Physics-Informed ML

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Overview

Physics-informed machine learning studies how to combine observations with governing equations, conservation laws, symmetries, and other scientific knowledge. Common approaches include physics-informed neural networks (PINNs) that penalize equation residuals, weak or variational formulations, neural operators for families of PDE solutions, and hybrid models that learn corrections to mechanistic simulators. These methods are used for forward simulation, inverse problems, parameter estimation, forecasting, and fast surrogates when measurements or computational resources are limited. A central challenge is that physical constraints can improve data efficiency and consistency but can also hinder training or harm predictions when the equations, discretization, or assumed structure do not match the system.

Latest papers

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843 papers75 papers in the last four weeks, down 13% on the four weeks before. 0.4% of all new papers.

Jan 28, 2026

Leveraging Differentiable PDE Solvers for Semi-Neural Spatial Reconstruction From Sparse Measurements

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art approaches to this problem either rely on spatial statistics that ignore the governing physics, integrate the physics into a multiple-objective optimization process, or require examples of the complete, fully-resolved simulation state during training, which are frequently unavailable outside of synthetic benchmarks. Here, we present a novel alternative that leverages recent advances in the integration of numerical simulators with data-driven models. Namely, we propose a hybrid modeling pipeline that couples Radial Basis Function (RBF) reconstruction with a Neural Network (NN) correction and a Partial Differential Equation (PDE) solver, so that the numerical simulator itself is embedded directly in the training loop of the learned component. Notably, the NN is trained without assuming availability of examples of the fully-resolved simulation state. This is made possible by implementing the PDE solver so that it is end-to-end differentiable, allowing gradients to be backpropagated through the simulation step during training. This grey-box methodology is evaluated on three standard benchmarks from fluid mechanics, where it achieves superior results over statistical and machine-learning-based reconstruction methods.

Ofek Aloni, Barak Fishbain

Department of Mathematics, Technion, Israel · Department of Civil and Environmental Engineering, Technion, Israel

Differentiable PhysicsPhysics-Informed MLComputational Fluid Dynamics
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Jan 19, 2026

Adaptively trained Physics-informed Radial Basis Function Neural Networks for Solving Multi-asset Option Pricing Problems

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.

Yan Ma, Yumeng Ren, Elisabeth Larsson

aCollege of Mathematics and Statistics, Northwest Normal University, Lanzhou, China · bDepartment of Mathematics, City University of Hong Kong, Hong Kong SAR, China · cDepartment of Information Technology, Uppsala University, Sweden

Quantitative FinancePDE SolvingPhysics-Informed ML
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Jan 7, 2026

Disentangling Aleatoric and Epistemic Uncertainty in Physics-Informed Neural Networks. Application to Insulation Material Degradation Prognostics

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, and the influence of measurement noise on aleatoric uncertainty. Overall, the findings highlight the capability of Bayesian physics-informed learning to support uncertainty-aware prognostics and informed decision-making in transformer asset management by tracking aleatoric and epistemic sources of uncertainty.

Ibai Ramirez, Jokin Alcibar, Joel Pino +2

Department of Electronics and Computer Science Mondragon University Arrasate-Mondragon, 20500, Spain · CARTIF Technology Centre Valladolid, 47151, Spain · Department of Physical Chemistry University of the Basque Country (EHU)[cs.LG] Basque Centre for Applied Mathematics (BCAM) Leioa, 20018, Spain +1

Bayesian Neural NetworksUncertainty QuantificationPredictive MaintenancePhysics-Informed MLAleatoric UncertaintyUncertainty Calibration
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Jan 1, 2026

Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning

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.

Abhisek Ganguly, Santosh Ansumali, Sauro Succi

Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, 560064, Karnataka, India · Italian Institute of Technology, Viale Regina Elena 291, 00161 Rome, Italy · University of Roma Tre, Italy +2

Dynamical SystemsNeural PDE SolversPhysics-Informed MLNeural Network Training Dynamics
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Dec 30, 2025

Evolutionary Physics-Informed Temporal Fusion for Lane-Change Intention Prediction

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.

