefield representation. Meta‑learning is employed to provide robust initialization, improving optimization stability and reducing training time. Additionally, we implement adaptive rank reduction and FEH pruning during the meta‑testing phase to further enhance efficiency. Numerical experiments, which are presented on multi‑frequency scattered wavefields for different velocity models, demonstrate that Meta‑LRPINN achieves much fast convergence speed and much high accuracy compared to baseline methods such as Meta‑PINN and vanilla PINN. Also, the proposed framework shows strong generalization to out‑of‑distribution frequencies while maintaining computational efficiency. These results highlight the potential of our Meta‑LRPINN for scalable and adaptable seismic wavefield modeling.

Abstract:
Physics‑informed neural networks (PINNs) have earned high expectations in solving partial differential equations (PDEs), but their optimization usually faces thorny challenges due to the unique derivative‑dependent loss function. By analyzing the loss distribution, previous research observed the propagation failure phenomenon of PINNs, intuitively described as the correct supervision for model outputs cannot ''propagate'' from initial states or boundaries to the interior domain. Going beyond intuitive understanding, this paper provides a formal and in‑depth study of propagation failure and its root cause. Based on a detailed comparison with classical finite element methods, we ascribe the failure to the conventional single‑point‑processing architecture of PINNs and further prove that propagation failure is essentially caused by the lower gradient correlation of PINN models on nearby collocation points. Compared to superficial loss maps, this new perspective provides a more precise quantitative criterion to identify where and why PINN fails. The theoretical finding also inspires us to present a new PINN architecture, named ProPINN, which can effectively unite the gradients of region points for better propagation. ProPINN can reliably resolve PINN failure modes and significantly surpass advanced Transformer‑based models with 46% relative promotion.

Abstract:
AI for PDEs has garnered significant attention, particularly Physics‑Informed Neural Networks (PINNs). However, PINNs are typically limited to solving specific problems, and any changes in problem conditions necessitate retraining. Therefore, we explore the generalization capability of transfer learning in the strong and energy form of PINNs across different boundary conditions, materials, and geometries. The transfer learning methods we employ include full finetuning, lightweight finetuning, and Low‑Rank Adaptation (LoRA). The results demonstrate that full finetuning and LoRA can significantly improve convergence speed while providing a slight enhancement in accuracy.

Abstract:
Multi‑task learning through composite loss functions is fundamental to modern deep learning, yet optimizing competing objectives remains challenging. We present new theoretical and practical approaches for addressing directional conflicts between loss terms, demonstrating their effectiveness in physics‑informed neural networks (PINNs) where such conflicts are particularly challenging to resolve. Through theoretical analysis, we demonstrate how these conflicts limit first‑order methods and show that second‑order optimization naturally resolves them through implicit gradient alignment. We prove that SOAP, a recently proposed quasi‑Newton method, efficiently approximates the Hessian preconditioner, enabling breakthrough performance in PINNs: state‑of‑the‑art results on 10 challenging PDE benchmarks, including the first successful application to turbulent flows with Reynolds numbers up to 10,000, with 2‑10x accuracy improvements over existing methods. We also introduce a novel gradient alignment score that generalizes cosine similarity to multiple gradients, providing a practical tool for analyzing optimization dynamics. Our findings establish frameworks for understanding and resolving gradient conflicts, with broad implications for optimization beyond scientific computing.

Abstract:
Our work aims at simulating and predicting the temperature conditions inside a power transformer using Physics‑Informed Neural Networks (PINNs). The predictions obtained are then used to determine the optimal placement for temperature sensors inside the transformer under the constraint of a limited number of sensors, enabling efficient performance monitoring. The method consists of combining PINNs with Mixed Integer Optimization Programming to obtain the optimal temperature reconstruction inside the transformer. First, we extend our PINN model for the thermal modeling of power transformers to solve the heat diffusion equation from 1D to 2D space. Finally, we construct an optimal sensor placement model inside the transformer that can be applied to problems in 1D and 2D.

Abstract:
Physics‑informed neural operators (PINOs) have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Recent research has demonstrated that incorporating Lie point symmetry information can significantly enhance the training efficiency of PINOs, primarily through techniques like data, architecture, and loss augmentation. In this work, we focus on the latter, highlighting that point symmetries oftentimes result in no training signal, limiting their effectiveness in many problems. To address this, we propose a novel loss augmentation strategy that leverages evolutionary representatives of point symmetries, a specific class of generalized symmetries of the underlying PDE. These generalized symmetries provide a richer set of generators compared to standard symmetries, leading to a more informative training signal. We demonstrate that leveraging evolutionary representatives enhances the performance of neural operators, resulting in improved data efficiency and accuracy during training.

Abstract:
Spectral bias, the tendency of neural networks to prioritize learning low‑frequency components of functions during the initial training stages, poses a significant challenge when approximating solutions with high‑frequency details. This issue is particularly pronounced in physics‑informed neural networks (PINNs), widely used to solve differential equations that describe physical phenomena. In the literature, contributions such as Wavelet Kolmogorov Arnold Networks (Wav‑KANs) have demonstrated promising results in capturing both low‑ and high‑frequency components. Similarly, Fourier features (FF) are often employed to address this challenge. However, the theoretical foundations of Wav‑KANs, particularly the relationship between the frequency of the mother wavelet and spectral bias, remain underexplored. A more in‑depth understanding of how Wav‑KANs manage high‑frequency terms could offer valuable insights for addressing oscillatory phenomena encountered in parabolic, elliptic, and hyperbolic differential equations. In this work, we analyze the eigenvalues of the neural tangent kernel (NTK) of Wav‑KANs to enhance their ability to converge on high‑frequency components, effectively mitigating spectral bias. Our theoretical findings are validated through numerical experiments, where we also discuss the limitations of traditional approaches, such as standard PINNs and Fourier features, in addressing multi‑frequency problems.

Abstract:
Railroad bridges are a crucial component of the U.S. freight rail system, which moves over 40 percent of the nation's freight and plays a critical role in the economy. However, aging bridge infrastructure and increasing train traffic pose significant safety hazards and risk service disruptions. The U.S. rail network includes over 100,000 railroad bridges, averaging one every 1.4 miles of track, with steel bridges comprising over 50% of the network's total bridge length. Early identification and assessment of damage in these bridges remain challenging tasks. This study proposes a physics‑informed neural network (PINN) based approach for damage identification in steel truss railroad bridges. The proposed approach employs an unsupervised learning approach, eliminating the need for large datasets typically required by supervised methods. The approach utilizes train wheel load data and bridge response during train crossing events as inputs for damage identification. The PINN model explicitly incorporates the governing differential equations of the linear time‑varying (LTV) bridge‑train system. Herein, this model employs a recurrent neural network (RNN) based architecture incorporating a custom Runge‑Kutta (RK) integrator cell, designed for gradient‑based learning. The proposed approach updates the bridge finite element model while also quantifying damage severity and localizing the affected structural members. A case study on the Calumet Bridge in Chicago, Illinois, with simulated damage scenarios, is used to demonstrate the model's effectiveness in identifying damage while maintaining low false‑positive rates. Furthermore, the damage identification pipeline is designed to seamlessly integrate prior knowledge from inspections and drone surveys, also enabling context‑aware updating and assessment of bridge's condition.

Abstract:
Koopman operator theory provides a powerful data‑driven technique for modeling nonlinear dynamical systems in a linear framework, in comparison to computationally expensive and highly nonlinear physics‑based simulations. However, Koopman operator‑based models for soft robots are very high dimensional and require considerable amounts of data to properly resolve. Inspired by physics‑informed techniques from machine learning, we present a novel physics‑informed Koopman operator identification method that improves simulation accuracy for small dataset sizes. Through Strang splitting, the method takes advantage of both continuous and discrete Koopman operator approximation to obtain information both from trajectory and phase space data. The method is validated on a tendon‑driven soft robotic arm, showing orders of magnitude improvement over standard methods in terms of the shape error. We envision this method can significantly reduce the data requirement of Koopman operators for systems with partially known physical models, and thus reduce the cost of obtaining data.

Abstract:
Radio map reconstruction is essential for enabling advanced applications, yet challenges such as complex signal propagation and sparse observational data hinder accurate reconstruction in practical scenarios. Existing methods often fail to align physical constraints with data‑driven features, particularly under sparse measurement conditions. To address these issues, we propose Physics‑Aligned Radio Map Diffusion Model (PhyRMDM), a novel framework that establishes cross‑domain representation alignment between physical principles and neural network features through dual learning pathways. The proposed model integrates Physics‑Informed Neural Networks (PINNs) with a representation alignment mechanism that explicitly enforces consistency between Helmholtz equation constraints and environmental propagation patterns. Experimental results demonstrate significant improvements over state‑of‑the‑art methods, achieving NMSE of 0.0031 under Static Radio Map (SRM) conditions, and NMSE of 0.0047 with Dynamic Radio Map (DRM) scenarios. The proposed representation alignment paradigm provides 37.2% accuracy enhancement in ultra‑sparse cases (1% sampling rate), confirming its effectiveness in bridging physics‑based modeling and deep learning for radio map reconstruction.

Abstract:
Physics‑informed machine learning is gaining significant traction for enhancing statistical performance and sample efficiency through the integration of physical knowledge. However, current theoretical analyses often presume complete prior knowledge in non‑hybrid settings, overlooking the crucial integration of observational data, and are frequently limited to linear systems, unlike the prevalent nonlinear nature of many real‑world applications. To address these limitations, we introduce a unified residual form that unifies collocation and variational methods, enabling the incorporation of incomplete and complex physical constraints in hybrid learning settings. Within this formulation, we establish that the generalization performance of physics‑informed regression in such hybrid settings is governed by the dimension of the affine variety associated with the physical constraint, rather than by the number of parameters. This enables a unified analysis that is applicable to both linear and nonlinear equations. We also present a method to approximate this dimension and provide experimental validation of our theoretical findings.

Abstract:
We investigate the training of Physics‑Informed Neural Networks (PINNs) from a control‑theoretic perspective. Using gradient descent with resampling, we interpret the training dynamics as asymptotically equivalent to a stochastic control‑affine system, where sampling effects act as process disturbances and measurement noise. Within this framework, we introduce two controllers for dynamically adapting the physics weight: an integral controller and a leaky integral controller. We theoretically analyze their asymptotic properties under the accuracy‑robustness trade‑off, and we evaluate them on a toy example. Numerical evidence suggests that the integral controller achieves accurate and robust convergence when the physical model is correct, whereas the leaky integrator provides improved performance in the presence of model mismatch. This work represents a first step toward convergence guarantees and principled training algorithms tailored to the distinct characteristics of PINN tasks.

Abstract:
A novel strategy is proposed to improve the accuracy of state estimation and reconstruction from low‑fidelity models and sparse data from sensors. This strategy combines ensemble Data Assimilation (DA) and Machine Learning (ML) tools, exploiting their complementary features. ML techniques rely on the data produced by DA methods during analysis phases to train physics‑informed corrective algorithms, which are then coupled with the low‑fidelity models when data from sensors is unavailable. The methodology is validated via the analysis of the turbulent plane channel flow test case for Re_τ\approx 550. Here, the low‑fidelity model consists of coarse‑grained simulations coupled with the Immersed Boundary Method (IBM), while observation is sampled by a highly refined body‑fitted calculation. The analysis demonstrates the capabilities of the algorithm based on DA and ML to accurately predict the flow features with significantly reduced computational costs. This approach exhibits potential for future synergistic applications of DA and ML, leveraging the robustness and efficiency of ML models alongside the physical interpretability ensured by DA algorithms.

Abstract:
Many physics‑informed machine learning methods for PDE‑based problems rely on Gaussian processes (GPs) or neural networks (NNs). However, both face limitations when data are scarce and the dimensionality is high. Although GPs are known for their robust uncertainty quantification in low‑dimensional settings, their computational complexity becomes prohibitive as the dimensionality increases. In contrast, while conventional NNs can accommodate high‑dimensional input, they often require extensive training data and do not offer uncertainty quantification. To address these challenges, we propose a PDE‑constrained Deep Kernel Learning (PDE‑DKL) framework that combines DL and GPs under explicit PDE constraints. Specifically, NNs learn a low‑dimensional latent representation of the high‑dimensional PDE problem, reducing the complexity of the problem. GPs then perform kernel regression subject to the governing PDEs, ensuring accurate solutions and principled uncertainty quantification, even when available data are limited. This synergy unifies the strengths of both NNs and GPs, yielding high accuracy, robust uncertainty estimates, and computational efficiency for high‑dimensional PDEs. Numerical experiments demonstrate that PDE‑DKL achieves high accuracy with reduced data requirements. They highlight its potential as a practical, reliable, and scalable solver for complex PDE‑based applications in science and engineering.

Abstract:
Thermal protection systems (TPS) of space vehicles are designed computationally rather than experimentally. They are validated using ground experiments, but all aspects of the flight cannot be replicated on ground. This ground‑to‑flight mapping introduces uncertainties which need to be accounted for while designing any thermal protection system. Thus, precise computational models along with uncertainty quantification in the models are required to design the TPS. The focus of this study is to estimate the thermal material parameters of TPS based on the target reliability requirements using statistical methods. To perform uncertainty quantification (UQ) of a system, a simulated model of the system needs to be solved many times on statistical samples, increasing the computational time and cost of the overall process. A physics‑informed neural network (PINN) model is used in the analysis instead of traditional physics based numerical solutions. The accuracy of PINN is comparable to that of the numerical solution. To find the parameter distribution, sampling of the parameter space is performed using Sequential Monte‑ Carlo (SMC) method. The sampling method is efficient as it generates samples based on the target distribution in parallel and it also generates diverse samples for proper UQ. Combining the use of both PINN predictive model and SMC sampling, the framework can approximate the parameter distributions that satisfy the TPS design reliability constraints. The framework achieved remarkable increases in the speed of performing the reliability analysis of the TPS. This reliability analysis can be used for design optimization of the TPS based on risk analysis along with other systems of the vehicle.

Abstract:
The importance and cost of time‑domain simulations when studying power systems have exponentially increased in the last decades. With the growing share of renewable energy sources, the slow and predictable responses from large turbines are replaced by the fast and unpredictable dynamics from power electronics. The current existing simulation tools require new solutions designed for faster dynamics. Physics‑Informed Neural Networks (PINNs) have recently emerged in power systems to accelerate such simulations. By incorporating knowledge during the up‑front training, PINNs provide more accurate results over larger time steps than traditional numerical methods. This paper introduces PINNs as an alternative approximation method that seamlessly integrates with the current simulation framework. We replace a synchronous machine for a trained PINN in the IEEE 9‑, 14‑, and 30‑bus systems and simulate several network disturbances. Including PINNs systematically boosts the simulations' accuracy, providing more accurate results for both the PINN‑modeled component and the whole multi‑machine system states.

Abstract:
Magnetron sputtered titanium nitride (TiN) thin films are widely used as protective coatings due to their high hardness, but suffer from inherent brittleness and low fracture toughness, limiting their applicability. The multilayering of TiN films with metallic titanium (Ti) interlayers in the form of bi‑layer and tri‑layer systems have been studied using microcantilever fracture tests. Plastic dissipation in the Ti layer is shown to lead to an increase in crack growth resistance. The effect of the elastic‑plastic mismatch between the two materials on the crack driving force, as well as the size of the fully developed plastic zone in Ti have been quantified in this work for the first time. It is shown that incorporating a Ti layer thickness of 250 nm can improve the fracture resistance by nearly ten times compared to the initiation fracture toughness in TiN, preventing catastrophic fracture of these multi‑layered films. These results will aid in physics informed design of optimised thickness of metallic interlayers in multi‑layered thin films.

Abstract:
Stiff differential equations are prevalent in various scientific domains, posing significant challenges due to the disparate time scales of their components. As computational power grows, physics‑informed neural networks (PINNs) have led to significant improvements in modeling physical processes described by differential equations. Despite their promising outcomes, vanilla PINNs face limitations when dealing with stiff systems, known as failure modes. In response, we propose a novel approach, stiff transfer learning for physics‑informed neural networks (STL‑PINNs), to effectively tackle stiff ordinary differential equations (ODEs) and partial differential equations (PDEs). Our methodology involves training a Multi‑Head‑PINN in a low‑stiff regime, and obtaining the final solution in a high stiff regime by transfer learning. This addresses the failure modes related to stiffness in PINNs while maintaining computational efficiency by computing "one‑shot" solutions. The proposed approach demonstrates superior accuracy and speed compared to PINNs‑based methods, as well as comparable computational efficiency with implicit numerical methods in solving stiff‑parameterized linear and polynomial nonlinear ODEs and PDEs under stiff conditions. Furthermore, we demonstrate the scalability of such an approach and the superior speed it offers for simulations involving initial conditions and forcing function reparametrization.

Abstract:
We consider the use of Gaussian Processes (GPs) or Neural Networks (NNs) to numerically approximate the solutions to nonlinear partial differential equations (PDEs) with rough forcing or source terms, which commonly arise as pathwise solutions to stochastic PDEs. Kernel methods have recently been generalized to solve nonlinear PDEs by approximating their solutions as the maximum a posteriori estimator of GPs that are conditioned to satisfy the PDE at a finite set of collocation points. The convergence and error guarantees of these methods, however, rely on the PDE being defined in a classical sense and its solution possessing sufficient regularity to belong to the associated reproducing kernel Hilbert space. We propose a generalization of these methods to handle roughly forced nonlinear PDEs while preserving convergence guarantees with an oversmoothing GP kernel that is misspecified relative to the true solution's regularity. This is achieved by conditioning a regular GP to satisfy the PDE with a modified source term in a weak sense (when integrated against a finite number of test functions). This is equivalent to replacing the empirical L^2‑loss on the PDE constraint by an empirical negative‑Sobolev norm. We further show that this loss function can be used to extend physics‑informed neural networks (PINNs) to stochastic equations, thereby resulting in a new NN‑based variant termed Negative Sobolev Norm‑PINN (NeS‑PINN).

Abstract:
Micro‑structured surfaces influence nucleation characteristics and bubble dynamics besides increasing the heat transfer surface area, thus enabling efficient nucleate boiling heat transfer. Modeling the pool boiling heat transfer characteristics of these surfaces under varied conditions is essential in diverse applications. A new empirical correlation for nucleate boiling on microchannel structured surfaces has been proposed with the data collected from various experiments in previous studies since the existing correlations are limited by their accuracy and narrow operating ranges. This study also examines various Machine Learning (ML) algorithms and Deep Neural Networks (DNN) on the microchannel structured surfaces dataset to predict the nucleate pool boiling Heat Transfer Coefficient (HTC). With the aim to integrate both the ML and domain knowledge, a Physics‑Informed Machine Learning Aided Framework (PIMLAF) is proposed. The proposed correlation in this study is employed as the prior physics‑based model for PIMLAF, and a DNN is employed to model the residuals of the prior model. This hybrid framework achieved the best performance in comparison to the other ML models and DNNs. This framework is able to generalize well for different datasets because the proposed correlation provides the baseline knowledge of the boiling behavior. Also, SHAP interpretation analysis identifies the critical parameters impacting the model predictions and their effect on HTC prediction. This analysis further makes the model more robust and reliable. Keywords: Pool boiling, Microchannels, Heat transfer coefficient, Correlation analysis, Machine learning, Deep neural network, Physics‑informed machine learning aided framework, SHAP analysis

Abstract:
Physics‑Informed Neural Networks (PINNs) have revolutionized the computation of PDE solutions by integrating partial differential equations (PDEs) into the neural network's training process as soft constraints, becoming an important component of the scientific machine learning (SciML) ecosystem. More recently, physics‑informed Kolmogorv‑Arnold networks (PIKANs) have also shown to be effective and comparable in accuracy with PINNs. In their current implementation, both PINNs and PIKANs are mainly optimized using first‑order methods like Adam, as well as quasi‑Newton methods such as BFGS and its low‑memory variant, L‑BFGS. However, these optimizers often struggle with highly non‑linear and non‑convex loss landscapes, leading to challenges such as slow convergence, local minima entrapment, and (non)degenerate saddle points. In this study, we investigate the performance of Self‑Scaled BFGS (SSBFGS), Self‑Scaled Broyden (SSBroyden) methods and other advanced quasi‑Newton schemes, including BFGS and L‑BFGS with different line search strategies. These methods dynamically rescale updates based on historical gradient information, thus enhancing training efficiency and accuracy. We systematically compare these optimizers using both PINNs and PIKANs on key challenging PDEs, including the Burgers, Allen‑Cahn, Kuramoto‑Sivashinsky, Ginzburg‑Landau, and Stokes equations. Additionally, we evaluate the performance of SSBFGS and SSBroyden for Deep Operator Network (DeepONet) architectures, demonstrating their effectiveness for data‑driven operator learning. Our findings provide state‑of‑the‑art results with orders‑of‑magnitude accuracy improvements without the use of adaptive weights or any other enhancements typically employed in PINNs.

Abstract:
In this paper, we present the Residual Integral Solver Network (RISN), a novel neural network architecture designed to solve a wide range of integral and integro‑differential equations, including one‑dimensional, multi‑dimensional, ordinary and partial integro‑differential, systems, fractional types, and Helmholtz‑type integral equations involving oscillatory kernels. RISN integrates residual connections with high‑accuracy numerical methods such as Gaussian quadrature and fractional derivative operational matrices, enabling it to achieve higher accuracy and stability than traditional Physics‑Informed Neural Networks (PINN). The residual connections help mitigate vanishing gradient issues, allowing RISN to handle deeper networks and more complex kernels, particularly in multi‑dimensional problems. Through extensive experiments, we demonstrate that RISN consistently outperforms not only classical PINNs but also advanced variants such as Auxiliary PINN (A‑PINN) and Self‑Adaptive PINN (SA‑PINN), achieving significantly lower Mean Absolute Errors (MAE) across various types of equations. These results highlight RISN's robustness and efficiency in solving challenging integral and integro‑differential problems, making it a valuable tool for real‑world applications where traditional methods often struggle.

Abstract:
A novel Trunk‑Branch (TB)‑net physics‑informed neural network (PINN) architecture is developed, which is a PINN‑based method incorporating trunk and branch nets to capture both global and local features. The aim is to solve four main classes of problems: forward flow problem, forward heat transfer problem, inverse heat transfer problem, and transfer learning problem within the porous medium, which are notoriously complex that could not be handled by origin PINN. In the proposed TB‑net PINN architecture, a Fully‑connected Neural Network (FNN) is used as the trunk net, followed by separated FNNs as the branch nets with respect to outputs, and automatic differentiation is performed for partial derivatives of outputs with respect to inputs by considering various physical loss. The effectiveness and flexibility of the novel TB‑net PINN architecture is demonstrated through a collection of forward problems, and transfer learning validates the feasibility of resource reuse. Combining with the superiority over traditional numerical methods in solving inverse problems, the proposed TB‑net PINN shows its great potential for practical engineering applications.

Abstract:
Physics‑Informed Neural Networks (PINNs) are a powerful deep learning method capable of providing solutions and parameter estimations of physical systems. Given the complexity of their neural network structure, the convergence speed is still limited compared to numerical methods, mainly when used in applications that model realistic systems. The network initialization follows a random distribution of the initial weights, as in the case of traditional neural networks, which could lead to severe model convergence bottlenecks. To overcome this problem, we follow current studies that deal with optimal initial weights in traditional neural networks. In this paper, we use a convex optimization model to improve the initialization of the weights in PINNs and accelerate convergence. We investigate two optimization models as a first training step, defined as pre‑training, one involving only the boundaries and one including physics. The optimization is focused on the first layer of the neural network part of the PINN model, while the other weights are randomly initialized. We test the methods using a practical application of the heat diffusion equation to model the temperature distribution of power transformers. The PINN model with boundary pre‑training is the fastest converging method at the current stage.

Abstract:
We present a novel class of Physics‑Informed Neural Networks that is formulated based on the principles of Evidential Deep Learning, where the model incorporates uncertainty quantification by learning parameters of a higher‑order distribution. The dependent and trainable variables of the PDE residual loss and data‑fitting loss terms are recast as functions of the hyperparameters of an evidential prior distribution. Our model is equipped with an information‑theoretic regularizer that contains the Kullback‑Leibler divergence between two inverse‑gamma distributions characterizing predictive uncertainty. Relative to Bayesian‑Physics‑Informed‑Neural‑Networks, our framework appeared to exhibit higher sensitivity to data noise, preserve boundary conditions more faithfully and yield empirical coverage probabilities closer to nominal ones. Toward examining its relevance for data mining in scientific discoveries, we demonstrate how to apply our model to inverse problems involving 1D and 2D nonlinear differential equations.

Abstract:
This work investigates data‑driven prediction and control of Hammerstein‑Wiener systems using physics‑informed Gaussian process (GP) models that encode the block‑oriented model structure. Data‑driven prediction algorithms have been developed for structured nonlinear systems based on Willems' fundamental lemma. However, existing frameworks do not apply to output nonlinearities in Wiener systems and rely on a finite‑dimensional dictionary of basis functions for Hammerstein systems. In this work, an implicit predictor structure is considered, leveraging the linearity for the dynamical part of the model. This implicit function is learned by GP regression, utilizing carefully designed structured kernel functions from linear model parameters and GP priors for the nonlinearities. Virtual derivative points are added to the regression by expectation propagation to encode monotonicity information of the nonlinearities. The linear model parameters are estimated as hyperparameters by assuming a stable spline hyperprior. The implicit GP model provides explicit output prediction by optimizing selected optimality criteria. The implicit model is also applied to receding horizon control with the expected control cost and chance constraint satisfaction guarantee. Numerical results demonstrate that the proposed prediction and control algorithms are superior to black‑box GP models without model structure knowledge.

Abstract:
We present a physics‑informed autoencoder designed to encode the equation of state of neutron stars into an interpretable latent space. In particular the input will be encoded in the mass, radius, and tidal deformability values of a neutron star. Unlike traditional black‑box models, our approach incorporates additional loss functions to enforce explainability in the encoded representations. This method enhances the transparency of machine learning models in physics, providing a robust proof‑of‑concept tool to study compact stars data. Our results demonstrate that the proposed autoencoder not only accurately estimates the EoS parameters and central density/pressure but also offers insights into the physical connection between equation of state and observable physical quantities. This framework conceptualizes the physical differential equations themselves as the ``encoders", allowing interpretability of the latent space.

Abstract:
In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.

Abstract:
Limited by the operation and measurement conditions, numerical simulation is often the only feasible approach for studying plasma behavior and mechanisms. Although artificial intelligence methods, especially physics‑informed neural network (PINN), have been widely applied in plasma simulation, the design of the neural network structures still largely relies on the experience of researchers. Meanwhile, existing neural architecture search methods tailored for PINN have encountered failures when dealing with complex plasma governing equations characterized by variable coefficients and strong nonlinearity. Therefore, we propose an improved neural architecture search‑guided method, namely NAS‑PINNv2, to address the limitations of existing methods. By analyzing the causes of failure, the sigmoid function is applied to calculate the architecture‑related weights, and a new loss term is introduced. The performance of NAS‑PINNv2 is verified in several numerical experiments including the Elenbaas‑Heller equation without and with radial velocity, the drift‑diffusion‑Poisson equation and the Boltzmann equation. The results again emphasize that larger neural networks do not necessarily perform better, and the discovered neural architecture with multiple neuron numbers in a single hidden layer imply a more flexible and sophisticated design rule for fully connected networks.

Abstract:
The recent advances in information technology and artificial intelligence have fueled a rapid expansion of the data center (DC) industry worldwide, accompanied by an immense appetite for electricity to power the DCs. In a typical DC, around 30~40% of the energy is spent on the cooling system rather than on computer servers, posing a pressing need for developing new energy‑saving optimization technologies for DC cooling systems. However, optimizing such real‑world industrial systems faces numerous challenges, including but not limited to a lack of reliable simulation environments, limited historical data, and stringent safety and control robustness requirements. In this work, we present a novel physics‑informed offline reinforcement learning (RL) framework for energy efficiency optimization of DC cooling systems. The proposed framework models the complex dynamical patterns and physical dependencies inside a server room using a purposely designed graph neural network architecture that is compliant with the fundamental time‑reversal symmetry. Because of its well‑behaved and generalizable state‑action representations, the model enables sample‑efficient and robust latent space offline policy learning using limited real‑world operational data. Our framework has been successfully deployed and verified in a large‑scale production DC for closed‑loop control of its air‑cooling units (ACUs). We conducted a total of 2000 hours of short and long‑term experiments in the production DC environment. The results show that our method achieves 14~21% energy savings in the DC cooling system, without any violation of the safety or operational constraints. Our results have demonstrated the significant potential of offline RL in solving a broad range of data‑limited, safety‑critical real‑world industrial control problems.

Abstract:
Recent advancements in machine learning‑based methods have demonstrated great potential for improved property prediction in material science. However, reliable estimation of the confidence intervals for the predicted values remains a challenge, due to the inherent complexities in material modeling. This study introduces a novel approach for uncertainty quantification in fatigue life prediction of metal materials based on integrating knowledge from physics‑based fatigue life models and machine learning models. The proposed approach employs physics‑based input features estimated using the Basquin fatigue model to augment the experimentally collected data of fatigue life. Furthermore, a physics‑informed loss function that enforces boundary constraints for the estimated fatigue life of considered materials is introduced for the neural network models. Experimental validation on datasets comprising collected data from fatigue life tests for Titanium alloys and Carbon steel alloys demonstrates the effectiveness of the proposed approach. The synergy between physics‑based models and data‑driven models enhances the consistency in predicted values and improves uncertainty interval estimates.

Abstract:
We present a physics‑informed deep learning framework to address common limitations in Confocal Laser Scanning Microscopy (CLSM), such as diffraction limited resolution, noise, and undersampling due to low laser power conditions. The optical system's point spread function (PSF) and common CLSM image degradation mechanisms namely photon shot noise, dark current noise, motion blur, speckle noise, and undersampling were modeled and were directly included into model architecture. The model reconstructs high fidelity images from heavily noisy inputs by using convolutional and transposed convolutional layers. Following the advances in compressed sensing, our approach significantly reduces data acquisition requirements without compromising image resolution. The proposed method was extensively evaluated on simulated CLSM images of diverse structures, including lipid droplets, neuronal networks, and fibrillar systems. Comparisons with traditional deconvolution algorithms such as Richardson‑Lucy (RL), non‑negative least squares (NNLS), and other methods like Total Variation (TV) regularization, Wiener filtering, and Wavelet denoising demonstrate the superiority of the network in restoring fine structural details with high fidelity. Assessment metrics like Structural Similarity Index (SSIM) and Peak Signal to Noise Ratio (PSNR), underlines that the AdaptivePhysicsAutoencoder achieved robust image enhancement across diverse CLSM conditions, helping faster acquisition, reduced photodamage, and reliable performance in low light and sparse sampling scenarios holding promise for applications in live cell imaging, dynamic biological studies, and high throughput material characterization.

Abstract:
The techno‑economic and safety concerns of battery capacity knee occurrence call for developing online knee detection and prediction methods as an advanced battery management system (BMS) function. To address this, a transferable physics‑informed framework that consists of a histogram‑based feature engineering method, a hybrid physics‑informed model, and a fine‑tuning strategy, is proposed for online battery degradation diagnosis and knee‑onset detection. The hybrid model is first developed and evaluated using a scenario‑aware pipeline in protocol cycling scenarios and then fine‑tuned to create local models deployed in a dynamic cycling scenario. A 2D histogram‑based 17‑feature set is found to be the best choice in both source and target scenarios. The fine‑tuning strategy is proven to be effective in improving battery degradation mode estimation and degradation phase detection performance in the target scenario. Again, a strong linear correlation was found between the identified knee‑onset and knee points. As a result, advanced BMS functions, such as online degradation diagnosis and prognosis, online knee‑onset detection and knee prediction, aging‑aware battery classification, and second‑life repurposing, can be enabled through a battery performance digital twin in the cloud.

Abstract:
Parameter estimation and trajectory reconstruction for data‑driven dynamical systems governed by ordinary differential equations (ODEs) are essential tasks in fields such as biology, engineering, and physics. These inverse problems ‑‑ estimating ODE parameters from observational data ‑‑ are particularly challenging when the data are noisy, sparse, and the dynamics are nonlinear. We propose the Eigen‑Fourier Physics‑Informed Gaussian Process (EFiGP), an algorithm that integrates Fourier transformation and eigen‑decomposition into a physics‑informed Gaussian Process framework. This approach eliminates the need for numerical integration, significantly enhancing computational efficiency and accuracy. Built on a principled Bayesian framework, EFiGP incorporates the ODE system through probabilistic conditioning, enforcing governing equations in the Fourier domain while truncating high‑frequency terms to achieve denoising and computational savings. The use of eigen‑decomposition further simplifies Gaussian Process covariance operations, enabling efficient recovery of trajectories and parameters even in dense‑grid settings. We validate the practical effectiveness of EFiGP on three benchmark examples, demonstrating its potential for reliable and interpretable modeling of complex dynamical systems while addressing key challenges in trajectory recovery and computational cost.

Abstract:
The novel neural networks show great potential in solving partial differential equations. For single‑phase flow problems in subsurface porous media with high‑contrast coefficients, the key is to develop neural operators with accurate reconstruction capability and strict adherence to physical laws. In this study, we proposed a hybrid two‑stage framework that uses multiscale basis functions and physics‑guided deep learning to solve the Darcy flow problem in high‑contrast fractured porous media. In the first stage, a data‑driven model is used to reconstruct the multiscale basis function based on the permeability field to achieve effective dimensionality reduction while preserving the necessary multiscale features. In the second stage, the physics‑informed neural network, together with Transformer‑based global information extractor is used to reconstruct the pressure field by integrating the physical constraints derived from the Darcy equation, ensuring consistency with the physical laws of the real world. The model was evaluated on datasets with different combinations of permeability and basis functions and performed well in terms of reconstruction accuracy. Specifically, the framework achieves R2 values above 0.9 in terms of basis function fitting and pressure reconstruction, and the residual indicator is on the order of 1× 10^‑4. These results validate the ability of the proposed framework to achieve accurate reconstruction while maintaining physical consistency.

Abstract:
Accurate prediction of rarefied gas dynamics is crucial for optimizing flows through microelectromechanical systems, air filtration devices, and shale gas extraction. Traditional methods, such as discrete velocity and direct simulation Monte Carlo (DSMC), demand intensive memory and computation, especially for microflows in non‑convex domains. Recently, physics‑informed neural networks emerged as a meshless and adaptable alternative for solving non‑linear partial differential equations. We trained a PINN using a limited number of DSMC‑generated rarefied gas microflows in the transition regime with Knudsen number from 0.1 to 3, incorporating continuity and Cauchy momentum exchange equations in the loss function. The PINN achieved under 2 percent error on these residuals and effectively filtered DSMC intrinsic statistical noise. Predictions remained strong for a tested flow field with Kn equal to 0.7, and showed limited extrapolation performance on a flow field with Kn equal to 5 with a local overshoot of about 20 percent, while maintaining physical consistency. Notably, each DSMC field required about 20 hours on 4 graphics processing units (GPU), while the PINN training took less than 2 hours on one GPU, with evaluations under 2 seconds.

Abstract:
Counterfactuals are a concept inherited from the field of logic and in general attain to the existence of causal relations between sentences or events. In particular, this concept has been introduced also in the context of interpretability in artificial intelligence, where counterfactuals refer to the minimum change to the feature values that changes the prediction of a classification model. The artificial intelligence framework of counterfactuals is mostly focused on machine learning approaches, typically neglecting the physics of the variables that determine a change in class. However, a theoretical formulation of counterfactuals in a control system framework ‑ i.e., able to account for the mechanisms underlying a change in class ‑ is lacking. To fill this gap, in this work we propose an original control system, physics‑informed, theoretical foundation for counterfactuals, by means of the formulation of an optimal control problem. We apply the proposed methodology to a general glucose‑insulin regulation model and results appear promising and pave the way to the possible integration with artificial intelligence techniques, with the aim of feeding machine learning models with the physics knowledge acquired through the system framework.

Abstract:
The masses of Population III stars are largely unconstrained since no simulations exist that take all relevant primordial star formation physics into account. We perform the first suite of radiation magnetohydrodynamics (RMHD) simulations of Population III star formation, with the POPSICLE project. Compared to control simulations that only include magnetic fields (MHD), protostellar ionizing and dissociating feedback, or neither, the RMHD simulation best resembles the MHD simulation during the earliest stages of collapse and star formation. In 5000\,\rmyrs, the mass of the most massive star is 65\,\rmM_\odot in the RMHD simulation, compared to 120\,\rmM_\odot in simulations without magnetic fields. This difference arises because magnetic fields act against gravity, suppress mass transport, and reduce compressional heating. The maximum stellar mass of Population III stars is thus already limited by magnetic fields, even before accretion rates drop to allow significant protostellar radiative feedback. Following classical main sequence stellar evolution with MESA reveals that it is difficult to create Population III stars with masses larger than 600\,\rmM_\odot in typical dark matter minihaloes at z \gtrsim 20, with maximum stellar masses ～ 100\,\rmM_\odot more likely due to expected negative feedback from both magnetic fields and stellar radiation. This work lays the first step in building a full physics‑informed mass function of Population III stars.

Abstract:
Off‑road navigation is critical for a wide range of field robotics applications from planetary exploration to disaster response. However, it remains a longstanding challenge due to unstructured environments and the inherently complex terrain‑vehicle interactions. Traditional physics‑based methods struggle to accurately capture the nonlinear dynamics underlying these interactions, while purely data‑driven approaches often overfit to specific motion patterns, vehicle geometries, or platforms, limiting their generalization in diverse, real‑world scenarios. To address these limitations, we introduce AnyNav, a vision‑based friction estimation and navigation framework grounded in neuro‑symbolic principles. Our approach integrates neural networks for visual perception with symbolic physical models for reasoning about terrain‑vehicle dynamics. To enable self‑supervised learning in real‑world settings, we adopt the imperative learning paradigm, employing bilevel optimization to train the friction network through physics‑based optimization. This explicit incorporation of physical reasoning substantially enhances generalization across terrains, vehicle types, and operational conditions. Leveraging the predicted friction coefficients, we further develop a physics‑informed navigation system capable of generating physically feasible, time‑efficient paths together with corresponding speed profiles. We demonstrate that AnyNav seamlessly transfers from simulation to real‑world robotic platforms, exhibiting strong robustness across different four‑wheeled vehicles and diverse off‑road environments.

Abstract:
We present approximation results and numerical experiments for the use of randomized neural networks within physics‑informed extreme learning machines to efficiently solve high‑dimensional PDEs, demonstrating both high accuracy and low computational cost. Specifically, we prove that RaNNs can approximate certain classes of functions, including Sobolev functions, in the H^2‑norm at dimension‑independent convergence rates, thereby alleviating the curse of dimensionality. Numerical experiments are provided for the high‑dimensional heat equation, the Black‑Scholes model, and the Heston model, demonstrating the accuracy and efficiency of randomized neural networks.

Abstract:
Non‑linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well‑established method to tackle stiff differential equations. Here we present a machine learning framework to facilitate the solution of nonlinear multiscale differential equations and, especially, inverse problems using Physics‑Informed Neural Networks (PINNs). This framework is based on what is called multi‑head (MH) training, which involves training the network to learn a general space of all solutions for a given set of equations with certain variability, rather than learning a specific solution of the system. This setup is used with a second novel technique that we call Unimodular Regularization (UR) of the latent space of solutions. We show that the multi‑head approach, combined with Unimodular Regularization, significantly improves the efficiency of PINNs by facilitating the transfer learning process thereby enabling the finding of solutions for nonlinear, coupled, and multiscale differential equations.

Abstract:
Solving partial differential equations (PDEs) using neural methods has been a long‑standing scientific and engineering research pursuit. Physics‑Informed Neural Networks (PINNs) have emerged as a promising alternative to traditional numerical methods for solving PDEs. However, the gap between domain‑specific knowledge and deep learning expertise often limits the practical application of PINNs. Previous works typically involve manually conducting extensive PINNs experiments and summarizing heuristic rules for hyperparameter tuning. In this work, we introduce PINNsAgent, a novel surrogation framework that leverages large language models (LLMs) and utilizes PINNs as a foundation to bridge the gap between domain‑specific knowledge and deep learning. Specifically, PINNsAgent integrates (1) Physics‑Guided Knowledge Replay (PGKR), which encodes the essential characteristics of PDEs and their associated best‑performing PINNs configurations into a structured format, enabling efficient knowledge transfer from solved PDEs to similar problems and (2) Memory Tree Reasoning, a strategy that effectively explores the search space for optimal PINNs architectures. By leveraging LLMs and exploration strategies, PINNsAgent enhances the automation and efficiency of PINNs‑based solutions. We evaluate PINNsAgent on 14 benchmark PDEs, demonstrating its effectiveness in automating the surrogation process and significantly improving the accuracy of PINNs‑based solutions.

Abstract:
Physics‑informed neural networks (PINNs) have shown remarkable prospects in solving forward and inverse problems involving partial differential equations (PDEs). However, PINNs still face the challenge of high computational cost in solving strongly nonlinear PDEs involving fluid dynamics. In this study, inspired by the input design in surrogate modeling, we propose a feature‑enhanced neural network. By introducing geometric features including distance and angle or physical features including the solution of the potential flow equation in the inputs of PINNs, FENN can more easily learn the flow, resulting in better performance in terms of both accuracy and efficiency. We establish the feature networks in advance to avoid the invalid PDE loss in FENN caused by neglecting the partial derivatives of the features with respect to space‑time coordinates. Through five numerical experiments involving forward, inverse, and parametric problems, we verify that FENN generally reduces the computational cost of PINNs by approximately four times. In addition, the numerical experiments also demonstrate that the proposed method can reduce the number of observed data for inverse problem and successfully solve the parametric problem where PINNs fail.

Abstract:
Mesh generation plays a crucial role in scientific computing. Traditional mesh generation methods, such as TFI and PDE‑based methods, often struggle to achieve a balance between efficiency and mesh quality. To address this challenge, physics‑informed intelligent learning methods have recently emerged, significantly improving generation efficiency while maintaining high mesh quality. However, physics‑informed methods fail to generalize when applied to previously unseen geometries, as even small changes in the boundary shape necessitate burdensome retraining to adapt to new geometric variations. In this paper, we introduce MeshONet, the first generalizable intelligent learning method for structured mesh generation. The method transforms the mesh generation task into an operator learning problem with multiple input and solution functions. To effectively overcome the multivariable mapping restriction of operator learning methods, we propose a dual‑branch, shared‑trunk architecture to approximate the mapping between function spaces based on input‑output pairs. Experimental results show that MeshONet achieves a speedup of up to four orders of magnitude in generation efficiency over traditional methods. It also enables generalization to different geometries without retraining, greatly enhancing the practicality of intelligent methods.

Abstract:
This paper proposes a novel learning approach for designing Kazantzis‑Kravaris/Luenberger (KKL) observers for autonomous nonlinear systems. The design of a KKL observer involves finding an injective map that transforms the system state into a higher‑dimensional observer state, whose dynamics is linear and stable. The observer's state is then mapped back to the original system coordinates via the inverse map to obtain the state estimate. However, finding this transformation and its inverse is quite challenging. We propose learning the forward mapping using a physics‑informed neural network, and then learning its inverse mapping with a conventional feedforward neural network. Theoretical guarantees for the robustness of state estimation against approximation error and system uncertainties are provided, including non‑asymptotic learning guarantees that link approximation quality to finite sample sizes. The effectiveness of the proposed approach is demonstrated through numerical simulations on benchmark examples, showing superior generalization capability outside the training domain compared to state‑of‑the‑art methods.

Abstract:
Solving the two‑dimensional shallow water equations is a fundamental problem in flood simulation technology. In recent years, physics‑informed neural networks (PINNs) have emerged as a novel methodology for addressing this problem. Given their advantages in parallel computing, the potential for data assimilation and parameter calibration, and the rapid advancement of artificial intelligence, it is crucial to investigate both the capabilities and limitations of PINNs. While current research has demonstrated the significant potential of PINNs, many aspects of this new approach remain to be explored. In this study, we employ PINNs enhanced by dimensional transformation and N‑LAAF techniques to validate their effectiveness in solving two‑dimensional free surface flow with rainfall on terrain topography. The shallow water equations primarily exist in two forms: the variables form and the conservative form. Through theoretical analysis and experimental validation, we demonstrate that a hybrid variable‑conservation form offers superior performance. Additionally, we find that incorporating the energy conservation law, specifically the entropy condition, does not yield substantial improvements and may even lead to training failure. Furthermore, we have developed an open‑source module on the PINNacle platform for solving shallow water equations using PINNs, which includes over ten case studies and various equation forms, to promote research and application in this field.