Jiazhao Shi, Qiyang Xie, Ziyu Wang +7

Tandon School of Engineering New York University New York, USA · Khoury College of Computer Science Northeastern University Boston, USA · School of Business Wake Forest University Winston Salem, USA +6

Autonomous DrivingPhysics-Informed ML
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Dec 8, 2025

KIGNet: Physics-Motivated Multi-Graph Representation Learning for Explainable Jet Tagging

Jet identification plays a central role in analyzing data from high-energy collider experiments. While deep learning has improved jet classification, it often lacks interpretability. We introduce the Kinematic Interaction Graph Network (KIGNet), a graph neural network that integrates kinematic variables into jet classification by constructing four graph representations per jet, each weighted by a distinct variable: angular separation (ΔΔ), relative transverse momentum (kTk_T), momentum fraction (zz), and invariant mass squared (m2m^2). Three of these (ΔΔ, kTk_T, zz) are motivated by the Lund jet plane, grounded in perturbative QCD factorization; the fourth (m2m^2) adds complementary mass-scale sensitivity for heavy-flavor identification. Using Gradient-weighted Class Activation Mapping (Grad-CAM), we determine which variables dominate classification. Angular separation and relative transverse momentum account for about 76% of the total Grad-CAM attribution (40.72% and 35.67%), with momentum fraction and invariant mass contributing the remaining 24%. This hierarchy is consistent with the soft-collinear structure of QCD radiation in the training data, showing that the network learns physically interpretable representations rather than spurious correlations. On the JetClass dataset, KIGNet achieves a macro-accuracy of 95.07%, macro-AUC of 96.61%, and macro-AUPR of 81.52%, relative improvements of 2.45%, 3.40%, and 19.11% over the state-of-the-art baseline. On the Aspen Open Jets dataset of real CMS collision data, KIGNet produces substantially more structured latent representations than the baseline, reducing the Davies-Bouldin Index by 52.15% (0.8395→0.40170.8395 \rightarrow 0.4017) and increasing the Dunn Index by 42.33% (0.0189→0.02690.0189 \rightarrow 0.0269), confirming that physics-informed kinematic encoding generalizes beyond idealized simulation to experimental detector conditions.

Md Raqibul Islam, Adrita Khan, Mir Sazzat Hossain +6

Center for Computational & Data Sciences, Independent University, Bangladesh, Dhaka-1229, Bangladesh · Department of Theoretical Physics, University of Dhaka, Dhaka-1000, Bangladesh · Department of Physical Sciences, Independent University, Bangladesh, Dhaka-1229, Bangladesh

Graph Representation LearningJet TaggingHigh-Energy PhysicsPhysics-Informed MLGNN Explainability
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Dec 3, 2025

Quantum-Classical Physics-Informed Neural Networks for Solving Reservoir Seepage Equations

Solving reservoir seepage partial differential equations (PDEs) is critical for hydrocarbon development optimization and production forecasting. Conventional numerical solvers suffer from mesh errors and high computational cost. Standard Physics-Informed Neural Networks (PINNs) are limited by inefficient parameters. They also face weak high-dimensional representation and poor nonlinear fitting ability. We adapt the Discrete Variable (DV)-Circuit Quantum-Classical Physics-Informed Neural Network (QCPINN) to tackle these drawbacks. This work applies the method to four representative reservoir seepage models. The models cover heterogeneous single-phase pressure diffusion. They also include nonlinear Buckley-Leverett two-phase waterflooding. Another case is adsorption-coupled convection-diffusion compositional flow. The last one refers to fully coupled pressure-saturation two-phase flow with exponential permeability heterogeneity. The QCPINN combines classical preprocessing and postprocessing networks with a DV quantum core. It leverages quantum entanglement and superposition to boost high-dimensional feature mapping. Physical constraints are embedded within the framework to guarantee physically consistent solutions. We evaluate three quantum circuit topologies including Cascade, Cross-mesh and Alternate. Numerical tests prove QCPINNs outperform classical PINNs in prediction accuracy, and the circuit topologies show case-specific performance with clear applicable ranges. This work verifies QCPINN's practicability for reservoir engineering. It bridges quantum computing research with oilfield industrial applications.

Xiang Rao, Yina Liu, Yuxuan Shen

School of Petroleum Engineering, Yangtze University, Wuhan 430100, China · College of Future Technology, Yangtze University, Wuhan 430100, China · State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University), Wuhan 430100, China +2

PDE SolvingPhysics-Informed MLHybrid Quantum-Classical ML
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Nov 19, 2025

Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation

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.

Victorita Dolean, Daria Hrebenshchykova, Stéphane Lanteri +1

Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands · Atlantis project-team, Inria Research Center at Université Côte d’Azur, France · MACARON project-team, Université de Strasbourg, CNRS, Inria, IRMA, France

Neural PDE SolversPDE SolvingDomain Decomposition MethodsPhysics-Informed ML
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Nov 8, 2025

Hard-constraint physics-residual networks for hydrogen crossover prediction and high-pressure extrapolation in PEM water electrolysis

Hydrogen crossover is a critical safety and efficiency constraint in high-pressure polymer electrolyte membrane water electrolysis (PEMWE), but accurate prediction remains difficult because data are limited, transport physics are strongly coupled, and industrial operation requires reliable extrapolation beyond observed conditions. This study develops a hard-constraint physics-residual network (PR-Net) for hydrogen crossover prediction in PEMWE and compares it with a purely data-driven neural network (NN) and a soft-constraint physics-informed neural network (PINN). PR-Net embeds Henry's, Fick's, and Faraday's laws as a deterministic backbone and learns only a residual correction for unmodelled nonlinear effects. The benchmark includes 184 observations from eight peer-reviewed sources across six membrane types, covering 1-200 bar, 25−85°C25-85°C, and 0.05−5.0Acm−20.05-5.0 A cm^{-2}. PR-Net achieves R2=99.57±0.16R^2 = 99.57 \pm 0.16%, with 9-fold lower prediction variability than NN and PINN. In pressure-axis extrapolation, PR-Net attains R2=94.02±0.92R^2 = 94.02 \pm 0.92% at 200 bar, 2.5 times beyond the training pressure range, compared with 68.06±5.5268.06 \pm 5.52% for PINN and 58.00±8.6058.00 \pm 8.60% for NN (p < 0.001). Residual analysis indicates that the learned correction captures part of the high-pressure gas-phase non-ideality and recovers a transport-regime transition near 0.23Acm−20.23 A cm^{-2} between Fickian diffusion-dominated and Faradaic production-dominated transport. With a computation time of 1.08±0.34ms1.08 \pm 0.34 ms on low-power embedded hardware, PR-Net provides a practical framework for real-time crossover monitoring, adaptive process control, and safer high-pressure green-hydrogen operation.

Yong-Woon Kim, Jihyeok Lee, Chulung Kang +1

Department of Computer Engineering, Jeju National University, Jeju, 63243, South Korea · Faculty of Applied Energy System, Jeju National University, Jeju, 63243, South Korea · Department of Mechanical System Engineering, Jeju National University, Jeju, 63243, South Korea +1

Neural Surrogate ModelingPhysics-Informed ML
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Nov 1, 2025

Physics-Informed Neural Networks for Speech Production

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.

Kazuya Yokota, Ryosuke Harakawa, Masaaki Baba +1

Department of Mechanical Engineering, Nagaoka University of Technology, 1603-1, Kamitomioka, Nagaoka, Niigata, Japan · Department of Electrical, Electronics and Information Engineering, Nagaoka University of Technology

Speech ProcessingPhysics-Informed MLInverse Problems
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Oct 29, 2025

LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries

Initial-boundary value problems (IBVPs) provide the essential framework for modelling a wide range of phenomena in physics and engineering. We introduce a novel method for efficiently solving IBVPs using Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction. By leveraging symmetry transformations, our model embeds the underlying physical laws and learns the solution solely from initial and boundary data. Consequently, the boundary loss directly quantifies domain-wide error, enabling rigorous error estimation for well-posed IBVPs. We implement LieSolver and demonstrate its application to linear homogeneous PDEs, showing that it outperforms physics-informed neural networks (PINNs) in both speed and accuracy while yielding compact models. Overall, our approach significantly enhances the efficiency and reliability of predictions for PDE-constrained problems.