Abstract:
Reconfigurable intelligent surface (RIS) is a two‑dimensional periodic structure integrated with a large number of reflective elements, which can manipulate electromagnetic waves in a digital way, offering great potentials for wireless communication and radar detection applications. However, conventional RIS designs highly rely on extensive full‑wave EM simulations that are extremely time‑consuming. To address this challenge, we propose a machine‑learning‑assisted approach for efficient RIS design. An accurate and fast model to predict the reflection coefficient of RIS element is developed by combining a multi‑layer perceptron neural network (MLP) and a dual‑port network, which can significantly reduce tedious EM simulations in the network training. A RIS has been practically designed based on the proposed method. To verify the proposed method, the RIS has also been fabricated and measured. The experimental results are in good agreement with the simulation results, which validates the efficacy of the proposed method in RIS design.

Abstract:
This paper introduces Residual‑based Smote (RSmote), an innovative local adaptive sampling technique tailored to improve the performance of Physics‑Informed Neural Networks (PINNs) through imbalanced learning strategies. Traditional residual‑based adaptive sampling methods, while effective in enhancing PINN accuracy, often struggle with efficiency and high memory consumption, particularly in high‑dimensional problems. RSmote addresses these challenges by targeting regions with high residuals and employing oversampling techniques from imbalanced learning to refine the sampling process. Our approach is underpinned by a rigorous theoretical analysis that supports the effectiveness of RSmote in managing computational resources more efficiently. Through extensive evaluations, we benchmark RSmote against the state‑of‑the‑art Residual‑based Adaptive Distribution (RAD) method across a variety of dimensions and differential equations. The results demonstrate that RSmote not only achieves or exceeds the accuracy of RAD but also significantly reduces memory usage, making it particularly advantageous in high‑dimensional scenarios. These contributions position RSmote as a robust and resource‑efficient solution for solving complex partial differential equations, especially when computational constraints are a critical consideration.

Abstract:
We derive the quasiconvex relaxation of the Biot‑type energy density \lVert\sqrt\operatornameDφ^T \operatornameDφ‑I_2\rVert^2 for planar mappings φ\colon\mathbbR^2\to \mathbbR^2 in two different scenarios. First, we consider the case \operatornameDφ\in\textrmGL^+(2), in which the energy can be expressed as the squared Euclidean distance \operatornamedist^2(\operatornameDφ,\textrmSO(2)) to the special orthogonal group \textrmSO(2). We then allow for planar mappings with arbitrary \operatornameDφ\in\mathbbR^2× 2; in the context of solid mechanics, this lack of determinant constraints on the deformation gradient would allow for self‑interpenetration of matter. We demonstrate that the two resulting relaxations do not coincide and compare the analytical findings to numerical results for different relaxation approaches, including a rank‑one sequential lamination algorithm, trust‑region FEM calculations of representative microstructures and physics‑informed neural networks.

Abstract:
Estimating the material properties of thermal protection films is crucial for their effective design and application, particularly in high‑temperature environments. This work presents a novel approach to determine the properties using uncertainty quantification simulations. We quantify uncertainty in the material properties for effective insulation by proposing a Bayesian distribution for them. Sampling from this distribution is performed using Monte Carlo simulations, which require repeatedly solving the predictive thermal model. To address the computational inefficiency of conventional numerical simulations, we develop a parametric Physics‑Informed Neural Network (PINN) to solve the heat transfer problem. The proposed PINN significantly reduces computational time while maintaining accuracy, as verified against traditional numerical solutions. Additionally, we used the Sequential Monte Carlo (SMC) method to enable vectorized and parallel computations, further enhancing computational speedup. Our results demonstrate that integrating MCMC with PINN decreases computational time substantially compared to using standard numerical methods. Moreover, combining the SMC method with PINN yields multifold computational speedup, making this approach highly effective for the rapid and accurate estimation of material properties.

Abstract:
We present a novel framework combining Deep Operator Networks (DeepONets) with Physics‑Informed Neural Networks (PINNs) to solve partial differential equations (PDEs) and estimate their unknown parameters. By integrating data‑driven learning with physical constraints, our method achieves robust and accurate solutions across diverse scenarios. Bayesian training is implemented through variational inference, allowing for comprehensive uncertainty quantification for both aleatoric and epistemic uncertainties. This ensures reliable predictions and parameter estimates even in noisy conditions or when some of the physical equations governing the problem are missing. The framework demonstrates its efficacy in solving forward and inverse problems, including the 1D unsteady heat equation and 2D reaction‑diffusion equations, as well as regression tasks with sparse, noisy observations. This approach provides a computationally efficient and generalizable method for addressing uncertainty quantification in PDE surrogate modeling.

Abstract:
Optimal transportation of raw material from suppliers to customers is an issue arising in logistics that is addressed here with a continuous model relying on optimal transport theory. A physics informed neuralnetwork method is advocated here for the solution of the corresponding generalized Monge‑Amp`ere equation. Convex neural networks are advocated to enforce the convexity of the solution to the Monge‑Ampère equation and obtain a suitable approximation of the optimal transport map. A particular focus is set on the enforcement of transport boundary conditions in the loss function. Numerical experiments illustrate the solution to the optimal transport problem in several configurations, and sensitivity analyses are performed.

Abstract:
Diffusion model shows remarkable potential on sparse‑view computed tomography (SVCT) reconstruction. However, when a network is trained on a limited sample space, its generalization capability may be constrained, which degrades performance on unfamiliar data. For image generation tasks, this can lead to issues such as blurry details and inconsistencies between regions. To alleviate this problem, we propose a Sinogram‑based Wavelet random decomposition And Random mask diffusion Model (SWARM) for SVCT reconstruction. Specifically, introducing a random mask strategy in the sinogram effectively expands the limited training sample space. This enables the model to learn a broader range of data distributions, enhancing its understanding and generalization of data uncertainty. In addition, applying a random training strategy to the high‑frequency components of the sinogram wavelet enhances feature representation and improves the ability to capture details in different frequency bands, thereby improving performance and robustness. Two‑stage iterative reconstruction method is adopted to ensure the global consistency of the reconstructed image while refining its details. Experimental results demonstrate that SWARM outperforms competing approaches in both quantitative and qualitative performance across various datasets.

Abstract:
Researchers have long debated which spatial arrangements and swimming synchronizations are beneficial for the hydrodynamic performance of fish in schools. In our previous work (Seo and Mittal, Bioinsp. Biomim., Vol. 17, 066020, 2022), we demonstrated using direct numerical simulations that hydrodynamic interactions with the wake of a leading body‑caudal fin carangiform swimmer could significantly enhance the swimming performance of a trailing swimmer by augmenting the leading‑edge vortex (LEV) on its caudal fin. In this study, we develop a model based on the phenomenology of LEV enhancement, which utilizes wake velocity data from direct numerical simulations of a leading fish to predict the trailing swimmer's hydrodynamic performance without additional simulations. This approach enables a comprehensive analysis of the effects of relative positioning, phase difference, flapping amplitude, Reynolds number, and the number of swimmers in the school on thrust enhancement. The results offer several insights regarding the effect of these parameters that have implications for fish schools as well as for bio‑inspired underwater vehicle applications.

Abstract:
Accurate forecasting of contagious diseases is critical for public health policymaking and pandemic preparedness. We propose a new infectious disease forecasting model based on physics‑informed neural networks (PINNs), an emerging scientific machine learning approach. By embedding a compartmental model into the loss function, our method integrates epidemiological theory with data, helping to prevent model overfitting. We further enhance the model with a sub‑network that accounts for covariates such as mobility and cumulative vaccine doses, which influence the transmission rate. Using state‑level COVID‑19 data from California, we demonstrate that the PINN model accurately predicts cases, deaths, and hospitalizations, aligning well with existing benchmarks. Notably, the PINN model outperforms naive baseline forecasts and several sequence deep learning models, including Recurrent Neural Networks (RNNs), Long Short‑Term Memory (LSTM) networks, Gated Recurrent Units (GRUs), and Transformers. It also achieves performance comparable to a sophisticated Gaussian infection state forecasting model that combines compartmental dynamics, a data observation model, and parameter regression. However, the PINN model features a simpler structure and is easier to implement. In summary, we systematically evaluate the PINN model's ability to forecast infectious disease dynamics, demonstrating its potential as an efficient computational tool to strengthen forecasting capabilities.

Abstract:
This review examines the current state of research on microscale cellular changes during the drying of plant‑based food materials (PBFM), with particular emphasis on computational modelling approaches. The review addresses the critical need for advanced computational methods in microscale investigations. We systematically analyse experimental studies in PBFM drying, highlighting their contributions and limitations in capturing cellular‑level phenomena, including challenges in data acquisition and measurement accuracy under varying drying conditions. The evolution of computational models for microstructural investigations is thoroughly examined, from traditional numerical methods to contemporary state‑of‑the‑art approaches, with specific focus on their ability to handle the complex, nonlinear properties of plant cellular materials. Special attention is given to the emergence of data‑driven models and their limitations in predicting microscale cellular behaviour during PBFM drying, particularly addressing challenges in dataset acquisition and model generalization. The review provides an in‑depth analysis of Physics‑Informed Machine Learning (PIML) frameworks, examining their theoretical foundations, current applications in related fields, and unique advantages in combining physical principles with neural network architectures. Through this comprehensive assessment, we identify critical gaps in existing methodologies, evaluate the trade‑offs between different modelling approaches, and provide insights into future research directions for improving our understanding of cellular‑level transformations during PBFM drying processes. The review concludes with recommendations for integrating experimental and computational approaches to advance the field of food preservation technology.

Abstract:
Uncertainty quantification (UQ) plays a pivotal role in scientific machine learning, especially when surrogate models are used to approximate complex systems. Although multilayer perceptions (MLPs) are commonly employed as surrogates, they often suffer from overfitting due to their large number of parameters. Kolmogorov‑Arnold networks (KANs) offer an alternative solution with fewer parameters. However, gradient‑based inference methods, such as Hamiltonian Monte Carlo (HMC), may result in computational inefficiency when applied to KANs, especially for large‑scale datasets, due to the high cost of back‑propagation. To address these challenges, we propose a novel approach, combining the dropout Tikhonov ensemble Kalman inversion (DTEKI) with Chebyshev KANs. This gradient‑free method effectively mitigates overfitting and enhances numerical stability. Additionally, we incorporate the active subspace method to reduce the parameter‑space dimensionality, allowing us to improve the accuracy of predictions and obtain more reliable uncertainty estimates. Extensive experiments demonstrate the efficacy of our approach in various test cases, including scenarios with large datasets and high noise levels. Our results show that the new method achieves comparable or better accuracy, much higher efficiency as well as stability compared to HMC, in addition to scalability. Moreover, by leveraging the low‑dimensional parameter subspace, our method preserves prediction accuracy while substantially reducing further the computational cost.

Abstract:
The assimilation and prediction of phase‑resolved surface gravity waves are critical challenges in ocean science and engineering. Potential flow theory (PFT) has been widely employed to develop wave models and numerical techniques for wave prediction. However, traditional wave prediction methods are often limited. For example, most simplified wave models have a limited ability to capture strong wave nonlinearity, while fully nonlinear PFT solvers often fail to meet the speed requirements of engineering applications. This computational inefficiency also hinders the development of effective data assimilation techniques, which are required to reconstruct spatial wave information from sparse measurements to initialize the wave prediction. To address these challenges, we propose a novel solver method that leverages physics‑informed neural networks (PINNs) that parameterize PFT solutions as neural networks. This provides a computationally inexpensive way to assimilate and predict wave data. The proposed PINN framework is validated through comparisons with analytical linear PFT solutions and experimental data collected in a laboratory wave flume. The results demonstrate that our approach accurately captures and predicts irregular, nonlinear, and dispersive wave surface dynamics. Moreover, the PINN can infer the fully nonlinear velocity potential throughout the entire fluid volume solely from surface elevation measurements, enabling the calculation of fluid velocities that are difficult to measure experimentally.

Abstract:
Deep operator network (DeepONet) has shown significant promise as surrogate models for systems governed by partial differential equations (PDEs), enabling accurate mappings between infinite‑dimensional function spaces. However, when applied to systems with high‑dimensional input‑output mappings arising from large numbers of spatial and temporal collocation points, these models often require heavily overparameterized networks, leading to long training times. Latent DeepONet addresses some of these challenges by introducing a two‑step approach: first learning a reduced latent space using a separate model, followed by operator learning within this latent space. While efficient, this method is inherently data‑driven and lacks mechanisms for incorporating physical laws, limiting its robustness and generalizability in data‑scarce settings. In this work, we propose PI‑Latent‑NO, a physics‑informed latent neural operator framework that integrates governing physics directly into the learning process. Our architecture features two coupled DeepONets trained end‑to‑end: a Latent‑DeepONet that learns a low‑dimensional representation of the solution, and a Reconstruction‑DeepONet that maps this latent representation back to the physical space. By embedding PDE constraints into the training via automatic differentiation, our method eliminates the need for labeled training data and ensures physics‑consistent predictions. The proposed framework is both memory and compute‑efficient, exhibiting near‑constant scaling with problem size and demonstrating significant speedups over traditional physics‑informed operator models. We validate our approach on a range of parametric PDEs, showcasing its accuracy, scalability, and suitability for real‑time prediction in complex physical systems.

Abstract:
We address the neutral inclusion problem with imperfect boundary conditions, focusing on designing interface functions for inclusions of arbitrary shapes. Traditional Physics‑Informed Neural Networks (PINNs) struggle with this inverse problem, leading to the development of Conformal Mapping Coordinates Physics‑Informed Neural Networks (CoCo‑PINNs), which integrate geometric function theory with PINNs. CoCo‑PINNs effectively solve forward‑inverse problems by modeling the interface function through neural network training, which yields a neutral inclusion effect. This approach enhances the performance of PINNs in terms of credibility, consistency, and stability.

Abstract:
Physics‑Informed Neural Networks (PINNs) solve partial differential equations (PDEs) by embedding governing equations and boundary/initial conditions into the loss function. However, enforcing Dirichlet boundary conditions accurately remains challenging, often leading to soft enforcement that compromises convergence and reliability in complex domains. We propose a hybrid approach, PINN‑FEM, which combines PINNs with finite element methods (FEM) to impose strong Dirichlet boundary conditions via domain decomposition. This method incorporates FEM‑based representations near the boundary, ensuring exact enforcement without compromising convergence. Through six experiments of increasing complexity, PINN‑FEM outperforms standard PINN models, showcasing superior accuracy and robustness. While distance functions and similar techniques have been proposed for boundary condition enforcement, they lack generality for real‑world applications. PINN‑FEM bridges this gap by leveraging FEM near boundaries, making it well‑suited for industrial and scientific problems.

Abstract:
Physics‑informed neural networks (PINNs) have gained significant attention for solving forward and inverse problems related to partial differential equations (PDEs). While advancements in loss functions and network architectures have improved PINN accuracy, the impact of collocation point sampling on their performance remains underexplored. Fixed sampling methods, such as uniform random sampling and equispaced grids, can fail to capture critical regions with high solution gradients, limiting their effectiveness for complex PDEs. Adaptive methods, inspired by adaptive mesh refinement from traditional numerical methods, address this by dynamically updating collocation points during training but may overlook residual dynamics between updates, potentially losing valuable information. To overcome this limitation, we propose two adaptive collocation point selection strategies utilizing the QR Discrete Empirical Interpolation Method (QR‑DEIM), a reduced‑order modeling technique for efficiently approximating nonlinear functions. Our results on benchmark PDEs demonstrate that our QR‑DEIM‑based approaches improve PINN accuracy compared to existing methods, offering a promising direction for adaptive collocation point strategies.

Abstract:
Accurate, interpretable, and real‑time modeling of multi‑body dynamical systems is essential for predicting behaviors and inferring physical properties in natural and engineered environments. Traditional physics‑based models face scalability challenges and are computationally demanding, while data‑driven approaches like Graph Neural Networks (GNNs) often lack physical consistency, interpretability, and generalization. In this paper, we propose Dynami‑CAL GraphNet, a Physics‑Informed Graph Neural Network that integrates the learning capabilities of GNNs with physics‑based inductive biases to address these limitations. Dynami‑CAL GraphNet enforces pairwise conservation of linear and angular momentum for interacting nodes using edge‑local reference frames that are equivariant to rotational symmetries, invariant to translations, and equivariant to node permutations. This design ensures physically consistent predictions of node dynamics while offering interpretable, edge‑wise linear and angular impulses resulting from pairwise interactions. Evaluated on a 3D granular system with inelastic collisions, Dynami‑CAL GraphNet demonstrates stable error accumulation over extended rollouts, effective extrapolations to unseen configurations, and robust handling of heterogeneous interactions and external forces. Dynami‑CAL GraphNet offers significant advantages in fields requiring accurate, interpretable, and real‑time modeling of complex multi‑body dynamical systems, such as robotics, aerospace engineering, and materials science. By providing physically consistent and scalable predictions that adhere to fundamental conservation laws, it enables the inference of forces and moments while efficiently handling heterogeneous interactions and external forces.

Abstract:
Deep learning models trained on finite data lack a complete understanding of the physical world. On the other hand, physics‑informed neural networks (PINNs) are infused with such knowledge through the incorporation of mathematically expressible laws of nature into their training loss function. By complying with physical laws, PINNs provide advantages over purely data‑driven models in limited‑data regimes and present as a promising route towards Physical AI. This feature has propelled them to the forefront of scientific machine learning, a domain characterized by scarce and costly data. However, the vision of accurate physics‑informed learning comes with significant challenges. This work examines PINNs in terms of model optimization and generalization, shedding light on the need for new algorithmic advances to overcome issues pertaining to the training speed, precision, and generalizability of today's PINN models. Of particular interest are gradient‑free evolutionary algorithms (EAs) for optimizing the uniquely complex loss landscapes arising in PINN training. Methods synergizing gradient descent and EAs for discovering bespoke neural architectures and balancing multiple terms in physics‑informed learning objectives are positioned as important avenues for future research. Another exciting track is to cast EAs as a meta‑learner of generalizable PINN models. To substantiate these proposed avenues, we further highlight results from recent literature to showcase the early success of such approaches in addressing the aforementioned challenges in PINN optimization and generalization.

Abstract:
This study aims to benchmark candidate strategies for embedding neural network (NN) surrogates in nonlinear model predictive control (NMPC) formulations that are subject to systems described with partial differential equations and that are solved via direct transcription (i.e., simultaneous methods). This study focuses on the use of physics‑informed NNs and physics‑informed convolutional NNs as the internal (surrogate) models within the NMPC formulation. One strategy embeds NN models as explicit algebraic constraints, leveraging the automatic differentiation (AD) of an algebraic modelling language (AML) to evaluate the derivatives. Alternatively, the solver can be provided with derivatives computed external to the AML via the AD routines of the machine learning environment the NN is trained in. The three numerical experiments considered in this work reveal that replacing mechanistic models with NN surrogates may not always offer computational advantages when smooth activation functions are used in conjunction with a local nonlinear solver (e.g., Ipopt), even with highly nonlinear systems. Moreover, in this context, the external function evaluation of the NN surrogates often outperforms the embedding strategies that rely on explicit algebraic constraints, likely due to the difficulty in initializing the auxiliary variables and constraints introduced by explicit algebraic reformulations.

Abstract:
Normal mode analysis (NMA) provides a mathematical framework for exploring the intrinsic global dynamics of molecules through the definition of an energy function, where normal modes correspond to the eigenvectors of the Hessian matrix derived from the second derivatives of this function. The energy required to 'trigger' each normal mode is proportional to the square of its eigenvalue, with six zero‑eigenvalue modes representing universal translation and rotation, common to all molecular systems. In contrast, modes associated with small non‑zero eigenvalues are more easily excited by external forces and are thus closely related to molecular functions. Inspired by the anisotropic network model (ANM), this work establishes a novel connection between normal mode analysis and sheaf theory by introducing a cellular sheaf structure, termed the anisotropic sheaf, defined on undirected, simple graphs, and identifying the conventional Hessian matrix as the sheaf Laplacian. By interpreting the global section space of the anisotropic sheaf as the kernel of the Laplacian matrix, we demonstrate a one‑to‑one correspondence between the zero‑eigenvalue‑related normal modes and a basis for the global section space. We further analyze the dimension of this global section space, representing the space of harmonic signals, under conditions typically considered in normal mode analysis. Additionally, we propose a systematic method to streamline the Delaunay triangulation‑based construction for more efficient graph generation while preserving the ideal number of normal modes with zero eigenvalues in ANM analysis.

Abstract:
This paper investigates data‑driven solutions and parameter discovery to (2+1)‑dimensional coupled nonlinear Schrödinger equations with variable coefficients (VC‑CNLSEs), which describe transverse effects in optical fiber systems under perturbed dispersion and nonlinearity. By setting different forms of perturbation coefficients, we aim to recover the dark and anti‑dark one‑ and two‑soliton structures by employing an enhanced physics‑based deep neural network algorithm, namely a physics‑informed neural network (PINN). The enhanced PINN algorithm leverages the locally adaptive activation function mechanism to improve convergence speed and accuracy. In the lack of data acquisition, the PINN algorithms will enhance the capability of the neural networks by incorporating physical information into the training phase. We demonstrate that applying PINN algorithms to (2+1)‑dimensional VC‑CNLSEs requires distinct distributions of physical information. To address this, we propose a region‑specific weighted loss function with the help of residual‑based adaptive refinement strategy. In the meantime, we perform data‑driven parameter discovery for the model equation, classified into two categories: constant coefficient discovery and variable coefficient discovery. For the former, we aim to predict the cross‑phase modulation constant coefficient under varying noise intensities using enhanced PINN with a single neural network. For the latter, we employ a dual‑network strategy to predict the dynamic behavior of the dispersion and nonlinearity perturbation functions. Our study demonstrates that the proposed framework holds significant potential for studying high‑dimensional and complex solitonic dynamics in optical fiber systems.

Abstract:
Adaptive physics‑informed super‑resolution diffusion is developed for non‑invasive virtual diagnostics of the 6D phase space density of charged particle beams. An adaptive variational autoencoder (VAE) embeds initial beam condition images and scalar measurements to a low‑dimensional latent space from which a 326 pixel 6D tensor representation of the beam's 6D phase space density is generated. Projecting from a 6D tensor generates physically consistent 2D projections. Physics‑guided super‑resolution diffusion transforms low‑resolution images of the 6D density to high resolution 256x256 pixel images. Un‑supervised adaptive latent space tuning enables tracking of time‑varying beams without knowledge of time‑varying initial conditions. The method is demonstrated with experimental data and multi‑particle simulations at the HiRES UED. The general approach is applicable to a wide range of complex dynamic systems evolving in high‑dimensional phase space. The method is shown to be robust to distribution shift without re‑training.

Abstract:
The present work is focused on exploring convergence of Physics‑informed Neural Networks (PINNs) when applied to a specific class of second‑order fully nonlinear Partial Differential Equations (PDEs). It is well‑known that as the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We show that such sequence converges to a unique viscosity solution of a certain class of second‑order fully nonlinear PDE's, provided the latter satisfies the comparison principle in the viscosity sense.

Abstract:
This paper broaches the peridynamic inverse problem of determining the horizon size of the kernel function in a one‑dimensional model of a linear microelastic material. We explore different kernel functions, including V‑shaped, distributed, and tent kernels. The paper presents numerical experiments using PINNs to learn the horizon parameter for problems in one and two spatial dimensions. The results demonstrate the effectiveness of PINNs in solving the peridynamic inverse problem, even in the presence of challenging kernel functions. We observe and prove a one‑sided convergence behavior of the Stochastic Gradient Descent method towards a global minimum of the loss function, suggesting that the true value of the horizon parameter is an unstable equilibrium point for the PINN's gradient flow dynamics.

Abstract:
The Large Hadron Collider at CERN produces immense volumes of complex data from high‑energy particle collisions, demanding sophisticated analytical techniques for effective interpretation. Neural Networks, including Graph Neural Networks, have shown promise in tasks such as event classification and object identification by representing collisions as graphs. However, while Graph Neural Networks excel in predictive accuracy, their "black box" nature often limits their interpretability, making it difficult to trust their decision‑making processes. In this paper, we propose a novel approach that combines a Graph Transformer model with Mixture‑of‑Expert layers to achieve high predictive performance while embedding interpretability into the architecture. By leveraging attention maps and expert specialization, the model offers insights into its internal decision‑making, linking predictions to physics‑informed features. We evaluate the model on simulated events from the ATLAS experiment, focusing on distinguishing rare Supersymmetric signal events from Standard Model background. Our results highlight that the model achieves competitive classification accuracy while providing interpretable outputs that align with known physics, demonstrating its potential as a robust and transparent tool for high‑energy physics data analysis. This approach underscores the importance of explainability in machine learning methods applied to high energy physics, offering a path toward greater trust in AI‑driven discoveries.

Abstract:
This paper proposes an innovative method for predicting deformation in architected lattice structures that combines Physics‑Informed Neural Networks (PINNs) with finite element analysis. A thorough study was carried out on FCC‑based lattice beams utilizing five different materials (Structural Steel, AA6061, AA7075, Ti6Al4V, and Inconel 718) under varied edge loads (1000‑10000 N). The PINN model blends data‑driven learning with physics‑based limitations via a proprietary loss function, resulting in much higher prediction accuracy than linear regression. PINN outperforms linear regression, achieving greater R‑square (0.7923 vs 0.5686) and lower error metrics (MSE: 0.00017417 vs 0.00036187). Among the materials examined, AA6061 had the highest displacement sensitivity (0.1014 mm at maximum load), while Inconel718 had better structural stability.

Abstract:
Partial Differential Equations (PDEs) are integral to modeling many scientific and engineering problems. Physics‑informed Neural Networks (PINNs) have emerged as promising tools for solving PDEs by embedding governing equations into the neural network loss function. However, when dealing with PDEs characterized by strong oscillatory dynamics over large computational domains, PINNs based on Multilayer Perceptrons (MLPs) often exhibit poor convergence and reduced accuracy. To address these challenges, this paper introduces Scaled‑cPIKAN, a physics‑informed architecture rooted in Kolmogorov‑Arnold Networks (KANs). Scaled‑cPIKAN integrates Chebyshev polynomial representations with a domain scaling approach that transforms spatial variables in PDEs into the standardized domain \([‑1,1]^d\), as intrinsically required by Chebyshev polynomials. By combining the flexibility of Chebyshev‑based KANs (cKANs) with the physics‑driven principles of PINNs, and the spatial domain transformation, Scaled‑cPIKAN enables efficient representation of oscillatory dynamics across extended spatial domains while improving computational performance. We demonstrate Scaled‑cPIKAN efficacy using four benchmark problems: the diffusion equation, the Helmholtz equation, the Allen‑Cahn equation, as well as both forward and inverse formulations of the reaction‑diffusion equation (with and without noisy data). Our results show that Scaled‑cPIKAN significantly outperforms existing methods in all test cases. In particular, it achieves several orders of magnitude higher accuracy and faster convergence rate, making it a highly efficient tool for approximating PDE solutions that feature oscillatory behavior over large spatial domains.

Abstract:
We propose an entropy‑enhanced Generative Pre‑Trained Physics‑Informed Neural Network with a transform layer (EGPT‑PINN) for solving parameterized nonlinear conservation laws. The EGPT‑PINN extends the traditional physics‑informed neural networks and its recently proposed generative pre‑trained strategy for linear model reduction to nonlinear model reduction and shock‑capturing domains. By utilizing an adaptive meta‑network, a simultaneously trained transform layer, entropy enhancement strategies, implementable shock interaction analysis, and a separable training process, the EGPT‑PINN efficiently captures complex parameter‑dependent shock formations and interactions. Numerical results of EGPT‑PINN applied to the families of inviscid Burgers' equation and the Euler equations, parameterized by their initial conditions, demonstrate the robustness and accuracy of the proposed technique. It accurately solves the viscosity solution via very few neurons without leveraging any \it a priori knowledge of the equations or its initial condition. Moreover, via a simple augmentation of the loss function by model‑data mismatch, we demonstrate the robustness of EGPT‑PINN in solving inverse problems more accurately than the vanilla and entropy‑enhanced versions of PINN.

Abstract:
Machine learning offers a powerful framework for validating and predicting atomic mass. We compare three improved neural network methods for representation and extrapolation for atomic mass prediction. The powerful method, adopting a macroscopic‑microscopic approach and treating complex nuclear effects as output labels, achieves superior accuracy in AME2020, yielding a much lower root‑mean‑square deviation of 0.122 MeV in the test set, significantly lower than alternative methods. It also exhibits a better extrapolation performance when predicting AME2020 from AME2016, with a root‑mean‑square deviation of 0.191 MeV. We further conduct sensitivity analyses against the model inputs to verify interpretable alignment beyond statistical metrics. Incorporating theoretical predictions of magic numbers and masses, our fully connected neural networks reproduce key nuclear phenomena including nucleon pairing correlation and magic number effects. The extrapolation capability of the framework is discussed and the accuracy of predicting new mass measurements for isotope chains has also been tested.

Abstract:
Recent years have witnessed increasing research interests of physics‑informed neural networks (PINNs) in solving forward, inverse, and parametric problems governed by partial differential equations (PDEs). Despite their promise, PINNs still face significant challenges in many scenarios due to ill‑conditioning. Time‑stepping‑oriented neural network (TSONN) addresses this by reformulating the ill‑conditioned optimization problem into a series of well‑conditioned sub‑problems, greatly improving its ability to handle complex scenarios. This paper presents a new solver for laminar flow around airfoils based on TSONN and mesh transformation, validated across various test cases. Specifically, the solver achieves mean relative errors of approximately 3.6% for lift coefficients and 1.4% for drag coefficients. Furthermore, this paper extends the solver to parametric problems involving flow conditions and airfoil shapes, covering nearly all laminar flow scenarios in engineering. The shape parameter space is defined as the union of 30% perturbations applied to each airfoil in the UIUC airfoil database, with Reynolds numbers ranging from 100 to 5000 and angles of attack spanning from ‑5° to 15°. The parametric solver solves all laminar flows within the parameter space in just 4.6 day, at approximately 40 times the computational cost of solving a single flow. The model training involves hundreds of millions of flow conditions and airfoil shapes, ultimately yielding a surrogate model with strong generalization capability that does not require labeled data. Specifically, the surrogate model achieves average errors of 4.6% for lift coefficients and 1.1% for drag coefficients, demonstrating its potential for high generalizability, cost‑effectiveness, and efficiency in addressing high‑dimensional parametric problems and surrogate modeling.

Abstract:
Flight test analysis often requires predefined test points with arbitrarily tight tolerances, leading to extensive and resource‑intensive experimental campaigns. To address this challenge, we propose a novel approach to flight test analysis using Gaussian processes (GPs) with physics‑informed mean functions to estimate aerodynamic quantities from arbitrary flight test data, validated using real T‑38 aircraft data collected in collaboration with the United States Air Force Test Pilot School. We demonstrate our method by estimating the pitching moment coefficient without requiring predefined or repeated flight test points, significantly reducing the need for extensive experimental campaigns. Our approach incorporates aerodynamic models as priors within the GP framework, enhancing predictive accuracy across diverse flight conditions and providing robust uncertainty quantification. Key contributions include the integration of physics‑based priors in a probabilistic model, which allows for precise computation from arbitrary flight test maneuvers, and the demonstration of our method capturing relevant dynamic characteristics such as short‑period mode behavior. The proposed framework offers a scalable and generalizable solution for efficient data‑driven flight test analysis and is able to accurately predict the short period frequency and damping for the T‑38 across several Mach and dynamic pressure profiles.

Abstract:
We introduce a novel meshless simulation method for the McKean‑Vlasov Stochastic Differential Equation (MV‑SDE) utilizing deep learning, applicable to both self‑interaction and interaction scenarios. Traditionally, numerical methods for this equation rely on the interacting particle method combined with techniques based on the Itô‑Taylor expansion. The convergence rate of this approach is determined by two parameters: the number of particles N and the time step size h for each Euler iteration. However, for extended time horizons or equations with larger Lipschitz coefficients, this method is often limited, as it requires a significant increase in Euler iterations to achieve the desired precision ε. To overcome the challenges posed by the difficulty of parallelizing the simulation of continuous interacting particle systems, which involve solving high‑dimensional coupled SDEs, we propose a meshless MV‑SDE solver grounded in Physics‑Informed Neural Networks (PINNs) that does not rely on the propagation of chaos result. Our method constructs a pseudo MV‑SDE using Itô calculus, then quantifies the discrepancy between this equation and the original MV‑SDE, with the error minimized through a loss function. This loss is controlled via an optimization algorithm, independent of the time step size, and we provide an error estimate for the loss function. The advantages of our approach are demonstrated through corresponding simulations.

Abstract:
In response to climate change, assessing crop productivity under extreme weather conditions is essential to enhance food security. Crop simulation models, which align with physical processes, offer explainability but often perform poorly. Conversely, machine learning (ML) models for crop modeling are powerful and scalable yet operate as black boxes and lack adherence to crop growths physical principles. To bridge this gap, we propose a novel method that combines the strengths of both approaches by estimating the water use and the crop sensitivity to water scarcity at the pixel level. This approach enables yield loss estimation grounded in physical principles by sequentially solving the equation for crop yield response to water scarcity, using an enhanced loss function. Leveraging Sentinel‑2 satellite imagery, climate data, simulated water use data, and pixel‑level yield data, our model demonstrates high accuracy, achieving an R2 of up to 0.77, matching or surpassing state‑of‑the‑art models like RNNs and Transformers. Additionally, it provides interpretable and physical consistent outputs, supporting industry, policymakers, and farmers in adapting to extreme weather conditions.

Abstract:
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics‑informed neural networks (PINNs) and kernel method. In this paper, we introduce a random feature based framework toward efficiently solving PDEs. Random feature method was originally proposed to approximate large‑scale kernel machines and can be viewed as a shallow neural network as well. We provide an error analysis for our proposed method along with comprehensive numerical results on several PDE benchmarks. In contrast to the state‑of‑the‑art solvers that face challenges with a large number of collocation points, our proposed method reduces the computational complexity. Moreover, the implementation of our method is simple and does not require additional computational resources. Due to the theoretical guarantee and advantages in computation, our approach is proven to be efficient for solving PDEs.

Abstract:
We present a physics‑informed Bayesian analysis of equation of state constraints using observational data for masses, radii and tidal deformability of pulsars and a generic class of hybrid neutron star equation of state with color superconducting quark matter on the basis of a recently developed nonlocal chiral quark model. The nuclear matter phase is described within a relativistic density functional model of the DD2 class and the phase transition is obtained by a Maxwell construction. We find the region in the two‑dimensional parameter space spanned by the vector meson coupling and the scalar diquark coupling, where three conditions are fulfilled: (1) the Maxwell construction can be performed, \mbox(2) the maximum mass of the hybrid neutron star is not smaller than \mbox2.0 M_\odot and (3) the onset density of the phase transition is not below the nuclear saturation density n_0=0.15 fm^‑3. The result of this study shows that the favorable neutron star equation of state has low onset masses for the occurrence of a color superconducting quark matter core between 0.5‑0.7 M_\odot and maximum masses in the range 2.15‑2.22 M_\odot. In the typical mass range of 1.2‑2.0 M_\odot, the radii of these stars are between 11.9 and 12.4 km, almost independent of the mass. In principle, hybrid stars would allow for larger maximum masses than provided by the hadronic reference equation of state.

Abstract:
Phase‑field modeling reformulates fracture problems as energy minimization problems and enables a comprehensive characterization of the fracture process, including crack nucleation, propagation, merging, and branching, without relying on ad‑hoc assumptions. However, the numerical solution of phase‑field fracture problems is characterized by a high computational cost. To address this challenge, in this paper, we employ a deep neural operator (DeepONet) consisting of a branch network and a trunk network to solve brittle fracture problems. We explore three distinct approaches that vary in their trunk network configurations. In the first approach, we demonstrate the effectiveness of a two‑step DeepONet, which results in a simplification of the learning task. In the second approach, we employ a physics‑informed DeepONet, whereby the mathematical expression of the energy is integrated into the trunk network's loss to enforce physical consistency. The integration of physics also results in a substantially smaller data size needed for training. In the third approach, we replace the neural network in the trunk with a Kolmogorov‑Arnold Network and train it without the physics loss. Using these methods, we model crack nucleation in a one‑dimensional homogeneous bar under prescribed end displacements, as well as crack propagation and branching in single edge‑notched specimens with varying notch lengths subjected to tensile and shear loading. We show that the networks predict the solution fields accurately, and the error in the predicted fields is localized near the crack.

Abstract:
Generating an accurate solution of the Navier‑‑Stokes equations using physics‑‑informed neural networks (PINNs) for higher Reynolds numbers in the corners of a lid‑‑driven cavity problem is challenging. In this paper, we improve the solution accuracy of the incompressible Navier‑‑Stokes equations in the region near the walls significantly and generate accurate secondary vortices in the corners of the lid‑‑driven cavity by solving the governing equations using finite difference‑‑based PINNs (FD‑‑PINNs) without employing the known solution. We adopt the domain decomposition method (DDM) and combine it with the FD‑‑PINNs to solve the lid‑‑driven cavity problem for the Reynolds numbers Re = 400 and Re=1000. A comparison of the mean square error (MSE) between the presented and standard FD‑‑PINNs using the reference solution is exhibited, showing the accuracy and effectiveness of the new approach.

Abstract:
Structural optimization is essential for designing safe, efficient, and durable components with minimal material usage. Traditional methods for vibration control often rely on active systems to mitigate unpredictable vibrations, which may lead to resonance and potential structural failure. However, these methods face significant challenges when addressing the nonlinear inverse eigenvalue problems required for optimizing structures subjected to a wide range of frequencies. As a result, no existing approach has effectively addressed the need for real‑time vibration suppression within this context, particularly in high‑performance environments such as automotive noise, vibration and harshness, where computational efficiency is crucial. This study introduces DeepF‑fNet, a novel neural network framework designed to replace traditional active systems in vibration‑based structural optimization. Leveraging DeepONets within the context of physics‑informed neural networks, DeepF‑fNet integrates both data and the governing physical laws. This enables rapid identification of optimal parameters to suppress critical vibrations at specific frequencies, offering a more efficient and real‑time alternative to conventional methods. The proposed framework is validated through a case study involving a locally resonant metamaterial used to isolate structures from user‑defined frequency ranges. The results demonstrate that DeepF‑fNet outperforms traditional genetic algorithms in terms of computational speed while achieving comparable results, making it a promising tool for vibration‑sensitive applications. By replacing active systems with machine learning techniques, DeepF‑fNet paves the way for more efficient and cost‑effective structural optimization in real‑world scenarios.

Abstract:
The article is devoted to the study of neural networks with one hidden layer and a modified activation function for solving physical problems. A rectified sigmoid activation function has been proposed to solve physical problems described by the ODE with neural networks. Algorithms for physics‑informed data‑driven initialization of a neural network and a neuron‑by‑neuron gradient‑free fitting method have been presented for the neural network with this activation function. Numerical experiments demonstrate the superiority of neural networks with a rectified sigmoid function over neural networks with a sigmoid function in the accuracy of solving physical problems (harmonic oscillator, relativistic slingshot, and Lorentz system).

Abstract:
In this paper we consider time‑dependent PDEs discretized by a special class of Physics Informed Neural Networks whose design is based on the framework of Runge‑‑Kutta and related time‑Galerkin discretizations. The primary motivation for using such methods is that alternative time‑discrete schemes not only enable higher‑order approximations but also have a crucial impact on the qualitative behavior of the discrete solutions. The design of the methods follows a novel training approach based on two key principles: (a) the discrete loss is designed using a time‑discrete framework, and (b) the final loss formulation incorporates Runge‑‑Kutta or time‑Galerkin discretization in a carefully structured manner. We then demonstrate that the resulting methods inherit the stability properties of the Runge‑‑Kutta or time‑Galerkin schemes, and furthermore, their computational behavior aligns with that of the original time discrete method used in their formulation. In our analysis, we focus on linear parabolic equations, demonstrating both the stability of the methods and the convergence of the discrete minimizers to solutions of the underlying evolution PDE. An important novel aspect of our work is the derivation of maximal regularity (MR) estimates for B‑stable Runge‑‑Kutta schemes and both continuous and discontinuous Galerkin time discretizations. This allows us to provide new energy‑based proofs for maximal regularity estimates previously established by Kovács, Li, and Lubich, now in the Hilbert space setting and with the flexibility of variable time steps.

Abstract:
Aberration‑corrected Scanning Transmission Electron Microscopy (STEM) has become an essential tool in understanding materials at the atomic scale. However, tuning the aberration corrector to produce a sub‑Ångström probe is a complex and time‑costly procedure, largely due to the difficulty of precisely measuring the optical state of the system. When measurements are both costly and noisy, Bayesian methods provide rapid and efficient optimization. To this end, we develop a Bayesian approach to fully automate the process by minimizing a new quality metric, beam emittance, which is shown to be equivalent to performing aberration correction. In part I, we derived several important properties of the beam emittance metric and trained a deep neural network to predict beam emittance growth from a single Ronchigram. Here we use this as the black box function for Bayesian Optimization and demonstrate automated tuning of simulated and real electron microscopes. We explore different surrogate functions for the Bayesian optimizer and implement a deep neural network kernel to effectively learn the interactions between different control channels without the need to explicitly measure a full set of aberration coefficients. Both simulation and experimental results show the proposed method outperforms conventional approaches by achieving a better optical state with a higher convergence rate.

Abstract:
Various methods for leveraging turbulent fluctuation measurements from fusion plasma experiments are introduced, along with selected application examples. These can be categorized into spectral methods, statistical methods, and physics informed neural network based methods, and they are most effective for two‑dimensional turbulence measurements, which are now widely accessible. Extracting more information from turbulence data would pave the way for a better understanding of plasma turbulence transport in fusion experiments.

Abstract:
The Zakharov‑Kuznetsov equation, originally a three dimensional mathematical model of plasma with a uniform magnetic field, is a direct extension of the KdV equation into higher dimensions and is a typical quasi‑integrable system. Physics‑Informed Neural Networks (PINNs) are used to study the collision of soliton solutions in the 2+1 dimensional Zakharov‑Kuznetsov equation. PINNs are able to successfully solve the equations in the forward process, and the solutions are obtained using a mesh‑free approach and automatic differentiation, taking into account conservation laws. In the inverse process, the proper form of the equation can be successfully derived from a given training data. However, the situation becomes intractable in the collision process. The forward analysis result no longer adheres to the laws of conservation, and is better described as a dynamically incompatible field configuration (DIFC) than a solution to the system. Conservative PINNs have thus been introduced for this purpose, and in this paper we succeed in obtaining solutions that satisfy conservation laws. The inverse analysis suggests a different equation in which the coefficients exhibit significant changes, implying an emergence of temporary interactions. With these modulated coefficients, we recalculate the equation and confirm that the adherence to the laws of conservation has unquestionably improved.

Abstract:
Differential equations (DEs) are crucial for modeling the evolution of natural or engineered systems. Traditionally, the parameters in DEs are adjusted to fit data from system observations. However, in fields such as politics, economics, and biology, available data are often independently collected at distinct time points from different subjects (i.e., repeated cross‑sectional (RCS) data). Conventional optimization techniques struggle to accurately estimate DE parameters when RCS data exhibit various heterogeneities, leading to a significant loss of information. To address this issue, we propose a new estimation method called the emulator‑informed deep‑generative model (EIDGM), designed to handle RCS data. Specifically, EIDGM integrates a physics‑informed neural network‑based emulator that immediately generates DE solutions and a Wasserstein generative adversarial network‑based parameter generator that can effectively mimic the RCS data. We evaluated EIDGM on exponential growth, logistic population models, and the Lorenz system, demonstrating its superior ability to accurately capture parameter distributions. Additionally, we applied EIDGM to an experimental dataset of Amyloid beta 40 and beta 42, successfully capturing diverse parameter distribution shapes. This shows that EIDGM can be applied to model a wide range of systems and extended to uncover the operating principles of systems based on limited data.

Abstract:
The article discusses the development of various methods and techniques for initializing and training neural networks with a single hidden layer, as well as training a separable physics‑informed neural network consisting of neural networks with a single hidden layer to solve physical problems described by ordinary differential equations (ODEs) and partial differential equations (PDEs). A method for strictly deterministic initialization of a neural network with one hidden layer for solving physical problems described by an ODE is proposed. Modifications to existing methods for weighting the loss function are given, as well as new methods developed for training strictly deterministic‑initialized neural networks to solve ODEs (detaching, additional weighting based on the second derivative, predicted solution‑based weighting, relative residuals). An algorithm for physics‑informed data‑driven initialization of a neural network with one hidden layer is proposed. A neural network with pronounced generalizing properties is presented, whose generalizing abilities of which can be precisely controlled by adjusting network parameters. A metric for measuring the generalization of such neural network has been introduced. A gradient‑free neuron‑by‑neuron fitting method has been developed for adjusting the parameters of a single‑hidden‑layer neural network, which does not require the use of an optimizer or solver for its implementation. The proposed methods have been extended to 2D problems using the separable physics‑informed neural networks approach. Numerous experiments have been carried out to develop the above methods and approaches. Experiments on physical problems, such as solving various ODEs and PDEs, have demonstrated that these methods for initializing and training neural networks with one or two hidden layers (SPINN) achieve competitive accuracy and, in some cases, state‑of‑the‑art results.

Abstract:
Understanding thermal stress evolution in metal additive manufacturing (AM) is crucial for producing high‑quality components. Recent advancements in machine learning (ML) have shown great potential for modeling complex multiphysics problems in metal AM. While physics‑based simulations face the challenge of high computational costs, conventional data‑driven ML models require large, labeled training datasets to achieve accurate predictions. Unfortunately, generating large datasets for ML model training through time‑consuming experiments or high‑fidelity simulations is highly expensive in metal AM. To address these challenges, this study introduces a physics‑informed neural network (PINN) framework that incorporates governing physical laws into deep neural networks (NNs) to predict temperature and thermal stress evolution during the laser metal deposition (LMD) process. The study also discusses the enhanced accuracy and efficiency of the PINN model when supplemented with small simulation data. Furthermore, it highlights the PINN transferability, enabling fast predictions with a set of new process parameters using a pre‑trained PINN model as an online soft sensor, significantly reducing computation time compared to physics‑based numerical models while maintaining accuracy.

Abstract:
Aircraft design optimization traditionally relies on computationally expensive simulation techniques such as Finite Element Method (FEM) and Finite Volume Method (FVM), which, while accurate, can significantly slow down the design iteration process. The challenge lies in reducing the computational complexity while maintaining high accuracy for quick evaluations of multiple design alternatives. This research explores advanced methods, including surrogate models, reduced‑order models (ROM), and multi‑fidelity machine learning techniques, to achieve more efficient aircraft design evaluations. Specifically, the study investigates the application of Multi‑fidelity Physics‑Informed Neural Networks (MPINN) and autoencoders for manifold alignment, alongside the potential of Generative Adversarial Networks (GANs) for refining design geometries. Through a proof‑of‑concept task, the research demonstrates the ability to predict high‑fidelity results from low‑fidelity simulations, offering a path toward faster and more cost effective aircraft design iterations.

Abstract:
Nonlinear mathematical models introduce the relation between various physical and biological interactions present in nature. One of the most famous models is the Lotka‑Volterra model which defined the interaction between predator and prey species present in nature. However, predators, scavengers, and prey populations coexist in a natural system where scavengers can additionally rely on the dead bodies of predators present in the system. Keeping this in mind, the formulation and simulation of the predator prey scavenger model is introduced in this paper. For the predation response, respective prey species are assumed to have Holling's functional response of type III. The proposed model is tested for various simulations and is found to be showing satisfactory results in different scenarios. After simulations, the American forest dataset is taken for parameter estimation which imitates the real‑world case. For parameter estimation, a physics‑informed deep neural network is used with the Adam backpropagation method which prevents the avalanche effect in trainable parameters updation. For neural networks, mean square error and physics‑informed informed error are considered. After the neural network, the hence‑found parameters are fine‑tuned using the Broyden‑Fletcher‑Goldfarb‑Shanno algorithm. Finally, the hence‑found parameters using a natural dataset are tested for stability using Jacobian stability analysis. Future research work includes minimization of error induced by parameters, bifurcation analysis, and sensitivity analysis of the parameters.

Abstract:
The Wind Storage Integrated System with Power Smoothing Control (PSC) has emerged as a promising solution to ensure both efficient and reliable wind energy generation. However, existing PSC strategies overlook the intricate interplay and distinct control frequencies between batteries and wind turbines, and lack consideration of wake effect and battery degradation cost. In this paper, a novel coordinated control framework with hierarchical levels is devised to address these challenges effectively, which integrates the wake model and battery degradation model. In addition, after reformulating the problem as a Markov decision process, the multi‑agent reinforcement learning method is introduced to overcome the bi‑level characteristic of the problem. Moreover, a Physics‑informed Neural Network‑assisted Multi‑agent Deep Deterministic Policy Gradient (PAMA‑DDPG) algorithm is proposed to incorporate the power fluctuation differential equation and expedite the learning process. The effectiveness of the proposed methodology is evaluated through simulations conducted in four distinct scenarios using WindFarmSimulator (WFSim). The results demonstrate that the proposed algorithm facilitates approximately an 11% increase in total profit and a 19% decrease in power fluctuation compared to the traditional methods, thereby addressing the dual objectives of economic efficiency and grid‑connected energy reliability.

Abstract:
Real‑time and accurate prediction of aerodynamic flow fields around airfoils is crucial for flow control and aerodynamic optimization. However, achieving this remains challenging due to the high computational costs and the non‑linear nature of flow physics. Traditional Computational Fluid Dynamics (CFD) methods face limitations in balancing computational efficiency and accuracy, hindering their application in real‑time scenarios. To address these challenges, this study presents AeroDiT, a novel surrogate model that integrates scalable diffusion models with transformer architectures to address these challenges. Trained on Reynolds‑Averaged Navier‑Stokes (RANS) simulation data for high Reynolds‑number airfoil flows, AeroDiT accurately captures complex flow patterns while enabling real‑time predictions. The model demonstrates impressive performance, with mean relative L_2 errors of 0.1, 0.025, and 0.050 for pressure p and velocity components u_x, u_y, confirming its reliability. To further enhance physical consistency, we incorporate explicit physics‑informed losses based on RANS residuals, including mass and momentum conservation constraints. The transformer‑based structure allows for real‑time predictions within seconds, enabling efficient aerodynamic simulations. This work underscores the potential of generative machine learning techniques to advance computational fluid dynamics, offering potential solutions to challenges in simulating high‑fidelity aerodynamic flows.

Abstract:
Nonlinear differential equations and systems play a crucial role in modeling systems where time‑dependent factors exhibit nonlinear characteristics. Due to their nonlinear nature, solving such systems often presents significant difficulties and challenges. In this study, we propose a method utilizing Physics‑Informed Neural Networks (PINNs) to solve the nonlinear energy supply‑demand (ESD) system. We design a neural network with four outputs, where each output approximates a function that corresponds to one of the unknown functions in the nonlinear system of differential equations describing the four‑dimensional ESD problem. The neural network model is then trained and the parameters are identified, optimized to achieve a more accurate solution. The solutions obtained from the neural network for this problem are equivalent when we compare and evaluate them against the Runge‑Kutta numerical method of order 4/5 (RK45). However, the method utilizing neural networks is considered a modern and promising approach, as it effectively exploits the superior computational power of advanced computer systems, especially in solving complex problems. Another advantage is that the neural network model, after being trained, can solve the nonlinear system of differential equations across a continuous domain. In other words, neural networks are not only trained to approximate the solution functions for the nonlinear ESD system but can also represent the complex dynamic relationships between the system's components. However, this approach requires significant time and computational power due to the need for model training.

Abstract:
Inspired by the Kolmogorov‑Arnold representation theorem and Kurkova's principle of using approximate representations, we propose the Kurkova‑Kolmogorov‑Arnold Network (KKAN), a new two‑block architecture that combines robust multi‑layer perceptron (MLP) based inner functions with flexible linear combinations of basis functions as outer functions. We first prove that KKAN is a universal approximator, and then we demonstrate its versatility across scientific machine‑learning applications, including function regression, physics‑informed machine learning (PIML), and operator‑learning frameworks. The benchmark results show that KKANs outperform MLPs and the original Kolmogorov‑Arnold Networks (KANs) in function approximation and operator learning tasks and achieve performance comparable to fully optimized MLPs for PIML. To better understand the behavior of the new representation models, we analyze their geometric complexity and learning dynamics using information bottleneck theory, identifying three universal learning stages, fitting, transition, and diffusion, across all types of architectures. We find a strong correlation between geometric complexity and signal‑to‑noise ratio (SNR), with optimal generalization achieved during the diffusion stage. Additionally, we propose self‑scaled residual‑based attention weights to maintain high SNR dynamically, ensuring uniform convergence and prolonged learning.

Abstract:
Estimating the evolution of the battery's State of Charge (SoC) in response to its usage is critical for implementing effective power management policies and for ultimately improving the system's lifetime. Most existing estimation methods are either physics‑based digital twins of the battery or data‑driven models such as Neural Networks (NNs). In this work, we propose two new contributions in this domain. First, we introduce a novel NN architecture formed by two cascaded branches: one to predict the current SoC based on sensor readings, and one to estimate the SoC at a future time as a function of the load behavior. Second, we integrate battery dynamics equations into the training of our NN, merging the physics‑based and data‑driven approaches, to improve the models' generalization over variable prediction horizons. We validate our approach on two publicly accessible datasets, showing that our Physics‑Informed Neural Networks (PINNs) outperform purely data‑driven ones while also obtaining superior prediction accuracy with a smaller architecture with respect to the state‑of‑the‑art.

Abstract:
We compare traditional finite volume and Physics Informed Neural Network (PINN) solvers for elliptic (Poisson), hyperbolic (advection), and parabolic (diffusion) equations in 2d settings. We describe the challenges of using traditional and PINN solvers for electron kinetic equations in collisional plasmas. The advantages and drawbacks of PINNs over state‑of‑the‑art traditional solvers are discussed. We also consider angular moments in spherical velocity space and the potential use of ML algorithms for reduced kinetic models in the coordinate‑energy phase space based on adaptive closure relations.

Abstract:
Traditional numerical methods, such as the finite element method and finite volume method, adress partial differential equations (PDEs) by discretizing them into algebraic equations and solving these iteratively. However, this process is often computationally expensive and time‑consuming. An alternative approach involves transforming PDEs into integral equations and solving them using Green's functions, which provide analytical solutions. Nevertheless, deriving Green's functions analytically is a challenging and non‑trivial task, particularly for complex systems. In this study, we introduce a novel framework, termed GreensONet, which is constructed based on the strucutre of deep operator networks (DeepONet) to learn embedded Green's functions and solve PDEs via Green's integral formulation. Specifically, the Trunk Net within GreensONet is designed to approximate the unknown Green's functions of the system, while the Branch Net are utilized to approximate the auxiliary gradients of the Green's function. These outputs are subsequently employed to perform surface integrals and volume integrals, incorporating user‑defined boundary conditions and source terms, respectively. The effectiveness of the proposed framework is demonstrated on three types of PDEs in bounded domains: 3D heat conduction equations, reaction‑diffusion equations, and Stokes equations. Comparative results in these cases demonstrate that GreenONet's accuracy and generalization ability surpass those of existing methods, including Physics‑Informed Neural Networks (PINN), DeepONet, Physics‑Informed DeepONet (PI‑DeepONet), and Fourier Neural Operators (FNO).

Abstract:
Model predictive control has emerged as an effective approach for real‑time optimal control of connected and automated vehicles. However, nonlinear dynamics of vehicle and traffic systems make accurate modeling and real‑time optimization challenging. Learning‑based control offer a promising alternative, as they adapt to environment without requiring an explicit model. For learning control framework, an augmented state space system design is necessary since optimal control depends on both the ego vehicle's state and predicted states of other vehicles. This work develops a traffic adaptive augmented state space system that allows the control strategy to intelligently adapt to varying traffic conditions. This design ensures that while different vehicle trajectories alter initial conditions, the system dynamics remain independent of specific trajectories. Additionally, a physics‑informed learning control framework is presented that combines value function from Bellman's equation with derivative of value functions from Pontryagin's Maximum Principle into a unified loss function. This method aims to reduce required training data and time while enhancing robustness and efficiency. The proposed control framework is applied to car‑following scenarios in real‑world data calibrated simulation environments. The results show that this learning control approach alleviates real‑time computational requirements while achieving car‑following behaviors comparable to model‑based methods, resulting in 9% energy savings in scenarios not previously seen in training dataset.

Abstract:
Radiation heat transfer in a graded‑index medium often suffers accuracy problems due to the gradual changes in the refractive index. The finite element method, meshfree, and other numerical methods often struggle to maintain accuracy when applied to this medium. To address this issue, we apply physics‑informed neural networks (PINNs)‑based machine learning algorithms to simulate forward and inverse problems for this medium. We also provide the theoretical upper bounds. This theoretical framework is validated through numerical experiments of predefined and newly developed models that demonstrate the accuracy and robustness of the algorithms in solving radiation transport problems in the medium. The simulations show that the novel algorithm goes on with numerical stability and effectively mitigates oscillatory errors, even in cases with more pronounced variations in the refractive index.

Abstract:
We investigate the numerical solution of multiscale transport equations using Physics Informed Neural Networks (PINNs) with ReLU activation functions. Therefore, we study the analogy between PINNs and Least‑Squares Finite Elements (LSFE) which lies in the shared approach to reformulate the PDE solution as a minimization of a quadratic functional. We prove that in the diffusive regime, the correct limit is not reached, in agreement with known results for first‑order LSFE. A diffusive scaling is introduced that can be applied to overcome this, again in full agreement with theoretical results for LSFE. We provide numerical results in the case of slab geometry that support our theoretical findings.

Abstract:
jinns is an open‑source Python library for physics‑informed neural networks, built to tackle both forward and inverse problems, as well as meta‑model learning. Rooted in the JAX ecosystem, it provides a versatile framework for efficiently prototyping real‑problems, while easily allowing extensions to specific needs. Furthermore, the implementation leverages existing popular JAX libraries such as equinox and optax for model definition and optimisation, bringing a sense of familiarity to the user. Many models are available as baselines, and the documentation provides reference implementations of different use‑cases along with step‑by‑step tutorials for extensions to specific needs. The code is available on Gitlab https://gitlab.com/mia_jinns/jinns.

Abstract:
In machine learning and statistical modeling, the mean square or absolute error is commonly used as an error metric, also called a "loss function." While effective in reducing the average error, this approach may fail to address localized outliers, leading to significant inaccuracies in regions with sharp gradients or discontinuities. This issue is particularly evident in physics‑informed neural networks (PINNs), where such localized errors are expected and affect the overall solution. To overcome this limitation, we propose a novel loss function that combines the mean and the standard deviation of the chosen error metric. By minimizing this combined loss function, the method ensures a more uniform error distribution and reduces the impact of localized high‑error regions. The proposed loss function is easy to implement and tested on problems of varying complexity: the 1D Poisson equation, the unsteady Burgers' equation, 2D linear elastic solid mechanics, and 2D steady Navier‑Stokes equations. Results demonstrate improved solution quality and lower maximum error compared to the standard mean‑based loss, with minimal impact on computational time.

Abstract:
Glioblastoma, the most aggressive primary brain tumor, poses a severe clinical challenge due to its diffuse microscopic infiltration, which remains largely undetected on standard MRI. As a result, current radiotherapy planning employs a uniform 15 mm margin around the resection cavity, failing to capture patient‑specific tumor spread. Tumor growth modeling offers a promising approach to reveal this hidden infiltration. However, methods based on partial differential equations or physics‑informed neural networks tend to be computationally intensive or overly constrained, limiting their clinical adaptability to individual patients. In this work, we propose a lightweight, rapid, and robust optimization framework that estimates the 3D tumor concentration by fitting it to MRI tumor segmentations while enforcing a smooth concentration landscape. This approach achieves superior tumor recurrence prediction on 192 brain tumor patients across two public datasets, outperforming state‑of‑the‑art baselines while reducing runtime from 30 minutes to less than one minute. Furthermore, we demonstrate the framework's versatility and adaptability by showing its ability to seamlessly integrate additional imaging modalities or physical constraints.

Abstract:
'A trustworthy representation of uncertainty is desirable and should be considered as a key feature of any machine learning method' (Huellermeier and Waegeman, 2021). This conclusion of Huellermeier et al. underpins the importance of calibrated uncertainties. Since AI‑based algorithms are heavily impacted by dataset shifts, the automotive industry needs to safeguard its system against all possible contingencies. One important but often neglected dataset shift is caused by optical aberrations induced by the windshield. For the verification of the perception system performance, requirements on the AI performance need to be translated into optical metrics by a bijective mapping. Given this bijective mapping it is evident that the optical system characteristics add additional information about the magnitude of the dataset shift. As a consequence, we propose to incorporate a physical inductive bias into the neural network calibration architecture to enhance the robustness and the trustworthiness of the AI target application, which we demonstrate by using a semantic segmentation task as an example. By utilizing the Zernike coefficient vector of the optical system as a physical prior we can significantly reduce the mean expected calibration error in case of optical aberrations. As a result, we pave the way for a trustworthy uncertainty representation and for a holistic verification strategy of the perception chain.

Abstract:
Modern machine learning often relies on optimizing a neural network's parameters using a loss function to learn complex features. Beyond training, examining the loss function with respect to a network's parameters (i.e., as a loss landscape) can reveal insights into the architecture and learning process. While the local structure of the loss landscape surrounding an individual solution can be characterized using a variety of approaches, the global structure of a loss landscape, which includes potentially many local minima corresponding to different solutions, remains far more difficult to conceptualize and visualize. To address this difficulty, we introduce LossLens, a visual analytics framework that explores loss landscapes at multiple scales. LossLens integrates metrics from global and local scales into a comprehensive visual representation, enhancing model diagnostics. We demonstrate LossLens through two case studies: visualizing how residual connections influence a ResNet‑20, and visualizing how physical parameters influence a physics‑informed neural network (PINN) solving a simple convection problem.

Abstract:
In this work, we address the challenges posed by the high nonlinearity of the Butler‑Volmer (BV) equation in forward and inverse simulations of the pseudo‑two‑dimensional (P2D) model using the physics‑informed neural network (PINN) framework. The BV equation presents significant challenges for PINNs, primarily due to the hyperbolic sine term, which renders the Hessian of the PINN loss function highly ill‑conditioned. To address this issue, we introduce a bypassing term that improves numerical stability by substantially reducing the condition number of the Hessian matrix. Furthermore, the small magnitude of the ionic flux \( j \) often leads to a common failure mode where PINNs converge to incorrect solutions. We demonstrate that incorporating a secondary conservation law for the solid‑phase potential \( ψ\) effectively prevents such convergence issues and ensures solution accuracy. The proposed methods prove effective for solving both forward and inverse problems involving the BV equation. Specifically, we achieve precise parameter estimation in inverse scenarios and reliable solution predictions for forward simulations.

Abstract:
Atomic resolution STEM images often suffer from noise due to low electron doses and instrument imperfections, hence it is challenging to obtain critical structural details required for material analysis. To address the problem, we propose a Physics‑Informed Neural Network (PINN) framework for denoising STEM images. Our method integrates spectral fidelity, total variation, and brightness/contrast consistency losses to ensure the preservation of fine structures, smooth regions, and physical signal intensities, maintaining the structural integrity of the denoised images. Our proposed method effectively balances noise reduction with the preservation of atomic resolution details and complements existing methods, seeking to enhance the utility of STEM images in material characterization and analysis.

Abstract:
An adjoint formulation leveraging a physics‑informed neural network (PINN) is employed to advance the density moment of a runaway electron (RE) distribution forward in time. A distinguishing feature of this approach is that once the adjoint problem is solved, its solution can be used to project the RE density forward in time for an arbitrary initial momentum space distribution of REs. Furthermore, by employing a PINN, a parametric solution to the adjoint problem can be learned. Thus, once trained, this adjoint‑deep learning framework is able to efficiently project the RE density forward in time across various plasma conditions while still including a fully kinetic description of RE dynamics. As an example application, the temporal evolution of the density of primary electrons is studied, with particular emphasis on evaluating the decay of a RE population when below threshold. Predictions from the adjoint‑deep learning framework are found to be in good agreement with a traditional relativistic electron Fokker‑Planck solver, for several distinct initial conditions, and across an array of physics parameters. Once trained the PINN thus provides a means of generating RE density time histories with exceptionally low online execution time.

Abstract:
Strongly lensed quasars provide valuable insights into the rate of cosmic expansion, the distribution of dark matter in foreground deflectors, and the characteristics of quasar hosts. However, detecting them in astronomical images is difficult due to the prevalence of non‑lensing objects. To address this challenge, we developed a generative deep learning model called VariLens, built upon a physics‑informed variational autoencoder. This model seamlessly integrates three essential modules: image reconstruction, object classification, and lens modeling, offering a fast and comprehensive approach to strong lens analysis. VariLens is capable of rapidly determining both (1) the probability that an object is a lens system and (2) key parameters of a singular isothermal ellipsoid (SIE) mass model ‑‑ including the Einstein radius (θ_\mathrmE), lens center, and ellipticity ‑‑ in just milliseconds using a single CPU. A direct comparison of VariLens estimates with traditional lens modeling for 20 known lensed quasars within the Subaru Hyper Suprime‑Cam (HSC) footprint shows good agreement, with both results consistent within 2σ for systems with θ_\mathrmE<3 arcsecs. To identify new lensed quasar candidates, we begin with an initial sample of approximately 80 million sources, combining HSC data with multiwavelength information from various surveys. After applying a photometric preselection aimed at locating z>1.5 sources, the number of candidates was reduced to 710,966. Subsequently, VariLens highlights 13,831 sources, each showing a high likelihood of being a lens. A visual assessment of these objects results in 42 promising candidates that await spectroscopic confirmation. These results underscore the potential of automated deep learning pipelines to efficiently detect and model strong lenses in large datasets.

Abstract:
Neural‑network‑based variational quantum states in general, and more recently autoregressive models in particular, have proven to be powerful tools to describe complex many‑body wave functions. However, their performance crucially depends on the computational basis chosen and they often lack physical interpretability. To mitigate these issues, we here propose a modified variational Monte‑Carlo framework which leverages prior physical information to construct a computational second‑quantized basis containing a reference state that serves as a rough approximation to the true ground state. In this basis, a Transformer is used to parametrize and autoregressively sample the corrections to the reference state, giving rise to a more interpretable and computationally efficient representation of the ground state. We demonstrate this approach using a non‑sparse fermionic model featuring a metal‑insulator transition and employing Hartree‑Fock and a strong‑coupling limit to define physics‑informed bases. We also show that the Transformer's hidden representation captures the natural energetic order of the different basis states. This work paves the way for more efficient and interpretable neural quantum‑state representations.

Abstract:
Improving diesel engine efficiency, reducing emissions, and enabling robust health monitoring have been critical research topics in engine modelling. While recent advancements in the use of neural networks for system monitoring have shown promising results, such methods often focus on component‑level analysis, lack generalizability, and physical interpretability. In this study, we propose a novel hybrid framework that combines physics‑informed neural networks (PINNs) with deep operator networks (DeepONet) to enable accurate and computationally efficient parameter identification in mean‑value diesel engine models. Our method leverages physics‑based system knowledge in combination with data‑driven training of neural networks to enhance model applicability. Incorporating offline‑trained DeepONets to predict actuator dynamics significantly lowers the online computation cost when compared to the existing PINN framework. To address the re‑training burden typical of PINNs under varying input conditions, we propose two transfer learning (TL) strategies: (i) a multi‑stage TL scheme offering better runtime efficiency than full online training of the PINN model and (ii) a few‑shot TL scheme that freezes a shared multi‑head network body and computes physics‑based derivatives required for model training outside the training loop. The second strategy offers a computationally inexpensive and physics‑based approach for predicting engine dynamics and parameter identification, offering computational efficiency over the existing PINN framework. Compared to existing health monitoring methods, our framework combines the interpretability of physics‑based models with the flexibility of deep learning, offering substantial gains in generalization, accuracy, and deployment efficiency for diesel engine diagnostics.

Abstract:
Machine learning (ML) has emerged as a powerful tool for tackling complex regression and classification tasks, yet its success often hinges on the quality of training data. This study introduces an ML paradigm inspired by domain knowledge of the structure of output function, akin to physics‑informed ML, but rooted in probabilistic principles rather than physical laws. The proposed approach integrates the probabilistic structure of the target variable (such as its cumulative distribution function) into the training process. This probabilistic information is obtained from historical data or estimated using structural reliability methods during experimental design. By embedding domain‑specific probabilistic insights into the learning process, the technique enhances model accuracy and mitigates risks of overfitting and underfitting. Applications in regression, image denoising, and classification demonstrate the approach's effectiveness in addressing real‑world problems.

Abstract:
In the recent years, Physics Informed Neural Networks (PINNs) have received strong interest as a method to solve PDE driven systems, in particular for data assimilation purpose. This method is still in its infancy, with many shortcomings and failures that remain not properly understood. In this paper we propose a natural gradient approach to PINNs which contributes to speed‑up and improve the accuracy of the training. Based on an in depth analysis of the differential geometric structures of the problem, we come up with two distinct contributions: (i) a new natural gradient algorithm that scales as \min(P^2S, S^2P), where P is the number of parameters, and S the batch size; (ii) a mathematically principled reformulation of the PINNs problem that allows the extension of natural gradient to it, with proved connections to Green's function theory.

Authors: Talon Chandler, Ivan E. Ivanov, Gabriel Sturm, Sheng Xiao, Xiang Zhao, Alexander Hillsley, Allyson Quinn Ryan, Ziwen Liu, Sricharan Reddy Varra, Ilan Theodoro, Eduardo Hirata-Miyasaki, Deepika Sundarraman, Amitabh Verma, Madhurya Sekhar, Chad Liu, Soorya Pradeep, See-Chi Lee, Shannon N. Rhoads, Maria Clara Zanellati, Sarah Cohen, Carolina Arias, Manuel D. Leonetti, Adrian Jacobo, Keir Balla, Loïc A. Royer, Shalin B. Mehta

Abstract:
Correlative computational microscopy can accelerate imaging and modeling of cellular dynamics by relaxing trade‑offs inherent to dynamic imaging. Existing computational microscopy frameworks are either specialized or overly generic, limiting use to fixed configurations or domain experts. We introduce WaveOrder, a generalist wave‑optical framework for imaging the architectural order of biomolecules. WaveOrder reconstructs diverse specimen properties from multi‑channel acquisitions, with or without fluorescence. It provides a unified representation of linear optical properties and differentiable physics‑based image formation models spanning widefield, confocal, light‑sheet, and oblique label‑free geometries. WaveOrder uses physics‑informed ML to auto‑tune model parameters and solve blind shift‑variant restoration problems. This open‑source, PyTorch‑based framework enables scalable quantitative imaging across scales from organelles to adult zebrafish, and improves restoration of cellular structures in high‑throughput experiments. We validate WaveOrder on diverse imaging applications, demonstrating its ability to recover biomolecular structure beyond the limits of existing approaches.

Abstract:
Twelve physics‑informed machine learning models have been trained to model binding energy residuals. Our approach begins with determining the difference between measured experimental binding energies and three different mass models. Then four machine learning approaches are used to train on each energy difference. The most successful ML technique, both in interpolation and extrapolation, is the least squares boosted ensemble of trees. The best model resulting from that technique utilizes eight physical features to model the difference between experimental atomic binding energy values in AME 2012 and the Duflo Zuker mass model. This resulted in a model that fit the training data with a standard deviation of 17 keV and that has a standard deviation of 92 keV when compared all of the values in the AME 2020. The extrapolation capability of each model is discussed, and the accuracy of predicting new mass measurements has also been tested.

Abstract:
PINN models have demonstrated capabilities in addressing fluid PDE problems, and their potential in solid mechanics is beginning to emerge. This study identifies two key challenges when using PINN to solve general solid mechanics problems. These challenges become evident when comparing the limitations of PINN with the well‑established numerical methods commonly used in solid mechanics, such as the finite element method (FEM). Specifically: a) PINN models generate solutions over an infinite domain, which conflicts with the finite boundaries typical of most solid structures; and b) the solution space utilised by PINN is Euclidean, which is inadequate for addressing the complex geometries often present in solid structures. This work presents a PINN architecture for general solid mechanics problems, referred to as the Finite‑PINN model. The model is designed to effectively tackle two key challenges, while retaining as much of the original PINN framework as possible. To this end, the Finite‑PINN incorporates finite geometric encoding into the neural network inputs, thereby transforming the solution space from a conventional Euclidean space into a hybrid Euclidean‑topological space. The model is comprehensively trained using both strong‑form and weak‑form loss formulations, enabling its application to a wide range of forward and inverse problems in solid mechanics. For forward problems, the Finite‑PINN model efficiently approximates solutions to solid mechanics problems when the geometric information of a given structure has been preprocessed. For inverse problems, it effectively reconstructs full‑field solutions from very sparse observations by embedding both physical laws and geometric information within its architecture.

Abstract:
This paper explores the application of physics‑informed neural networks (PINNs) to tackle forward problems in 3D contact mechanics, focusing on small deformation elasticity. We utilize a mixed‑variable formulation, enhanced with output transformations, to enforce Dirichlet and Neumann boundary conditions as hard constraints. The inherent inequality constraints in contact mechanics, particularly the Karush‑Kuhn‑Tucker (KKT) conditions, are addressed as soft constraints by integrating them into the network's loss function. To enforce the KKT conditions, we leverage the nonlinear complementarity problem (NCP) approach, specifically using the Fischer‑Burmeister function, which is known for its advantageous properties in optimization. We investigate two benchmark examples of PINNs in 3D contact mechanics: a single contact patch test and the Hertzian contact problem.

Abstract:
Deep learning‑based techniques have potential to optimize scan and post‑processing times required for MRI‑based fat quantification, but they are constrained by the lack of large training datasets. Generative models are a promising tool to perform data augmentation by synthesizing realistic datasets. However no previous methods have been specifically designed to generate datasets for quantitative MRI (q‑MRI) tasks, where reference quantitative maps and large variability in scanning protocols are usually required. We propose a Physics‑Informed Latent Diffusion Model (PI‑LDM) to synthesize quantitative parameter maps jointly with customizable MR images by incorporating the signal generation model. We assessed the quality of PI‑LDM's synthesized data using metrics such as the Fréchet Inception Distance (FID), obtaining comparable scores to state‑of‑the‑art generative methods (FID: 0.0459). We also trained a U‑Net for the MRI‑based fat quantification task incorporating synthetic datasets. When we used a few real (10 subjects, ~200 slices) and numerous synthetic samples (>3000), fat fraction at specific liver ROIs showed a low bias on data obtained using the same protocol than training data (0.10% at \hboxROI_1, 0.12% at \hboxROI_2) and on data acquired with an alternative protocol (0.14% at \hboxROI_1, 0.62% at \hboxROI_2). Future work will be to extend PI‑LDM to other q‑MRI applications.

Abstract:
Learning dynamics governing physical and spatiotemporal processes is a challenging problem, especially in scenarios where states are partially measured. In this work, we tackle the problem of learning dynamics governing these systems when parts of the system's states are not measured, specifically when the dynamics generating the non‑measured states are unknown. Inspired by state estimation theory and Physics Informed Neural ODEs, we present a sequential optimization framework in which dynamics governing unmeasured processes can be learned. We demonstrate the performance of the proposed approach leveraging numerical simulations and a real dataset extracted from an electro‑mechanical positioning system. We show how the underlying equations fit into our formalism and demonstrate the improved performance of the proposed method when compared with baselines.

Abstract:
Maxwell's equations are the fundamental equations for understanding electric and magnetic field interactions and play a crucial role in designing and optimizing sensor systems like capacitive touch sensors, which are widely prevalent in automotive switches and smartphones. Ensuring robust functionality and stability of the sensors in dynamic environments necessitates profound domain expertise and computationally intensive multi‑physics simulations. This paper introduces a novel approach using a Physics‑Informed Neural Network (PINN) based surrogate model to accelerate the design process. The PINN model solves the governing electrostatic equations describing the interaction between a finger and a capacitive sensor. Inputs include spatial coordinates from a 3D domain encompassing the finger, sensor, and PCB, along with finger distances. By incorporating the electrostatic equations directly into the neural network's loss function, the model captures the underlying physics. The learned model thus serves as a surrogate sensor model on which inference can be carried out in seconds for different experimental setups without the need to run simulations. Efficacy results evaluated on unseen test cases demonstrate the significant potential of PINNs in accelerating the development and design optimization of capacitive touch sensors.

Abstract:
Our method proposes the efficient generation of samples from an unnormalized Boltzmann density by solving the underlying continuity equation in the low‑rank tensor train (TT) format. It is based on the annealing path commonly used in MCMC literature, which is given by the linear interpolation in the space of energies. Inspired by Sequential Monte Carlo, we alternate between deterministic time steps from the TT representation of the flow field and stochastic steps, which include Langevin and resampling steps. These adjust the relative weights of the different modes of the target distribution and anneal to the correct path distribution. We showcase the efficiency of our method on multiple numerical examples.

Abstract:
Vector‑borne diseases continue to pose a significant health threat globally with more than 3 billion people at risk each year. Despite some limitations, mechanistic dynamic models are a popular approach to representing biological processes using ordinary differential equations where the parameters describe the different development and survival rates. Recent advances in population modelling have seen the combination of these mechanistic models with machine learning. One approach is physics‑informed neural networks (PINNs) whereby the machine learning framework embeds physical, biological, or chemical laws into neural networks trained on observed or measured data. This enables forward simulations, predicting system behaviour from given parameters and inputs, and inverse modelling, improving parameterisation of existing parameters and estimating unknown or latent variables. In this paper, we focus on improving the parameterisation of biological processes in mechanistic models using PINNs to determine inverse parameters. In comparing mechanistic and PINN models, our experiments offer important insights into the strengths and weaknesses of both approaches but demonstrated that the PINN approach generally outperforms the dynamic model. For a deeper understanding of the performance of PINN models, a final validation was used to investigate how modifications to PINN architectures affect the performance of the framework. By varying only a single component at a time and keeping all other factors constant, we are able to observe the effect of each change.

Abstract:
We present a method whereby the finite element method is used to train physics‑informed neural networks that are suitable for surrogate modeling. The method is based on a custom convolutional operation called stencil convolution which leverages the inverse isoparametric map of the finite element method. We demonstrate the performance of the method in several training and testing scenarios with linear boundary‑value problems of varying geometries. The resulting neural networks show reasonable accuracy when tested on unseen geometries that are similar to those used for training. Furthermore, when the number of training geometries is increased the testing errors systematically decrease, demonstrating that the neural networks learn how to generalize as the training set becomes larger. Further extending the method to allow for variable boundary conditions, properties, and body forces will lead to a general‑purpose surrogate modeling framework that can leverage existing finite element codes for training.

Abstract:
Physics‑informed neural networks (PINNs) commonly address ill‑posed inverse problems by uncovering unknown physics. This study presents a novel unsupervised learning framework that identifies spatial subdomains with specific governing physics. It uses the partition of unity networks (POUs) to divide the space into subdomains, assigning unique nonlinear model parameters to each, which are integrated into the physics model. A vital feature of this method is a physics residual‑based loss function that detects variations in physical properties without requiring labeled data. This approach enables the discovery of spatial decompositions and nonlinear parameters in partial differential equations (PDEs), optimizing the solution space by dividing it into subdomains and improving accuracy. Its effectiveness is demonstrated through applications in porous media thermal ablation and ice‑sheet modeling, showcasing its potential for tackling real‑world physics challenges.

Abstract:
Control synthesis for continuously‑parameterized families of quantum gates can enable critical advantages for mid‑sized quantum computing applications in advance of fault‑tolerance. We combine quantum optimal control with physics‑informed machine learning to efficiently synthesize control surfaces that interpolate among continuously‑parameterized gate families. Using optimal control as an active learning strategy to guide pretraining, we bootstrap a physics‑informed neural network to achieve rapid convergence to nonlinear control surfaces sufficient for our desired gates. We find our approach is critical for enabling an expressiveness beyond linear interpolation, which is important in cases of hard quantum control. We show in simulation that by adapting our pretraining to use a few reference pulse calibrations, we can apply transfer learning to quickly calibrate our learned control surfaces when devices fluctuate over time. We demonstrate synthesis for one and two qubit gates with one or two parameters, focusing on gate families for variational quantum algorithm (VQA) ansatz. By avoiding the inefficient decomposition of VQA ansatz into basis gate sets, continuous gate families are a potential method to improve the noise robustness of VQAs in the near term. Our framework shows how accessible optimal control tools can be combined with simple machine learning to enable practitioners to achieve 3x speedups for their algorithms by going beyond the standard gate sets.

Abstract:
In this paper, we study the neural tangent kernel (NTK) for general partial differential equations (PDEs) based on physics‑informed neural networks (PINNs). As we all know, the training of an artificial neural network can be converted to the evolution of NTK. We analyze the initialization of NTK and the convergence conditions of NTK during training for general PDEs. The theoretical results show that the homogeneity of differential operators plays a crucial role for the convergence of NTK. Moreover, based on the PINNs, we validate the convergence conditions of NTK using the initial value problems of the sine‑Gordon equation and the initial‑boundary value problem of the KdV equation.

Abstract:
Neural operators are aiming at approximating operators mapping between Banach spaces of functions, achieving much success in the field of scientific computing. Compared to certain deep learning‑based solvers, such as Physics‑Informed Neural Networks (PINNs), Deep Ritz Method (DRM), neural operators can solve a class of Partial Differential Equations (PDEs). Although much work has been done to analyze the approximation and generalization error of neural operators, there is still a lack of analysis on their training error. In this work, we conduct the convergence analysis of gradient descent for the wide shallow neural operators and physics‑informed shallow neural operators within the framework of Neural Tangent Kernel (NTK). The core idea lies on the fact that over‑parameterization and random initialization together ensure that each weight vector remains near its initialization throughout all iterations, yielding the linear convergence of gradient descent. In this work, we demonstrate that under the setting of over‑parametrization, gradient descent can find the global minimum regardless of whether it is in continuous time or discrete time.

Abstract:
Advancements in computational fluid mechanics have largely relied on Newtonian frameworks, particularly through the direct simulation of Navier‑Stokes equations. In this work, we propose an alternative computational framework that employs variational methods, specifically by leveraging the principle of minimum pressure gradient, which turns the fluid mechanics problem into a minimization problem whose solution can be used to predict the flow field in unsteady incompressible viscous flows. This method exhibits two particulary intriguing properties. First, it circumvents the chronic issues of pressure‑velocity coupling in incompressible flows, which often dominates the computational cost in computational fluid dynamics (CFD). Second, this method eliminates the reliance on unphysical assumptions at the outflow boundary, addressing another longstanding challenge in CFD. We apply this framework to three benchmark examples across a range of Reynolds numbers: (i) unsteady flow field in a lid‑driven cavity, (ii) Poiseuille flow, and (iii) flow past a circular cylinder. The minimization framework is carried out using a physics‑informed neural network (PINN), which integrates the underlying physical principles directly into the training of the model. The results from the proposed method are validated against high‑fidelity CFD simulations, showing an excellent agreement. Comparison of the proposed variational method to the conventional method, wherein PINNs is directly applied to solve Navier‑Stokes Equations, reveals that the proposed method outperforms conventional PINNs in terms of both convergence rate and time, demonstrating its potential for solving complex fluid mechanics problems.

Abstract:
This study presents the conditional neural fields for reduced‑order modeling (CNF‑ROM) framework to approximate solutions of parametrized partial differential equations (PDEs). The approach combines a parametric neural ODE (PNODE) for modeling latent dynamics over time with a decoder that reconstructs PDE solutions from the corresponding latent states. We introduce a physics‑informed learning objective for CNF‑ROM, which includes two key components. First, the framework uses coordinate‑based neural networks to calculate and minimize PDE residuals by computing spatial derivatives via automatic differentiation and applying the chain rule for time derivatives. Second, exact initial and boundary conditions (IC/BC) are imposed using approximate distance functions (ADFs) [Sukumar and Srivastava, CMAME, 2022]. However, ADFs introduce a trade‑off as their second‑ or higher‑order derivatives become unstable at the joining points of boundaries. To address this, we introduce an auxiliary network inspired by [Gladstone et al., NeurIPS ML4PS workshop, 2022]. Our method is validated through parameter extrapolation and interpolation, temporal extrapolation, and comparisons with analytical solutions.

Abstract:
Distance functions are crucial in robotics for representing spatial relationships between a robot and its environment. They provide an implicit, continuous, and differentiable representation that integrates seamlessly with control, optimization, and learning. While standard distance fields rely on the Euclidean metric, many robotic tasks inherently involve non‑Euclidean structures. To this end, we generalize Euclidean distance fields to more general metric spaces by solving the Riemannian eikonal equation, a first‑order partial differential equation whose solution defines a distance field and its associated gradient flow on the manifold, enabling the computation of geodesics and globally length‑minimizing paths. We demonstrate that geodesic distance fields, the classical Riemannian distance function represented as a global, continuous, and queryable field, are effective for a broad class of robotic problems where Riemannian geometry naturally arises. To realize this, we present a neural Riemannian eikonal solver (NES) that solves the equation as a mesh‑free implicit representation without grid discretization, scaling to high‑dimensional robot manipulators. Training leverages a physics‑informed neural network (PINN) objective that constrains spatial derivatives via the PDE residual and boundary and metric conditions, so the model is supervised by the governing equation and requires no labeled distances or geodesics. We propose two NES variants, conditioned on boundary data and on spatially varying Riemannian metrics, underscoring the flexibility of the neural parameterization. We validate the effectiveness of our approach through extensive examples, yielding minimal‑length geodesics across diverse robot tasks involving Riemannian geometry.

Abstract:
Discovering hidden physical laws and identifying governing system parameters from sparse observations are central challenges in computational science and engineering. Existing data‑driven methods, such as physics‑informed neural networks (PINNs) and sparse regression, are limited by their need for extensive retraining, sensitivity to noise, or inability to generalize across families of partial differential equations (PDEs). In this work, we introduce two complementary frameworks based on deep operator networks (DeepONet) to address these limitations. The first, termed the Deep Hidden Physics Operator (DHPO), extends hidden‑physics modeling into the operator‑learning paradigm, enabling the discovery of unknown PDE terms across diverse equation families by identifying the mapping of unknown physical operators. The second is a parameter identification framework that combines pretrained DeepONet with physics‑informed inverse modeling to infer system parameters directly from sparse sensor data. We demonstrate the effectiveness of these approaches on benchmark problems, including the Reaction‑Diffusion system, Burgers' equation, the 2D Heat equation, and 2D Helmholtz equation. Across all cases, the proposed methods achieve high accuracy, with relative solution errors on the order of O(10^‑2) and parameter estimation errors on the order of O(10^‑3), even under limited and noisy observations. By uniting operator learning with physics‑informed modeling, this work offers a unified and data‑efficient framework for physics discovery and parameter identification, paving the way for robust inverse modeling in complex dynamical systems.

Abstract:
We introduce a novel algorithm for controlling linear time invariant systems in a tracking problem. The controller is based on a Gaussian Process (GP) whose realizations satisfy a system of linear ordinary differential equations with constant coefficients. Control inputs for tracking are determined by conditioning the prior GP on the setpoints, i.e. control as inference. The resulting Model Predictive Control scheme incorporates pointwise soft constraints by introducing virtual setpoints to the posterior Gaussian process. We show theoretically that our controller satisfies open‑loop stability for the optimal control problem by leveraging general results from Bayesian inference and demonstrate this result in a numerical example.

Abstract:
Computational biomechanical analysis plays a pivotal role in understanding and improving human movements and physical functions. Although physics‑based modeling methods can interpret the dynamic interaction between the neural drive to muscle dynamics and joint kinematics, they suffer from high computational latency. In recent years, data‑driven methods have emerged as a promising alternative due to their fast execution speed, but label information is still required during training, which is not easy to acquire in practice. To tackle these issues, this paper presents a novel physics‑informed deep learning method to predict muscle forces without any label information during model training. In addition, the proposed method could also identify personalized muscle‑tendon parameters. To achieve this, the Hill muscle model‑based forward dynamics is embedded into the deep neural network as the additional loss to further regulate the behavior of the deep neural network. Experimental validations on the wrist joint from six healthy subjects are performed, and a fully connected neural network (FNN) is selected to implement the proposed method. The predicted results of muscle forces show comparable or even lower root mean square error (RMSE) and higher coefficient of determination compared with baseline methods, which have to use the labeled surface electromyography (sEMG) signals, and it can also identify muscle‑tendon parameters accurately, demonstrating the effectiveness of the proposed physics‑informed deep learning method.