René P. Klausen, Ivan Timofeev, Jonas Naujoks +4

Department of Artificial Intelligence, Fraunhofer Heinrich Hertz Institute, Berlin, Germany · Department of Electrical Engineering and Computer Science, Technische Universität Berlin, Germany · BIFOLD Berlin Institute for the Foundations of Learning and Data, Berlin, Germany +1

Neural PDE SolversPDE SolvingPhysics-Informed ML
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Oct 27, 2025

Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Method

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: (A) what is the trade-off between soft penalties and hard constraints, and (B) 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 classes of functions whose convex hull satisfies a small-ball condition, in particular convex classes, under physical knowledge expressed as linear equations by developing appropriate complexity dependent error rates based on the small-ball method. We specialise them to linear classes of finite dimension qq in which the space of solutions of the equation has dimension k<qk < q, and show that (A) the physics-informed estimator attains the error bound of the hard-constrained empirical error minimiser whenever the penalty parameter is above an explicit threshold, and that (B) the penalty improves the error rate from q/n\sqrt{q/n} to k/n\sqrt{k/n} under a second moment condition on the noise only, having the effect of a dimensionality reduction. We also quantify how these conclusions degrade when the equation is misspecified, in terms of the excess residual of the target function. This work establishes a theoretical framework for evaluating the statistical properties of physics-informed estimators in such classes, contributing to closing the gap between statistical theory and practical PISL.

Diego Marcondes

Physics-Informed MLStatistical Learning TheoryEmpirical Risk Minimization
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Oct 16, 2025

Inverse Problem for Partial Differential Equations with Jump Discontinuities in Coefficients by Two-stage Physics-Informed Deep Learning and Statistical Mixture Models

This work proposes a two-stage physics-informed deep learning framework that combines neural-network-based sampling with statistical inference and constrained parameter refinement. In the first stage, a dual-network physics-informed architecture is used, where a main-network approximates the PDE solution and an auxiliary coefficient sub-network provides a relaxed continuous soft approximation of the true discontinuous coefficient field. A gradient-adaptive weighting strategy is incorporated to improve residual training and enhance sampling reliability near possible discontinuity regions. The sampled coefficient values are then analyzed using Bayesian learning for Gaussian mixture models and birth-death Markov chain model selection, which identify the number of coefficient regimes and provide candidate intervals for coefficient values and transition regions. In the second stage, the inverse problem is reformulated as a constrained physics-informed estimator, in which the coefficient is replaced by a form-consistent hard approximation explicitly represented as a piecewise-constant function over the spatiotemporal domain. Comprehensive numerical experiments on PDEs with jump-discontinuous coefficients demonstrate that the proposed framework achieves adaptability and accurate parameter identification with acceptable computational costs compared to existing methods. Applications to solution reconstruction further illustrate its practical potential. This work provides a generalizable computational approach for inverse problems governed by PDEs with discontinuous parameter structures, particularly in non-stationary and heterogeneous systems.

Zhikun Zhang, Guanyu Pan, Xiangjun Wang +2

School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710072, China · School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China · SandGold AI Research, Guangzhou 510642, China +3

Bayesian Mixture ModelsPhysics-Informed MLPDE Inverse Problems
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Oct 2, 2025

Neural non-canonical Hamiltonian dynamics for long-time simulations

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.

Clémentine Courtès, Emmanuel Franck, Michael Kraus +2

IRMA, MACARON · Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS, 7 rue René Descartes 672000 Strasbourg, France. · INRIA Nancy-Grand Est, MACARON Project, Strasbourg, France. +5

Dynamical SystemsNonlinear System IdentificationHamiltonian Neural NetworksPhysics-Informed MLHamiltonian Dynamics
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Sep 28, 2025

MAD-PINN: A Decentralized Physics-Informed Machine Learning Framework for Safe and Optimal Multi-Agent Control

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.