Abstract:
Virtual models of human gait, or digital twins, offer a promising solution for studying mobility without the need for labor‑intensive data collection. However, challenges such as the sim‑to‑real gap and limited adaptability to diverse walking conditions persist. To address these, we developed and validated a framework to create a skeletal humanoid agent capable of adapting to varying walking speeds while maintaining biomechanically realistic motions. The framework combines a synthetic data generator, which produces biomechanically plausible gait kinematics from open‑source biomechanics data, and a training system that uses adversarial imitation learning to train the agent's walking policy. We conducted comprehensive analyses comparing the agent's kinematics, synthetic data, and the original biomechanics dataset. The agent achieved a root mean square error of 5.24 +‑ 0.09 degrees at varying speeds compared to ground‑truth kinematics data, demonstrating its adaptability. This work represents a significant step toward developing a digital twin of human locomotion, with potential applications in biomechanics research, exoskeleton design, and rehabilitation.

Abstract:
This paper develops a physics‑informed scenario approach for safety verification of nonlinear systems using barrier certificates (BCs) to ensure that system trajectories remain within safe regions over an infinite time horizon. Designing BCs often relies on an accurate dynamics model; however, such models are often imprecise due to the model complexity involved, particularly when dealing with highly nonlinear systems. In such cases, while scenario approaches effectively address the safety problem using collected data to construct a guaranteed BC for the unknown dynamical system, they often require solving an optimization problem with substantial amounts of data. To address this, we propose a physics‑informed scenario approach that selects data samples such that the outputs of the physics‑based model and the observed data are sufficiently close. This approach guides the scenario optimization process to eliminate redundant samples and potentially reduce the required dataset size. We validate our approach through three case studies, showcasing its practical application in reducing the required data.

Abstract:
Dynamic Mode Decomposition (DMD) has received increasing research attention due to its capability to analyze and model complex dynamical systems. However, it faces challenges in computational efficiency, noise sensitivity, and difficulty adhering to physical laws, which negatively affect its performance. Addressing these issues, we present Online Physics‑informed DMD (OPIDMD), a novel adaptation of DMD into a convex optimization framework. This approach not only ensures convergence to a unique global optimum, but also enhances the efficiency and accuracy of modeling dynamical systems in an online setting. Leveraging the Bayesian DMD framework, we propose a probabilistic interpretation of Physics‑informed DMD (piDMD), examining the impact of physical constraints on the DMD linear operator. Further, we implement online proximal gradient descent and formulate specific algorithms to tackle problems with different physical constraints, enabling real‑time solutions across various scenarios. Compared with existing algorithms such as Exact DMD, Online DMD, and piDMD, OPIDMD achieves the best prediction performance in short‑term forecasting, e.g. an R^2 value of 0.991 for noisy Lorenz system. The proposed method employs a time‑varying linear operator, offering a promising solution for the real‑time simulation and control of complex dynamical systems.

Abstract:
Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely on large amounts of labeled training data, which is impractical for most real‑world applications. Moreover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose a novel architecture called Physics‑Informed Deep Inverse Operator Networks (PI‑DIONs), which can learn the solution operator of PDE‑based inverse problems without labeled training data. We extend the stability estimates established in the inverse problem literature to the operator learning framework, thereby providing a robust theoretical foundation for our method. These estimates guarantee that the proposed model, trained on a finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI‑DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.

Abstract:
Koopman operator theory has gained significant attention in recent years for identifying discrete‑time nonlinear systems by embedding them into an infinite‑dimensional linear vector space. However, providing stability guarantees while learning the continuous‑time dynamics, especially under conditions of relatively low observation frequency, remains a challenge within the existing Koopman‑based learning frameworks. To address this challenge, we propose an algorithmic framework to simultaneously learn the vector field and Lyapunov functions for unknown nonlinear systems, using a limited amount of data sampled across the state space and along the trajectories at a relatively low sampling frequency. The proposed framework builds upon recently developed high‑accuracy Koopman generator learning for capturing transient system transitions and physics‑informed neural networks for training Lyapunov functions. We show that the learned Lyapunov functions can be formally verified using a satisfiability modulo theories (SMT) solver and provide less conservative estimates of the region of attraction compared to existing methods.

Abstract:
This paper presents a physics‑informed deep learning approach for predicting the replicator equation, allowing accurate forecasting of population dynamics. This methodological innovation allows us to derive governing differential or difference equations for systems that lack explicit mathematical models. We used the SINDy model first introduced by Fasel, Kaiser, Kutz, Brunton, and Brunt 2016a to get the replicator equation, which will significantly advance our understanding of evolutionary biology, economic systems, and social dynamics. By refining predictive models across multiple disciplines, including ecology, social structures, and moral behaviours, our work offers new insights into the complex interplay of variables shaping evolutionary outcomes in dynamic systems

Abstract:
Design exploration or optimization using computational fluid dynamics (CFD) is commonly used in the industry. Geometric variation is a key component of such design problems, especially in turbulent flow scenarios, which involves running costly simulations at every design iteration. While parametric RANS‑PINN type approaches have been proven to make effective turbulent surrogates, as a means of predicting unknown Reynolds number flows for a given geometry at near real‑time, geometry aware physics informed surrogates with the ability to predict varying geometries are a relatively less studied topic. A novel geometry aware parametric PINN surrogate model has been created, which can predict flow fields for NACA 4 digit airfoils in turbulent conditions, for unseen shapes as well as inlet flow conditions. A local+global approach for embedding has been proposed, where known global design parameters for an airfoil as well as local SDF values can be used as inputs to the model along with velocity inlet/Reynolds number (\mathcalR_e) to predict the flow fields. A RANS formulation of the Navier‑Stokes equations with a 2‑equation k‑epsilon turbulence model has been used for the PDE losses, in addition to limited CFD data from 8 different NACA airfoils for training. The models have then been validated with unknown NACA airfoils at unseen Reynolds numbers.

Abstract:
We demonstrate that including the second viscosity of an electron gas in the hydrodynamic model allows for highly accurate modeling of the optical response of heavily doped semiconductors. In our setup, which improves resonance visibility compared to previous approaches, plasmon resonances become more distinct, allowing for a detailed analysis of the underlying physics. With advanced fitting techniques based on a physics‑informed cost function and a tailored optimization algorithm, we obtain a close agreement between simulations and experimental data across different sample thicknesses. This enhanced resonance visibility, combined with our integrated approach, shows that key parameters such as doping level and effective electron mass, as well as the second viscosity of the electron gas, can be retrieved from a single optical measurement. The spatial dispersion taken into account in the hydrodynamic framework is essential for accurately describing the optical response of plasmonic materials in this frequency range and is likely to become a standard modeling approach.

Abstract:
In this work we develop a novel algorithm, termed as mixed least‑squares deep neural network (MLS‑DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least‑squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. We provide error bounds on the approximations obtained via both population and empirical losses. The analysis relies on the canonical source condition, approximation theory of deep neural networks and statistical learning theory. We also present multiple numerical experiments to illustrate the performance of the method, and conduct a comparative study with the standard Galerkin finite element method and physics informed neural network. The results indicate that the method can accurately recover the anisotropic conductivity in both two‑ and three‑dimensional cases, up to 10% noise in the data.

Abstract:
This paper proposes a new way to learn Physics‑Informed Neural Network loss functions using Generalized Additive Models. We apply our method by meta‑learning parametric partial differential equations, PDEs, on Burger's and 2D Heat Equations. The goal is to learn a new loss function for each parametric PDE using meta‑learning. The derived loss function replaces the traditional data loss, allowing us to learn each parametric PDE more efficiently, improving the meta‑learner's performance and convergence.

Abstract:
Traditional electrostatic simulation are meshed‑based methods which convert partial differential equations into an algebraic system of equations and their solutions are approximated through numerical methods. These methods are time consuming and any changes in their initial or boundary conditions will require solving the numerical problem again. Newer computational methods such as the physics informed neural net (PINN) similarly require re‑training when boundary conditions changes. In this work, we propose an end‑to‑end deep learning approach to model parameter changes to the boundary conditions. The proposed method is demonstrated on the test problem of a long air‑filled capacitor structure. The proposed approach is compared to plain vanilla deep learning (NN) and PINN. It is shown that our method can significantly outperform both NN and PINN under dynamic boundary condition as well as retaining its full capability as a forward model.

Abstract:
Optimizing neural networks with loss that contain high‑dimensional and high‑order differential operators is expensive to evaluate with back‑propagation due to \mathcalO(d^k) scaling of the derivative tensor size and the \mathcalO(2^k‑1L) scaling in the computation graph, where d is the dimension of the domain, L is the number of ops in the forward computation graph, and k is the derivative order. In previous works, the polynomial scaling in d was addressed by amortizing the computation over the optimization process via randomization. Separately, the exponential scaling in k for univariate functions (d=1) was addressed with high‑order auto‑differentiation (AD). In this work, we show how to efficiently perform arbitrary contraction of the derivative tensor of arbitrary order for multivariate functions, by properly constructing the input tangents to univariate high‑order AD, which can be used to efficiently randomize any differential operator. When applied to Physics‑Informed Neural Networks (PINNs), our method provides >1000× speed‑up and >30× memory reduction over randomization with first‑order AD, and we can now solve \emph1‑million‑dimensional PDEs in 8 minutes on a single NVIDIA A100 GPU. This work opens the possibility of using high‑order differential operators in large‑scale problems.

Abstract:
Rapid reconstruction of 2D plasma profiles from line‑integral measurements is important in nuclear fusion. This paper introduces a physics‑informed model architecture called Onion, that can enhance the performance of models and be adapted to various backbone networks. The model under Onion incorporates physical information by a multiplication process and applies the physics‑informed loss function according to the principle of line integration. Prediction results demonstrate that the additional input of physical information improves the deep learning model's ability, leading to a reduction in the average relative error E_1 between the reconstruction profiles and the target profiles by approximately 0.84x10^(‑2) on synthetic datasets and about 0.06x10^(‑2) on experimental datasets. Furthermore, the implementation of the Softplus activation function in the final two fully connected layers improves model performance. This enhancement results in a reduction in the E_1 by approximately 1.06x10^(‑2) on synthetic datasets and about 0.11x10^(‑2) on experimental datasets. The incorporation of the physics‑informed loss function has been shown to correct the model's predictions, bringing the back‑projections closer to the actual inputs and reducing the errors associated with inversion algorithms. Besides, we have developed a synthetic data model to generate customized line‑integral diagnostic datasets and have also collected soft x‑ray diagnostic datasets from EAST and HL‑2A. This study achieves reductions in reconstruction errors, and accelerates the development of surrogate models in fusion research.

Abstract:
We introduce a framework for molecular structure optimization using denoising model on a physics‑informed Riemannian manifold (R‑DM). Unlike conventional approaches operating in Euclidean space, our method leverages a Riemannian metric that better aligns with molecular energy change, enabling more robust modeling of potential energy surfaces. By incorporating internal coordinates reflective of energetic properties, R‑DM achieves chemical accuracy with an energy error below 1 kcal/mol. Comparative evaluations on QM9, QM7‑X, and GEOM datasets demonstrate improvements in both structural and energetic accuracy, surpassing conventional Euclidean‑based denoising models. This approach highlights the potential of physics‑informed coordinates for tackling complex molecular optimization problems, with implications for tasks in computational chemistry and materials science.

Abstract:
Stiff systems of ordinary differential equations (ODEs) arise in a wide range of scientific and engineering disciplines and are traditionally solved using implicit integration methods due to their stability and efficiency. However, these methods are computationally expensive, particularly for applications requiring repeated integration, such as parameter estimation, Bayesian inference, neural ODEs, physics‑informed neural networks, and MeshGraphNets. Explicit exponential integration methods have been proposed as a potential alternative, leveraging the matrix exponential to address stiffness without requiring nonlinear solvers. This study evaluates several state‑of‑the‑art explicit single‑step exponential schemes against classical implicit methods on benchmark stiff ODE problems, analyzing their accuracy, stability, and scalability with step size. Despite their initial appeal, our results reveal that explicit exponential methods significantly lag behind implicit schemes in accuracy and scalability for stiff ODEs. The backward Euler method consistently outperformed higher‑order exponential methods in accuracy at small step sizes, with none surpassing the accuracy of the first‑order integrating factor Euler method. Exponential methods fail to improve upon first‑order accuracy, revealing the integrating factor Euler method as the only reliable choice for repeated, inexpensive integration in applications such as neural ODEs and parameter estimation. This study exposes the limitations of explicit exponential methods and calls for the development of improved algorithms.

Abstract:
Mesh generation is essential for accurate and efficient computational fluid dynamics simulations. To resolve critical features in the flow, adaptive mesh refinement (AMR) is routinely employed in certain regions of the computational domain, where gradients or error estimates of the solution are often considered as the refining criteria. In many scenarios, however, these indicators can lead to unnecessary refinement over a large region, making the process a matter of trial and error and resulting in slow convergence of the computation. To this end, we propose a heuristic strategy that employs the residuals of the governing partial differential equations (PDEs) as a novel criterion to adaptively guide the mesh refining process. In particular, we leverage on the physics‑informed neural networks (PINNs) to integrate imprecise data obtained on a coarse mesh and the governing PDEs. Once trained, PINNs are capable of identifying regions of highest residuals of the Navier‑Stokes/Euler equations and suggesting new potential vertices for the coarse mesh cells. Moreover, we put forth two schemes to maintain the quality of the refined mesh through the strategic insertion of vertices and the implementation of Delaunay triangulation. By applying the residuals‑guided AMR to address a multitude of typical incompressible/compressible flow problems and comparing the outcomes with those of gradient‑based methods, we illustrate that the former effectively attains a favorable balance between the computational accuracy and cost.

Abstract:
Physics‑informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs). However, their generalization capabilities across varying scenarios remain limited. To overcome this limitation, we propose PIDO, a novel physics‑informed neural PDE solver designed to generalize effectively across diverse PDE configurations, including varying initial conditions, PDE coefficients, and training time horizons. PIDO exploits the shared underlying structure of dynamical systems with different properties by projecting PDE solutions into a latent space using auto‑decoding. It then learns the dynamics of these latent representations, conditioned on the PDE coefficients. Despite its promise, integrating latent dynamics models within a physics‑informed framework poses challenges due to the optimization difficulties associated with physics‑informed losses. To address these challenges, we introduce a novel approach that diagnoses and mitigates these issues within the latent space. This strategy employs straightforward yet effective regularization techniques, enhancing both the temporal extrapolation performance and the training stability of PIDO. We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier‑Stokes equations. Additionally, we demonstrate the transferability of its learned representations to downstream applications such as long‑term integration and inverse problems.

Abstract:
The growing adoption of machine learning (ML) in modelling atmospheric and oceanic processes offers a promising alternative to traditional numerical methods. It is essential to benchmark the performance of both ML and physics‑informed ML (PINN) models to evaluate their predictive skill, particularly for short‑ to medium‑term forecasting. In this study, we utilize gridded sea surface temperature (SST) data and six atmospheric predictors (cloud cover, relative humidity, solar radiation, surface pressure, u‑component of velocity, and v‑component of velocity) to capture both spatial and temporal patterns in SST predictions.

Abstract:
We propose a physics‑informed neural network (PINN) model to efficiently predict the self‑energy of Anderson impurity models (AIMs) based on the Lehmann representation. As an example, we apply the PINN model to a single‑orbital AIM (SAIM) for a noninteracting electron bath with a semicircular density of states. Trained across a wide range of onsite Coulomb interactions U and hybridization strengths V, the PINN model demonstrates high accuracy in both U‑V and Matsubara‑frequency spaces. Additionally, we investigate the effectiveness of physical constraints implemented in the PINN model. For example, We show that the Lehmann representation allows the PINN model to reduce the maximum test error in an electron filling by a factor of approximately 7.8.

Abstract:
Physics‑informed deep operator networks (DeepONets) have emerged as a promising approach toward numerically approximating the solution of partial differential equations (PDEs). In this work, we aim to develop further understanding of what is being learned by physics‑informed DeepONets by assessing the universality of the extracted basis functions and demonstrating their potential toward model reduction with spectral methods. Results provide clarity about measuring the performance of a physics‑informed DeepONet through the decays of singular values and expansion coefficients. In addition, we propose a transfer learning approach for improving training for physics‑informed DeepONets between parameters of the same PDE as well as across different, but related, PDEs where these models struggle to train well. This approach results in significant error reduction and learned basis functions that are more effective in representing the solution of a PDE.

Abstract:
In recent years, Physics‑Informed Neural Networks (PINNs) have become a representative method for solving partial differential equations (PDEs) with neural networks. PINNs provide a novel approach to solving PDEs through optimization algorithms, offering a unified framework for solving both forward and inverse problems. However, some limitations in terms of solution accuracy and generality have also been revealed. This paper systematically summarizes the limitations of PINNs and identifies three root causes for their failure in solving PDEs: (1) Poor multiscale approximation ability and ill‑conditioning caused by PDE losses; (2) Insufficient exploration of convergence and error analysis, resulting in weak mathematical rigor; (3) Inadequate integration of physical information, causing mismatch between residuals and iteration errors. By focusing on addressing these limitations in PINNs, we outline the future directions and prospects for the intelligent computing of PDEs: (1) Analysis of ill‑conditioning in PINNs and mitigation strategies; (2) Improvements to PINNs by enforcing temporal causality; (3) Empowering PINNs with classical numerical methods.

Abstract:
Electricity grid's resiliency and climate change strongly impact one another due to an array of technical and policy‑related decisions that impact both. This paper introduces a physics‑informed machine learning‑based framework to enhance grid's resiliency. Specifically, when encountering disruptive events, this paper designs remedial control actions to prevent blackouts. The proposed Physics‑Guided Reinforcement Learning (PG‑RL) framework determines effective real‑time remedial line‑switching actions, considering their impact on power balance, system security, and grid reliability. To identify an effective blackout mitigation policy, PG‑RL leverages power‑flow sensitivity factors to guide the RL exploration during agent training. Comprehensive evaluations using the Grid2Op platform demonstrate that incorporating physical signals into RL significantly improves resource utilization within electric grids and achieves better blackout mitigation policies ‑ both of which are critical in addressing climate change.

Abstract:
In earthquake source inversions aimed at understanding diverse fault activities on earthquake faults using seismic observation data, uncertainties in velocity structure models are typically not considered. As a result, biases and underestimations of uncertainty can occur in source inversion. This article provides an overview of the author's efforts to address this issue by quantitatively evaluating the uncertainty in velocity structure models and appropriately accounting for its propagation into source inversion. First, the Bayesian multi‑model source inversion method that can incorporate such uncertainties as probability distributions in the form of ensembles is explained. Next, a Bayesian traveltime tomography technique utilizing physics‑informed neural networks (PINN) to quantify uncertainties in velocity structure models is introduced. Furthermore, the author's recent efforts to integrate these methods and apply them to hypocenter determination in the Nankai Trough region are briefly discussed. The article also outlines future prospects of source inversions considering uncertainties in velocity structure models and the anticipated role of the emerging scientific machine learning (SciML) methods such as PINN.

Abstract:
Physics‑Informed Neural Networks (PINN) has evolved into a powerful tool for solving partial differential equations, which has been applied to various fields such as energy, environment, en‑gineering, etc. When utilizing PINN to solve partial differential equations, it is common to rely on Automatic Differentiation (AD) to compute the residuals of the governing equations. This can lead to certain precision losses, thus affecting the accuracy of the network prediction. This paper pro‑poses a Finite Volume Physics‑Informed Neural Network (FV‑PINN), designed to address steady‑state problems of incompressible flow. This method divides the solution domain into mul‑tiple grids. Instead of calculating the residuals of the Navier‑Stokes equations at collocation points within the grid, as is common in traditional PINNs, this approach evaluates them at Gaussian in‑tegral points on the grid boundaries using Gauss's theorem. The loss function is constructed using the Gaussian integral method, and the differentiation order for velocity is reduced. To validate the effectiveness of this approach, we predict the velocity and pressure fields for two typical examples in fluid topology optimization. The results are compared with commercial software COMSOL, which indicates that FVI‑PINN significantly improves the prediction accuracy of both the velocity and pressure fields while accelerating the training speed of the network.

Abstract:
Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low‑regularity or high‑dimensional problems. We propose a novel deep learning framework based on deterministic number‑theoretic sampling points, which is a robust approach specifically designed to handle partial differential equations with rough solutions or in high dimensions. The sample points are generated by the generating vector to achieve the smallest discrepancy. The architecture integrates Physics‑Informed Neural Networks (PINNs) with rigorous mathematical guarantees demonstrating lower error bounds compared to conventional uniform random sampling. Numerical validation includes low‑regularity Poisson equations, two‑dimensional inverse Helmholtz problems, and high‑dimensional linear/nonlinear PDEs, systematically demonstrating the algorithm's superior performance and generalization capabilities.

Abstract:
We propose a novel design paradigm for arbitrarily capable deep photonic networks of cascaded Mach‑Zehnder Interferometers (MZIs) for on‑chip universal polarization handling. Using a device architecture made of cascaded Mach‑Zehnder interferometers, we modify and train the phase difference between interferometer arms for both polarizations through wide operation bandwidths. Three proof‑of‑concept polarization handling devices are illustrated using a software‑defined, physics‑informed neural framework, to achieve user‑specified target device responses as functions of polarization and wavelength. These devices include a polarization splitter, a polarization‑independent power splitter, and an arbitrary polarization‑dependent splitter to illustrate the capabilities of the design framework. The performance for all three devices is optimized using transfer matrix calculations; and their final responses are verified through 3D‑FDTD simulations. All devices demonstrate state‑of‑the‑art performance metrics with over 20 dB extinction, and flat‑top transmission bands through bandwidths of 120 nm. In addition to the functional diversity enabled, the optimization for each device is completed in under a minute, highlighting the computational efficiency of the design paradigm presented. These results demonstrate the versatility of the deep photonic network design ecosystem in polarization management, unveiling promising prospects for advanced on‑chip applications in optical communications, sensing, and computing.

Abstract:
Two‑phase flow phenomena underpin critical technologies such as hydrogen fuel cells, spray cooling, and combustion, where droplet dynamics govern performance and efficiency. Conventional optical diagnostics, including shadowgraphy and particle image velocimetry, provide valuable insights but are limited to two‑dimensional projections of inherently three‑dimensional flows. We employ a specialized optical technique that encodes droplet surface information through color‑coded glare points, enabling enhanced reconstruction of gas‑liquid interfaces. To interpret these measurements, we introduce video‑conditioned physics‑informed neural networks (VcPINNs), which integrate experimental observations with governing fluid dynamics equations. This hybrid framework leverages the strengths of both data‑driven learning and physical constraints, allowing accurate volumetric flow reconstruction from limited input images. Applied to droplet impingement experiments, our method yields highly resolved and physically consistent 3D interface and flow dynamics. The combined imaging and PINN reconstruction strategy provides a powerful platform for advancing multiphase‑flow analysis, with broad potential impact across energy, cooling, and propulsion applications.

Abstract:
We present a new method for enhancing symbolic regression for differential equations via dimensional analysis, specifically Ipsen's and Buckingham pi methods. Since symbolic regression often suffers from high computational costs and overfitting, non‑dimensionalizing datasets reduces the number of input variables, simplifies the search space, and ensures that derived equations are physically meaningful. As our main contribution, we integrate Ipsen's method of dimensional analysis with Universal Physics‑Informed Neural Networks. We also combine dimensional analysis with the AI Feynman symbolic regression algorithm to show that dimensional analysis significantly improves the accuracy of the recovered equation. The results demonstrate that transforming data into a dimensionless form significantly decreases computation time and improves accuracy of the recovered hidden term. For algebraic equations, using the Buckingham pi theorem reduced complexity, allowing the AI Feynman model to converge faster with fewer data points and lower error rates. For differential equations, Ipsen's method was combined with Universal Physics‑Informed Neural Networks (UPINNs) to identify hidden terms more effectively. These findings suggest that integrating dimensional analysis with symbolic regression can significantly lower computational costs, enhance model interpretability, and increase accuracy, providing a robust framework for automated discovery of governing equations in complex systems when data is limited.

Abstract:
In this paper, we present a novel approach for fluid dynamic simulations by harnessing the capabilities of Physics‑Informed Neural Networks (PINNs) guided by the newly unveiled principle of minimum pressure gradient (PMPG). In a PINN formulation, the physics problem is converted into a minimization problem (typically least squares). The PMPG asserts that for incompressible flows, the total magnitude of the pressure gradient over the domain must be minimum at every time instant, turning fluid mechanics into minimization problems, making it an excellent choice for PINNs formulation. Following the PMPG, the proposed PINN formulation seeks to construct a neural network for the flow field that minimizes Nature's cost function for incompressible flows in contrast to traditional PINNs that minimize the residuals of the Navier‑Stokes equations. This technique eliminates the need to train a separate pressure model, thereby reducing training time and computational costs. We demonstrate the effectiveness of this approach through a case study of inviscid flow around a cylinder, showing its ability to capture the underlying physics, while reducing computational cost and training time. The proposed approach outperforms the traditional PINNs approach in terms of Root Mean Square Error, training time, convergence rate, and compliance with physical metrics. While demonstrated on a simple geometry, the methodology is extendable to more complex flow fields (e.g., Three‑Dimensional, unsteady, viscous flows) within the incompressible realm, which is the region of applicability of the PMPG.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs). However, they face challenges related to spectral bias (the tendency to learn low‑frequency components while struggling with high‑frequency features) and unstable convergence dynamics (mainly stemming from the multi‑objective nature of the PINN loss function). These limitations impact their accuracy for problems involving rapid oscillations, sharp gradients, and complex boundary behaviors. We systematically investigate learnable activation functions as a solution to these challenges, comparing Multilayer Perceptrons (MLPs) using fixed and learnable activation functions against Kolmogorov‑Arnold Networks (KANs) that employ learnable basis functions. Our evaluation spans diverse PDE types, including linear and non‑linear wave problems, mixed‑physics systems, and fluid dynamics. Using empirical Neural Tangent Kernel (NTK) analysis and Hessian eigenvalue decomposition, we assess spectral bias and convergence stability of the models. Our results reveal a trade‑off between expressivity and training convergence stability. While learnable activation functions work well in simpler architectures, they encounter scalability issues in complex networks due to the higher functional dimensionality. Counterintuitively, we find that low spectral bias alone does not guarantee better accuracy, as functions with broader NTK eigenvalue spectra may exhibit convergence instability. We demonstrate that activation function selection remains inherently problem‑specific, with different bases showing distinct advantages for particular PDE characteristics. We believe these insights will help in the design of more robust neural PDE solvers.

Abstract:
Topological solitons, which are stable, localized solutions of nonlinear differential equations, are crucial in various fields of physics and mathematics, including particle physics and cosmology. However, solving these solitons presents significant challenges due to the complexity of the underlying equations and the computational resources required for accurate solutions. To address this, we have developed a novel method using neural network (NN) to efficiently solve solitons. A similar NN approach is Physics‑Informed Neural Networks (PINN). In a comparative analysis between our method and PINN, we find that our method achieves shorter computation times while maintaining the same level of accuracy. This advancement in computational efficiency not only overcomes current limitations but also opens new avenues for studying topological solitons and their dynamical behavior.

Abstract:
An onboard prediction of dynamic parameters (e.g. Aerodynamic drag, rolling resistance) enables accurate path planning for EVs. This paper presents EV‑PINN, a Physics‑Informed Neural Network approach in predicting instantaneous battery power and cumulative energy consumption during cruising while generalizing to the nonlinear dynamics of an EV. Our method learns real‑world parameters such as motor efficiency, regenerative braking efficiency, vehicle mass, coefficient of aerodynamic drag, and coefficient of rolling resistance using automatic differentiation based on dynamics and ensures consistency with ground truth vehicle data. EV‑PINN was validated using 15 and 35 minutes of in‑situ battery log data from the Tesla Model 3 Long Range and Tesla Model S, respectively. With only vehicle speed and time as inputs, our model achieves high accuracy and generalization to dynamics, with validation losses of 0.002195 and 0.002292, respectively. This demonstrates EV‑PINN's effectiveness in estimating parameters and predicting battery usage under actual driving conditions without the need for additional sensors.

Abstract:
LLM‑based autonomous agents have recently shown strong capabilities in solving complex industrial design tasks. However, in domains aiming for carbon neutrality and high‑performance renewable energy systems, current AI‑assisted design automation methods face critical challenges in explainability, scalability, and practical usability. To address these limitations, we introduce PHIA (Physics‑Informed Autonomous Agent), an LLM‑driven system that automates modulation design for power converters in Power Electronics Systems with minimal human intervention. In contrast to traditional pipeline‑based methods, PHIA incorporates an LLM‑based planning module that interactively acquires and verifies design requirements via a user‑friendly chat interface. This planner collaborates with physics‑informed simulation and optimization components to autonomously generate and iteratively refine modulation designs. The interactive interface also supports interpretability by providing textual explanations and visual outputs throughout the design process. Experimental results show that PHIA reduces standard mean absolute error by 63.2% compared to the second‑best benchmark and accelerates the overall design process by over 33 times. A user study involving 20 domain experts further confirms PHIA's superior design efficiency and usability, highlighting its potential to transform industrial design workflows in power electronics.

Abstract:
The use of neural networks to solve differential equations, as an alternative to traditional numerical solvers, has increased recently. However, error bounds for the obtained solutions have only been developed for certain equations. In this work, we report important progress in calculating error bounds of physics‑informed neural networks (PINNs) solutions of nonlinear first‑order ODEs. We give a general expression that describes the error of the solution that the PINN‑based method provides for a nonlinear first‑order ODE. In addition, we propose a technique to calculate an approximate bound for the general case and an exact bound for a particular case. The error bounds are computed using only the residual information and the equation structure. We apply the proposed methods to particular cases and show that they can successfully provide error bounds without relying on the numerical solution.

Abstract:
In machine learning, a loss function measures the difference between model predictions and ground‑truth (or target) values. For neural network models, visualizing how this loss changes as model parameters are varied can provide insights into the local structure of the so‑called loss landscape (e.g., smoothness) as well as global properties of the underlying model (e.g., generalization performance). While various methods for visualizing the loss landscape have been proposed, many approaches limit sampling to just one or two directions, ignoring potentially relevant information in this extremely high‑dimensional space. This paper introduces a new representation based on topological data analysis that enables the visualization of higher‑dimensional loss landscapes. After describing this new topological landscape profile representation, we show how the shape of loss landscapes can reveal new details about model performance and learning dynamics, highlighting several use cases, including image segmentation (e.g., UNet) and scientific machine learning (e.g., physics‑informed neural networks). Through these examples, we provide new insights into how loss landscapes vary across distinct hyperparameter spaces: we find that the topology of the loss landscape is simpler for better‑performing models; and we observe greater variation in the shape of loss landscapes near transitions from low to high model performance.

Abstract:
Machine learning for scientific discovery is increasingly becoming popular because of its ability to extract and recognize the nonlinear characteristics from the data. The black‑box nature of deep learning methods poses difficulties in interpreting the identified model. There is a dire need to interpret the machine learning models to develop a physical understanding of dynamic systems. An interpretable form of neural network called Kolmogorov‑Arnold networks (KAN) or Multiplicative KAN (MultKAN) offers critical features that help recognize the nonlinearities in the governing ordinary/partial differential equations (ODE/PDE) of various dynamic systems and find their equation structures. In this study, an equation discovery framework is proposed that includes i) sequentially regularized derivatives for denoising (SRDD) algorithm to denoise the measure data to obtain accurate derivatives, ii) KAN to identify the equation structure and suggest relevant nonlinear functions that are used to create a small overcomplete library of functions, and iii) physics‑informed spline fitting (PISF) algorithm to filter the excess functions from the library and converge to the correct equation. The framework was tested on the forced Duffing oscillator, Van der Pol oscillator (stiff ODE), Burger's equation, and Bouc‑Wen model (coupled ODE). The proposed method converged to the true equation for the first three systems. It provided an approximate model for the Bouc‑Wen model that could acceptably capture the hysteresis response. Using KAN maintains low complexity, which helps the user interpret the results throughout the process and avoid the black‑box‑type nature of machine learning methods.

Abstract:
Physics Informed Machine Learning has emerged as a popular approach for modeling and simulation in digital twins, enabling the generation of accurate models of processes and behaviors in real‑world systems. However, existing methods either rely on simple loss regularizations that offer limited physics integration or employ highly specialized architectures that are difficult to generalize across diverse physical systems. This paper presents a generic approach based on a novel physics‑encoded residual neural network (PERNN) architecture that seamlessly combines data‑driven and physics‑based analytical models to overcome these limitations. Our method integrates differentiable physics blocks‑implementing mathematical operators from physics‑based models with feed‑forward learning blocks, while intermediate residual blocks ensure stable gradient flow during training. Consequently, the model naturally adheres to the underlying physical principles even when prior physics knowledge is incomplete, thereby improving generalizability with low data requirements and reduced model complexity. We investigate our approach in two application domains. The first is a steering model for autonomous vehicles in a simulation environment, and the second is a digital twin for climate modeling using an ordinary differential equation (ODE)‑based model of Net Ecosystem Exchange (NEE) to enable gap‑filling in flux tower data. In both cases, our method outperforms conventional neural network approaches as well as state‑of‑the‑art Physics Informed Machine Learning methods.

Abstract:
Rigid body interactions are fundamental to numerous scientific disciplines, but remain challenging to simulate due to their abrupt nonlinear nature and sensitivity to complex, often unknown environmental factors. These challenges call for adaptable learning‑based methods capable of capturing complex interactions beyond explicit physical models and simulations. While graph neural networks can handle simple scenarios, they struggle with complex scenes and long‑term predictions. We introduce a novel framework for modeling rigid body dynamics and learning collision interactions, addressing key limitations of existing graph‑based methods. Our approach extends the traditional representation of meshes by incorporating higher‑order topology complexes, offering a physically consistent representation. Additionally, we propose a physics‑informed message‑passing neural architecture, embedding physical laws directly in the model. Our method demonstrates superior accuracy, even during long rollouts, and exhibits strong generalization to unseen scenarios. Importantly, this work addresses the challenge of multi‑entity dynamic interactions, with applications spanning diverse scientific and engineering domains.

Abstract:
Physics‑Informed Neural Networks (PINNs) solve forward PDEs by minimizing residual losses from the governing equations with initial and boundary conditions, but they often struggle with discontinuities such as shocks. In contrast, finite volume methods (FVM) handle discontinuities by enforcing integral conservation, which admits weak solutions. Motivated by this, we propose a Coupled Integral PINN (CI‑PINN) that augments a standard PINN with an auxiliary network for integral potentials and coupled integral constraints. This improves robustness near shocks while avoiding meshing and the numerical flux integration/reconstruction used in classical schemes. We validate CI‑PINN on forward benchmarks including Burgers, Buckley‑‑Leverett, the Euler system, and the Shallow‑Water equations.

Abstract:
Scientific machine learning (SciML) represents a significant advancement in integrating machine learning (ML) with scientific methodologies. At the forefront of this development are Physics‑Informed Neural Networks (PINNs), which offer a promising approach by incorporating physical laws directly into the learning process, thereby reducing the need for extensive datasets. However, when data is limited or the system becomes more complex, PINNs can face challenges, such as instability and difficulty in accurately fitting the training data. In this article, we explore the capabilities and limitations of the DeepXDE framework, a tool specifically designed for implementing PINNs, in addressing both forward and inverse problems related to dielectric properties. Using RC circuit models to represent dielectric materials in HVDC systems, we demonstrate the effectiveness of PINNs in analyzing and improving system performance. Additionally, we show that applying a logarithmic transformation to the current (ln(I)) significantly enhances the stability and accuracy of PINN predictions, especially in challenging scenarios with sparse data or complex models. In inverse mode, however, we faced challenges in estimating key system parameters, such as resistance and capacitance, in more complex scenarios with longer time domains. This highlights the potential for future work in improving PINNs through transformations or other methods to enhance performance in inverse problems. This article provides pedagogical insights for those looking to use PINNs in both forward and inverse modes, particularly within the DeepXDE framework.

Abstract:
This paper introduces an adaptive physics‑guided neural network (APGNN) framework for predicting quality attributes from image data by integrating physical laws into deep learning models. The APGNN adaptively balances data‑driven and physics‑informed predictions, enhancing model accuracy and robustness across different environments. Our approach is evaluated on both synthetic and real‑world datasets, with comparisons to conventional data‑driven models such as ResNet. For the synthetic data, 2D domains were generated using three distinct governing equations: the diffusion equation, the advection‑diffusion equation, and the Poisson equation. Non‑linear transformations were applied to these domains to emulate complex physical processes in image form. In real‑world experiments, the APGNN consistently demonstrated superior performance in the diverse thermal image dataset. On the cucumber dataset, characterized by low material diversity and controlled conditions, APGNN and PGNN showed similar performance, both outperforming the data‑driven ResNet. However, in the more complex thermal dataset, particularly for outdoor materials with higher environmental variability, APGNN outperformed both PGNN and ResNet by dynamically adjusting its reliance on physics‑based versus data‑driven insights. This adaptability allowed APGNN to maintain robust performance across structured, low‑variability settings and more heterogeneous scenarios. These findings underscore the potential of adaptive physics‑guided learning to integrate physical constraints effectively, even in challenging real‑world contexts with diverse environmental conditions.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as an influential technology, merging the swift and automated capabilities of machine learning with the precision and dependability of simulations grounded in theoretical physics. PINNs are often employed to solve algebraic or differential equations to replace some or even all steps of multi‑stage computational workflows, leading to their significant speed‑up. However, wide adoption of PINNs is still hindered by reliability issues, particularly at extreme ends of the input parameter ranges. In this study, we demonstrate this in the context of a system of coupled non‑linear differential reaction‑diffusion and heat transfer equations related to Fischer‑Tropsch synthesis, which are solved by a finite‑difference method with a PINN used in evaluating their source terms. It is shown that the testing strategies traditionally used to assess the accuracy of neural networks as function approximators can overlook the peculiarities which ultimately cause instabilities of the finite‑difference solver. We propose a domain knowledge‑based modifications to the PINN architecture ensuring its correct asymptotic behavior. When combined with an improved numerical scheme employed as an initial guess generator, the proposed modifications are shown to recover the overall stability of the simulations, while preserving the speed‑up brought by PINN as the workflow component. We discuss the possible applications of the proposed hybrid transport equation solver in context of chemical reactors simulations.

Abstract:
With the popularity of electric vehicles, the demand for lithium‑ion batteries is increasing. Temperature significantly influences the performance and safety of batteries. Battery thermal management systems can effectively control the temperature of batteries; therefore, the performance and safety can be ensured. However, the development process of battery thermal management systems is time‑consuming and costly due to the extensive training dataset needed by data‑driven models requiring enormous computational costs for finite element analysis. Therefore, a new approach to constructing surrogate models is needed in the era of AI. Physics‑informed machine learning enforces the physical laws in surrogate models, making it the perfect candidate for estimating battery pack temperature distribution. In this study, we first developed a 21700 battery pack indirect liquid cooling system with cold plates on the top and bottom with thermal paste surrounding the battery cells. Then, the simplified finite element model was built based on experiment results. Due to the high coolant flow rate, the cold plates can be considered as constant temperature boundaries, while battery cells are the heat sources. The physics‑informed convolutional neural network served as a surrogate model to estimate the temperature distribution of the battery pack. The loss function was constructed considering the heat conduction equation based on the finite difference method. The physics‑informed loss function helped the convergence of the training process with less data. As a result, the physics‑informed convolutional neural network showed more than 15 percents improvement in accuracy compared to the data‑driven method with the same training data.

Abstract:
Characterizing the loss of a neural network with respect to model parameters, i.e., the loss landscape, can provide valuable insights into properties of that model. Various methods for visualizing loss landscapes have been proposed, but less emphasis has been placed on quantifying and extracting actionable and reproducible insights from these complex representations. Inspired by powerful tools from topological data analysis (TDA) for summarizing the structure of high‑dimensional data, here we characterize the underlying shape (or topology) of loss landscapes, quantifying the topology to reveal new insights about neural networks. To relate our findings to the machine learning (ML) literature, we compute simple performance metrics (e.g., accuracy, error), and we characterize the local structure of loss landscapes using Hessian‑based metrics (e.g., largest eigenvalue, trace, eigenvalue spectral density). Following this approach, we study established models from image pattern recognition (e.g., ResNets) and scientific ML (e.g., physics‑informed neural networks), and we show how quantifying the shape of loss landscapes can provide new insights into model performance and learning dynamics.

Abstract:
Numerical modeling errors are unavoidable in finite element analysis. The presence of model errors inherently reflects both model accuracy and uncertainty. To date there have been few methods for explicitly quantifying errors at points of interest (e.g. at finite element nodes). The lack of explicit model error approximators has been addressed recently with the emergence of machine learning (ML), which closes the loop between numerical model features/solutions and explicit model error approximations. In this paper, we propose physics‑informed neural networks (PINNs) for simultaneous numerical model error approximation and superresolution. To test our approach, numerical data was generated using finite element simulations on a two‑dimensional elastic plate with a central opening. Four‑ and eight‑node quadrilateral elements were used in the discretization to represent the reduced‑order and higher‑order models, respectively. It was found that the developed PINNs effectively predict model errors in both x and y displacement fields with small differences between predictions and ground truth. Our findings demonstrate that the integration of physics‑informed loss functions enables neural networks (NNs) to surpass a purely data‑driven approach for approximating model errors.

Abstract:
Non‑linear convection‑reaction‑diffusion (CRD) partial differential equations (PDEs) are crucial for modeling complex phenomena in fields such as biology, ecology, population dynamics, physics, and engineering. Numerical approximation of these non‑linear systems is essential due to the challenges of obtaining exact solutions. Traditionally, the Galerkin finite element method (GFEM) has been the standard computational tool for solving these PDEs. With the advancements in machine learning, Physics‑Informed Neural Network (PINN) has emerged as a promising alternative for approximating non‑linear PDEs. In this study, we compare the performance of PINN and GFEM by solving four distinct one‑dimensional CRD problems with varying initial and boundary conditions and evaluate the performance of PINN over GFEM. This evaluation metrics includes error estimates, and visual representations of the solutions, supported by statistical methods such as the root mean squared error (RMSE), the standard deviation of error, the the Wilcoxon Signed‑Rank Test and the coefficient of variation (CV) test. Our findings reveal that while both methods achieve solutions close to the analytical results, PINN demonstrate superior accuracy and efficiency. PINN achieved significantly lower RMSE values and smaller standard deviations for Burgers' equation, Fisher's equation, and Newell‑Whitehead‑Segel equation, indicating higher accuracy and greater consistency. While GFEM shows slightly better accuracy for the Burgers‑Huxley equation, its performance was less consistent over time. In contrast, PINN exhibit more reliable and robust performance, highlighting their potential as a cutting‑edge approach for solving non‑linear PDEs.

Abstract:
This paper proposes and studies two extensions of applying hp‑variational physics‑informed neural networks, more precisely the FastVPINNs framework, to convection‑dominated convection‑diffusion‑reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard‑constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature.

Abstract:
Contraction analysis offers, through elegant mathematical developments, a unified way of designing observers for a general class of nonlinear systems, where the observer correction term is obtained by solving an infinite dimensional inequality that guarantees global exponential convergence. However, solving the matrix partial differential inequality involved in contraction analysis design is both analytically and numerically challenging and represents a long‑lasting challenge that prevented its wide use. Therefore, the present paper proposes a novel approach that relies on an unsupervised Physics Informed Neural Network (PINN) to design the observer's correction term by enforcing the partial differential inequality in the loss function. The performance of the proposed PINN‑based nonlinear observer is assessed in numerical simulation as well as its robustness to measurement noise and neural network approximation error.

Abstract:
The integration of machine learning techniques into Inertial Confinement Fusion (ICF) simulations has emerged as a powerful approach for enhancing computational efficiency. By replacing the costly Non‑Local Thermodynamic Equilibrium (NLTE) model with machine learning models, significant reductions in calculation time have been achieved. However, determining how to optimize machine learning‑based NLTE models in order to match ICF simulation dynamics remains challenging, underscoring the need for physically relevant error metrics and strategies to enhance model accuracy with respect to these metrics. Thus, we propose novel physics‑informed transformations designed to emphasize energy transport, use these transformations to establish new error metrics, and demonstrate that they yield smaller errors within reduced principal component spaces compared to conventional transformations.

Abstract:
This paper investigates a numerical solution of Allen‑Cahn equation with constant and degenerate mobility, with polynomial and logarithmic energy functionals, with deterministic and random initial functions, and with advective term in one, two, and three spatial dimensions, based on the physics‑informed neural network (PINN). To improve the learning capacity of the PINN, we incorporate the energy dissipation property of the Allen‑Cahn equation as a penalty term into the loss function of the network. To facilitate the learning process of random initials, we employ a continuous analogue of the initial random condition by utilizing the Fourier series expansion. Adaptive methods from traditional numerical analysis are also integrated to enhance the effectiveness of the proposed PINN. Numerical results indicate a consistent decrease in the discrete energy, while also revealing phenomena such as phase separation and metastability.

Abstract:
We introduce a deep learning‑based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics‑informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary conditions, even for singularly perturbed problems. We provide a comprehensive mathematical analysis demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm through numerical experiments, including high‑dimensional, singularly perturbed problems and those posed over non‑convex domains.

Abstract:
We introduce Neural Conjugate Flows (NCF), a class of neural network architectures equipped with exact flow structure. By leveraging topological conjugation, we prove that these networks are not only naturally isomorphic to a continuous group, but are also universal approximators for flows of ordinary differential equation (ODEs). Furthermore, topological properties of these flows can be enforced by the architecture in an interpretable manner. We demonstrate in numerical experiments how this topological group structure leads to concrete computational gains over other physics informed neural networks in estimating and extrapolating latent dynamics of ODEs, while training up to five times faster than other flow‑based architectures.

Abstract:
Soft‑ and hard‑constrained Physics Informed Neural Networks (PINNs) have achieved great success in solving partial differential equations (PDEs). However, these methods still face great challenges when solving the Navier‑Stokes equations (NSEs) with complex boundary conditions. To address these challenges, this paper introduces a novel complementary scheme combining soft and hard constraint PINN methods. The soft‑constrained part is thus formulated to obtain the preliminary results with a lighter training burden, upon which refined results are then achieved using a more sophisticated hard‑constrained mechanism with a primary network and a distance metric network. Specifically, the soft‑constrained part focuses on boundary points, while the primary network emphasizes inner domain points, primarily through PDE loss. Additionally, the novel distance metric network is proposed to predict the power function of the distance from a point to the boundaries, which serves as the weighting factor for the first two components. This approach ensures accurate predictions for both boundary and inner domain areas. The effectiveness of the proposed method on the NSEs problem with complex boundary conditions is demonstrated by solving a 2D cylinder wake problem and a 2D blocked cavity flow with a segmented inlet problem, achieving significantly higher accuracy compared to traditional soft‑ and hard‑constrained PINN approaches. Given PINN's inherent advantages in solving the inverse and the large‑scale problems, which are challenging for traditional computational fluid dynamics (CFD) methods, this approach holds promise for the inverse design of required flow fields by specifically‑designed boundary conditions and the reconstruction of large‑scale flow fields by adding a limited number of training input points. The code for our approach will be made publicly available.

Abstract:
Mueller matrix polarimetry captures essential information about polarized light interactions with a sample, presenting unique challenges for data augmentation in deep learning due to its distinct structure. While augmentations are an effective and affordable way to enhance dataset diversity and reduce overfitting, standard transformations like rotations and flips do not preserve the polarization properties in Mueller matrix images. To this end, we introduce a versatile simulation framework that applies physically consistent rotations and flips to Mueller matrices, tailored to maintain polarization fidelity. Our experimental results across multiple datasets reveal that conventional augmentations can lead to falsified results when applied to polarimetric data, underscoring the necessity of our physics‑based approach. In our experiments, we first compare our polarization‑specific augmentations against real‑world captures to validate their physical consistency. We then apply these augmentations in a semantic segmentation task, achieving substantial improvements in model generalization and performance. This study underscores the necessity of physics‑informed data augmentation for polarimetric imaging in deep learning (DL), paving the way for broader adoption and more robust applications across diverse research in the field. In particular, our framework unlocks the potential of DL models for polarimetric datasets with limited sample sizes. Our code implementation is available at github.com/hahnec/polar_augment.

Abstract:
We propose a novel fine‑tuning method to achieve multi‑operator learning through training a distributed neural operator with diverse function data and then zero‑shot fine‑tuning the neural network using physics‑informed losses for downstream tasks. Operator learning effectively approximates solution operators for PDEs and various PDE‑related problems, yet it often struggles to generalize to new tasks. To address this, we investigate fine‑tuning a pretrained model, while carefully selecting an initialization that enables rapid adaptation to new tasks with minimal data. Our approach combines distributed learning to integrate data from various operators in pre‑training, while physics‑informed methods enable zero‑shot fine‑tuning, minimizing the reliance on downstream data. We investigate standard fine‑tuning and Low‑Rank Adaptation fine‑tuning, applying both to train complex nonlinear target operators that are difficult to learn only using random initialization. Through comprehensive numerical examples, we demonstrate the advantages of our approach, showcasing significant improvements in accuracy. Our findings provide a robust framework for advancing multi‑operator learning and highlight the potential of transfer learning techniques in this domain.

Abstract:
Fetal brain tissue segmentation in magnetic resonance imaging (MRI) is a crucial tool that supports understanding of neurodevelopment, yet it faces challenges due to the heterogeneity of data coming from different scanners and settings, as well as data scarcity. Recent approaches based on domain randomization, like SynthSeg, have shown great potential for single‑source domain generalization by simulating images with randomized contrast and image resolution from the label maps. In this work, we investigate how to maximize the out‑of‑domain (OOD) generalization potential of SynthSegbased methods in fetal brain MRI. Specifically, we demonstrate that the simple Gaussian mixture models employed in FetalSynthSeg outperform physics‑informed generation methods in terms of OOD generalization. We further show that incorporating intensity clustering significantly enhances generalization in settings with limited label classes by producing more realistic synthetic data. By combining synthetic pretraining with fine‑tuning on real images and applying weight‑space interpolation between the two models, we propose DRIFTS as an effective and practical solution for single‑source domain generalization. DRIFTS consistently outperforms current state‑of‑the‑art models across multiple benchmarks and is, to our knowledge, the first method to achieve accurate brain tissue segmentation on fetal T1‑weighted images. We validate our approach on 308 subjects from four datasets acquired at three different sites, covering a range of scanner field strengths (0.55T to 3T) and both T1w and T2w modalities. We conclude with five practical recommendations to guide the development of SynthSeg‑based methods for other organs and imaging modalities.

Abstract:
Forecasting temporal processes such as virus spreading in epidemics often requires more than just observed time‑series data, especially at the beginning of a wave when data is limited. Traditional methods employ mechanistic models like the SIR family, which make strong assumptions about the underlying spreading process, often represented as a small set of compact differential equations. Data‑driven methods such as deep neural networks make no such assumptions and can capture the generative process in more detail, but fail in long‑term forecasting due to data limitations. We propose a new hybrid method called MP‑PINN (Multi‑Phase Physics‑Informed Neural Network) to overcome the limitations of these two major approaches. MP‑PINN instils the spreading mechanism into a neural network, enabling the mechanism to update in phases over time, reflecting the dynamics of the epidemics due to policy interventions. Experiments on COVID‑19 waves demonstrate that MP‑PINN achieves superior performance over pure data‑driven or model‑driven approaches for both short‑term and long‑term forecasting.

Abstract:
Accurately characterizing migration velocity models is crucial for a wide range of geophysical applications, from hydrocarbon exploration to monitoring of CO2 sequestration projects. Traditional velocity model building methods such as Full‑Waveform Inversion (FWI) are powerful but often struggle with the inherent complexities of the inverse problem, including noise, limited bandwidth, receiver aperture and computational constraints. To address these challenges, we propose a scalable methodology that integrates generative modeling, in the form of Diffusion networks, with physics‑informed summary statistics, making it suitable for complicated imaging problems including field datasets. By defining these summary statistics in terms of subsurface‑offset image volumes for poor initial velocity models, our approach allows for computationally efficient generation of Bayesian posterior samples for migration velocity models that offer a useful assessment of uncertainty. To validate our approach, we introduce a battery of tests that measure the quality of the inferred velocity models, as well as the quality of the inferred uncertainties. With modern synthetic datasets, we reconfirm gains from using subsurface‑image gathers as the conditioning observable. For complex velocity model building involving salt, we propose a new iterative workflow that refines amortized posterior approximations with salt flooding and demonstrate how the uncertainty in the velocity model can be propagated to the final product reverse time migrated images. Finally, we present a proof of concept on field datasets to show that our method can scale to industry‑sized problems.

Abstract:
Biophysical model fitting plays a key role in obtaining quantitative parameters from physiological signals and images. However, the model complexity for molecular magnetic resonance imaging (MRI) often translates into excessive computation time, which makes clinical use impractical. Here, we present a generic computational approach for solving the parameter extraction inverse problem posed by ordinary differential equation (ODE) modeling coupled with experimental measurement of the system dynamics. This is achieved by formulating a numerical ODE solver to function as a step‑wise analytical one, thereby making it compatible with automatic differentiation‑based optimization. This enables efficient gradient‑based model fitting, and provides a new approach to parameter quantification based on self‑supervised learning from a single data observation. The neural‑network‑based train‑by‑fit pipeline was used to quantify semisolid magnetization transfer (MT) and chemical exchange saturation transfer (CEST) amide proton exchange parameters in the human brain, in an in‑vivo molecular MRI study (n = 4). The entire pipeline of the first whole brain quantification was completed in 18.3 \pm 8.3 minutes. Reusing the single‑subject‑trained network for inference in new subjects took 1.0 \pm 0.2 s, to provide results in agreement with literature values and scan‑specific fit results.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) in scientific computing. While PINNs typically use multilayer perceptrons (MLPs) as their underlying architecture, recent advancements have explored alternative neural network structures. One such innovation is the Kolmogorov‑Arnold Network (KAN), which has demonstrated benefits over traditional MLPs, including faster neural scaling and better interpretability. The application of KANs to physics‑informed learning has led to the development of Physics‑Informed KANs (PIKANs), enabling the use of KANs to solve PDEs. However, despite their advantages, KANs often suffer from slower training speeds, particularly in higher‑dimensional problems where the number of collocation points grows exponentially with the dimensionality of the system. To address this challenge, we introduce Separable Physics‑Informed Kolmogorov‑Arnold Networks (SPIKANs). This novel architecture applies the principle of separation of variables to PIKANs, decomposing the problem such that each dimension is handled by an individual KAN. This approach drastically reduces the computational complexity of training without sacrificing accuracy, facilitating their application to higher‑dimensional PDEs. Through a series of benchmark problems, we demonstrate the effectiveness of SPIKANs, showcasing their superior scalability and performance compared to PIKANs and highlighting their potential for solving complex, high‑dimensional PDEs in scientific computing.

Abstract:
We propose a scalable preconditioned primal‑dual hybrid gradient algorithm for solving partial differential equations (PDEs). We multiply the PDE with a dual test function to obtain an inf‑sup problem whose loss functional involves lower‑order differential operators. The Primal‑Dual Hybrid Gradient (PDHG) algorithm is then leveraged for this saddle point problem. By introducing suitable precondition operators to the proximal steps in the PDHG algorithm, we obtain an alternative natural gradient ascent‑descent optimization scheme for updating the neural network parameters. We apply the Krylov subspace method (MINRES) to evaluate the natural gradients efficiently. Such treatment readily handles the inversion of precondition matrices via matrix‑vector multiplication. An a posteriori convergence analysis is established for the time‑continuous version of the proposed algorithm for general linear PDEs. By incorporating appropriate boundary loss terms, we further obtain a refined a priori convergence result for elliptic equations in divergence form. The algorithm is tested on various types of PDEs with dimensions ranging from 1 to 50, including linear and nonlinear elliptic equations, reaction‑diffusion equations, and Monge‑Ampère equations stemming from the L^2 optimal transport problems. We compare the performance of the proposed method with several commonly used deep learning algorithms such as physics‑informed neural networks (PINNs), the DeepRitz method and weak adversarial networks (WANs) using either the Adam or the L‑BFGS optimizer. The numerical results suggest that the proposed method performs efficiently and robustly and converges more stably with higher accuracy.

Abstract:
This paper presents a novel neural operator learning framework for designing boundary control to mitigate stop‑and‑go congestion on freeways. The freeway traffic dynamics are described by second‑order coupled hyperbolic partial differential equations (PDEs). The proposed framework learns feedback boundary control strategies from the closed‑loop PDE solution using backstepping controllers, which are widely employed for boundary stabilization of PDE systems. The PDE backstepping control design is time‑consuming and requires intensive depth of expertise, since it involves constructing and solving backstepping control kernels. To address these challenges, we present neural operator (NO) learning schemes for the ARZ traffic system that not only ensure closed‑loop stability robust to parameter and initial condition variations but also accelerate boundary controller computation. The stability guarantee of the NO‑approximated control laws is obtained using Lyapunov analysis. We further propose the physics‑informed neural operator (PINO) to reduce the reliance on extensive training data. The performance of the NO schemes is evaluated by simulated and real traffic data, compared with the benchmark backstepping controller, a Proportional Integral (PI) controller, and a PINN‑based controller. The NO‑approximated methods achieve a computational speedup of approximately 300 times with only a 1% error trade‑off compared to the backstepping controller, while outperforming the other two controllers in both accuracy and computational efficiency. The robustness of the NO schemes is validated using real traffic data, and tested across various initial traffic conditions and demand scenarios. The results show that neural operators can significantly expedite and simplify the process of obtaining controllers for traffic PDE systems with great potential application for traffic management.

Abstract:
In this study, we propose a high‑performance disparity (depth) estimation method using dual‑pixel (DP) images with few parameters. Conventional end‑to‑end deep‑learning methods have many parameters but do not fully exploit disparity constraints, which limits their performance. Therefore, we propose a lightweight disparity estimation method based on a completion‑based network that explicitly constrains disparity and learns the physical and systemic disparity properties of DP. By modeling the DP‑specific disparity error parametrically and using it for sampling during training, the network acquires the unique properties of DP and enhances robustness. This learning also allows us to use a common RGB‑D dataset for training without a DP dataset, which is labor‑intensive to acquire. Furthermore, we propose a non‑learning‑based refinement framework that efficiently handles inherent disparity expansion errors by appropriately refining the confidence map of the network output. As a result, the proposed method achieved state‑of‑the‑art results while reducing the overall system size to 1/5 of that of the conventional method, even without using the DP dataset for training, thereby demonstrating its effectiveness. The code and dataset are available on our project site.

Abstract:
Accurate hypocenter determination in the Nankai Trough subduction zone is essential for hazard assessment and advancing our understanding of seismic activity in the region. A handy hypocenter determination tool incorporating a realistic 3D velocity structure, accessible to the scientific community, is beneficial. In this study, we developed HypoNet Nankai, a rapid hypocenter determination tool based on a physics‑informed neural network (PINN) emulator (surrogate model) for travel time calculations. This tool leverages a PINN trained to predict P‑wave travel times between arbitrary underground sources and surface receivers with a realistic 3D P‑wave velocity structure model of the Nankai Trough subduction zone that incorporates marine seismic survey data. The PINN embeds physical laws, namely, the Eikonal equation, directly into the loss function of training and circumvents the need for labeled training data. To address the training challenges posed by small‑scale features in the velocity model, we employed a simple domain decomposition approach and Fourier feature embedding. Once trained, the PINN immediately infers the P‑wave travel time, enabling rapid hypocenter determination. The data size required to store NNs for travel time calculations is significantly smaller than those of conventional travel‑time tables. HypoNet Nankai provides high flexibility for addition of new observation points. We verified HypoNet Nankai by comparing its performance with a widely used grid‑based numerical method for forward travel time calculations and synthetic hypocenter determination. In both tests, HypoNet Nankai provided results consistent with those for the conventional method. HypoNet Nankai offers a rapid, accurate, and easy‑to‑use hypocenter determination method for the Nankai Trough subduction zone, with greater data efficiency and extendibility compared to conventional approaches.

Abstract:
Solving partial differential equations (PDEs) is essential in scientific forecasting and fluid dynamics. Traditional approaches often incur expensive computational costs and trade‑offs in efficiency and accuracy. Recent deep neural networks have improved the accuracy but require high‑quality training data. Physics‑informed neural networks (PINNs) effectively integrate physical laws to reduce the data reliance in limited sample scenarios. A novel machine‑learning framework, Chebyshev physics‑informed Kolmogorov‑‑Arnold network (ChebPIKAN), is proposed to integrate the robust architectures of Kolmogorov‑‑Arnold networks (KAN) with physical constraints to enhance the calculation accuracy of PDEs for fluid mechanics. We study the fundamentals of KAN, take advantage of the orthogonality of Chebyshev polynomial basis functions in spline fitting, and integrate physics‑informed loss functions that are tailored to specific PDEs in fluid dynamics, including Allen‑‑Cahn equation, nonlinear Burgers equation, Helmholtz equations, Kovasznay flow, cylinder wake flow, and cavity flow. Extensive experiments demonstrate that the proposed ChebPIKAN model significantly outperforms the standard KAN architecture in solving various PDEs by effectively embedding essential physical information. These results indicate that augmenting KAN with physical constraints can alleviate the overfitting issues of KAN and improve the extrapolation performance. Consequently, this study highlights the potential of ChebPIKAN as a powerful tool in computational fluid dynamics and propose a path toward fast and reliable predictions in fluid mechanics and beyond.

Abstract:
Acquisition of large datasets for three‑dimensional (3D) partial differential equations (PDE) is usually very expensive. Physics‑informed neural operator (PINO) eliminates the high costs associated with generation of training datasets, and shows great potential in a variety of partial differential equations. In this work, we employ physics‑informed neural operator, encoding the large‑eddy simulation (LES) equations directly into the neural operator for simulating three‑dimensional incompressible turbulent flows. We develop the LESnets (Large‑Eddy Simulation nets) by adding large‑eddy simulation equations to two different data‑driven models, including Fourier neural operator (FNO) and implicit Fourier neural operator (IFNO) without using label data. Notably, by leveraging only PDE constraints to learn the spatio‑temporal dynamics, LESnets models retain the computational efficiency of data‑driven approaches while obviating the necessity for data. Meanwhile, using LES equations as PDE constraints makes it possible to efficiently predict complex turbulence at coarse grids. We investigate the performance of the LESnets models with two standard three‑dimensional turbulent flows: decaying homogeneous isotropic turbulence and temporally evolving turbulent mixing layer. In the numerical experiments, the LESnets models show similar accuracy as compared to traditional large‑eddy simulation and data‑driven models including FNO and IFNO, and exhibits a robust generalization ability to unseen regime of flow fields. By integrating a single set of flow data, the LESnets models can automatically learn the coefficient of the subgrid scale (SGS) model during the training of the neural operator. Moreover, the well‑trained LESnets models are significantly faster than traditional LES, and exhibits comparable computational efficiency to the data‑driven FNO and IFNO models.

Abstract:
The automatic differentiation (AD) in the vanilla physics‑informed neural networks (PINNs) is the computational bottleneck for the high‑efficiency analysis. The concept of derivative discretization in smoothed particle hydrodynamics (SPH) can provide an accelerated training method for PINNs. In this paper, smoothing kernel physics‑informed neural networks (SK‑PINNs) are established, which solve differential equations using smoothing kernel discretization. It is a robust framework capable of solving problems in the computational mechanics of complex domains. When the number of collocation points gradually increases, the training speed of SK‑PINNs significantly surpasses that of vanilla PINNs. In cases involving large collocation point sets or higher‑order problems, SK‑PINN training can be up to tens of times faster than vanilla PINN. Additionally, analysis using neural tangent kernel (NTK) theory shows that the convergence rates of SK‑PINNs are consistent with those of vanilla PINNs. The superior performance of SK‑PINNs is demonstrated through various examples, including regular and complex domains, as well as forward and inverse problems in fluid dynamics and solid mechanics.

Abstract:
Physics‑informed neural networks (PINNs) are employed to solve the Dyson‑‑Schwinger equations of quantum electrodynamics (QED) in Euclidean space, with a focus on the non‑perturbative generation of the fermion's dynamical mass function in the Landau gauge. By inserting the integral equation directly into the loss function, our PINN framework enables a single neural network to learn a continuous and differentiable representation of the mass function over a spectrum of momenta. Also, we benchmark our approach against a traditional numerical algorithm showing the main differences among them. Our novel strategy, which is expected to be extended to other quantum field theories, is the first step towards forefront applications of machine learning in high‑level theoretical physics.

Abstract:
Reconstructing unknown external source functions is an important perception capability for a large range of robotics domains including manipulation, aerial, and underwater robotics. In this work, we propose a Physics‑Informed Neural Network (PINN [1]) based approach for solving the inverse source problems in robotics, jointly identifying unknown source functions and the complete state of a system given partial and noisy observations. Our approach demonstrates several advantages over prior works (Finite Element Methods (FEM) and data‑driven approaches): it offers flexibility in integrating diverse constraints and boundary conditions; eliminates the need for complex discretizations (e.g., meshing); easily accommodates gradients from real measurements; and does not limit performance based on the diversity and quality of training data. We validate our method across three simulation and real‑world scenarios involving up to 4th order partial differential equations (PDEs), constraints such as Signorini and Dirichlet, and various regression losses including Chamfer distance and L2 norm.

Abstract:
Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real‑world systems remains challenging due to issues in mathematical modeling, nonlinearity, and high dimensionality. In this work, we introduce a data‑driven computational framework to derive low‑dimensional linear models for nonlinear dynamical systems directly from raw experimental data. This framework enables global stability analysis through interpretable linear models that capture the underlying system structure. Our approach employs time‑delay embedding, physics‑informed deep autoencoders, and annealing‑based regularization to identify novel low‑dimensional coordinate representations, unlocking insights across a variety of simulated and previously unstudied experimental dynamical systems. These new coordinate representations enable accurate long‑horizon predictions and automatic identification of intricate invariant sets while providing empirical stability guarantees. Our method offers a promising pathway to analyze complex dynamical behaviors across fields such as physics, climate science, and engineering, with broad implications for understanding nonlinear systems in the real world.

Abstract:
Reconstructing brain sources is a fundamental challenge in neuroscience, crucial for understanding brain function and dysfunction. Electroencephalography (EEG) signals have a high temporal resolution. However, identifying the correct spatial location of brain sources from these signals remains difficult due to the ill‑posed structure of the problem. Traditional methods predominantly rely on manually crafted priors, missing the flexibility of data‑driven learning, while recent deep learning approaches focus on end‑to‑end learning, typically using the physical information of the forward model only for generating training data. We propose the novel hybrid method 3D‑PIUNet for EEG source localization that effectively integrates the strengths of traditional and deep learning techniques. 3D‑PIUNet starts from an initial physics‑informed estimate by using the pseudo inverse to map from measurements to source space. Secondly, by viewing the brain as a 3D volume, we use a 3D convolutional U‑Net to capture spatial dependencies and refine the solution according to the learned data prior. Training the model relies on simulated pseudo‑realistic brain source data, covering different source distributions. Trained on this data, our model significantly improves spatial accuracy, demonstrating superior performance over both traditional and end‑to‑end data‑driven methods. Additionally, we validate our findings with real EEG data from a visual task, where 3D‑PIUNet successfully identifies the visual cortex and reconstructs the expected temporal behavior, thereby showcasing its practical applicability.

Abstract:
The anisotropic structure of the myocardium is a key determinant of the cardiac function. To date, there is no imaging modality to assess in‑vivo the cardiac fiber structure. We recently proposed Fibernet, a method for the automatic identification of the anisotropic conduction ‑‑ and thus fibers ‑‑ in the atria from local electrical recordings. Fibernet uses cardiac activation as recorded during electroanatomical mappings to infer local conduction properties using physics‑informed neural networks. In this work, we extend Fibernet to cope with the uncertainty in the estimated fiber field. Specifically, we use an ensemble of neural networks to produce multiple samples, all fitting the observed data, and compute posterior statistics. We also introduce a methodology to select the best fiber orientation members and define the input of the neural networks directly on the atrial surface. With these improvements, we outperform the previous methodology in terms of fiber orientation error in 8 different atrial anatomies. Currently, our approach can estimate the fiber orientation and conduction velocities in under 7 minutes with quantified uncertainty, which opens the door to its application in clinical practice. We hope the proposed methodology will enable further personalization of cardiac digital twins for precision medicine.

Abstract:
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable solutions, their accuracy is often tied to the use of computationally intensive fine meshes. Moreover, they do not naturally account for measurements or prior solutions, and any change in the problem parameters requires results to be fully recomputed. Neural network‑based approaches, such as physics‑informed neural networks and neural operators, offer a mesh‑free alternative by directly fitting those models to the PDE solution. They can also integrate prior knowledge and tackle entire families of PDEs by simply aggregating additional training losses. Nevertheless, they are highly sensitive to hyperparameters such as collocation points and the weights associated with each loss. This paper addresses these challenges by developing a science‑constrained learning (SCL) framework. It demonstrates that finding a (weak) solution of a PDE is equivalent to solving a constrained learning problem with worst‑case losses. This explains the limitations of previous methods that minimize the expected value of aggregated losses. SCL also organically integrates structural constraints (e.g., invariances) and (partial) measurements or known solutions. The resulting constrained learning problems can be tackled using a practical algorithm that yields accurate solutions across a variety of PDEs, neural network architectures, and prior knowledge levels without extensive hyperparameter tuning and sometimes even at a lower computational cost.

Abstract:
Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the Fokker‑Planck partial differential equation (FP‑PDE). However, it is generally infeasible to solve the FP‑PDE in closed form. In this work, we show that physics‑informed neural networks (PINNs) can be trained to approximate the solution PDF. Our main contribution is the analysis of PINN approximation error: we develop a theoretical framework to construct tight error bounds using PINNs. In addition, we derive a practical error bound that can be efficiently constructed with standard training methods. We discuss that this error‑bound framework generalizes to approximate solutions of other linear PDEs. Empirical results on nonlinear, high‑dimensional, and chaotic systems validate the correctness of our error bounds while demonstrating the scalability of PINNs and their significant computational speedup in obtaining accurate PDF solutions compared to the Monte Carlo approach.

Abstract:
Physics‑Informed Neural Operators provide efficient, high‑fidelity simulations for systems governed by partial differential equations (PDEs). However, most existing studies focus only on multi‑scale, multi‑physics systems within a single spatial region, neglecting the case with multiple interconnected sub‑regions, such as gas and thermal systems. To address this, this paper proposes a Physics‑Informed Partitioned Coupled Neural Operator (PCNO) to enhance the simulation performance of such networks. Compared to the existing Fourier Neural Operator (FNO), this method designs a joint convolution operator within the Fourier layer, enabling global integration capturing all sub‑regions. Additionally, grid alignment layers are introduced outside the Fourier layer to help the joint convolution operator accurately learn the coupling relationship between sub‑regions in the frequency domain. Experiments on gas networks demonstrate that the proposed operator not only accurately simulates complex systems but also shows good generalization and low model complexity.

Abstract:
We propose a workflow based on physics‑informed neural networks (PINNs) to model multiphase fluid flow in fractured porous media. After validating the workflow in forward and inverse modeling of a synthetic problem of flow in fractured porous media, we applied it to a real experimental dataset in which brine is injected at a constant pressure drop into a CO2 saturated naturally fractured shale core plug. The exact spatial positions of natural fractures and the dynamic in‑situ distribution of fluids were imaged using a CT‑scan setup. To model the targeted system, we followed a domain decomposition approach for matrix and fractures and a multi‑network architecture for the separate calculation of water saturation and pressure. The flow equations in the matrix, fractures and interplay between them were solved during training. Prior to fully‑coupled simulations, we proposed pre‑training the model. This aided in a more efficient and successful training of the coupled system. Both for the synthetic and experimental inverse problems, we determined flow parameters within the matrix and the fractures. Multiple random initializations of network and system parameters were performed to assess the uncertainty and uniqueness of the results. The results confirmed the precision of the inverse calculated parameters in retrieving the main flow characteristics of the system. The consideration of multiscale matrix‑fracture impacts is commonly overlooked in existing workflows. Accounting for them led to several orders of magnitude variations in the calculated flow properties compared to not accounting for them. To the best of our knowledge, the proposed PINNs‑based workflow is the first to offer a reliable and computationally efficient solution for inverse modeling of multiphase flow in fractured porous media, achieved through history‑matching noisy and multi‑fidelity experimental measurements.

Abstract:
Alternative Current Optimal Power Flow (AC‑OPF) is essential for efficient power system planning and real‑time operation but remains an NP‑hard and non‑convex optimization problem with significant computational challenges. This paper proposes a novel hybrid classical‑quantum deep learning framework for AC‑OPF problem, integrating parameterized quantum circuits (PQCs) for feature extraction with classical deep learning for data encoding and decoding. The proposed framework integrates two types of residual connection structures to mitigate the ``barren plateau" problem in quantum circuits, enhancing training stability and convergence. Furthermore, a physics‑informed neural network (PINN) module is incorporated to guarantee tolerable constraint violation, improving the physical consistency and reliability of AC‑OPF solutions. Experimental evaluations on multiple IEEE test systems demonstrate that the proposed approach achieves superior accuracy, generalization, and robustness to quantum noise while requiring minimal quantum resources.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a powerful tool for solving physical systems described by partial differential equations (PDEs). However, their accuracy in dynamical systems, particularly those involving sharp moving boundaries with complex initial morphologies, remains a challenge. This study introduces an approach combining residual‑based adaptive refinement (RBAR) with causality‑informed training to enhance the performance of PINNs in solving spatio‑temporal PDEs. Our method employs a three‑step iterative process: initial causality‑based training, RBAR‑guided domain refinement, and subsequent causality training on the refined mesh. Applied to the Allen‑‑Cahn equation, a widely used model in phase field simulations, our approach demonstrates significant improvements in solution accuracy and computational efficiency over traditional PINNs. Notably, we observe an overshoot and relocate phenomenon in dynamic cases with complex morphologies, showcasing the method's adaptive error correction capabilities. This synergistic interaction between RBAR and causality training enables accurate capture of interface evolution, even in challenging scenarios where traditional PINNs fail. Our framework not only resolves the limitations of uniform refinement strategies but also provides a generalizable methodology for solving a broad range of spatio‑temporal PDEs. The enhanced performance of the RBAR‑‑causality combined framework demonstrates its strong potential for advancing PINN‑based modeling of physical systems characterized by complex, evolving interfaces.

Abstract:
Accurately predicting the dynamic responses of building structures under seismic loads is essential for ensuring structural safety and minimizing potential damage. This critical aspect of structural analysis allows engineers to evaluate how structures perform under various loading conditions, facilitating informed design and safety decisions. Traditional methods, which rely on complex finite element models often struggle with balancing computational efficiency and accuracy. To address this challenge, we introduce SeisGPT, a data‑driven, large physics‑informed model that leverages deep neural networks based on the Generative Pre‑trained Transformer (GPT) architecture. SeisGPT is designed to predict, in real‑time the dynamic behavior of building structures under seismic forces. Trained on a diverse corpus of seismic data and structural engineering principles, it instantly generates predictive responses, including displacement, acceleration, and inter‑story drift, with high accuracy and computational efficiency. Its adaptability across various building typologies and seismic intensities makes this framework a valuable tool for designing robust structures and assessing seismic risk. Through comprehensive validation, this approach exhibits superior performance, offering engineers and researchers a powerful tool for assessing seismic response and informing resilient design strategies. This innovative framework represents a significant advancement in seismic engineering practice, with potential applications in mitigating seismic hazards and enhancing structural resilience.

Abstract:
In recent years, Artificial intelligence (AI) has become ubiquitous, empowering various fields, especially integrating artificial intelligence and traditional science (AI for Science: Artificial intelligence for science), which has attracted widespread attention. In AI for Science, using artificial intelligence algorithms to solve partial differential equations (AI for PDEs: Artificial intelligence for partial differential equations) has become a focal point in computational mechanics. The core of AI for PDEs is the fusion of data and partial differential equations (PDEs), which can solve almost any PDEs. Therefore, this article provides a comprehensive review of the research on AI for PDEs, summarizing the existing algorithms and theories. The article discusses the applications of AI for PDEs in computational mechanics, including solid mechanics, fluid mechanics, and biomechanics. The existing AI for PDEs algorithms include those based on Physics‑Informed Neural Networks (PINNs), Deep Energy Methods (DEM), Operator Learning, and Physics‑Informed Neural Operator (PINO). AI for PDEs represents a new method of scientific simulation that provides approximate solutions to specific problems using large amounts of data, then fine‑tuning according to specific physical laws, avoiding the need to compute from scratch like traditional algorithms. Thus, AI for PDEs is the prototype for future foundation models in computational mechanics, capable of significantly accelerating traditional numerical algorithms.

Abstract:
Modeling distributions that depend on external control parameters is a common scenario in diverse applications like molecular simulations, where system properties like temperature affect molecular configurations. Despite the relevance of these applications, existing solutions are unsatisfactory as they require severely restricted model architectures or rely on energy‑based training, which is prone to instability. We introduce TRADE, which overcomes these limitations by formulating the learning process as a boundary value problem. By initially training the model for a specific condition using either i.i.d.~samples or backward KL training, we establish a boundary distribution. We then propagate this information across other conditions using the gradient of the unnormalized density with respect to the external parameter. This formulation, akin to the principles of physics‑informed neural networks, allows us to efficiently learn parameter‑dependent distributions without restrictive assumptions. Experimentally, we demonstrate that TRADE achieves excellent results in a wide range of applications, ranging from Bayesian inference and molecular simulations to physical lattice models.

Abstract:
A novel physics‑informed operator learning technique based on spectral methods is introduced to model the complex behavior of heterogeneous materials. The Lippmann‑Schwinger operator in Fourier space is employed to construct physical constraints with minimal computational overhead, effectively eliminating the need for automatic differentiation. The introduced methodology accelerates the training process by enabling gradient construction on a fixed, finite discretization in Fourier space. Later, the spectral physics‑informed finite operator learning (SPiFOL) framework is built based on this discretization and trained to map the arbitrary shape of microstructures to their mechanical responses (strain fields) without relying on labeled data. The training is done by minimizing equilibrium in Fourier space concerning the macroscopic loading condition, which also guarantees the periodicity. SPiFOL, as a physics‑informed operator learning method, enables rapid predictions through forward inference after training. To ensure accuracy, we incorporate physical constraints and diversify the training data. However, performance may still degrade for out‑of‑distribution microstructures. SPiFOL is further enhanced by integrating a Fourier Neural Operator (FNO). Compared to the standard data‑driven FNO, SPiFOL shows higher accuracy in predicting stress fields and provides nearly resolution‑independent results. Additionally, its zero‑shot super‑resolution capabilities are explored in heterogeneous domains. Finally, SPiFOL is extended to handle 3D problems and further adapted to finite elasticity, demonstrating the robustness of the framework in handling nonlinear mechanical behavior. The framework shows great potential for efficient and scalable prediction of mechanical responses in complex material systems while also reducing the training time required for training physics‑informed neural operators.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a powerful tool for analyzing nonlinear partial differential equations and identifying governing equations from observational data. In this study, we apply PINNs to investigate vortex‑type solutions of quasi‑integrable equations in two spatial dimensions, specifically the Zakharov‑Kuznetsov (ZK) and the Regularized Long‑Wave (RLW) equations. These equations are toy models for geostrophic shallow water dynamics in planetary atmospheres. We first demonstrate that PINNs can successfully solve these equations in the forward process using a mesh‑free approach with automatic differentiation. However, in the inverse process, substantial misidentification occurs due to the structural similarities between the ZK and the RLW equations. To address this issue, we then introduce conservation law‑enhanced PINNs, initial condition variations, and a friction‑based perturbation approach to improve identification accuracy. Our results show that incorporating small perturbations while preserving conservation laws significantly enhances the resolution of equation identification. These findings may contribute to the broader goal of using deep learning techniques for discovering governing equations in complex fluid dynamical systems, such as Jupiter's Great Red Spot.

Abstract:
Computational fluid dynamics (CFD) solvers employing two‑equation eddy viscosity models are the industry standard for simulating turbulent flows using the Reynolds‑averaged Navier‑Stokes (RANS) formulation. While these methods are computationally less expensive than direct numerical simulations, they can still incur significant computational costs to achieve the desired accuracy. In this context, physics‑informed neural networks (PINNs) offer a promising approach for developing parametric surrogate models that leverage both existing, but limited CFD solutions and the governing differential equations to predict simulation outcomes in a computationally efficient, differentiable, and near real‑time manner. In this work, we build upon the previously proposed RANS‑PINN framework, which only focused on predicting flow over a cylinder. To investigate the efficacy of RANS‑PINN as a viable approach to building parametric surrogate models, we investigate its accuracy in predicting relevant turbulent flow variables for both internal and external flows. To ensure training convergence with a more complex loss function, we adopt a novel sampling approach that exploits the domain geometry to ensure a proper balance among the contributions from various regions within the solution domain. The effectiveness of this framework is then demonstrated for two scenarios that represent a broad class of internal and external flow problems.

Abstract:
The advent of wurtzite ferroelectrics is enabling a new generation of ferroelectric devices for computer memory that has the potential to bypass the von Neumann bottleneck, due to their robust polarization and silicon compatibility. However, the microscopic switching mechanism of wurtzites is still undetermined due to the limitations of density functional theory simulation size and experimental temporal and spatial resolution. Thus, physics‑informed materials engineering to reduce coercive field and breakdown in these devices has been limited. Here, the atomistic mechanism of domain wall migration and domain growth in wurtzites is uncovered using molecular dynamics and Monte Carlo simulations of aluminum nitride. We reveal the anomalous switching mechanism of fast 1D single columns of atoms propagating from a slow‑moving 2D fractal‑like domain wall. We find that the critical nucleus in wurtzites is a single aluminum ion that breaks its bond with one nitrogen and bonds to another nitrogen; this creates a cascade that only flips atoms directly in the same column, due to the extreme locality (sharpness) of the domain walls in wurtzites. We further show how the fractal shape of the domain wall in the 2D plane breaks assumptions in the KAI model and leads to the anomalously fast switching in wurtzite structured ferroelectrics.

Abstract:
The Matrix‑Element Method (MEM) has long been a cornerstone of data analysis in high‑energy physics. It leverages theoretical knowledge of parton‑level processes and symmetries to evaluate the likelihood of observed events. In parallel, the advent of geometric deep learning has enabled neural network architectures that incorporate known symmetries directly into their design, leading to more efficient learning. This paper presents a novel approach that combines MEM‑inspired symmetry considerations with equivariant neural network design for particle physics analysis. Even though Lorentz invariance and permutation invariance overall reconstructed objects are the largest and most natural symmetry in the input domain, we find that they are sub‑optimal in most practical search scenarios. We propose a longitudinal boost‑equivariant message‑passing neural network architecture that preserves relevant discrete symmetries. We present numerical studies demonstrating MEM‑inspired architectures achieve new state‑of‑the‑art performance in distinguishing di‑Higgs decays to four bottom quarks from the QCD background, with enhanced sample and parameter efficiencies. This synergy between MEM and equivariant deep learning opens new directions for physics‑informed architecture design, promising more powerful tools for probing physics beyond the Standard Model.

Abstract:
In cellular mobile networks, wireless channel quality (CQ) is a crucial factor in determining communication performance and user's network experience. Accurately predicting CQ based on real environmental characteristics, specific base station configurations and user trajectories can help network operators optimize base station deployment, improving coverage and capacity. The Received Signal Reference Power (RSRP) and Signal‑to‑Interference‑plus‑Noise Ratio (SINR) of user equipment (UE) are key indicators of CQ in wireless communication. However, existing researches have limitations in terms of generation accuracy. Regression methods such as statistical inference and random forests fail to effectively capture the unique characteristics of wireless environments; theoretical derivations relying on specific communication protocols lack generalization capability; data‑driven machine learning (ML) methods like Long Short‑Term Memory (LSTM) Network often suffer from a lack of interpretability. To overcome these limitations, we propose physics‑informed diffusion models, which accurately generate RSRP and SINR at UE based on the wireless environment, base station configurations, and user trajectories. The model adopts a modular and end‑to‑end design, employing a teacher‑student framework to achieve knowledge distillation. This method integrates expert knowledge into the training of diffusion models, enhancing both the interpretability and accuracy, while also facilitating faster convergence of the model parameters. Furthermore, it allows for self‑adaptation in various scenarios through few‑shot learning. This approach provides valuable guidance for optimizing base station deployment, predicting user network experience, and building real‑world simulators.

Abstract:
Physics‑informed neural networks (PINNs) incorporate physical laws into their training to efficiently solve partial differential equations (PDEs) with minimal data. However, PINNs fail to guarantee adherence to conservation laws, which are also important to consider in modeling physical systems. To address this, we proposed PINN‑Proj, a PINN‑based model that uses a novel projection method to enforce conservation laws. We found that PINN‑Proj substantially outperformed PINN in conserving momentum and lowered prediction error by three to four orders of magnitude from the best benchmark tested. PINN‑Proj also performed marginally better in the separate task of state prediction on three PDE datasets.

Abstract:
Topological plasmonics offers new ways to manipulate light by combining concepts from topology and plasmonics, similar to topological edge states in photonics. However, designing such topological states remains challenging due to the complexity of the high dimensional design space. We present a novel method that uses supervised, physics informed deep learning and surrogate modeling to design topological devices for specific wavelengths. By embedding physical constrains in the neural network training, our model efficiently explores the design space, significantly reducing simulation time. Additionally, we use non planar wavefront excitations via electron beams to probe topologically protected plasmonic modes, making the design and training process nonlinear. Using this approach, we design a topological device with unidirectional edge modes in a ring resonator at specific operational frequencies. Our method reduces computational cost and time while maintaining high accuracy, highlighting the potential of combining machine learning and advanced techniques for photonic device innovation.

Abstract:
At the forefront of control techniques is Model Predictive Control (MPC). While MPCs are effective, their requisite to recompute an optimal control given a new state leads to sparse response to the system and may make their implementation infeasible in small systems with low computational resources. To address these limitations in stability control, this research presents a small deterministic Physics‑Informed MPC Surrogate model (PI‑MPCS). PI‑MPCS was developed to approximate the control by an MPC while encouraging stability and robustness through the integration of the system dynamics and the formation of a Lyapunov stability profile. Empirical results are presented on the task of 2D quadcopter landing. They demonstrate a rapid and precise MPC approximation on a non‑linear system along with an estimated two times speed up on the computational requirements when compared against an MPC. PI‑MPCS, in addition, displays a level of stable control for in‑ and out‑of‑distribution states as encouraged by the discrete dynamics residual and Lyapunov stability loss functions. PI‑MPCS is meant to serve as a surrogate to MPC on situations in which the computational resources are limited.

Abstract:
Traditional rule‑based physical models are limited by their reliance on singular physical formulas and parameters, making it difficult to effectively tackle the intricate tasks associated with crowd simulation. Recent research has introduced deep learning methods to tackle these issues, but most current approaches focus primarily on generating pedestrian trajectories, often lacking interpretability and failing to provide real‑time dynamic simulations.To address the aforementioned issues, we propose a novel data‑driven crowd simulation framework that integrates Physics‑informed Machine Learning (PIML) with navigation potential fields. Our approach leverages the strengths of both physical models and PIML. Specifically, we design an innovative Physics‑informed Spatio‑temporal Graph Convolutional Network (PI‑STGCN) as a data‑driven module to predict pedestrian movement trends based on crowd spatio‑temporal data. Additionally, we construct a physical model of navigation potential fields based on flow field theory to guide pedestrian movements, thereby reinforcing physical constraints during the simulation. In our framework, navigation potential fields are dynamically computed and updated based on the movement trends predicted by the PI‑STGCN, while the updated crowd dynamics, guided by these fields, subsequently feed back into the PI‑STGCN. Comparative experiments on two publicly available large‑scale real‑world datasets across five scenes demonstrate that our proposed framework outperforms existing rule‑based methods in accuracy and fidelity. The similarity between simulated and actual pedestrian trajectories increases by 10.8%, while the average error is reduced by 4%. Moreover, our framework exhibits greater adaptability and better interpretability compared to methods that rely solely on deep learning for trajectory generation.