Manan Tayal, Aditya Singh, Shishir Kolathaya +1

Center for Cyber-Physical Systems, Indian Institute of Science, Bangalore, India · Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, USA · Department of Aeronautics and Astronautics, Stanford University, USA

Multi-Agent ControlConstrained OptimizationPhysics-Informed ML
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Sep 26, 2025

Physics-informed GNN for medium-high voltage AC power flow with edge-aware attention and line search correction operator

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.

Changhun Kim, Timon Conrad, Redwanul Karim +6

Pattern Recognition Lab, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany · Institute of Electrical Energy Systems, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Efficient Neural Network InferenceGraph Attention NetworksPhysics-Informed MLPhysics-Informed GNNsPower Systems
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Sep 19, 2025

A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulations

This work aims to predict blood flow with non-Newtonian viscosity in stenosed arteries using convolutional neural network (CNN) surrogate models. An alternating Schwarz domain decomposition method is proposed which uses CNN-based subdomain solvers. A universal subdomain solver (USDS) is trained on a single, fixed geometry and then applied for each subdomain solve in the Schwarz method. Results for two-dimensional stenotic arteries of varying shape and length for different inflow conditions are presented and statistically evaluated. One key finding, when using a limited amount of training data, is that incorporating a physics-aware constraint, as, in our case, flow rate conservation, into the USDS improves the prediction accuracy and convergence behavior of the Schwarz method compared to a purely data-driven USDS. As the USDS is a data-driven, inexact subdomain solver, admissible parameter ranges for the geometry and inflow configurations must be defined and tested.

Simon Klaes, Axel Klawonn, Natalie Kubicki +4

Department of Mathematics and Computer Science, University of Cologne, Germany · Center for Data and Simulation Science, University of Cologne, Germany · Institute for Advanced Simulation (IAS-2), Forschungszentrum Jülich, Germany +1

PDE SolvingPDE Surrogate ModelingDomain Decomposition MethodsPhysics-Informed MLConvolutional Neural NetworksComputational Fluid Dynamics
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Aug 29, 2025

Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equation

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 with a general class of activation functions for solving one model second-order elliptic problem, i.e., the Poisson equation, in the sense of high probability. These results extend the existing result [20] 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 physics informed neural networks trained by stochastic algorithms.

Bangti Jin, Longjun Wu

Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China

Neural PDE SolversPDE SolvingStochastic Optimization ConvergencePhysics-Informed MLStochastic Gradient Descent
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Aug 25, 2025

InSituRes: A Physics-Informed Same-Grid Model for Enhanced Dynamic X-ray Micro-CT Reconstructions

X-ray micro-computed tomography (micro-CT) provides non-destructive three-dimensional (3D) imaging of porous material microstructures. In situ experiments, including mechanical loading and reactive transport, increasingly require dynamic four-dimensional (4D) imaging with volumes repeatedly acquired during experiments. However, rapid acquisition typically requires fewer projections, shorter exposures, or reduced fields of view, producing reconstructions with noise, blur, and artifacts that obscure pores, microcracks, and interfaces. To address this challenge, this study introduces InSituRes, a physics-informed same-grid volumetric enhancement framework for fast dynamic X-ray micro-CT imaging of temporally evolving materials. InSituRes maps fast-acquisition volumes to higher-quality long-acquisition reconstructions using paired scans of the same specimens. The model integrates 3D convolutional feature extraction with slice-wise transformer attention to capture local and broader in-plane context. A learnable forward degradation model approximates rapid acquisition effects, including spatial blurring, intensity scaling differences, and signal-dependent noise. During training, reconstructed volumes should match high-quality reference scans and reproduce observed fast acquisition data after propagation through the forward model, imposing a physics-guided consistency constraint. Experiments on unseen micro-CT datasets demonstrate improved reconstruction fidelity and enhanced visibility of fine microstructural features relative to conventional interpolation and learning-based enhancement approaches. The framework supports quantitative interpretation of fast 4D X-ray micro-CT scans of evolving materials.