Abstract:
Recent success of diffusion models has inspired a surge of interest in developing sampling techniques using reverse diffusion processes. However, accurately estimating the drift term in the reverse stochastic differential equation (SDE) solely from the unnormalized target density poses significant challenges, hindering existing methods from achieving state‑of‑the‑art performance. In this paper, we introduce the Diffusion‑PINN Sampler (DPS), a novel diffusion‑based sampling algorithm that estimates the drift term by solving the governing partial differential equation of the log‑density of the underlying SDE marginals via physics‑informed neural networks (PINN). We prove that the error of log‑density approximation can be controlled by the PINN residual loss, enabling us to establish convergence guarantees of DPS. Experiments on a variety of sampling tasks demonstrate the effectiveness of our approach, particularly in accurately identifying mixing proportions when the target contains isolated components.

Abstract:
Control in fluid environments is an important research area with numerous applications across various domains, including underwater robotics, aerospace engineering, and biomedical systems. However, in practice, control methods often face challenges due to sparse or missing observations, stemming from sensor limitations and faults. These issues result in observations that are not only sparse but also inconsistent in their number and modalities (e.g., velocity and pressure sensors). In this work, we propose a Physics‑Informed Representation (PIR) algorithm for multi‑modal policies of control to leverage the sparse and random observations in complex fluid environments. PIR integrates sparse observational data with the Partial Differential Equation (PDE) information to distill a unified representation of fluid systems. The main idea is that PDE solutions are determined by three elements: the equation, initial conditions, and boundary conditions. Given the equation, we only need to learn the representation of the initial and boundary conditions, which define a trajectory of a specific fluid system. Specifically, it leverages PDE loss to fit the neural network and data loss calculated on the observations with random quantities and multi‑modalities to propagate the information with initial and boundary conditions into the representations. The representations are the learnable parameters or the output of the encoder. In the experiments, the PIR illustrates the superior consistency with the features of the ground truth compared with baselines, even when there are missing modalities. Furthermore, PIR combined with Reinforcement Learning has been successfully applied in control tasks where the robot leverages the learned state by PIR faster and more accurately, passing through the complex vortex street from a random starting location to reach a random target.

Abstract:
Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least‑Squares‑based Neural Network (LS‑Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least‑squares solver. In this approach, the output of the deep neural network consists of a vector‑valued function, interpreted as basis functions for the parametric solution space, and the least‑squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS‑Net method requires a quadratic loss function for the least‑squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics‑Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS‑net method by solving one‑ and two‑dimensional problems. Numerical results clearly demonstrate the method's ability to approximate parametric solutions.

Abstract:
We develop a method for multifidelity Kolmogorov‑Arnold networks (KANs), which use a low‑fidelity model along with a small amount of high‑fidelity data to train a model for the high‑fidelity data accurately. Multifidelity KANs (MFKANs) reduce the amount of expensive high‑fidelity data needed to accurately train a KAN by exploiting the correlations between the low‑ and high‑fidelity data to give accurate and robust predictions in the absence of a large high‑fidelity dataset. In addition, we show that multifidelity KANs can be used to increase the accuracy of physics‑informed KANs (PIKANs), without the use of training data.

Abstract:
In an era increasingly focused on green computing and explainable AI, revisiting traditional approaches in theoretical and phenomenological particle physics is paramount. This project evaluates various machine learning (ML) algorithms‑including Nearest Neighbors, Decision Trees, Random Forest, AdaBoost, Naive Bayes, Quadratic Discriminant Analysis (QDA), and XGBoost‑alongside standard neural networks and a novel Physics‑Informed Neural Network (PINN) for physics data analysis. We apply these techniques to a binary classification task that distinguishes the experimental viability of simulated scenarios based on Higgs observables and essential parameters. Through this comprehensive analysis, we aim to showcase the capabilities and computational efficiency of each model in binary classification tasks, thereby contributing to the ongoing discourse on integrating ML and Deep Neural Networks (DNNs) into physics research. In this study, XGBoost emerged as the preferred choice among the evaluated machine learning algorithms for its speed and effectiveness, especially in the initial stages of computation with limited datasets. However, while standard Neural Networks and Physics‑Informed Neural Networks (PINNs) demonstrated superior performance in terms of accuracy and adherence to physical laws, they require more computational time. These findings underscore the trade‑offs between computational efficiency and model sophistication.

Abstract:
The use of machine learning (ML) algorithms in molecular simulations has become commonplace in recent years. There now exists, for instance, a multitude of ML force field algorithms that have enabled simulations approaching ab initio level accuracy at time scales and system sizes that significantly exceed what is otherwise possible with traditional methods. Far fewer algorithms exist for predicting rotationally equivariant, tensorial properties such as the electric polarizability. Here, we introduce a kernel ridge regression algorithm for machine learning of the polarizability tensor. This algorithm is based on the bond polarizability model and allows prediction of the tensor components at the cost similar to that of scalar quantities. We subsequently show the utility of this algorithm by simulating gas phase Raman spectra of biphenyl and malonaldehyde using classical molecular dynamics simulations of these systems performed with the recently developed MACE‑OFF23 potential. The calculated spectra are shown to agree very well with the experiments and thus confirm the expediency of our algorithm as well as the accuracy of the used force field. More generally, this work demonstrates the potential of physics‑informed approaches to yield simple yet effective machine learning algorithms for molecular properties.

Abstract:
Markov processes serve as a universal model for many real‑world random processes. This paper presents a data‑driven approach for learning these models through the spectral decomposition of the infinitesimal generator (IG) of the Markov semigroup. The unbounded nature of IGs complicates traditional methods such as vector‑valued regression and Hilbert‑Schmidt operator analysis. Existing techniques, including physics‑informed kernel regression, are computationally expensive and limited in scope, with no recovery guarantees for transfer operator methods when the time‑lag is small. We propose a novel method that leverages the IG's resolvent, characterized by the Laplace transform of transfer operators. This approach is robust to time‑lag variations, ensuring accurate eigenvalue learning even for small time‑lags. Our statistical analysis applies to a broader class of Markov processes than current methods while reducing computational complexity from quadratic to linear in the state dimension. Finally, we illustrate the behaviour of our method in two experiments.

Abstract:
We propose a novel dual physics‑informed neural network for topology optimization (DPNN‑TO), which merges physics‑informed neural networks (PINNs) with the traditional SIMP‑based topology optimization (TO) algorithm. This approach leverages two interlinked neural networks‑a displacement network and an implicit density network‑connected through an energy‑minimization‑based loss function derived from the variational principles of the governing equations. By embedding deep learning within the physical constraints of the problem, DPNN‑TO eliminates the need for large‑scale data and analytical sensitivity analysis, addressing key limitations of traditional methods. The framework efficiently minimizes compliance through energy‑based objectives while enforcing volume fraction constraints, producing high‑resolution designs for both 2D and 3D optimization problems. Extensive numerical validation demonstrates that DPNN‑TO outperforms conventional methods, solving complex structural optimization scenarios with greater flexibility and computational efficiency, while addressing challenges such as multiple load cases and three‑dimensional problems without compromising accuracy.

Abstract:
Our proposal is on a new stochastic optimizer for non‑convex and possibly non‑smooth objective functions typically defined over large dimensional design spaces. Towards this, we have tried to bridge noise‑assisted global search and faster local convergence, the latter being the characteristic feature of a Newton‑like search. Our specific scheme ‑‑ acronymed FINDER (Filtering Informed Newton‑like and Derivative‑free Evolutionary Recursion), exploits the nonlinear stochastic filtering equations to arrive at a derivative‑free update that has resemblance with the Newton search employing the inverse Hessian of the objective function. Following certain simplifications of the update to enable a linear scaling with dimension and a few other enhancements, we apply FINDER to a range of problems, starting with some IEEE benchmark objective functions to a couple of archetypal data‑driven problems in deep networks to certain cases of physics‑informed deep networks. The performance of the new method vis‑á‑vis the well‑known Adam and a few others bears evidence to its promise and potentialities for large dimensional optimization problems of practical interest.

Abstract:
We study numerical algorithms to solve a specific Partial Differential Equation (PDE), namely the Stefan problem, using Physics Informed Neural Networks (PINNs). This problem describes the heat propagation in a liquid‑solid phase change system. It implies a heat equation and a discontinuity at the interface where the phase change occurs. In the context of PINNs, this model leads to difficulties in the learning process, especially near the interface of phase change. We present different strategies that can be used in this context. We illustrate our results and compare with classical solvers for PDEs (finite differences).

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as powerful tools for solving partial differential equations (PDEs). However, training PINNs from scratch is often computationally intensive and time‑consuming. To address this problem, we propose a parameter‑efficient approach that fine‑tunes pre‑trained DeepONet models within the PINN framework (FTO‑PINN), enabling more efficient meshless PDE solving. Specifically, we freeze the weights of the pre‑trained DeepONet model and fine‑tune the output of the branch net by incorporating a small number of new trainable parameters, which can be quickly determined using least‑squares techniques. Additionally, we introduce trunk net expansions and low‑rank adaptation strategies to further enhance the performance of FTO‑PINN. The effectiveness of our proposed method is demonstrated through a series of numerical experiments across various types of PDEs. FTO‑PINN significantly reduces the training time of vanilla PINNs while maintaining comparable accuracy, and outperforms DeepONet, which is pre‑trained on general function data, in both fidelity and generalization capabilities.

Abstract:
Tele‑operated driving (ToD) systems are special types of cyber‑physical systems (CPSs) where the operator remotely controls the steering, acceleration, and braking actions of the vehicle. Malicious actors may inject false data in communication channels to manipulate the tele‑operators driving commands to cause harm. Hence, protection of this communication is necessary for the safe operation of the target vehicle. However, according to the National Institute of Standards and Technology (NIST) cybersecurity framework, protection merely is not enough and the detection of an attack is necessary. Moreover, UN R155 mandates that security incidents across vehicle fleets be detected and logged. Thus, cyber‑physical threats of ToD are modeled with an attack‑centric approach in this paper. Then, an attack model with false data injection (FDI) on steering control commands is created from real vehicle data. The risk of this attack model is assessed for a last‑mile delivery (LMD) application. Finally, a physics‑informed context‑aware anomaly detection system (PCADS) is proposed to detect such false injection attacks, and preliminary experimental results are presented to validate the model.

Abstract:
For the 3D localization problem using point spread function (PSF) engineering, we propose a novel enhancement of our previously introduced localization neural network, LocNet. The improved network is a physics‑informed neural network (PINN) that we call PiLocNet. Previous works on the localization problem may be categorized separately into model‑based optimization and neural network approaches. Our PiLocNet combines the unique strengths of both approaches by incorporating forward‑model‑based information into the network via a data‑fitting loss term that constrains the neural network to yield results that are physically sensible. We additionally incorporate certain regularization terms from the variational method, which further improves the robustness of the network in the presence of image noise, as we show for the Poisson and Gaussian noise models. This framework accords interpretability to the neural network, and the results we obtain show its superiority. Although the paper focuses on the use of single‑lobe rotating PSF to encode the full 3D source location, we expect the method to be widely applicable to other PSFs and imaging problems that are constrained by known forward processes.

Abstract:
Physics‑Informed Neural Networks (PINNs) have emerged as a key tool in Scientific Machine Learning since their introduction in 2017, enabling the efficient solution of ordinary and partial differential equations using sparse measurements. Over the past few years, significant advancements have been made in the training and optimization of PINNs, covering aspects such as network architectures, adaptive refinement, domain decomposition, and the use of adaptive weights and activation functions. A notable recent development is the Physics‑Informed Kolmogorov‑Arnold Networks (PIKANS), which leverage a representation model originally proposed by Kolmogorov in 1957, offering a promising alternative to traditional PINNs. In this review, we provide a comprehensive overview of the latest advancements in PINNs, focusing on improvements in network design, feature expansion, optimization techniques, uncertainty quantification, and theoretical insights. We also survey key applications across a range of fields, including biomedicine, fluid and solid mechanics, geophysics, dynamical systems, heat transfer, chemical engineering, and beyond. Finally, we review computational frameworks and software tools developed by both academia and industry to support PINN research and applications.

Abstract:
By leveraging neural networks, the emerging field of scientific machine learning (SciML) offers novel approaches to address complex problems governed by partial differential equations (PDEs). In practical applications, challenges arise due to the distributed essence of data, concerns about data privacy, or the impracticality of transferring large volumes of data. Federated learning (FL), a decentralized framework that enables the collaborative training of a global model while preserving data privacy, offers a solution to the challenges posed by isolated data pools and sensitive data issues. Here, this paper explores the integration of FL and SciML to approximate complex functions and solve differential equations. We propose two novel models: federated physics‑informed neural networks (FedPINN) and federated deep operator networks (FedDeepONet). We further introduce various data generation methods to control the degree of non‑independent and identically distributed (non‑iid) data and utilize the 1‑Wasserstein distance to quantify data heterogeneity in function approximation and PDE learning. We systematically investigate the relationship between data heterogeneity and federated model performance. Additionally, we propose a measure of weight divergence and develop a theoretical framework to establish growth bounds for weight divergence in federated learning compared to traditional centralized learning. To demonstrate the effectiveness of our methods, we conducted 10 experiments, including 2 on function approximation, 5 PDE problems on FedPINN, and 3 PDE problems on FedDeepONet. These experiments demonstrate that proposed federated methods surpass the models trained only using local data and achieve competitive accuracy of centralized models trained using all data.

Abstract:
Physics‑informed neural networks (PINN) is a machine learning (ML)‑based method to solve partial differential equations that has gained great popularity due to the fast development of ML libraries in the last few years. The Poisson‑Boltzmann equation (PBE) is widely used to model mean‑field electrostatics in molecular systems, and in this work we present a detailed investigation of the use of PINN to solve the PBE. Starting from a multidomain PINN for the PBE with an interface, we assess the importance of incorporating different features into the neural network architecture. Our findings indicate that the most accurate architecture utilizes input and output scaling layers, a random Fourier features layer, trainable activation functions, and a loss balancing algorithm. The accuracy of our implementation is of the order of 10^‑2 ‑‑ 10^‑3, which is similar to previous work using PINN to solve other differential equations. We also explore the possibility of incorporating experimental information into the model, and discuss challenges and future work, especially regarding the nonlinear PBE. Along with this manuscript, we are providing an open‑source implementation to easily perform computations from a PDB file. We hope this work will motivate application scientists into using PINN to study molecular electrostatics, as ML technology continues to evolve at a high pace.

Abstract:
This paper presents a scalable method for friction identification in robots equipped with electric motors and high‑ratio harmonic drives, utilizing Physics‑Informed Neural Networks (PINN). This approach eliminates the need for dedicated setups and joint torque sensors by leveraging the roboťs intrinsic model and state data. We present a comprehensive pipeline that includes data acquisition, preprocessing, ground truth generation, and model identification. The effectiveness of the PINN‑based friction identification is validated through extensive testing on two different joints of the humanoid robot ergoCub, comparing its performance against traditional static friction models like the Coulomb‑viscous and Stribeck‑Coulomb‑viscous models. Integrating the identified PINN‑based friction models into a two‑layer torque control architecture enhances real‑time friction compensation. The results demonstrate significant improvements in control performance and reductions in energy losses, highlighting the scalability and robustness of the proposed method, also for application across a large number of joints as in the case of humanoid robots.

Abstract:
Precision measurements of molecules offer an unparalleled paradigm to probe physics beyond the Standard Model. The rich internal structure within these molecules makes them exquisite sensors for detecting fundamental symmetry violations, local position invariance, and dark matter. While trapping and control of diatomic and a few very simple polyatomic molecules have been experimentally demonstrated, leveraging the complex rovibrational structure of more general polyatomics demands the development of robust and efficient quantum control schemes. In this study, we present reinforcement‑learning quantum‑logic spectroscopy (RL‑QLS), a general, reinforcement‑learning‑designed, quantum logic approach to prepare molecular ions in single, pure quantum states. The reinforcement learning agent optimizes the pulse sequence, each followed by a projective measurement, and probabilistically manipulates the collapse of the quantum system to a single state. The performance of the control algorithm is numerically demonstrated for the polyatomic molecule H_3O^+ with 130 thermally populated eigenstates and degenerate transitions within inversion doublets, where quantum Markov decision process modeling and a physics‑informed reward function play a key role, as well as for CaH^+ under the disturbance of environmental thermal radiation. The developed theoretical framework cohesively integrates techniques from quantum chemistry, AMO physics, and artificial intelligence, and we expect that the results can be readily implemented for quantum control of polyatomic molecular ions with densely populated structures, thereby enabling new experimental tests of fundamental theories.

Abstract:
Multilayer perceptron (MLP) networks are predominantly used to develop data‑driven constitutive models for granular materials. They offer a compelling alternative to traditional physics‑based constitutive models in predicting nonlinear responses of these materials, e.g., elasto‑plasticity, under various loading conditions. To attain the necessary accuracy, MLPs often need to be sufficiently deep or wide, owing to the curse of dimensionality inherent in these problems. To overcome this limitation, we present an elasto‑plasticity informed Chebyshev‑based Kolmogorov‑Arnold network (EPi‑cKAN) in this study. This architecture leverages the benefits of KANs and augmented Chebyshev polynomials, as well as integrates physical principles within both the network structure and the loss function. The primary objective of EPi‑cKAN is to provide an accurate and generalizable function approximation for non‑linear stress‑strain relationships, using fewer parameters compared to standard MLPs. To evaluate the efficiency, accuracy, and generalization capabilities of EPi‑cKAN in modeling complex elasto‑plastic behavior, we initially compare its performance with other cKAN‑based models, which include purely data‑driven parallel and serial architectures. Furthermore, to differentiate EPi‑cKAN's distinct performance, we also compare it against purely data‑driven and physics‑informed MLP‑based methods. Lastly, we test EPi‑cKAN's ability to predict blind strain‑controlled paths that extend beyond the training data distribution to gauge its generalization and predictive capabilities. Our findings indicate that, even with limited data and fewer parameters compared to other approaches, EPi‑cKAN provides superior accuracy in predicting stress components and demonstrates better generalization when used to predict sand elasto‑plastic behavior under blind triaxial axisymmetric strain‑controlled loading paths.

Abstract:
All numerical solutions of the pulsar magnetosphere over the past 25 years show closed‑line regions that end a significant distance inside the light cylinder, and manifest thick strongly dissipative separatrix surfaces instead of thin current sheets, with a tip that has a distinct pointed Y shape instead of a T shape. We need to understand the origin of these results which were not predicted by our early theories of the pulsar magnetosphere. In order to gain new intuition on this problem, we set out to obtain the theoretical steady‑state solution of the 3D ideal force‑free magnetosphere with zero dissipation along the separatrix and equatorial current sheets. In order to achieve our goal, we needed to develop a novel numerical method. We solve two independent magnetospheric problems without current sheet discontinuities in the domains of open and closed field lines, and adjust the shape of their interface (the separatrix) to satisfy pressure balance between the two regions. The solution is obtained with meshless Physics Informed Neural Networks (PINNs). In this paper we present our first results for an inclined dipole rotator using the new methodology. We are able to zoom‑in around the Y‑point and inside the closed‑line region, and we observe new interesting features. This is the first time the steady‑state 3D problem is addressed directly, and not through a time‑dependent simulation that eventually relaxes to a steady‑state. We have trained a Neural Network that instantaneously yields the three components of the magnetic field and their spatial derivatives at any given point. Our results demonstrate the potential of the new method to generate new solutions of the ideal pulsar magnetosphere.

Abstract:
We develop Riemannian approaches to variational autoencoders (VAEs) for PDE‑type ambient data with regularizing geometric latent dynamics, which we refer to as VAE‑DLM, or VAEs with dynamical latent manifolds. We redevelop the VAE framework such that manifold geometries, subject to our geometric flow, embedded in Euclidean space are learned in the intermediary latent space developed by encoders and decoders. By tailoring the geometric flow in which the latent space evolves, we induce latent geometric properties of our choosing, which are reflected in empirical performance. We reformulate the traditional evidence lower bound (ELBO) loss with a considerate choice of prior. We develop a linear geometric flow with a steady‑state regularizing term. This flow requires only automatic differentiation of one time derivative, and can be solved in moderately high dimensions in a physics‑informed approach, allowing more expressive latent representations. We discuss how this flow can be formulated as a gradient flow, and maintains entropy away from metric singularity. This, along with an eigenvalue penalization condition, helps ensure the manifold is sufficiently large in measure, nondegenerate, and a canonical geometry, which contribute to a robust representation. Our methods focus on the modified multi‑layer perceptron architecture with tanh activations for the manifold encoder‑decoder. We demonstrate, on our datasets of interest, our methods perform at least as well as the traditional VAE, and oftentimes better. Our methods can outperform this and a VAE endowed with our proposed architecture, frequently reducing out‑of‑distribution (OOD) error between 15% to 35% on select datasets. We highlight our method on ambient PDEs whose solutions maintain minimal variation in late times. We provide empirical justification towards how we can improve robust learning for external dynamics with VAEs.

Abstract:
Sparse system identification of nonlinear dynamic systems is still challenging, especially for stiff and high‑order differential equations for noisy measurement data. The use of highly correlated functions makes distinguishing between true and false functions difficult, which limits the choice of functions. In this study, an equation discovery method has been proposed to tackle these problems. The key elements include a) use of B‑splines for data fitting to get analytical derivatives superior to numerical derivatives, b) sequentially regularized derivatives for denoising (SRDD) algorithm, highly effective in removing noise from signal without system information loss, c) uncorrelated component analysis (UCA) algorithm that identifies and eliminates highly correlated functions while retaining the true functions, and d) physics‑informed spline fitting (PISF) where the spline fitting is updated gradually while satisfying the governing equation with a dictionary of candidate functions to converge to the correct equation sequentially. The complete framework is built on a unified deep‑learning architecture that eases the optimization process. The proposed method is demonstrated to discover various differential equations at various noise levels, including three‑dimensional, fourth‑order, and stiff equations. The parameter estimation converges accurately to the true values with a small coefficient of variation, suggesting robustness to the noise.

Abstract:
In this note, we establish some connections between standard (data‑driven) neural network‑based solvers for PDE and eigenvalue problems developed on one side in the applied mathematics and engineering communities (e.g. Deep‑Ritz and Physics Informed Neural Networks (PINN)), and on the other side in quantum chemistry (e.g. Variational Monte Carlo algorithms, \tt Ferminet or \tt Paulinet following the pioneer work of \it Carleo et. al. In particular, we re‑formulate a PINN algorithm as a \it fitting problem with data corresponding to the solution to a standard Diffusion Monte Carlo algorithm initialized thanks to neural network‑based Variational Monte Carlo. Connections at the level of the optimization algorithms are also established.

Abstract:
Kolmogorov‑‑Arnold Networks (KANs), a recently proposed neural network architecture, have gained significant attention in the deep learning community, due to their potential as a viable alternative to multi‑layer perceptrons (MLPs) and their broad applicability to various scientific tasks. Empirical investigations demonstrate that KANs optimized via stochastic gradient descent (SGD) are capable of achieving near‑zero training loss in various machine learning (e.g., regression, classification, and time series forecasting, etc.) and scientific tasks (e.g., solving partial differential equations). In this paper, we provide a theoretical explanation for the empirical success by conducting a rigorous convergence analysis of gradient descent (GD) and SGD for two‑layer KANs in solving both regression and physics‑informed tasks. For regression problems, we establish using the neural tangent kernel perspective that GD achieves global linear convergence of the objective function when the hidden dimension of KANs is sufficiently large. We further extend these results to SGD, demonstrating a similar global convergence in expectation. Additionally, we analyze the global convergence of GD and SGD for physics‑informed KANs, which unveils additional challenges due to the more complex loss structure. This is the first work establishing the global convergence guarantees for GD and SGD applied to optimize KANs and physics‑informed KANs.

Abstract:
Reconstructing fields from sparsely observed data is an ill‑posed problem that arises in many engineering and science applications. Here, we investigate the use of physics‑informed neural networks (PINNs) to reconstruct complete temperature, velocity and pressure fields from sparse and noisy experimental temperature data obtained through single‑color laser‑induced fluorescence (LIF). The PINNs are applied to the laminar mixed convection system, a complex but fundamentally important phenomenon characterized by the simultaneous presence of transient forced and natural convection behaviors. To enhance computation efficiency, this study also explores transfer learning (TL) as a mean of significantly reducing the time required for field reconstruction. Our findings demonstrate that PINNs are effective, capable of eliminating most experimental noise that does not conform to governing physics laws. Additionally, we show that the TL method achieves errors within 5% compared to the regular training scheme while reducing computation time by a factor of 9.9. We validate the PINN reconstruction results using non‑simultaneous particle image velocimetry (PIV) and finite volume method (FVM) simulations. The reconstructed velocity fields from the PINN closely match those obtained from PIV. When using FVM data as a reference, the average temperature errors are below 1%, while the pressure and velocity errors are below 10%. This research provides insights into the feasibility of using PINNs for solving ill‑posed problems with experimental data and highlights the potential of TL to enable near real‑time field reconstruction.

Abstract:
We develop improved physics‑informed neural networks (PINNs) for high‑order and high‑dimensional power system models described by nonlinear ordinary differential equations. We propose some novel enhancements to improve PINN training and accuracy and also implement several other recently proposed ideas from the literature. We successfully apply these to study the transient dynamics of synchronous generators. We also make progress towards applying PINNs to advanced inverter models. Such enhanced PINNs can allow us to accelerate high‑fidelity simulations needed to ensure a stable and reliable renewables‑rich future grid.

Abstract:
We address an inverse problem in modeling holographic superconductors. We focus our research on the critical temperature behavior depicted by experiments. We use a physics‑informed neural network method to find a mass function M(F^2), which is necessary to understand phase transition behavior. This mass function describes a nonlinear interaction between superconducting order and charge carrier density. We introduce positional embedding layers to improve the learning process in our algorithm, and the Adam optimization is used to predict the critical temperature data via holographic calculation with appropriate accuracy. Consideration of the positional embedding layers is motivated by the transformer model of natural‑language processing in the artificial intelligence (AI) field. We obtain holographic models that reproduce borderlines of the normal and superconducting phases provided by actual data. Our work is the first holographic attempt to match phase transition data quantitatively obtained from experiments. Also, the present work offers a new methodology for data‑based holographic models.

Abstract:
In climate science, models for global warming and weather prediction face significant challenges due to the limited availability of high‑quality data and the difficulty in obtaining it, making data efficiency crucial. In the past few years, Scientific Machine Learning (SciML) models have gained tremendous traction as they can be trained in a data‑efficient manner, making them highly suitable for real‑world climate applications. Despite this, very little attention has been paid to chaotic climate system modeling utilizing SciML methods. In this paper, we have integrated SciML methods into foundational weather models, where we have enhanced large‑scale climate predictions with a physics‑informed approach that achieves high accuracy with reduced data. We successfully demonstrate that by combining the interpretability of physical climate models with the computational power of neural networks, SciML models can prove to be a reliable tool for modeling climate. This indicates a shift from the traditional black box‑based machine learning modeling of climate systems to physics‑informed decision‑making, leading to effective climate policy implementation.

Abstract:
Large language models (LLMs), like ChatGPT, have shown that even trained with noisy prior data, they can generalize effectively to new tasks through in‑context learning (ICL) and pre‑training techniques. Motivated by this, we explore whether a similar approach can be applied to scientific foundation models (SFMs). Our methodology is structured as follows: (i) we collect low‑cost physics‑informed neural network (PINN)‑based approximated prior data in the form of solutions to partial differential equations (PDEs) constructed through an arbitrary linear combination of mathematical dictionaries; (ii) we utilize Transformer architectures with self and cross‑attention mechanisms to predict PDE solutions without knowledge of the governing equations in a zero‑shot setting; (iii) we provide experimental evidence on the one‑dimensional convection‑diffusion‑reaction equation, which demonstrate that pre‑training remains robust even with approximated prior data, with only marginal impacts on test accuracy. Notably, this finding opens the path to pre‑training SFMs with realistic, low‑cost data instead of (or in conjunction with) numerical high‑cost data. These results support the conjecture that SFMs can improve in a manner similar to LLMs, where fully cleaning the vast set of sentences crawled from the Internet is nearly impossible.

Abstract:
Learning complex physical dynamics purely from data is challenging due to the intrinsic properties of systems to be satisfied. Incorporating physics‑informed priors, such as in Hamiltonian Neural Networks (HNNs), achieves high‑precision modeling for energy‑conservative systems. However, real‑world systems often deviate from strict energy conservation and follow different physical priors. To address this, we present a framework that achieves high‑precision modeling for a wide range of dynamical systems from the numerical aspect, by enforcing Time‑Reversal Symmetry (TRS) via a novel regularization term. It helps preserve energies for conservative systems while serving as a strong inductive bias for non‑conservative, reversible systems. While TRS is a domain‑specific physical prior, we present the first theoretical proof that TRS loss can universally improve modeling accuracy by minimizing higher‑order Taylor terms in ODE integration, which is numerically beneficial to various systems regardless of their properties, even for irreversible systems. By integrating the TRS loss within neural ordinary differential equation models, the proposed model TREAT demonstrates superior performance on diverse physical systems. It achieves a significant 11.5% MSE improvement in a challenging chaotic triple‑pendulum scenario, underscoring TREAT's broad applicability and effectiveness.

Abstract:
Iterative methods are widely used for solving partial differential equations (PDEs). However, the difficulty in eliminating global low‑frequency errors significantly limits their convergence speed. In recent years, neural networks have emerged as a novel approach for solving PDEs, with studies revealing that they exhibit faster convergence for low‑frequency components. Building on this complementary frequency convergence characteristics of iterative methods and neural networks, we draw inspiration from multigrid methods and propose a hybrid solving framework that combining iterative methods and neural network‑based solvers, termed PINN‑MG (PMG). In this framework, the iterative method is responsible for eliminating local high‑frequency oscillation errors, while Physics‑Informed Neural Networks (PINNs) are employed to correct global low‑frequency errors. Throughout the solving process, high‑ and low‑frequency components alternately dominate the error, with each being addressed by the iterative method and PINNs respectively, thereby accelerating the convergence. We tested the proposed PMG framework on the linear Poisson equation and the nonlinear Helmholtz equation, and the results demonstrated significant acceleration of the PMG when built on Gauss‑Seidel, pseudo‑time, and GMRES methods. Furthermore, detailed analysis of the convergence process further validates the rationality of the framework. We proposed that the PMG framework is a hybrid solving approach that does not rely on training data, achieving an organic integration of neural network methods with iterative methods.

Abstract:
This study employs physics‑informed neural networks (PINNs) to reconstruct multiple flow fields in a transient natural convection system solely based on instantaneous temperature data at an arbitrary moment. Transient convection problems present reconstruction challenges due to the temporal variability of fields across different flow phases. In general, large reconstruction errors are observed during the incipient phase, while the quasi‑steady phase exhibits relatively smaller errors, reduced by a factor of 2 to 4. We hypothesize that reconstruction errors vary across different flow phases due to the changing solution space of a PINN, inferred from the temporal gradients of the fields. Furthermore, we find that reconstruction errors tend to accumulate in regions where the spatial gradients are smaller than the order of 10^‑6, likely due to the vanishing gradient phenomenon. In convection phenomena, field variations often manifest across multiple scales in space. However, PINN‑based reconstruction tends to preserve larger‑scale variations, while smaller‑scale variations become less pronounced due to the vanishing gradient problem. To mitigate the errors associated with vanishing gradients, we introduce a multi‑scale approach that determines scaling constants for the PINN inputs and reformulates inputs across multiple scales. This approach improves the maximum and mean errors by 72.2% and 6.4%, respectively. Our research provides insights into the behavior of PINNs when applied to transient convection problems with large solution space and field variations across multiple scales.

Abstract:
When the physics is wrong, physics‑informed machine learning becomes physics‑misinformed machine learning. A powerful black‑box model should not be able to conceal misconceived physics. We propose two criteria that can be used to assert integrity that a hybrid (physics plus black‑box) model: 0) the black‑box model should be unable to replicate the physical model, and 1) any best‑fit hybrid model has the same physical parameter as a best‑fit standalone physics model. We demonstrate them for a sample nonlinear mechanical system approximated by its small‑signal linearization.

Abstract:
The energy transition is driving the integration of large shares of intermittent power sources in the electric power grid. Therefore, addressing the AC optimal power flow (AC‑OPF) effectively becomes increasingly essential. The AC‑OPF, which is a fundamental optimization problem in power systems, must be solved more frequently to ensure the safe and cost‑effective operation of power systems. Due to its non‑linear nature, AC‑OPF is often solved in its linearized form, despite inherent inaccuracies. Non‑linear solvers, such as the interior point method, are typically employed to solve the full OPF problem. However, these iterative methods may not converge for large systems and do not guarantee global optimality. This work explores a physics‑informed graph neural network, PINCO, to solve the AC‑OPF. We demonstrate that this method provides accurate solutions in a fraction of the computational time when compared to the established non‑linear programming solvers. Remarkably, PINCO generalizes effectively across a diverse set of loading conditions in the power system. We show that our method can solve the AC‑OPF without violating inequality constraints. Furthermore, it can function both as a solver and as a hybrid universal function approximator. Moreover, the approach can be easily adapted to different power systems with minimal adjustments to the hyperparameters, including systems with multiple generators at each bus. Overall, this work demonstrates an advancement in the field of power system optimization to tackle the challenges of the energy transition. The code and data utilized in this paper are available at https://anonymous.4open.science/r/opf_pinn_iclr‑B83E/.

Abstract:
Integrated energy systems (IESs) are complex systems consisting of diverse operating units spanning multiple domains. To address its operational challenges, we propose a physics‑informed hybrid time‑series neural network (NN) surrogate to predict the dynamic performance of IESs across multiple time scales. This neural network‑based modeling approach develops time‑series multi‑layer perceptrons (MLPs) for the operating units and integrates them with prior process knowledge about system structure and fundamental dynamics. This integration forms three hybrid NNs (long‑term, slow, and fast MLPs) that predict the entire system dynamics across multiple time scales. Leveraging these MLPs, we design an NN‑based scheduler and an NN‑based economic model predictive control (NEMPC) framework to meet global operational requirements: rapid electrical power responsiveness to operators requests, adequate cooling supply to customers, and increased system profitability, while addressing the dynamic time‑scale multiplicity present in IESs. The proposed day‑ahead scheduler is formulated using the ReLU network‑based MLP, which effectively represents IES performance under a broad range of conditions from a long‑term perspective. The scheduler is then exactly recast into a mixed‑integer linear programming problem for efficient evaluation. The real‑time NEMPC, based on slow and fast MLPs, comprises two sequential distributed control agents: a slow NEMPC for the cooling‑dominant subsystem with slower transient responses and a fast NEMPC for the power‑dominant subsystem with faster responses. Extensive simulations demonstrate that the developed scheduler and NEMPC schemes outperform their respective benchmark scheduler and controller by about 25% and 40%. Together, they enhance overall system performance by over 70% compared to benchmark approaches.

Abstract:
In this paper, we investigate the applications of operator learning, specifically DeepONet, for solving nonlinear partial differential equations (PDEs). Unlike conventional function learning methods that require training separate neural networks for each PDE, operator learning enables generalization across different PDEs without retraining. This study examines the performance of DeepONet in physics‑informed training, focusing on two key aspects: (1) the approximation capabilities of deep branch and trunk networks, and (2) the generalization error in Sobolev norms. Our results show that complex branch networks provide substantial performance gains, while trunk networks are most effective when kept relatively simple. Furthermore, we derive a bound on the generalization error of DeepONet for solving nonlinear PDEs by analyzing the Rademacher complexity of its derivatives in terms of pseudo‑dimension. This work bridges a critical theoretical gap by delivering rigorous error estimates. This paper fills a theoretical gap by providing error estimates for a wide range of physics‑informed machine learning models and applications.

Abstract:
Transformer Neural Networks are driving an explosion of activity and discovery in the field of Large Language Models (LLMs). In contrast, there have been only a few attempts to apply Transformers in engineering physics. Aiming to offer an easy entry point to physics‑centric Transformers, we introduce a physics‑informed Transformer model for solving the heat conduction problem in a 2D plate with Dirichlet boundary conditions. The model is implemented in the machine learning framework MLX and leverages the unified memory of Apple M‑series processors. The use of MLX means that the models can be trained and perform predictions efficiently on personal machines with only modest memory requirements. To train, validate and test the Transformer model we solve the 2D heat conduction problem using central finite differences. Each finite difference solution in these sets is initialized with four random Dirichlet boundary conditions, a uniform but random internal temperature distribution and a randomly selected thermal diffusivity. Validation is performed in‑line during training to monitor against over‑fitting. The excellent performance of the trained model is demonstrated by predicting the evolution of the temperature field to steady state for the unseen test set of conditions.

Abstract:
This work introduces a new neural model which follows the transport equation by design. A physical problem, the Taylor‑Green vortex, defined on a bi‑periodic domain, is used as a benchmark to evaluate the performance of both the standard physics‑informed neural network and our model (transport‑embedded neural network). Results exhibit that while the standard physics‑informed neural network fails to predict the solution accurately and merely returns the initial condition for the entire time span, our model successfully captures the temporal changes in the physics, particularly for high Reynolds numbers of the flow. Additionally, the ability of our model to prevent false minima can pave the way for addressing multiphysics problems, which are more prone to false minima, and help them accurately predict complex physics.

Abstract:
In this paper, we propose to use Sinc interpolation in the context of Kolmogorov‑Arnold Networks, neural networks with learnable activation functions, which recently gained attention as alternatives to Multilayer Perceptron. Many different function representations have already been tried, but we show that Sinc interpolation proposes a viable alternative, since it is known in numerical analysis to effectively represent both smooth functions and functions with singularities. This is important not only for function approximation but also for solving the partial differential equations with physics‑informed neural networks. Through a series of experiments, we show that SincKANs provide better results in almost all of the examples we have considered.

Abstract:
We introduce Sparse Physics Informed Backpropagation (SPInProp), a new class of methods for accelerating backpropagation for a specialized neural network architecture called Low Rank Neural Representation (LRNR). The approach exploits the low rank structure within LRNR and constructs a reduced neural network approximation that is much smaller in size. We call the smaller network FastLRNR. We show that backpropagation of FastLRNR can be substituted for that of LRNR, enabling a significant reduction in complexity. We apply SPInProp to a physics informed neural networks framework and demonstrate how the solution of parametrized partial differential equations is accelerated.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a powerful approach for solving partial differential equations (PDEs) by training neural networks with loss functions that incorporate physical constraints. In this work, we introduce HyResPINNs, a two‑level convex‑gated architecture designed to maximize approximation expressiveness for a fixed number of degrees of freedom (DoF). The first level involves a trainable, per‑block combination of smooth basis functions with trainable sparsity, and deep neural networks; the second involves the ability to gate entire blocks (much like in ResNets or Highway Nets), allowing for expressivity along the depth dimension of the architecture. Our empirical evaluation on a diverse set of challenging PDE problems demonstrates that HyResPINNs consistently achieve superior accuracy to baseline methods while remaining competitive relative to training times. These results highlight the potential of HyResPINNs to combine desirable features from traditional scientific computing methods and modern machine learning, paving the way for more robust and expressive approaches to physics‑informed modeling.

Abstract:
Interest is rising in Physics‑Informed Neural Networks (PINNs) as a mesh‑free alternative to traditional numerical solvers for partial differential equations (PDEs). However, PINNs often struggle to learn high‑frequency and multi‑scale target solutions. To tackle this problem, we first study a strong Boundary Condition (BC) version of PINNs for Dirichlet BCs and observe a consistent decline in relative error compared to the standard PINNs. We then perform a theoretical analysis based on the Fourier transform and convolution theorem. We find that strong BC PINNs can better learn the amplitudes of high‑frequency components of the target solutions. However, constructing the architecture for strong BC PINNs is difficult for many BCs and domain geometries. Enlightened by our theoretical analysis, we propose Fourier PINNs ‑‑ a simple, general, yet powerful method that augments PINNs with pre‑specified, dense Fourier bases. Our proposed architecture likewise learns high‑frequency components better but places no restrictions on the particular BCs or problem domains. We develop an adaptive learning and basis selection algorithm via alternating neural net basis optimization, Fourier and neural net basis coefficient estimation, and coefficient truncation. This scheme can flexibly identify the significant frequencies while weakening the nominal frequencies to better capture the target solution's power spectrum. We show the advantage of our approach through a set of systematic experiments.

Abstract:
The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics‑Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their lack of physical invariances, coupled with other significant weaknesses, such as an inability to handle complex geometries or their lack of generalization capabilities, make them unable to compete with classical numerical solvers in industrial settings. In this work, a limitation regarding the use of automatic differentiation in the context of physics‑informed learning is highlighted. A hybrid approach combining physics‑informed graph neural networks with numerical kernels from finite elements is introduced. After studying the theoretical properties of our model, we apply it to complex geometries, in two and three dimensions. Our choices are supported by an ablation study, and we evaluate the generalisation capacity of the proposed approach.

Abstract:
The quest for robust and generalizable machine learning models has driven recent interest in exploiting symmetries through equivariant neural networks. In the context of PDE solvers, recent works have shown that Lie point symmetries can be a useful inductive bias for Physics‑Informed Neural Networks (PINNs) through data and loss augmentation. Despite this, directly enforcing equivariance within the model architecture for these problems remains elusive. This is because many PDEs admit non‑compact symmetry groups, oftentimes not studied beyond their infinitesimal generators, making them incompatible with most existing equivariant architectures. In this work, we propose Lie aLgebrA Canonicalization (LieLAC), a novel approach that exploits only the action of infinitesimal generators of the symmetry group, circumventing the need for knowledge of the full group structure. To achieve this, we address existing theoretical issues in the canonicalization literature, establishing connections with frame averaging in the case of continuous non‑compact groups. Operating within the framework of canonicalization, LieLAC can easily be integrated with unconstrained pre‑trained models, transforming inputs to a canonical form before feeding them into the existing model, effectively aligning the input for model inference according to allowed symmetries. LieLAC utilizes standard Lie group descent schemes, achieving equivariance in pre‑trained models. Finally, we showcase LieLAC's efficacy on tasks of invariant image classification and Lie point symmetry equivariant neural PDE solvers using pre‑trained models.

Abstract:
We propose a physics‑informed data‑driven framework for urban wind estimation. This framework validates and incorporates the Reynolds number independence for flows under various working conditions, thus allowing the extrapolation for wind conditions far beyond the training data. Another key enabler is a machine‑learned non‑dimensionalized manifold from snapshot data. The velocity field is modeled using a double encoder‑decoder approach. The first encoder normalizes data using the oncoming wind speed, while the second encoder projects this normalized data onto the isometric feature mapping manifold. The decoders reverse this process, with k‑nearest neighbor performing the first decoding and the second undoing the normalization. The manifold is coarse‑grained by clustering to reduce the computational load for de‑ and encoding. The sensor‑based flow estimation is based on the estimate of the oncoming wind speed and a mapping from sensor signal to the manifold latent variables. The proposed machine‑learned flow estimation framework is exemplified for the flow above an Unmanned Aerial Vehicle vertiport. The wind estimation is shown to generalize well for rare wind conditions, not included in the original database.

Abstract:
This paper explores alternative formulations of the Kolmogorov Superposition Theorem (KST) as a foundation for neural network design. The original KST formulation, while mathematically elegant, presents practical challenges due to its limited insight into the structure of inner and outer functions and the large number of unknown variables it introduces. Kolmogorov‑Arnold Networks (KANs) leverage KST for function approximation, but they have faced scrutiny due to mixed results compared to traditional multilayer perceptrons (MLPs) and practical limitations imposed by the original KST formulation. To address these issues, we introduce ActNet, a scalable deep learning model that builds on the KST and overcomes many of the drawbacks of Kolmogorov's original formulation. We evaluate ActNet in the context of Physics‑Informed Neural Networks (PINNs), a framework well‑suited for leveraging KST's strengths in low‑dimensional function approximation, particularly for simulating partial differential equations (PDEs). In this challenging setting, where models must learn latent functions without direct measurements, ActNet consistently outperforms KANs across multiple benchmarks and is competitive against the current best MLP‑based approaches. These results present ActNet as a promising new direction for KST‑based deep learning applications, particularly in scientific computing and PDE simulation tasks.

Abstract:
We enhance machine learning algorithms for learning model parameters in complex systems represented by ordinary differential equations (ODEs) with domain decomposition methods. The study evaluates the performance of two approaches, namely (vanilla) Physics‑Informed Neural Networks (PINNs) and Finite Basis Physics‑Informed Neural Networks (FBPINNs), in learning the dynamics of test models with a quasi‑stationary longtime behavior. We test the approaches for data sets in different dynamical regions and with varying noise level. As results, we find a better performance for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi‑stationary time domain with few dynamics.

Abstract:
The accurate modelling of structural dynamics is crucial across numerous engineering applications, such as Structural Health Monitoring (SHM), seismic analysis, and vibration control. Often, these models originate from physics‑based principles and can be derived from corresponding governing equations, often of differential equation form. However, complex system characteristics, such as nonlinearities and energy dissipation mechanisms, often imply that such models are approximative and often imprecise. This challenge is further compounded in SHM, where sensor data is often sparse, making it difficult to fully observe the system's states. To address these issues, this paper explores the use of Physics‑Informed Neural Networks (PINNs), a class of physics‑enhanced machine learning (PEML) techniques, for the identification and estimation of dynamical systems. PINNs offer a unique advantage by embedding known physical laws directly into the neural network's loss function, allowing for simple embedding of complex phenomena, even in the presence of uncertainties. This study specifically investigates three key applications of PINNs: state estimation in systems with sparse sensing, joint state‑parameter estimation, when both system response and parameters are unknown, and parameter estimation within a Bayesian framework to quantify uncertainties. The results demonstrate that PINNs deliver an efficient tool across all aforementioned tasks, even in presence of modelling errors. However, these errors tend to have a more significant impact on parameter estimation, as the optimization process must reconcile discrepancies between the prescribed model and the true system behavior. Despite these challenges, PINNs show promise in dynamical system modeling, offering a robust approach to handling uncertainties.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a versatile and widely applicable concept across various science and engineering domains over the past decade. This article offers a comprehensive overview of the fundamentals of PINNs, tracing their evolution, modifications, and various variants. It explores the impact of different parameters on PINNs and the optimization algorithms involved. The review also delves into the theoretical advancements related to the convergence, consistency, and stability of numerical solutions using PINNs, while highlighting the current state of the art. Given their ability to address equations involving complex physics, the article discusses various applications of PINNs, with a particular focus on their utility in computational fluid dynamics problems. Additionally, it identifies current gaps in the research and outlines future directions for the continued development of PINNs.

Abstract:
Volatility, which indicates the dispersion of returns, is a crucial measure of risk and is hence used extensively for pricing and discriminating between different financial investments. As a result, accurate volatility prediction receives extensive attention. The Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model and its succeeding variants are well established models for stock volatility forecasting. More recently, deep learning models have gained popularity in volatility prediction as they demonstrated promising accuracy in certain time series prediction tasks. Inspired by Physics‑Informed Neural Networks (PINN), we constructed a new, hybrid Deep Learning model that combines the strengths of GARCH with the flexibility of a Long Short‑Term Memory (LSTM) Deep Neural Network (DNN), thus capturing and forecasting market volatility more accurately than either class of models are capable of on their own. We refer to this novel model as a GARCH‑Informed Neural Network (GINN). When compared to other time series models, GINN showed superior out‑of‑sample prediction performance in terms of the Coefficient of Determination (R^2), Mean Squared Error (MSE), and Mean Absolute Error (MAE).

Abstract:
We propose a method utilizing physics‑informed neural networks (PINNs) to solve Poisson equations that serve as control variates in the computation of transport coefficients via fluctuation formulas, such as the Green‑‑Kubo and generalized Einstein‑like formulas. By leveraging approximate solutions to the Poisson equation constructed through neural networks, our approach significantly reduces the variance of the estimator at hand. We provide an extensive numerical analysis of the estimators and detail a methodology for training neural networks to solve these Poisson equations. The approximate solutions are then incorporated into Monte Carlo simulations as effective control variates, demonstrating the suitability of the method for moderately high‑dimensional problems where fully deterministic solutions are computationally infeasible.

Abstract:
Control Lyapunov functions are a central tool in the design and analysis of stabilizing controllers for nonlinear systems. Constructing such functions, however, remains a significant challenge. In this paper, we investigate physics‑informed learning and formal verification of neural network control Lyapunov functions. These neural networks solve a transformed Hamilton‑Jacobi‑Bellman equation, augmented by data generated using Pontryagin's maximum principle. Similar to how Zubov's equation characterizes the domain of attraction for autonomous systems, this equation characterizes the null‑controllability set of a controlled system. This principled learning of neural network control Lyapunov functions outperforms alternative approaches, such as sum‑of‑squares and rational control Lyapunov functions, as demonstrated by numerical examples. As an intermediate step, we also present results on the formal verification of quadratic control Lyapunov functions, which, aided by satisfiability modulo theories solvers, can perform surprisingly well compared to more sophisticated approaches and efficiently produce global certificates of null‑controllability.

Abstract:
Physical models in the form of partial differential equations serve as important priors for many under‑constrained problems. One such application is tumor treatment planning, which relies on accurately estimating the spatial distribution of tumor cells within a patient's anatomy. While medical imaging can detect the bulk of a tumor, it cannot capture the full extent of its spread, as low‑concentration tumor cells often remain undetectable, particularly in glioblastoma, the most common primary brain tumor. Machine learning approaches struggle to estimate the complete tumor cell distribution due to a lack of appropriate training data. Consequently, most existing methods rely on physics‑based simulations to generate anatomically and physiologically plausible estimations. However, these approaches face challenges with complex and unknown initial conditions and are constrained by overly rigid physical models. In this work, we introduce a novel method that integrates data‑driven and physics‑based cost functions, akin to Physics‑Informed Neural Networks (PINNs). However, our approach parametrizes the solution directly on a dynamic discrete mesh, allowing for the effective modeling of complex biomechanical behaviors. Specifically, we propose a unique discretization scheme that quantifies how well the learned spatiotemporal distributions of tumor and brain tissues adhere to their respective growth and elasticity equations. This quantification acts as a regularization term, offering greater flexibility and improved integration of patient data compared to existing models. We demonstrate enhanced coverage of tumor recurrence areas using real‑world data from a patient cohort, highlighting the potential of our method to improve model‑driven treatment planning for glioblastoma in clinical practice.

Abstract:
Physics‑informed neural networks (PINNs) have recently emerged as effective methods for solving partial differential equations (PDEs) in various problems. Substantial research focuses on the failure modes of PINNs due to their frequent inaccuracies in predictions. However, most are based on the premise that minimizing the loss function to zero causes the network to converge to a solution of the governing PDE. In this study, we prove that PINNs encounter a fundamental issue that the premise is invalid. We also reveal that this issue stems from the inability to regulate the behavior of the derivatives of the predicted solution. Inspired by the derivative pathology of PINNs, we propose a variable splitting strategy that addresses this issue by parameterizing the gradient of the solution as an auxiliary variable. We demonstrate that using the auxiliary variable eludes derivative pathology by enabling direct monitoring and regulation of the gradient of the predicted solution. Moreover, we prove that the proposed method guarantees convergence to a generalized solution for second‑order linear PDEs, indicating its applicability to various problems.

Abstract:
Physics‑Informed Neural Networks (PINNs) solve partial differential equations using deep learning. However, conventional PINNs perform pointwise predictions that neglect dependencies within a domain, which may result in suboptimal solutions. We introduce SetPINNs, a framework that effectively captures local dependencies. With a finite element‑inspired sampling scheme, we partition the domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide a rigorous theoretical analysis showing that SetPINNs yield unbiased, lower‑variance estimates of residual energy and its gradients, ensuring improved domain coverage and reduced residual error. Extensive experiments on synthetic and real‑world tasks show improved accuracy, efficiency, and robustness.

Abstract:
Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi‑analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid‑based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation ‑‑ the Legendre equation, the hypergeometric equation, the solution for black hole space‑times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

Abstract:
Ultrafast optics is driven by a myriad of complex nonlinear dynamics. The ubiquitous presence of governing equations in the form of partial integro‑differential equations (PIDE) necessitates the need for advanced computational tools to understand the underlying physical mechanisms. From the experimental perspective, signal‑to‑noise ratio and availability of measurable data, accounts for a bottle neck in numerical and data‑driven modeling methods. In this paper we extend the application of the physics informed neural network (PINN) architecture to include prior knowledge in both the physical and Fourier domain. We demonstrate our Fourier Domain PINN (FD‑PINN) in two distinct forms. The Continuous time FD‑PINN is used to predict accurate solutions to the Generalized Pulse Propagation Equation, which includes the complete delayed nonlinear response, in the data‑starved and noisy regime. We extend the architecture to the Discrete time FD‑PINN to recover the delayed‑response physics from spatially separated measurement points. We conducted the first systematic study of the effect of SNR on the spatiotemporal field prediction as well as physics discovery. Our architecture ensures high fidelity predictive modeling and hidden physics recovery for applications such as image reconstruction, pulse characterization and shaping, as well as hidden parameter discovery. The benefits of the FD‑PINN for ultrafast nonlinear optics make it immediately experimentally deployable. FD‑PINN represents the next generation of tools to study optical phenomena both through modeling and measurements for both forward and inverse problems.

Abstract:
In this paper, a mathematical model is presented to infer the wave free surface elevation from the horizontal velocity components using Physics Informed Neural Network (PINN). PINN is a deep learning framework to solve forward and inverse Ordinary/Partial Differential Equations (ODEs/PDEs). The model is verified by measuring a numerically generated Kelvin waves downstream of a KRISO Container Ship (KCS). The KCS Kelvin waves are generated using two phase Volume of Fluid (VoF) Computational Fluid Dynamics (CFD) simulation with OpenFOAM. In addition, the paper presented the use of the Fourier Features decomposition of the Neural Network inputs to avoid the spectral bias phenomena; Spectral bias is the tendency of Neural Network to converge towards the low frequency solution faster than the high frequency one. Fourier Features decomposition layer showed an improvement for the model learning, as the model was able to learn the high and low frequency components simultaneously.

Abstract:
Accurate dynamic modeling is critical for autonomous racing vehicles, especially during high‑speed and agile maneuvers where precise motion prediction is essential for safety. Traditional parameter estimation methods face limitations such as reliance on initial guesses, labor‑intensive fitting procedures, and complex testing setups. On the other hand, purely data‑driven machine learning methods struggle to capture inherent physical constraints and typically require large datasets for optimal performance. To address these challenges, this paper introduces the Fine‑Tuning Hybrid Dynamics (FTHD) method, which integrates supervised and unsupervised Physics‑Informed Neural Networks (PINNs), combining physics‑based modeling with data‑driven techniques. FTHD fine‑tunes a pre‑trained Deep Dynamics Model (DDM) using a smaller training dataset, delivering superior performance compared to state‑of‑the‑art methods such as the Deep Pacejka Model (DPM) and outperforming the original DDM. Furthermore, an Extended Kalman Filter (EKF) is embedded within FTHD (EKF‑FTHD) to effectively manage noisy real‑world data, ensuring accurate denoising while preserving the vehicle's essential physical characteristics. The proposed FTHD framework is validated through scaled simulations using the BayesRace Physics‑based Simulator and full‑scale real‑world experiments from the Indy Autonomous Challenge. Results demonstrate that the hybrid approach significantly improves parameter estimation accuracy, even with reduced data, and outperforms existing models. EKF‑FTHD enhances robustness by denoising real‑world data while maintaining physical insights, representing a notable advancement in vehicle dynamics modeling for high‑speed autonomous racing.

Abstract:
We have developed a novel activation function, named the Cauchy Activation Function. This function is derived from the Cauchy Integral Theorem in complex analysis and is specifically tailored for problems requiring high precision. This innovation has led to the creation of a new class of neural networks, which we call (Comple)XNet, or simply XNet. We will demonstrate that XNet is particularly effective for high‑dimensional challenges such as image classification and solving Partial Differential Equations (PDEs). Our evaluations show that XNet significantly outperforms established benchmarks like MNIST and CIFAR‑10 in computer vision, and offers substantial advantages over Physics‑Informed Neural Networks (PINNs) in both low‑dimensional and high‑dimensional PDE scenarios.

Abstract:
Echo State Networks (ESNs) are recurrent neural networks usually employed for modeling nonlinear dynamic systems with relatively ease of training. By incorporating physical laws into the training of ESNs, Physics‑Informed ESNs (PI‑ESNs) were proposed initially to model chaotic dynamic systems without external inputs. They require less data for training since Ordinary Differential Equations (ODEs) of the considered system help to regularize the ESN. In this work, the PI‑ESN is extended with external inputs to model controllable nonlinear dynamic systems. Additionally, an existing self‑adaptive balancing loss method is employed to balance the contributions of the residual regression term and the physics‑informed loss term in the total loss function. The experiments with two nonlinear systems modeled by ODEs, the Van der Pol oscillator and the four‑tank system, and with one differential‑algebraic (DAE) system, an electric submersible pump, revealed that the proposed PI‑ESN outperforms the conventional ESN, especially in scenarios with limited data availability, showing that PI‑ESNs can regularize an ESN model with external inputs previously trained on just a few datapoints, reducing its overfitting and improving its generalization error (up to 92% relative reduction in the test error). Further experiments demonstrated that the proposed PI‑ESN is robust to parametric uncertainties in the ODE equations and that model predictive control using PI‑ESN outperforms the one using plain ESN, particularly when training data is scarce.

Abstract:
Computer simulation of "virtual interventions" may inform optimal valve repair for a given patient prior to intervention. However, the paucity of noninvasive methods to determine in vivo mechanical parameters of valves limits the accuracy of computer prediction and their clinical application. To address this, we propose ADEPT: A noninvasive method for Determining Elastic Parameters of valve Tissue. In this work, we demonstrated its application to the tricuspid valve of a child. We first tracked valve displacements from open to closed frames within a 3D echocardiogram time sequence using image registration. Physics‑informed neural networks were subsequently applied to estimate the nonlinear mechanical properties from first principles and reference displacements. The simulated model using these patient‑specific parameters closely aligned with the reference image segmentation, achieving a mean symmetric distance of less than 1 mm. Our approach doubled the accuracy of the simulated model compared to the generic parameters reported in the literature.

Abstract:
Early diagnosis of Autism Spectrum Disorder (ASD) is an effective and favorable step towards enhancing the health and well‑being of children with ASD. Manual ASD diagnosis testing is labor‑intensive, complex, and prone to human error due to several factors contaminating the results. This paper proposes a novel framework that exploits the laws of physics for ASD severity recognition. The proposed physics‑informed neural network architecture encodes the behaviour of the subject extracted by observing a part of the skeleton‑based motion trajectory in a higher dimensional latent space. Two decoders, namely physics‑based and non‑physics‑based decoder, use this latent embedding and predict the future motion patterns. The physics branch leverages the laws of physics that apply to a skeleton sequence in the prediction process while the non‑physics‑based branch is optimised to minimise the difference between the predicted and actual motion of the subject. A classifier also leverages the same latent space embeddings to recognise the ASD severity. This dual generative objective explicitly forces the network to compare the actual behaviour of the subject with the general normal behaviour of children that are governed by the laws of physics, aiding the ASD recognition task. The proposed method attains state‑of‑the‑art performance on multiple ASD diagnosis benchmarks. To illustrate the utility of the proposed framework beyond the task ASD diagnosis, we conduct a third experiment using a publicly available benchmark for the task of fall prediction and demonstrate the superiority of our model.

Abstract:
Physics‑informed neural networks (PINNs) have emerged as a prominent approach for solving partial differential equations (PDEs) by minimizing a combined loss function that incorporates both boundary loss and PDE residual loss. Despite their remarkable empirical performance in various scientific computing tasks, PINNs often fail to generate reasonable solutions, and such pathological behaviors remain difficult to explain and resolve. In this paper, we identify that PINNs can be adversely trained when gradients of each loss function exhibit a significant imbalance in their magnitudes and present a negative inner product value. To address these issues, we propose a novel optimization framework, Dual Cone Gradient Descent (DCGD), which adjusts the direction of the updated gradient to ensure it falls within a dual cone region. This region is defined as a set of vectors where the inner products with both the gradients of the PDE residual loss and the boundary loss are non‑negative. Theoretically, we analyze the convergence properties of DCGD algorithms in a non‑convex setting. On a variety of benchmark equations, we demonstrate that DCGD outperforms other optimization algorithms in terms of various evaluation metrics. In particular, DCGD achieves superior predictive accuracy and enhances the stability of training for failure modes of PINNs and complex PDEs, compared to existing optimally tuned models. Moreover, DCGD can be further improved by combining it with popular strategies for PINNs, including learning rate annealing and the Neural Tangent Kernel (NTK).

Abstract:
Perceiving the global field from sparse sensors has been a grand challenge in the monitoring, analysis, and design of physical systems. In this context, sensor placement optimization is a crucial issue. Most existing works require large and sufficient data to construct data‑based criteria, which are intractable in data‑free scenarios without numerical and experimental data. To this end, we propose a novel physics‑driven sensor placement optimization (PSPO) method for temperature field reconstruction using a physics‑based criterion to optimize sensor locations. In our methodological framework, we firstly derive the theoretical upper and lower bounds of the reconstruction error under noise scenarios by analyzing the optimal solution, proving that error bounds correlate with the condition number determined by sensor locations. Furthermore, the condition number, as the physics‑based criterion, is used to optimize sensor locations by the genetic algorithm. Finally, the best sensors are validated by reconstruction models, including non‑invasive end‑to‑end models, non‑invasive reduced‑order models, and physics‑informed models. Experimental results, both on a numerical and an application case, demonstrate that the PSPO method significantly outperforms random and uniform selection methods, improving the reconstruction accuracy by nearly an order of magnitude. Moreover, the PSPO method can achieve comparable reconstruction accuracy to the existing data‑driven placement optimization methods.

Abstract:
Scientific machine learning (SciML) is an interdisciplinary research field that integrates machine learning, particularly deep learning, with physics theory to understand and predict complex natural phenomena. By incorporating physical knowledge, SciML reduces the dependency on observational data, which is often limited in the natural sciences. In this article, the fundamental concepts of SciML, its applications in seismology, and prospects are described. Specifically, two popular methods are mainly discussed: physics‑informed neural networks (PINNs) and neural operators (NOs). PINNs can address both forward and inverse problems by incorporating governing laws into the loss functions. The use of PINNs is expanding into areas such as simultaneous solutions of differential equations, inference in underdetermined systems, and regularization based on physics. These research directions would broaden the scope of deep learning in natural sciences. NOs are models designed for operator learning, which deals with relationships between infinite‑dimensional spaces. NOs show promise in modeling the time evolution of complex systems based on observational or simulation data. Since large amounts of data are often required, combining NOs with physics‑informed learning holds significant potential. Finally, SciML is considered from a broader perspective beyond deep learning: statistical (or mathematical) frameworks that integrate observational data with physical principles to model natural phenomena. In seismology, mathematically rigorous Bayesian statistics has been developed over the past decades, whereas more flexible and scalable deep learning has only emerged recently. Both approaches can be considered as part of SciML in a broad sense. Theoretical and practical insights in both directions would advance SciML methodologies and thereby deepen our understanding of earthquake phenomena.

Abstract:
Light field microscopy (LFM) has been widely utilized in various fields for its capability to efficiently capture high‑resolution 3D scenes. Despite the rapid advancements in neural representations, there are few methods specifically tailored for microscopic scenes. Existing approaches often do not adequately address issues such as the loss of high‑frequency information due to defocus and sample aberration, resulting in suboptimal performance. In addition, existing methods, including RLD, INR, and supervised U‑Net, face challenges such as sensitivity to initial estimates, reliance on extensive labeled data, and low computational efficiency, all of which significantly diminish the practicality in complex biological scenarios. This paper introduces PNR (Physics‑informed Neural Representation), a method for high‑resolution LFM reconstruction that significantly enhances performance. Our method incorporates an unsupervised and explicit feature representation approach, resulting in a 6.1 dB improvement in PSNR than RLD. Additionally, our method employs a frequency‑based training loss, enabling better recovery of high‑frequency details, which leads to a reduction in LPIPS by at least half compared to SOTA methods (1.762 V.S. 3.646 of DINER). Moreover, PNR integrates a physics‑informed aberration correction strategy that optimizes Zernike polynomial parameters during optimization, thereby reducing the information loss caused by aberrations and improving spatial resolution. These advancements make PNR a promising solution for long‑term high‑resolution biological imaging applications. Our code and dataset will be made publicly available.

Abstract:
We consider the subcritical nonlinear Schrödinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a new PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided a reasonable integration scheme is available. Applications to traveling waves, breathers and solitons, as well as numerical experiments confirming the validity of the approximation are also presented as well.

Abstract:
Lattice quantum chromodynamics (QCD) calculations share a defining challenge by requiring a small finite range of spatial separation z between quark/gluon bilinears for controllable power corrections in the perturbative QCD factorization, and a large hadron boost p_z for a successful determination of collinear parton distribution functions (PDFs). However, these two requirements make the determination of PDFs from lattice data very challenging. We present the application of generative machine learning algorithms to estimate the polarized and unpolarized gluon correlation functions utilizing short‑distance data and extending the correlation up to zp_z \lesssim 14, surpassing the current capabilities of lattice QCD calculations. We train physics‑informed machine learning algorithms to learn from the short‑distance correlation at z\lesssim 0.36 fm and take the limit, p_z \to \infty, thereby minimizing possible contamination from the higher‑twist effects for a successful reconstruction of the polarized gluon PDF. We also expose the bias and problems with underestimating uncertainties associated with the use of model‑dependent and overly constrained functional forms, such as x^α(1‑x)^β and its variants to extract PDFs from the lattice data. We propose the use of generative machine learning algorithms to mitigate these issues and present our determination of the polarized and unpolarized gluon PDFs in the nucleon.

Abstract:
We present a computational framework for obtaining multidimensional phase‑space solutions of systems of non‑linear coupled differential equations, using high‑order implicit Runge‑Kutta Physics‑ Informed Neural Networks (IRK‑PINNs) schemes. Building upon foundational work originally solving differential equations for fields depending on coordinates [J. Comput. Phys. 378, 686 (2019)], we adapt the scheme to a context where the coordinates are treated as functions. This modification enables us to efficiently solve equations of motion for a particle in an external field. Our scheme is particularly useful for explicitly time‑independent and periodic fields. We apply this approach to successfully solve the equations of motion for a mass particle placed in a central force field and a charged particle in a periodic electric field.

Abstract:
This paper introduces a Transformer‑Enhanced Physics‑Informed Neural Network (TE‑PINN) designed for accurate quaternion‑based orientation estimation in high‑dynamic environments, particularly within the field of robotics. By integrating transformer networks with physics‑informed learning, our approach innovatively captures temporal dependencies in sensor data while enforcing the fundamental physical laws governing rotational motion. TE‑PINN leverages a multi‑head attention mechanism to handle sequential data from inertial sensors, such as accelerometers and gyroscopes, ensuring temporal consistency. Simultaneously, the model embeds quaternion kinematics and rigid body dynamics into the learning process, aligning the network's predictions with mechanical principles like Euler's laws of motion. The physics‑informed loss function incorporates the dynamics of angular velocity and external forces, enhancing the network's ability to generalize in complex scenarios. Our experimental evaluation demonstrates that TE‑PINN consistently outperforms traditional methods such as Extended Kalman Filters (EKF) and LSTM‑based estimators, particularly in scenarios characterized by high angular velocities and noisy sensor data. The results show a significant reduction in mean quaternion error and improved gyroscope bias estimation compared to the state‑of‑the‑art. An ablation study further isolates the contributions of both the transformer architecture and the physics‑informed constraints, highlighting the synergistic effect of both components in improving model performance. The proposed model achieves real‑time performance on embedded systems typical of mobile robots, offering a scalable and efficient solution for orientation estimation in autonomous systems.

Abstract:
In this paper, we restructure the Neural Interconnection and Damping Assignment ‑ Passivity Based Control (Neural IDA‑PBC) design methodology, and we formally analyze its closed‑loop properties. Neural IDA‑PBC redefines the IDA‑PBC design approach as an optimization problem by building on the framework of Physics Informed Neural Networks (PINNs). However, the closed‑loop stability and robustness properties under Neural IDA‑PBC remain unexplored. To address the issue, we study the behavior of classical IDA‑PBC under approximations. Our theoretical analysis allows deriving conditions for practical and asymptotic stability of the desired equilibrium point. Moreover, it extends the Neural IDA‑PBC applicability to port‑Hamiltonian systems where the matching conditions cannot be solved exactly. Our renewed optimization‑based design introduces three significant aspects: i) it involves a novel optimization objective including stability and robustness constraints issued from our theoretical analysis; ii) it employs separate Neural Networks (NNs), which can be structured to reduce the search space to relevant functions; iii) it does not require knowledge about the port‑Hamiltonian formulation of the system's model. Our methodology is validated with simulations on three standard benchmarks: a double pendulum, a nonlinear mass‑spring‑damper and a cartpole. Notably, classical IDA‑PBC designs cannot be analytically derived for the latter.

Abstract:
The main focus of this paper is to analyze the behavior of a numerical solution of the Timoshenko system coupled with Thermoelasticity and incorporating second sound effects. In order to address this target, we employ the Physics‑Informed Neural Networks (PINNs) framework to derive an approximate solution for the system. Our investigation delves into the extent to which this approximate solution can accurately capture the asymptotic behavior of the discrete energy, contingent upon the stability number χ. Interestingly, the PINNs overcome the major difficulties encountered while using the standard numerical methods.

Abstract:
Linear control models have gained extensive application in vehicle control due to their simplicity, ease of use, and support for stability analysis. However, these models lack adaptability to the changing environment and multi‑objective settings. Reinforcement learning (RL) models, on the other hand, offer adaptability but suffer from a lack of interpretability and generalization capabilities. This paper aims to develop a family of RL‑based controllers enhanced by physics‑informed policies, leveraging the advantages of both physics‑based models (data‑efficient and interpretable) and RL methods (flexible to multiple objectives and fast computing). We propose the Physics‑Enhanced Residual Policy Learning (PERPL) framework, where the physics component provides model interpretability and stability. The learning‑based Residual Policy adjusts the physics‑based policy to adapt to the changing environment, thereby refining the decisions of the physics model. We apply our proposed model to decentralized control to mixed traffic platoon of Connected and Automated Vehicles (CAVs) and Human‑driven Vehicles (HVs) using a constant time gap (CTG) strategy for cruising and incorporating actuator and communication delays. Experimental results demonstrate that our method achieves smaller headway errors and better oscillation dampening than linear models and RL alone in scenarios with artificially extreme conditions and real preceding vehicle trajectories. At the macroscopic level, overall traffic oscillations are also reduced as the penetration rate of CAVs employing the PERPL scheme increases.

Abstract:
Accurate estimation of the speed‑of‑sound (SoS) is important for ultrasound (US) image reconstruction techniques and tissue characterization. Various approaches have been proposed to calculate SoS, ranging from tomography‑inspired algorithms like CUTE to convolutional networks, and more recently, physics‑informed optimization frameworks based on differentiable beamforming. In this work, we utilize implicit neural representations (INRs) for SoS estimation in US. INRs are a type of neural network architecture that encodes continuous functions, such as images or physical quantities, through the weights of a network. Implicit networks may overcome the current limitations of SoS estimation techniques, which mainly arise from the use of non‑adaptable and oversimplified physical models of tissue. Moreover, convolutional networks for SoS estimation, usually trained using simulated data, often fail when applied to real tissues due to out‑of‑distribution and data‑shift issues. In contrast, implicit networks do not require extensive training datasets since each implicit network is optimized for an individual data case. This adaptability makes them suitable for processing US data collected from varied tissues and across different imaging protocols. We evaluated the proposed SoS estimation method based on INRs using data collected from a tissue‑mimicking phantom containing four cylindrical inclusions, with SoS values ranging from 1480 m/s to 1600 m/s. The inclusions were immersed in a material with an SoS value of 1540 m/s. In experiments, the proposed method achieved strong performance, clearly demonstrating the usefulness of implicit networks for quantitative US applications.

Abstract:
Differential equations are important mechanistic models that are integral to many scientific and engineering applications. With the abundance of available data there has been a growing interest in data‑driven physics‑informed models. Gaussian processes (GPs) are particularly suited to this task as they can model complex, non‑linear phenomena whilst incorporating prior knowledge and quantifying uncertainty. Current approaches have found some success but are limited as they either achieve poor computational scalings or focus only on the temporal setting. This work addresses these issues by introducing a variational spatio‑temporal state‑space GP that handles linear and non‑linear physical constraints while achieving efficient linear‑in‑time computation costs. We demonstrate our methods in a range of synthetic and real‑world settings and outperform the current state‑of‑the‑art in both predictive and computational performance.

Abstract:
Physics‑informed machine learning typically integrates physical priors into the learning process by minimizing a loss function that includes both a data‑driven term and a partial differential equation (PDE) regularization. Building on the formulation of the problem as a kernel regression task, we use Fourier methods to approximate the associated kernel, and propose a tractable estimator that minimizes the physics‑informed risk function. We refer to this approach as physics‑informed kernel learning (PIKL). This framework provides theoretical guarantees, enabling the quantification of the physical prior's impact on convergence speed. We demonstrate the numerical performance of the PIKL estimator through simulations, both in the context of hybrid modeling and in solving PDEs. In particular, we show that PIKL can outperform physics‑informed neural networks in terms of both accuracy and computation time. Additionally, we identify cases where PIKL surpasses traditional PDE solvers, particularly in scenarios with noisy boundary conditions.

Abstract:
The physics of strongly correlated systems offers some of the most intriguing physics challenges such as competing orders or the emergence of dynamical composite degrees of freedom. Often, the resolution of these physics challenges is computationally hard, but can be simplified enormously by a formulation in terms of the dynamical degrees of freedom and within an expansion about the physical ground state. Importantly, such a formulation does not only reduce or minimise the computational challenges, it also facilitates the access to the physics mechanisms at play. The tasks of finding the dynamical degrees of freedom and the physical ground state can be systematically addressed within the functional renormalisation group approach with flowing fields which accommodates both, emergent composites as well as the physical ground state. In the present work we use this approach to set up physics‑informed renormalisation group flows (PIRG flows): Scale‑dependent coordinate transformations in field space induce emergent composites, and the respective flows for the effective action generate a large set of target actions, formulated in these emergent composite fields. This novel perspective on RG flows bears a great potential both for conceptual as well as computational applications: to begin with, PIRG flows allow for a systematic search of the dynamical degrees of freedom and the respective ground state that leads to the most rapid convergence of expansion schemes, thus minimising the computational effort. Secondly, the resolution of the remaining computational tasks within a given expansion scheme can be further reduced by optimising the physics content within a given approximation. Thirdly, the maximal variability of PIRG flows can be used to reduce the analytic and numerical effort of solving the flows within a given approximation.

Abstract:
We present a non‑overlapping, Schwarz‑type domain decomposition method with a generalized interface condition, designed for physics‑informed machine learning of partial differential equations (PDEs) in both forward and inverse contexts. Our approach employs physics and equality‑constrained artificial neural networks (PECANN) within each subdomain. Unlike the original PECANN method, which relies solely on initial and boundary conditions to constrain PDEs, our method uses both boundary conditions and the governing PDE to constrain a unique interface loss function for each subdomain. This modification improves the learning of subdomain‑specific interface parameters while reducing communication overhead by delaying information exchange between neighboring subdomains. To address the constrained optimization in each subdomain, we apply an augmented Lagrangian method with a conditionally adaptive update strategy, transforming the problem into an unconstrained dual optimization. A distinct advantage of our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations, even in cases with high‑wavenumber and complex‑valued solutions. Through numerical experiments with up to 64 subdomains, we demonstrate that our method consistently generalizes well as the number of subdomains increases.

Abstract:
We explore the possibilities of using energy minimization for the numerical modeling of strain localization in solids as a sharp discontinuity in the displacement field. For this purpose, we consider (regularized) strong discontinuity kinematics in elastoplastic solids. The corresponding mathematical model is discretized using Artificial Neural Networks (ANNs), aiming to predict both the magnitude and location of the displacement jump from energy minimization, i.e., within a variational setting. The architecture takes care of the kinematics, while the loss function takes care of the variational statement of the boundary value problem. The main idea behind this approach is to solve both the equilibrium problem and the location of the localization band by means of trainable parameters in the ANN. As a proof of concept, we show through both 1D and 2D numerical examples that the computational modeling of strain localization for elastoplastic solids using energy minimization is feasible.

Abstract:
Solving Singularly Perturbed Differential Equations (SPDEs) presents challenges due to the rapid change of their solutions at the boundary layer. In this manuscript, We propose Asymptotic Physics‑Informed Neural Networks (ASPINN), a generalization of Physics‑Informed Neural Networks (PINN) and General‑Kindred Physics‑Informed Neural Networks (GKPINN) approaches. This is a decomposition method based on the idea of asymptotic analysis. Compared to PINN, the ASPINN method has a strong fitting ability for solving SPDEs due to the placement of exponential layers at the boundary layer. Unlike GKPINN, ASPINN lessens the number of fully connected layers, thereby reducing the training cost more effectively. Moreover, ASPINN theoretically approximates the solution at the boundary layer more accurately, which accuracy is also improved compared to GKPINN. We demonstrate the effect of ASPINN by solving diverse classes of SPDEs, which clearly shows that the ASPINN method is promising in boundary layer problems. Furthermore, we introduce Chebyshev Kolmogorov‑Arnold Networks (Chebyshev‑KAN) instead of MLP, achieving better performance in various experiments.

Abstract:
In the field of pharmacokinetics and pharmacodynamics (PKPD) modeling, which plays a pivotal role in the drug development process, traditional models frequently encounter difficulties in fully encapsulating the complexities of drug absorption, distribution, and their impact on targets. Although multi‑compartment models are frequently utilized to elucidate intricate drug dynamics, they can also be overly complex. To generalize modeling while maintaining simplicity, we propose an innovative approach that enhances PK and integrated PK‑PD modeling by incorporating fractional calculus or time‑varying parameter(s), combined with constant or piecewise constant parameters. These approaches effectively model anomalous diffusion, thereby capturing drug trapping and escape rates in heterogeneous tissues, which is a prevalent phenomenon in drug dynamics. Furthermore, this method provides insight into the dynamics of drug in cancer in multi‑dose administrations. Our methodology employs a Physics‑Informed Neural Network (PINN) and fractional Physics‑Informed Neural Networks (fPINNs), integrating ordinary differential equations (ODEs) with integer/fractional derivative order from compartmental modeling with neural networks. This integration optimizes parameter estimation for variables that are time‑variant, constant, piecewise constant, or related to the fractional derivative order. The results demonstrate that this methodology offers a robust framework that not only markedly enhances the model's depiction of drug absorption rates and distributed delayed responses but also unlocks different drug‑effect dynamics, providing new insights into absorption rates, anomalous diffusion, drug resistance, peristance and pharmacokinetic tolerance, all within a system of just two (fractional) ODEs with explainable results.

Abstract:
In this work, we explore advanced machine learning techniques for minimizing Gibbs free energy in full 3D micromagnetic simulations. Building on Brown's bounds for magnetostatic self‑energy, we revisit their application in the context of variational formulations of the transmission problems for the scalar and vector potential. To overcome the computational challenges posed by whole‑space integrals, we reformulate these bounds on a finite domain, making the method more efficient and scalable for numerical simulation. Our approach utilizes an alternating optimization scheme for joint minimization of Brown's energy bounds and the Gibbs free energy. The Cayley transform is employed to rigorously enforce the unit norm constraint, while R‑functions are used to impose essential boundary conditions in the computation of magnetostatic fields. Our results highlight the potential of mesh‑free Physics‑Informed Neural Networks (PINNs) and Extreme Learning Machines (ELMs) when integrated with hard constraints, providing highly accurate approximations. These methods exhibit competitive performance compared to traditional numerical approaches, showing significant promise in computing magnetostatic fields and the application for energy minimization, such as the computation of hysteresis curves. This work opens the path for future directions of research on more complex geometries, such as grain structure models, and the application to large scale problem settings which are intractable with traditional numerical methods.

Abstract:
The slow microstructural evolution of materials often plays a key role in determining material properties. When the unit steps of the evolution process are slow, direct simulation approaches such as molecular dynamics become prohibitive and Kinetic Monte‑Carlo (kMC) algorithms, where the state‑to‑state evolution of the system is represented in terms of a continuous‑time Markov chain, are instead frequently relied upon to efficiently predict long‑time evolution. The accuracy of kMC simulations however relies on the complete and accurate knowledge of reaction pathways and corresponding kinetics. This requirement becomes extremely stringent in complex systems such as concentrated alloys where the astronomical number of local atomic configurations makes the a priori tabulation of all possible transitions impractical. Machine learning models of transition kinetics have been used to mitigate this problem by enabling the efficient on‑the‑fly prediction of kinetic parameters. In this study, we show how physics‑informed ML architectures can exactly enforce the detailed balance condition, by construction. Using the diffusion of a vacancy in a concentrated alloy as an example, we show that such ML architectures also exhibit superior performance in terms of prediction accuracy, demonstrating that the imposition of physical constraints can facilitate the accurate learning of barriers at no increase in computational cost.

Abstract:
Physics‑informed neural networks (PINNs) are a class of deep learning models that utilize physics in the form of differential equations to address complex problems, including those with limited data availability. However, solving differential equations with rapid oscillations, steep gradients, or singular behavior remains challenging for PINNs. To address this, we propose an efficient wavelet‑based physics‑informed neural network (W‑PINN) that learns solutions in wavelet space. Here, we represent the solution using localized wavelets. This framework represents the solution of a differential equation with significantly fewer degrees of freedom while retaining the dynamics of complex physical phenomena. The proposed architecture enables training to search for solutions within the wavelet domain, where multiscale characteristics are less pronounced compared to the physical domain. This facilitates more efficient training for such problems. Furthermore, the proposed model does not rely on automatic differentiation for derivatives in the loss function and does not require prior information regarding the behavior of the solution, such as the location of abrupt features. The removal of AD significantly reduces training time while maintaining accuracy. Thus, through a strategic fusion of wavelets with PINNs, W‑PINNs capture localized nonlinear information, making them well‑suited for problems with abrupt behavior, such as singularly perturbed and other multiscale problems. We further analyze the convergence behavior of W‑PINN through a comparative study using Neural Tangent Kernel theory. The efficiency and accuracy of the proposed model are demonstrated across various problems, including the FitzHugh‑‑Nagumo (FHN) model, Helmholtz equation, Maxwell equation, Allen‑‑Cahn equation, and lid‑driven cavity flow, along with other highly singularly perturbed nonlinear differential equations.

Abstract:
Energy efficiency of buildings is considered to be one of the major means of achieving the net‑zero carbon goal around the world. The big part of the energy savings are supposed to be coming from optimizing the operation of the building heating, ventilation, and air conditioning (HVAC) systems. There is a natural trade‑off between the energy efficiency and the indoor comfort level, and finding an optimal operating schedule/regime requires knowing the occupancy of different spaces inside of the building. Moreover, the COVID‑19 pandemic has also revealed the need to sustain the high quality of the indoor air in order to reduce the risk of spread of infection. Occupancy detection from indoor sensors is thus an important practical problem. In the present paper, we propose detection of occupancy based on the carbon dioxide measurements inside the building. In particular, a new approach based on the, so‑called, switching auto‑regressive process with Markov regime is presented and justified by the physical model of the carbon dioxide concentration dynamics. We demonstrate the efficiency of the method compared to simple Hidden Markov approaches on simulated and real‑life data. We also show that the model is flexible and can be generalized to account for different ventilation regimes, simultaneously detecting the occupancy and the ventilation rate.

Abstract:
Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and in particular Hamiltonian Neural Networks have emerged as a mechanism to incorporate structural inductive bias into the NN model. By ensuring physical invariances are conserved, the models exhibit significantly better sample complexity and out‑of‑distribution accuracy than standard NNs. Learning the Hamiltonian as a function of its canonical variables, typically position and velocity, from sample observations of the system thus becomes a critical task in system identification and long‑term prediction of system behavior. However, to truly preserve the long‑run physical conservation properties of Hamiltonian systems, one must use symplectic integrators for a forward pass of the system's simulation. While symplectic schemes have been used in the literature, they are thus far limited to situations when they reduce to explicit algorithms, which include the case of separable Hamiltonians or augmented non‑separable Hamiltonians. We extend it to generalized non‑separable Hamiltonians, and noting the self‑adjoint property of symplectic integrators, we bypass computationally intensive backpropagation through an ODE solver. We show that the method is robust to noise and provides a good approximation of the system Hamiltonian when the state variables are sampled from a noisy observation. In the numerical results, we show the performance of the method concerning Hamiltonian reconstruction and conservation, indicating its particular advantage for non‑separable systems.

Abstract:
The high‑pressure transportation process of pipeline necessitates an accurate hydraulic transient simulation tool to prevent slack line flow and over‑pressure, which can endanger pipeline operations. However, current numerical solution methods often face difficulties in balancing computational efficiency and accuracy. Additionally, few studies attempt to reform physics‑informed learning architecture for pipeline transient simulation with magnitude different in outputs and imbalanced gradient in loss function. To address these challenges, a Knowledge‑Inspired Hierarchical Physics‑Informed Neural Network is proposed for hydraulic transient simulation of multi‑product pipelines. The proposed model integrates governing equations, boundary conditions, and initial conditions into the training process to ensure consistency with physical laws. Furthermore, magnitude conversion of outputs and equivalent conversion of governing equations are implemented to enhance the training performance of the neural network. To further address the imbalanced gradient of multiple loss terms with fixed weights, a hierarchical training strategy is designed. Numerical simulations demonstrate that the proposed model outperforms state‑of‑the‑art models and can still produce accurate simulation results under complex hydraulic transient conditions, with mean absolute percentage errors reduced by 87.8% and 92.7 % in pressure prediction. Thus, the proposed model can conduct accurate and effective hydraulic transient analysis, ensuring the safe operation of pipelines.

Abstract:
Physics‑Informed Neural Network (PINN) is a novel multi‑task learning framework useful for solving physical problems modeled using differential equations (DEs) by integrating the knowledge of physics and known constraints into the components of deep learning. A large class of physical problems in materials science and mechanics involve moving boundaries, where interface flux balance conditions are to be satisfied while solving DEs. Examples of such systems include free surface flows, shock propagation, solidification of pure and alloy systems etc. While recent research works have explored applicability of PINNs for an uncoupled system (such as solidification of pure system), the present work reports a PINN‑based approach to solve coupled systems involving multiple governing parameters (energy and species, along with multiple interface balance equations). This methodology employs an architecture consisting of a separate network for each variable with a separate treatment of each phase, a training strategy which alternates between temporal learning and adaptive loss weighting, and a scheme which progressively reduces the optimisation space. While solving the benchmark problem of binary alloy solidification, it is distinctly successful at capturing the complex composition profile, which has a characteristic discontinuity at the interface and the resulting predictions align well with the analytical solutions. The procedure can be generalised for solving other transient multiphysics problems especially in the low‑data regime and in cases where measurements can reveal new physics.

Abstract:
Physics‑Informed Neural Networks (PINNs) represent a significant advancement in Scientific Machine Learning (SciML), which integrate physical domain knowledge into an empirical loss function as soft constraints and apply existing machine learning methods to train the model. However, recent research has noted that PINNs may fail to learn relatively complex Partial Differential Equations (PDEs). This paper addresses the failure modes of PINNs by introducing a novel, hard‑constrained deep learning method ‑‑ trust‑region Sequential Quadratic Programming (trSQP‑PINN). In contrast to directly training the penalized soft‑constrained loss as in PINNs, our method performs a linear‑quadratic approximation of the hard‑constrained loss, while leveraging the soft‑constrained loss to adaptively adjust the trust‑region radius. We only trust our model approximations and make updates within the trust region, and such an updating manner can overcome the ill‑conditioning issue of PINNs. We also address the computational bottleneck of second‑order SQP methods by employing quasi‑Newton updates for second‑order information, and importantly, we introduce a simple pretraining step to further enhance training efficiency of our method. We demonstrate the effectiveness of trSQP‑PINN through extensive experiments. Compared to existing hard‑constrained methods for PINNs, such as penalty methods and augmented Lagrangian methods, trSQP‑PINN significantly improves the accuracy of the learned PDE solutions, achieving up to 1‑3 orders of magnitude lower errors. Additionally, our pretraining step is generally effective for other hard‑constrained methods, and experiments have shown the robustness of our method against both problem‑specific parameters and algorithm tuning parameters.