Qinyi Tian, Andrea Bisciotti, Soniya Tiwari +2

Department of Civil & Environmental Engineering, Duke University, Durham, North Carolina, 27708, USA · Department of Physics and Earth Sciences, University of Ferrara, Via Saragat 1, Ferrara 44141, Italy · Department of Mechanical Engineering & Materials Science, Duke University, Durham, North Carolina, 27708, USA

Image RestorationPhysics-Informed MLComputed Tomography
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Aug 5, 2025

Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems

We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems. Starting from a model trained on low-fidelity or observational data, we apply a differentiable post-training procedure that minimizes weak-form residuals of governing partial differential equations (PDEs), promoting physical consistency and adherence to boundary conditions without distorting the underlying learned distribution. To infer unknown physical inputs, such as source terms, material parameters, or boundary data, we augment the generative process with a learnable latent parameter predictor and propose a joint optimization strategy. The resulting model produces physically valid field solutions alongside plausible estimates of hidden parameters, effectively addressing ill-posed inverse problems in a data-driven yet physicsaware manner. We validate our method on canonical PDE benchmarks, demonstrating improved satisfaction of PDE constraints and accurate recovery of latent coefficients. Our approach bridges generative modelling and scientific inference, opening new avenues for simulation-augmented discovery and data-efficient modelling of physical systems.

Jan Tauberschmidt, Sophie Fellenz, Sebastian J. Vollmer +1

DSA, German Research Center for Artificial Intelligence (DFKI) · Department of Computer Science, University of Kaiserslautern–Landau (RPTU) · Department of Mathematics, Imperial College London

Physics-Informed Generative ModelingFlow MatchingGenerative ModelingPhysics-Informed MLPDE Inverse Problems
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Jun 23, 2025

DPG loss functions for learning parameter-to-solution maps by neural networks

We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is on rigorous accuracy certification to enhance the prediction capability of the resulting deep neural network reduced models. This is achieved by the use of variationally correct loss functions. Through one specific example of an elliptic PDE, details for establishing the variational correctness of a loss function from an ultraweak Discontinuous Petrov Galerkin (DPG) discretization are worked out. Despite the focus on the example, the proposed concepts apply to a much wider scope of problems, namely problems for which stable DPG formulations are available. The issue of high-contrast diffusion fields and ensuing difficulties with degrading ellipticity are discussed. Both numerical results and theoretical arguments illustrate that for high-contrast diffusion parameters the proposed DPG loss functions deliver much more robust performance than simpler least-squares losses.

Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan

PDE SolvingPDE Surrogate ModelingRobust Loss FunctionsPDE Operator LearningPhysics-Informed ML
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Jun 23, 2025

GeNeRT: A Physics-Informed Approach to Intelligent Wireless Channel Modeling via Generalizable Neural Ray Tracing

Neural ray tracing (RT) has emerged as a promising paradigm for channel modeling by integrating physical propagation principles with neural networks. However, existing neural RT methods remain limited by strong spatial dependence and weak adherence to electromagnetic laws. We propose GeNeRT, a generalizable neural RT framework that improves generalization and accuracy through relative geometric features, scatterer semantics, and a Fresnel-inspired polarization-driven architecture. GeNeRT is trained through a three-stage strategy: polarization-specific module-wise pre-training captures general ray-surface interaction behavior; system-wise end-to-end training uses only receiver-side channel impulse responses to learn site-specific propagation characteristics; and measurement-based fine-tuning employs sparse measured multipath components (MPCs) to adapt polarization-related modules to real-world environments. Extensive outdoor simulations demonstrate robust intra-scenario transferability and inter-scenario zero-shot generalization. In an unseen scenario, GeNeRT achieves an overall error of −35.36-35.36 dB and an average-delay error of 4.91 ns, compared with −10.85-10.85 dB and 32.38 ns for the best baseline. With only 75 measured reflected MPCs, fine-tuning further reduces the overall error from −14.48-14.48 to −22.90-22.90 dB and the average-delay error from 6.28 to 3.58 ns. Ablation studies confirm the effectiveness of the proposed architecture and training strategy.