Abstract:
Context. Computational astronomy has reached the stage where running a gravitational N‑body simulation of a stellar system, such as a Milky Way star cluster, is computationally feasible, but a major limiting factor that remains is the ability to set up physically realistic initial conditions. Aims. We aim to obtain realistic initial conditions for N‑body simulations by taking advantage of machine learning, with emphasis on reproducing small‑scale interstellar distance distributions. Methods. The computational bottleneck for obtaining such distance distributions is the hydrodynamics of star formation, which ultimately determine the features of the stars, including positions, velocities, and masses. To mitigate this issue, we introduce a new method for sampling physically realistic initial conditions from a limited set of simulations using Gaussian processes. Results. We evaluated the resulting sets of initial conditions based on whether they meet tests for physical realism. We find that direct sampling based on the learned distribution of the star features fails to reproduce binary systems. Consequently, we show that physics‑informed sampling algorithms solve this issue, as they are capable of generating realisations closer to reality.

Abstract:
The star cluster surrounding the supermassive black hole in the center of Milky Way is probed using the data on the S2 star. The value of precession found at the physics‑informed neural networks (PINN) analysis of the S2 data is used to consider the role of the scattering of S2 star on stars of the cluster, described by a random force given by the Holtsmark distribution. The critical value for the star density of the core cluster for which the observed precession value by PINN lies inside 70% confidence interval (between 15% and 85% quantiles) around the median of precession due to scattering, is obtained as n_crit \approx 8.3 10^6 pc^‑3, that is at higher star densities the perturbation of the orbit of S2 would exceed the observed one.

Abstract:
Solving unsteady partial differential equations (PDEs) using recurrent neural networks (RNNs) typically requires numerical derivatives between each block of the RNN to form the physics informed loss function. However, this introduces the complexities of numerical derivatives into the training process of these models. In this study, we propose modifying the structure of the traditional RNN to enable the prediction of each block over a time interval, making it possible to calculate the derivative of the output with respect to time using the backpropagation algorithm. To achieve this, the time intervals of these blocks are overlapped, defining a mutual loss function between them. Additionally, the employment of conditional hidden states enables us to achieve a unique solution for each block. The forget factor is utilized to control the influence of the conditional hidden state on the prediction of the subsequent block. This new model, termed the Mutual Interval RNN (MI‑RNN), is applied to solve three different benchmarks: the Burgers equation, unsteady heat conduction in an irregular domain, and the Green vortex problem. Our results demonstrate that MI‑RNN can find the exact solution more accurately compared to existing RNN models. For instance, in the second problem, MI‑RNN achieved one order of magnitude less relative error compared to the RNN model with numerical derivatives.

Abstract:
Learning operators for parametric partial differential equations (PDEs) using neural networks has gained significant attention in recent years. However, standard approaches like Deep Operator Networks (DeepONets) require extensive labeled data, and physics‑informed DeepONets encounter training challenges. In this paper, we introduce a novel physics‑informed tailored finite point operator network (PI‑TFPONet) method to solve parametric interface problems without the need for labeled data. Our method fully leverages the prior physical information of the problem, eliminating the need to include the PDE residual in the loss function, thereby avoiding training challenges. The PI‑TFPONet is specifically designed to address certain properties of the problem, allowing us to naturally obtain an approximate solution that closely matches the exact solution. Our method is theoretically proven to converge if the local mesh size is sufficiently small and the training loss is minimized. Notably, our approach is uniformly convergent for singularly perturbed interface problems. Extensive numerical studies show that our unsupervised PI‑TFPONet is comparable to or outperforms existing state‑of‑the‑art supervised deep operator networks in terms of accuracy and versatility.

Abstract:
Power flow analysis plays a critical role in the control and operation of power systems. The high computational burden of traditional solution methods led to a shift towards data‑driven approaches, exploiting the availability of digital metering data. However, data‑driven approaches, such as deep learning, have not yet won the trust of operators as they are agnostic to the underlying physical model and have poor performances in regimes with limited observability. To address these challenges, this paper proposes a new, physics‑informed model. More specifically, a novel physics‑informed loss function is developed that can be used to train (deep) neural networks aimed at power flow simulation. The loss function is not only based on the theoretical AC power flow equations that govern the problem but also incorporates real physical line losses, resulting in higher loss accuracy and increased learning potential. The proposed model is used to train a Graph Neural Network (GNN) and is evaluated on a small 3‑bus test case both against another physics‑informed GNN that does not incorporate physical losses and against a model‑free technique. The validation results show that the proposed model outperforms the conventional physics‑informed network on all used performance metrics. Even more interesting is that the model shows strong prediction capabilities when tested on scenarios outside the training sample set, something that is a substantial deficiency of model‑free techniques.

Abstract:
Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite‑dimensional function spaces and perform zero‑shot super‑resolution. However, these models often require large datasets for effective training. While physics‑informed operators offer a data‑agnostic learning approach, they introduce additional training complexities and convergence issues, especially in highly nonlinear systems. To overcome these challenges, we introduce Finite Basis Physics‑Informed HyperDeepONet (FB‑HyDON), an advanced operator architecture featuring intrinsic domain decomposition. By leveraging hypernetworks and finite basis functions, FB‑HyDON effectively mitigates the training limitations associated with existing physics‑informed operator learning methods. We validated our approach on the high‑frequency harmonic oscillator, Burgers' equation at different viscosity levels, and Allen‑Cahn equation demonstrating substantial improvements over other operator learning models.

Abstract:
Recently, physics‑informed neural networks (PINNs) have emerged as a flexible and promising application of deep learning to partial differential equations in the physical sciences. While offering strong performance and competitive inference speeds on forward and inverse problems, their black‑box nature limits interpretability, particularly regarding alignment with expected physical behavior. In the present work, we explore the application of influence functions (IFs) to validate and debug PINNs post‑hoc. Specifically, we apply variations of IF‑based indicators to gauge the influence of different types of collocation points on the prediction of PINNs applied to a 2D Navier‑Stokes fluid flow problem. Our results demonstrate how IFs can be adapted to PINNs to reveal the potential for further studies. The code is publicly available at https://github.com/aleks‑krasowski/PINNfluence.

Abstract:
Graph convolutional neural networks have been instrumental in machine learning of material properties. When representing tensorial properties, weights and descriptors of a physics‑informed network must obey certain transformation rules to ensure the independence of the property on the choice of the reference frame. Here we explicitly encode such properties using an equivariant graph convolutional neural network. The network respects rotational symmetries of the crystal throughout by using equivariant weights and descriptors and provides a tensorial output of the target value. Applications to tensors of atomic Born effective charges in diverse materials including perovskite oxides, Li3PO4, and ZrO2, are demonstrated, and good performance and generalization ability is obtained.

Abstract:
The domain of laser fusion presents a unique and challenging predictive modeling application landscape for machine learning methods due to high problem complexity and limited training data. Data‑driven approaches utilizing prescribed functional forms, inductive biases and physics‑informed learning (PIL) schemes have been successful in the past for achieving desired generalization ability and model interpretation that aligns with physics expectations. In complex multi‑physics application domains, however, it is not always obvious how architectural biases or discriminative penalties can be formulated. In this work, focusing on nuclear fusion energy using high powered lasers, we present the use of Kolmogorov‑Arnold Networks (KANs) as an alternative to PIL for developing a new type of data‑driven predictive model which is able to achieve high prediction accuracy and physics interpretability. A KAN based model, a MLP with PIL, and a baseline MLP model are compared in generalization ability and interpretation with a domain expert‑derived symbolic regression model. Through empirical studies in this high physics complexity domain, we show that KANs can potentially provide benefits when developing predictive models for data‑starved physics applications.

Abstract:
In this paper, we present two computational methods for performing simulations of pollution propagation described by advection‑diffusion equations. The first method employs graph grammars to describe the generation process of the computational mesh used in simulations with the meshless solver of the three‑dimensional finite element method. The graph transformation rules express the three‑dimensional Rivara longest‑edge refinement algorithm. This solver is used for an exemplary application: performing three‑dimensional simulations of pollution generation by the coal‑burning power plant and its propagation in the city of Longyearbyen, the capital of Spitsbergen. The second computational code is based on the Physics Informed Neural Networks method. It is used to calculate the dissipation of the pollution along the valley in which the city of Longyearbyen is located. We discuss the instantiation and execution of the PINN method using Google Colab implementation. We discuss the benefits and limitations of the PINN implementation.

Abstract:
On 2024 May 10/11, the strongest geomagnetic storm since November 2003 has occurred, with a peak Dst index of ‑412 nT. The storm was caused by NOAA Active Region (AR) 13664, which was the source of a large number of coronal mass ejections and flares, including 12 X‑class flares. Starting from about May 7, AR 13664 showed a steep increase in its size and (free) magnetic energy, along with increased flare activity. In this study, we perform 3D magnetic field extrapolations with the NF2 nonlinear‑force free code based on physics informed neural networks (Jarolim et al. 2023). In addition, we introduce the computation of the vector potential to achieve divergence‑free solutions. We extrapolate vector magnetograms from SDO/HMI at the full 12 minute cadence from 2024 May 5‑00:00 to 11‑04:36 UT, in order to understand the active regions magnetic evolution and the large eruptions it produced. The computed change in magnetic energy and free magnetic energy shows a clear correspondence to the flaring activity. Regions of free magnetic energy and depleted magnetic energy indicate the flare origin and are in good correspondence with observations in Extreme Ultraviolet. Our results suggest that the modeled solar flares are related to significant topological reconfigurations. We provide a detailed analysis of the X4.0‑class flare on May 10, where we show that the interaction between separated magnetic domains is directly linked to major flaring events. With this study, we provide a comprehensive data set of the magnetic evolution of AR 13664 and make it publicly available for further analysis.

Abstract:
This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and infinity. A traditional numerical method for solving the resulting high‑dimensional coupled system of PDEs is replaced by deep learning techniques. In order to achieve this, physics‑informed neural networks (PINNs), which typically operate on the strong residual of the PDE and can therefore be applied in a wide range of settings, are considered. As a second approach, the Deep Ritz method, which is a neural network that minimizes the Ritz energy functional to find the weak solution, is employed. While the second approach only works in special cases, it overcomes the necessity of testing in variational problems while maintaining mathematical rigor and ensuring the existence of a unique solution. Furthermore, the residual is of a lower differentiation order, reducing the training cost considerably. The efficiency of the method is demonstrated on several model problems.

Abstract:
Singularly perturbed problems are known to have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods (FDMs), require a refined mesh to obtain stable and accurate solutions. As Physics‑Informed Neural Networks (PINNs) have been shown to successfully approximate solutions to differential equations from various fields, it is natural to examine their performance on singularly perturbed problems. The convection‑diffusion equation is a representative example of such a class of problems, and we consider the use of PINNs to produce numerical solutions of this equation. We study two ways to use PINNS: as a method for correcting oscillatory discrete solutions obtained using FDMs, and as a method for modifying reduced solutions of unperturbed problems. For both methods, we also examine the use of input transformation to enhance accuracy, and we explain the behavior of input transformations analytically, with the help of neural tangent kernels.

Abstract:
We present a versatile framework that employs Physics‑Informed Neural Networks (PINNs) to discover the entropic contribution that leads to the constitutive equation for the extra‑stress in rheological models of polymer solutions. In this framework the training of the Neural Network is guided by an evolution equation for the conformation tensor which is GENERIC‑compliant. We compare two training methodologies for the data‑driven PINN constitutive models: one trained on data from the analytical solution of the Oldroyd‑B model under steady‑state rheometric flows (PINN‑rheometric), and another trained on in‑silico data generated from complex flow CFD simulations around a cylinder that use the Oldroyd‑B model (PINN‑complex). The capacity of the PINN models to provide good predictions are evaluated by comparison with CFD simulations using the underlying Oldroyd‑B model as a reference. Both models are capable of predicting flow behavior in transient and complex conditions; however, the PINN‑complex model, trained on a broader range of mixed flow data, outperforms the PINN‑rheometric model in complex flow scenarios. The geometry agnostic character of our methodology allows us to apply the learned PINN models to flows with different topologies than the ones used for training.

Abstract:
This paper introduces a dynamic, error‑bounded hierarchical matrix (H‑matrix) compression method tailored for Physics‑Informed Neural Networks (PINNs). The proposed approach reduces the computational complexity and memory demands of large‑scale physics‑based models while preserving the essential properties of the Neural Tangent Kernel (NTK). By adaptively refining hierarchical matrix approximations based on local error estimates, our method ensures efficient training and robust model performance. Empirical results demonstrate that this technique outperforms traditional compression methods, such as Singular Value Decomposition (SVD), pruning, and quantization, by maintaining high accuracy and improving generalization capabilities. Additionally, the dynamic H‑matrix method enhances inference speed, making it suitable for real‑time applications. This approach offers a scalable and efficient solution for deploying PINNs in complex scientific and engineering domains, bridging the gap between computational feasibility and real‑world applicability.

Abstract:
In this paper, we introduce the KANtrol framework, which utilizes Kolmogorov‑Arnold Networks (KANs) to solve optimal control problems involving continuous time variables. We explain how Gaussian quadrature can be employed to approximate the integral parts within the problem, particularly for integro‑differential state equations. We also demonstrate how automatic differentiation is utilized to compute exact derivatives for integer‑order dynamics, while for fractional derivatives of non‑integer order, we employ matrix‑vector product discretization within the KAN framework. We tackle multi‑dimensional problems, including the optimal control of a 2D heat partial differential equation. The results of our simulations, which cover both forward and parameter identification problems, show that the KANtrol framework outperforms classical MLPs in terms of accuracy and efficiency.

Abstract:
Variational inference (VI) is a computationally efficient and scalable methodology for approximate Bayesian inference. It strikes a balance between accuracy of uncertainty quantification and practical tractability. It excels at generative modelling and inversion tasks due to its built‑in Bayesian regularisation and flexibility, essential qualities for physics related problems. For such problems, the underlying physical model determines the dependence between variables of interest, which in turn will require a tailored derivation for the central VI learning objective. Furthermore, in many physical inference applications this structure has rich meaning and is essential for accurately capturing the dynamics of interest. In this paper, we provide an accessible and thorough technical introduction to VI for forward and inverse problems, guiding the reader through standard derivations of the VI framework and how it can best be realized through deep learning. We then review and unify recent literature exemplifying the flexibility allowed by VI. This paper is designed for a general scientific audience looking to solve physics‑based problems with an emphasis on uncertainty quantification

Abstract:
In this work, we study the Galerkin‑Boltzmann formulation within a physics‑informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin‑Boltzmann equations are discretized with second‑order Hermite polynomials in microscopic velocity space, which leads to a first‑order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non‑equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. Our findings show the potential of utilizing the Galerkin‑Boltzmann formulation in PINN for weakly compressible flow problems.

Abstract:
This study introduces \ttPI‑AstroDeconv, a physics‑informed semi‑supervised learning method specifically designed for removing beam effects in astronomical telescope observation systems. The method utilizes an encoder‑decoder network architecture and combines the telescope's point spread function or beam as prior information, while integrating fast Fourier transform accelerated convolution techniques into the deep learning network. This enables effective removal of beam effects from astronomical observation images. \ttPI‑AstroDeconv can handle multiple PSFs or beams, tolerate imprecise measurements to some extent, and significantly improve the efficiency and accuracy of image deconvolution. Therefore, this architecture is particularly suitable for astronomical data processing that does not rely on annotated data. To validate the reliability of the architecture, we used the SKA Science Data Challenge 3a datasets and compared it with the \ttCLEAN deconvolution method at the 21‑cm power spectrum level. The results demonstrate that our algorithm not only restores details and reduces blurriness in celestial images at the pixel level but also more accurately recovers the true neutral hydrogen power spectrum at the power spectrum level.

Abstract:
Neural network approaches have been demonstrated to work quite well to solve partial differential equations in practice. In this context approaches like physics‑informed neural networks and the Deep Ritz method have become popular. In this paper, we propose a similar approach to solve an infinite‑dimensional total variation minimization problem using neural networks. We illustrate that the resulting neural network problem does not have a solution in general. To circumvent this theoretic issue, we consider an auxiliary neural network problem, which indeed has a solution, and show that it converges in the sense of Γ‑convergence to the original problem. For computing a numerical solution we further propose a discrete version of the auxiliary neural network problem and again show its Γ‑convergence to the original infinite‑dimensional problem. In particular, the Γ‑convergence proof suggests a particular discretization of the total variation. Moreover, we connect the discrete neural network problem to a finite difference discretization of the infinite‑dimensional total variation minimization problem. Numerical experiments are presented supporting our theoretical findings.

Abstract:
Full‑Stokes polarimetric datasets, originating from slit‑spectrograph or narrow‑band filtergrams, are routinely acquired nowadays. The data rate is increasing with the advent of bi‑dimensional spectropolarimeters and observing techniques that allow long‑time sequences of high‑quality observations. There is a clear need to go beyond the traditional pixel‑by‑pixel strategy in spectropolarimetric inversions by exploiting the spatiotemporal coherence of the inferred physical quantities. We explore the potential of neural networks as a continuous representation of the physical quantities over time and space (also known as neural fields), for spectropolarimetric inversions. We have implemented and tested a neural field to perform the inference of the magnetic field vector (approach also known as physics‑informed neural networks) under the weak‑field approximation (WFA). By using a neural field to describe the magnetic field vector, we can regularize the solution in the spatial and temporal domain by assuming that the physical quantities are continuous functions of the coordinates. We investigated the results in synthetic and real observations of the Ca II 8542 A line. We also explored the impact of other explicit regularizations, such as using the information of an extrapolated magnetic field, or the orientation of the chromospheric fibrils. Compared to the traditional pixel‑by‑pixel inversion, the neural field approach improves the fidelity of the reconstruction of the magnetic field vector, especially the transverse component. This implicit regularization is a way of increasing the effective signal‑to‑noise of the observations. Although it is slower than the pixel‑wise WFA estimation, this approach shows a promising potential for depth‑stratified inversions, by reducing the number of free parameters and inducing spatio‑temporal constraints in the solution.

Abstract:
We establish a unified theoretical framework addressing the stability, consistency, and convergence of neural networks under realistic training conditions, specifically, in the presence of non‑IID data, geometric constraints, and embedded physical laws. For standard supervised learning with dependent data, we derive uniform stability bounds for gradient‑based methods using mixing coefficients and dynamic learning rates. In federated learning with heterogeneous data and non‑Euclidean parameter spaces, we quantify model inconsistency via curvature‑aware aggregation and information‑theoretic divergence. For Physics‑Informed Neural Networks (PINNs), we rigorously prove perturbation stability, residual consistency, Sobolev convergence, energy stability for conservation laws, and convergence under adaptive multi‑domain refinements. Each result is grounded in variational analysis, compactness arguments, and universal approximation theorems in Sobolev spaces. Our theoretical guarantees are validated across parabolic, elliptic, and hyperbolic PDEs, confirming that residual minimization aligns with physical solution accuracy. This work offers a mathematically principled basis for designing robust, generalizable, and physically coherent neural architectures across diverse learning environments.

Abstract:
Reliability analysis is a formidable task, particularly in systems with a large number of stochastic parameters. Conventional methods for quantifying reliability often rely on extensive simulations or experimental data, which can be costly and time‑consuming, especially when dealing with systems governed by complex physical laws which necessitates computationally intensive numerical methods such as finite element or finite volume techniques. On the other hand, surrogate‑based methods offer an efficient alternative for computing reliability by approximating the underlying model from limited data. Neural operators have recently emerged as effective surrogates for modelling physical systems governed by partial differential equations. These operators can learn solutions to PDEs for varying inputs and parameters. Here, we investigate the efficacy of the recently developed physics‑informed wavelet neural operator in solving reliability analysis problems. In particular, we investigate the possibility of using physics‑informed operator for solving high‑dimensional reliability analysis problems, while bypassing the need for any simulation. Through four numerical examples, we illustrate that physics‑informed operator can seamlessly solve high‑dimensional reliability analysis problems with reasonable accuracy, while eliminating the need for running expensive simulations.

Abstract:
Physics‑informed neural networks (PINNs) are able to solve partial differential equations (PDEs) by incorporating the residuals of the PDEs into their loss functions. Variational Physics‑Informed Neural Networks (VPINNs) and hp‑VPINNs use the variational form of the PDE residuals in their loss function. Although hp‑VPINNs have shown promise over traditional PINNs, they suffer from higher training times and lack a framework capable of handling complex geometries, which limits their application to more complex PDEs. As such, hp‑VPINNs have not been applied in solving the Navier‑Stokes equations, amongst other problems in CFD, thus far. FastVPINNs was introduced to address these challenges by incorporating tensor‑based loss computations, significantly improving the training efficiency. Moreover, by using the bilinear transformation, the FastVPINNs framework was able to solve PDEs on complex geometries. In the present work, we extend the FastVPINNs framework to vector‑valued problems, with a particular focus on solving the incompressible Navier‑Stokes equations for two‑dimensional forward and inverse problems, including problems such as the lid‑driven cavity flow, the Kovasznay flow, and flow past a backward‑facing step for Reynolds numbers up to 200. Our results demonstrate a 2x improvement in training time while maintaining the same order of accuracy compared to PINNs algorithms documented in the literature. We further showcase the framework's efficiency in solving inverse problems for the incompressible Navier‑Stokes equations by accurately identifying the Reynolds number of the underlying flow. Additionally, the framework's ability to handle complex geometries highlights its potential for broader applications in computational fluid dynamics. This implementation opens new avenues for research on hp‑VPINNs, potentially extending their applicability to more complex problems.

Abstract:
Active grids operated with random protocols are a standard way to generate large Reynolds number turbulence in wind and water tunnels. But anomalies in the decay and third‑order scaling of active‑grid turbulence have been reported. We combine Laser Doppler Velocimetry and hot‑wire anemometry measurements in a wind tunnel, with machine learning techniques and numerical simulations, to gain further understanding on the reasons behind these anomalies. Numerical simulations that incorporate the statistical anomalies observed in the experimental velocity field near the active grid can reproduce the experimental anomalies observed later in the decay. The results indicate that anomalies in experiments near the active grid introduce correlations in the flow that can persist for long times.

Abstract:
Physics‑informed neural networks (PINNs) have great potential for flexibility and effectiveness in forward modeling and inversion of seismic waves. However, coordinate‑based neural networks (NNs) commonly suffer from the "spectral bias" pathology, which greatly limits their ability to model high‑frequency wave propagation in sharp and complex media. We propose a unified framework of Fourier feature physics‑informed neural networks (FF‑PINNs) for solving the time‑domain wave equations. The proposed framework combines the stochastic gradient descent (SGD) strategy with an independently pre‑trained wave velocity surrogate model to mitigate the singularity at the point source. The performance of the activation functions and gradient descent strategies are discussed through ablation experiments. In addition, we evaluate the accuracy comparison of Fourier feature mappings sampled from different families of distributions (Gaussian, Laplace, and uniform). The second‑order paraxial approximation‑based boundary conditions are incorporated into the loss function as a soft regularizer to eliminate spurious boundary reflections. Through the non‑smooth Marmousi and Overthrust model cases, we emphasized the necessity of the absorbing boundary conditions (ABCs) constraints. The results of a series of numerical experiments demonstrate the accuracy and effectiveness of the proposed method for modeling high‑frequency wave propagation in sharp and complex media.

Abstract:
This paper introduces a novel methodology for solving distributed‑order fractional differential equations using a physics‑informed machine learning framework. The core of this approach involves extending the support vector regression (SVR) algorithm to approximate the unknown solutions of the governing equations during the training phase. By embedding the distributed‑order functional equation into the SVR framework, we incorporate physical laws directly into the learning process. To further enhance computational efficiency, Gegenbauer orthogonal polynomials are employed as the kernel function, capitalizing on their fractional differentiation properties to streamline the problem formulation. Finally, the resulting optimization problem of SVR is addressed either as a quadratic programming problem or as a positive definite system in its dual form. The effectiveness of the proposed approach is validated through a series of numerical experiments on Caputo‑based distributed‑order fractional differential equations, encompassing both ordinary and partial derivatives.

Abstract:
Physics‑Informed Neural Networks (PINNs) are regarded as state‑of‑the‑art tools for addressing highly nonlinear problems based on partial differential equations. Despite their broad range of applications, PINNs encounter several performance challenges, including issues related to efficiency, minimization of computational cost, and enhancement of accuracy. Burgers' equation, a fundamental equation in fluid dynamics that is extensively used in PINNs, provides flexible results with the Adam optimizer that does not account for past gradients. This paper introduces a novel strategy for solving Burgers' equation by incorporating DiffGrad with PINNs, a method that leverages the difference between current and immediately preceding gradients to enhance performance. A comprehensive computational analysis is conducted using optimizers such as Adam, Adamax, RMSprop, and DiffGrad to evaluate and compare their effectiveness. Our approach includes visualizing the solutions over space at various time intervals to demonstrate the accuracy of the network. The results show that DiffGrad not only improves the accuracy of the solution but also reduces training time compared to the other optimizers.

Abstract:
This paper proposes a novel learning‑based framework for autonomous driving based on the concept of maximal safety probability. Efficient learning requires rewards that are informative of desirable/undesirable states, but such rewards are challenging to design manually due to the difficulty of differentiating better states among many safe states. On the other hand, learning policies that maximize safety probability does not require laborious reward shaping but is numerically challenging because the algorithms must optimize policies based on binary rewards sparse in time. Here, we show that physics‑informed reinforcement learning can efficiently learn this form of maximally safe policy. Unlike existing drift control methods, our approach does not require a specific reference trajectory or complex reward shaping, and can learn safe behaviors only from sparse binary rewards. This is enabled by the use of the physics loss that plays an analogous role to reward shaping. The effectiveness of the proposed approach is demonstrated through lane keeping in a normal cornering scenario and safe drifting in a high‑speed racing scenario.

Abstract:
Self‑supervised learning is a powerful approach for developing traversability models for off‑road navigation, but these models often struggle with inputs unseen during training. Existing methods utilize techniques like evidential deep learning to quantify model uncertainty, helping to identify and avoid out‑of‑distribution terrain. However, always avoiding out‑of‑distribution terrain can be overly conservative, e.g., when novel terrain can be effectively analyzed using a physics‑based model. To overcome this challenge, we introduce Physics‑Informed Evidential Traversability (PIETRA), a self‑supervised learning framework that integrates physics priors directly into the mathematical formulation of evidential neural networks and introduces physics knowledge implicitly through an uncertainty‑aware, physics‑informed training loss. Our evidential network seamlessly transitions between learned and physics‑based predictions for out‑of‑distribution inputs. Additionally, the physics‑informed loss regularizes the learned model, ensuring better alignment with the physics model. Extensive simulations and hardware experiments demonstrate that PIETRA improves both learning accuracy and navigation performance in environments with significant distribution shifts.

Abstract:
Physics‑informed neural networks (PINNs) are an emerging technique to solve partial differential equations (PDEs). In this work, we propose a simple but effective PINN approach for the phase‑field model of ferroelectric microstructure evolution. This model is a time‑dependent, nonlinear, and high‑order PDE system of multi‑physics, challenging to be solved using a baseline PINN. Considering that the acquisition of steady microstructures is one of the primary focuses in simulations of ferroelectric microstructure evolution, we simplify the time‑dependent PDE system to be a static problem. This static problem, however, is ill‑posed. To overcome this issue, a term originated from the law of energy dissipation is embedded into the loss function as an extra constraint for the PINN. With this modification, the PINN successfully predicts the steady ferroelectric microstructure without tracking the evolution process. In addition, although the proposed PINN approach cannot tackle the dynamic problem in a straightforward fashion, it is of benefit to the PINN prediction of the evolution process by providing labeled data. These data are crucial because they help the PINN avoid the propagation failure, a common failure mode of PINNs when predicting dynamic behaviors. The above mentioned advantages of the proposed PINN approach are demonstrated through a number of examples.

Abstract:
This paper addresses integrated design of engineering systems, where physical structure of the plant and controller design are optimized simultaneously. To cope with uncertainties due to noises acting on the dynamics and modeling errors, an Uncertain Control Co‑design (UCCD) problem formulation is proposed. Existing UCCD methods usually rely on uncertainty propagation analyses using Monte Calro methods for open‑loop solutions of optimal control, which suffer from stringent trade‑offs among accuracy, time horizon, and computational time. The proposed method utilizes closed‑loop solutions characterized by the Hamilton‑Jacobi‑Bellman equation, a Partial Differential Equation (PDE) defined on the state space. A solution algorithm for the proposed UCCD formulation is developed based on PDE solutions of Physics‑informed Neural Networks (PINNs). Numerical examples of regulator design problems are provided, and it is shown that simultaneous update of PINN weights and the design parameters effectively works for solving UCCD problems.

Abstract:
In this paper, we develop a systematic deep learning approach to solve two‑dimensional (2D) stationary quantum droplets (QDs) and investigate their wave propagation in the 2D amended Gross‑Pitaevskii equation with Lee‑Huang‑Yang correction and two kinds of potentials. Firstly, we use the initial‑value iterative neural network (IINN) algorithm for 2D stationary quantum droplets of stationary equations. Then the learned stationary QDs are used as the initial value conditions for physics‑informed neural networks (PINNs) to explore their evolutions in the some space‑time region. Especially, we consider two types of potentials, one is the 2D quadruple‑well Gaussian potential and the other is the PT‑symmetric HO‑Gaussian potential, which lead to spontaneous symmetry breaking and the generation of multi‑component QDs. The used deep learning method can also be applied to study wave propagations of other nonlinear physical models.

Abstract:
Domain‑decomposed variants of physics‑informed neural networks (PINNs) such as finite basis PINNs (FBPINNs) mitigate some of PINNs' issues like slow convergence and spectral bias through localisation, but still rely on iterative nonlinear optimisation within each subdomain. In this work, we propose a hybrid approach that combines multilevel domain decomposition and partition‑of‑unity constructions with random feature models, yielding a method referred to as multilevel ELM‑FBPINN. By replacing trainable subdomain networks with extreme learning machines, the resulting formulation eliminates backpropagation entirely and reduces training to a structured linear least‑squares problem. We provide a systematic numerical study comparing ELM‑FBPINNs and multilevel ELM‑FBPINNs with standard PINNs and FBPINNs on representative benchmark problems, demonstrating that ELM‑FBPINNs and multilevel ELM‑FBPINNs achieve competitive accuracy while significantly accelerating convergence and improving robustness with respect to architectural and optimisation parameters. Through ablation studies, we further clarify the distinct roles of domain decomposition and random feature enrichment in controlling expressivity, conditioning, and scalability.

Abstract:
We propose Gradient Informed Neural Networks (GradINNs), a methodology inspired by Physics Informed Neural Networks (PINNs) that can be used to efficiently approximate a wide range of physical systems for which the underlying governing equations are completely unknown or cannot be defined, a condition that is often met in complex engineering problems. GradINNs leverage prior beliefs about a system's gradient to constrain the predicted function's gradient across all input dimensions. This is achieved using two neural networks: one modeling the target function and an auxiliary network expressing prior beliefs, e.g., smoothness. A customized loss function enables training the first network while enforcing gradient constraints derived from the auxiliary network. We demonstrate the advantages of GradINNs, particularly in low‑data regimes, on diverse problems spanning non time‑dependent systems (Friedman function, Stokes Flow) and time‑dependent systems (Lotka‑Volterra, Burger's equation). Experimental results showcase strong performance compared to standard neural networks and PINN‑like approaches across all tested scenarios.

Abstract:
Studying the magnetic field properties on the solar surface is crucial for understanding the solar and heliospheric activities, which in turn shape space weather in the solar system. Surface Flux Transport (SFT) modeling helps us to simulate and analyse the transport and evolution of magnetic flux on the solar surface, providing valuable insights into the mechanisms responsible for solar activity. In this work, we demonstrate the use of machine learning techniques in solving magnetic flux transport, making it accurate. We have developed a novel Physics‑Informed Neural Networks (PINN)‑based model to study the evolution of Bipolar Magnetic Regions (BMRs) using SFT in one‑dimensional azimuthally averaged and also in two‑dimensions. We demonstrate the efficiency and computational feasibility of our PINN‑based model by comparing its performance and accuracy with that of a numerical model implemented using the Runge‑Kutta Implicit‑Explicit (RK‑IMEX) scheme. The mesh‑independent PINN method can be used to reproduce the observed polar magnetic field with better flux conservation. This advancement is important for accurately reproducing observed polar magnetic fields, thereby providing insights into the strength of future solar cycles. This work paves the way for more efficient and accurate simulations of solar magnetic flux transport and showcases the applicability of PINN in solving advection‑diffusion equations with a particular focus on heliophysics.

Abstract:
The growing computational demands of artificial intelligence (AI) in addressing climate change raise significant concerns about inefficiencies and environmental impact, as highlighted by the Jevons paradox. We propose an attention‑enhanced quantum physics‑informed neural networks model (AQ‑PINNs) to tackle these challenges. This approach integrates quantum computing techniques into physics‑informed neural networks (PINNs) for climate modeling, aiming to enhance predictive accuracy in fluid dynamics governed by the Navier‑Stokes equations while reducing the computational burden and carbon footprint. By harnessing variational quantum multi‑head self‑attention mechanisms, our AQ‑PINNs achieve a 51.51% reduction in model parameters compared to classical multi‑head self‑attention methods while maintaining comparable convergence and loss. It also employs quantum tensor networks to enhance representational capacity, which can lead to more efficient gradient computations and reduced susceptibility to barren plateaus. Our AQ‑PINNs represent a crucial step towards more sustainable and effective climate modeling solutions.

Abstract:
Compared to purely data‑driven methods, a key feature of physics‑informed neural networks (PINNs) ‑ a proven powerful tool for solving partial differential equations (PDEs) ‑ is the embedding of PDE constraints into the loss function. The selection and distribution of collocation points for evaluating PDE residuals are critical to the performance of PINNs. Furthermore, the causal training is currently a popular training mode. In this work, we propose the causality‑guided adaptive sampling (Causal AS) method for PINNs. Given the characteristics of causal training, we use the weighted PDE residuals as the indicator for the selection of collocation points to focus on areas with larger PDE residuals within the regions being trained. For the hyper‑parameter p involved, we develop the temporal alignment driven update (TADU) scheme for its dynamic update beyond simply fixing it as a constant. The collocation points selected at each time will be released before the next adaptive sampling step to avoid the cumulative effects caused by previously chosen collocation points and reduce computational costs. To illustrate the effectiveness of the Causal AS method, we apply it to solve time‑dependent equations, including the Allen‑Cahn equation, the NLS equation, the KdV equation and the mKdV equation. During the training process, we employe a time‑marching technique and strictly impose the periodic boundary conditions by embedding the input coordinates into Fourier expansion to mitigate optimization challenges. Numerical results indicate that the predicted solution achieves an excellent agreement with the ground truth. Compared to a similar work, the causal extension of R3 sampling (Causal R3), our proposed Causal AS method demonstrates a significant advantage in accuracy.

Abstract:
Dataset distillation (DD) is an increasingly important technique that focuses on constructing a synthetic dataset capable of capturing the core information in training data to achieve comparable performance in models trained on the latter. While DD has a wide range of applications, the theory supporting it is less well evolved. New methods of DD are compared on a common set of benchmarks, rather than oriented towards any particular learning task. In this work, we present a formal model of DD, arguing that a precise characterization of the underlying optimization problem must specify the inference task associated with the application of interest. Without this task‑specific focus, the DD problem is under‑specified, and the selection of a DD algorithm for a particular task is merely heuristic. Our formalization reveals novel applications of DD across different modeling environments. We analyze existing DD methods through this broader lens, highlighting their strengths and limitations in terms of accuracy and faithfulness to optimal DD operation. Finally, we present numerical results for two case studies important in contemporary settings. Firstly, we address a critical challenge in medical data analysis: merging the knowledge from different datasets composed of intersecting, but not identical, sets of features, in order to construct a larger dataset in what is usually a small sample setting. Secondly, we consider out‑of‑distribution error across boundary conditions for physics‑informed neural networks (PINNs), showing the potential for DD to provide more physically faithful data. By establishing this general formulation of DD, we aim to establish a new research paradigm by which DD can be understood and from which new DD techniques can arise.

Abstract:
This work builds off the manifold‑constrained Gaussian process inference (MAGI) method for Bayesian parameter inference and trajectory reconstruction of ODE‑based dynamical systems, focusing primarily on sparse and noisy data conditions. First, we introduce Pilot MAGI (pMAGI), a novel methodological upgrade on the base MAGI method that confers significantly‑improved numerical stability, parameter inference, and trajectory reconstruction. Second, we demonstrate, for the first time to our knowledge, how one can combine MAGI‑based methods with dynamical systems theory to provide probabilistic classifications of whether a system is stable or chaotic. Third, we demonstrate how pMAGI performs favorably in many settings against much more computationally‑expensive and overparameterized methods. Fourth, we introduce Pilot MAGI Sequential Prediction (PMSP), a novel method building upon pMAGI that allows one to predict the trajectory of ODE‑based dynamical systems multiple time steps into the future, given only sparse and noisy observations. We show that PMSP can output accurate future predictions even on chaotic dynamical systems and significantly outperform PINN‑based methods. Overall, we contribute to the literature two novel methods, pMAGI and PMSP, that serve as Bayesian, uncertainty‑quantified competitors to the Physics‑Informed Neural Network.

Abstract:
The coordinated dispatch of power and gas in the electricity‑gas integrated energy system (EG‑IES) is fundamental for ensuring operational security. However, the gas dynamics in the natural gas system (NGS) are governed by the nonlinear partial differential equations (PDE), making the dispatch problem of the EG‑IES a complicated optimization model constrained by nonlinear PDE. To address it, we propose a globally linearized gas network model based on the Koopman operator theory, avoiding the commonly used local linearization and spatial discretization. Particularly, we propose a data‑driven Koopman operator approximation approach for the globally linearized gas network model based on the extended dynamic mode decomposition, in which a physics‑informed stability constraint is derived and embedded to improve the generalization ability and accuracy of the model. Based on this, we develop an optimal dispatch model for the EG‑IES that first considers the nonlinear gas dynamics in the NGS. The case study verifies the effectiveness of this work. Simulation results reveal that the commonly used locally linearized gas network model fails to accurately capture the dynamic characteristics of NGS, bringing potential security threats to the system.

Abstract:
We propose a new two‑stage initial‑value iterative neural network (IINN) algorithm for solitary wave computations of nonlinear wave equations based on traditional numerical iterative methods and physics‑informed neural networks (PINNs). Specifically, the IINN framework consists of two subnetworks, one of which is used to fit a given initial value, and the other incorporates physical information and continues training on the basis of the first subnetwork. Importantly, the IINN method does not require any additional data information including boundary conditions, apart from the given initial value. Corresponding theoretical guarantees are provided to demonstrate the effectiveness of our IINN method. The proposed IINN method is efficiently applied to learn some types of solutions in different nonlinear wave equations, including the one‑dimensional (1D) nonlinear Schrödinger equations (NLS) equation (with and without potentials), the 1D saturable NLS equation with PT symmetric optical lattices, the 1D focusing‑defocusing coupled NLS equations, the KdV equation, the two‑dimensional (2D) NLS equation with potentials, the 2D amended GP equation with a potential, the (2+1)‑dimensional KP equation, and the 3D NLS equation with a potential. These applications serve as evidence for the efficacy of our method. Finally, by comparing with the traditional methods, we demonstrate the advantages of the proposed IINN method.

Abstract:
Finite element modeling is a well‑established tool for structural analysis, yet modeling complex structures often requires extensive pre‑processing, significant analysis effort, and considerable time. This study addresses this challenge by introducing an innovative method for real‑time prediction of structural static responses using DeepOnet which relies on a novel approach to physics‑informed networks driven by structural balance laws. This approach offers the flexibility to accurately predict responses under various load classes and magnitudes. The trained DeepONet can generate solutions for the entire domain, within a fraction of a second. This capability effectively eliminates the need for extensive remodeling and analysis typically required for each new case in FE modeling. We apply the proposed method to two structures: a simple 2D beam structure and a comprehensive 3D model of a real bridge. To predict multiple variables with DeepONet, we utilize two strategies: a split branch/trunk and multiple DeepONets combined into a single DeepONet. In addition to data‑driven training, we introduce a novel physics‑informed training approaches. This method leverages structural stiffness matrices to enforce fundamental equilibrium and energy conservation principles, resulting in two novel physics‑informed loss functions: energy conservation and static equilibrium using the Schur complement. We use various combinations of loss functions to achieve an error rate of less than 5% with significantly reduced training time. This study shows that DeepONet, enhanced with hybrid loss functions, can accurately and efficiently predict displacements and rotations at each mesh point, with reduced training time.

Abstract:
Digital Image Correlation (DIC) is a key technique in experimental mechanics for full‑field deformation measurement, traditionally relying on subset matching to determine displacement fields. However, selecting optimal parameters like shape functions and subset size can be challenging in non‑uniform deformation scenarios. Recent deep learning‑based DIC approaches, both supervised and unsupervised, use neural networks to map speckle images to deformation fields, offering precise measurements without manual tuning. However, these methods require complex network architectures to extract speckle image features, which does not guarantee solution accuracy This paper introduces PINN‑DIC, a novel DIC method based on Physics‑Informed Neural Networks (PINNs). Unlike traditional approaches, PINN‑DIC uses a simple fully connected neural network that takes the coordinate domain as input and outputs the displacement field. By integrating the DIC governing equation into the loss function, PINN‑DIC directly extracts the displacement field from reference and deformed speckle images through iterative optimization. Evaluations on simulated and real experiments demonstrate that PINN‑DIC maintains the accuracy of deep learning‑based DIC in non‑uniform fields while offering three distinct advantages: 1) enhanced precision with a simpler network by directly fitting the displacement field from coordinates, 2) effective handling of irregular boundary displacement fields with minimal parameter adjustments, and 3) easy integration with other neural network‑based mechanical analysis methods for comprehensive DIC result analysis.

Abstract:
In this study, we explore the application of Physics‑Informed Neural Networks (PINNs) to the analysis of bifurcation phenomena in ecological migration models. By integrating the fundamental principles of diffusion‑advection‑reaction equations with deep learning techniques, we address the complexities of species migration dynamics, particularly focusing on the detection and analysis of Hopf bifurcations. Traditional numerical methods for solving partial differential equations (PDEs) often involve intricate calculations and extensive computational resources, which can be restrictive in high‑dimensional problems. In contrast, PINNs offer a more flexible and efficient alternative, bypassing the need for grid discretization and allowing for mesh‑free solutions. Our approach leverages the DeepXDE framework, which enhances the computational efficiency and applicability of PINNs in solving high‑dimensional PDEs. We validate our results against conventional methods and demonstrate that PINNs not only provide accurate bifurcation predictions but also offer deeper insights into the underlying dynamics of diffusion processes. Despite these advantages, the study also identifies challenges such as the high computational costs and the sensitivity of PINN performance to network architecture and hyperparameter settings. Future work will focus on optimizing these algorithms and expanding their application to other complex systems involving bifurcations. The findings from this research have significant implications for the modeling and analysis of ecological systems, providing a powerful tool for predicting and understanding complex dynamical behaviors.

Abstract:
Physics‑informed deep learning (PIDL)‑based models have recently garnered remarkable success in traffic state estimation (TSE). However, the prior knowledge used to guide regularization training in current mainstream architectures is based on deterministic physical models. The drawback is that a solely deterministic model fails to capture the universally observed traffic flow dynamic scattering effect, thereby yielding unreliable outcomes for traffic control. This study, for the first time, proposes stochastic physics‑informed deep learning (SPIDL) for traffic state estimation. The idea behind such SPIDL is simple and is based on the fact that a stochastic fundamental diagram provides the entire range of possible speeds for any given density with associated probabilities. Specifically, we select percentile‑based fundamental diagram and distribution‑based fundamental diagram as stochastic physics knowledge, and design corresponding physics‑uninformed neural networks for effective fusion, thereby realizing two specific SPIDL models, namely \textα‑SPIDL and \text\cal B‑SPIDL. The main contribution of SPIDL lies in addressing the "overly centralized guidance" caused by the one‑to‑one speed‑density relationship in deterministic models during neural network training, enabling the network to digest more reliable knowledge‑based constraints.Experiments on the real‑world dataset indicate that proposed SPIDL models achieve accurate traffic state estimation in sparse data scenarios. More importantly, as expected, SPIDL models reproduce well the scattering effect of field observations, demonstrating the effectiveness of fusing stochastic physics model knowledge with deep learning frameworks.