Kejia Bian, Meixia Tao, Shu Sun +2

School of Information Science and Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · Shanghai Innovation Institute, Shanghai 200231, China

Neural Network GeneralizationWireless CommunicationsRadio Propagation ModelingPhysics-Informed ML
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Jun 18, 2025

Interpretability and Generalization Bounds for Learning Spatial Physics

While there are many applications of ML to scientific problems that look promising, visuals can be deceiving. Using numerical analysis techniques, we rigorously quantify the accuracy, convergence rates, and generalization bounds of certain ML models applied to linear differential equations for parameter discovery or solution finding. Beyond the quantity and discretization of data, we identify that the function space of the data is critical to the generalization of the model. A similar lack of generalization is empirically demonstrated for commonly used models, including physics-specific techniques. Counterintuitively, we find that different classes of models can exhibit opposing generalization behaviors. Based on our theoretical analysis, we also introduce a new mechanistic interpretability lens on scientific models whereby Green's function representations can be extracted from the weights of black-box models. Our results inform a new cross-validation technique for measuring generalization in physical systems, which can serve as a benchmark.

Alejandro Francisco Queiruga, Theo Gutman-Solo, Shuai Jiang

OpenAI, San Francisco, CA, USA · Google, Mountain View, CA, USA · Sandia National Laboratories, Albuquerque, NM, USA

Neural Network GeneralizationMechanistic InterpretabilityScientific MLPhysics-Informed MLGeneralization Bounds
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May 28, 2025

Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation

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 xt\boldsymbol{x}_t at intermediate steps, making it infeasible to directly enforce constraints on the clean sample x0\boldsymbol{x}_0 at each noisy level. As a workaround, constraints are typically applied to the expectation of clean samples E[x0∣xt]\mathbb{E}[\boldsymbol{x}_0|\boldsymbol{x}_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.

Yi Zhang, Peng Wang, Difan Zou

Institute of Data Science · Department of Computer and Information Science · Institute of Data Science & School of Computing and Data Science

Physics-Informed Diffusion ModelsDiffusion Model DistillationGenerative ModelingPhysics-Informed MLPDE Inverse Problems
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May 23, 2025

Data-driven multi-agent modelling of calcium interactions in cell culture: PINN vs Regularized Least-squares

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.

Aurora Poggi, Giuseppe Alessio D'Inverno, Hjalmar Brismar +3

KTH Royal Institute of Technology, Department of Mathematics, Stockholm, Sweden · International School for Advanced Studies (SISSA), Trieste, Italy · KTH Royal Institute of Technology, Department of Biophysics, Stockholm, Sweden +1

Nonlinear System IdentificationParameter EstimationPhysics-Informed ML
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May 22, 2025

Repulsive Ensembles for Bayesian Inference in Physics-informed Neural Networks

Physics-informed neural networks (PINNs) have proven an effective tool for solving differential equations, in particular when considering the inverse problem. 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 for a selection of basic differential equations and employ repulsive ensembles of PINNs (RE-PINN) to obtain 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 available, 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. The experimental results demonstrate that RE-PINN can accurately infer the Bayesian posterior in low-dimensional problems, establishing repulsive ensembles as a promising framework for uncertainty quantification in PINNs.

Philipp Pilar, Markus Heinonen, Niklas Wahlström

Department of Information Technology Uppsala University, Uppsala, Sweden · Department of Computer Science Aalto University, Espoo, Finland

Uncertainty QuantificationPhysics-Informed MLBayesian Inverse Problems
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May 16, 2025

Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra

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 algorithms demonstrate substantial performance improvements over existing NGD-based methods on a range of PDE problems discretized using neural networks, and offer competitive results compared to other state-of-the-art optimizers.

Ivan Bioli, Carlo Marcati, Giancarlo Sangalli

Department of Mathematics, University of Pavia, 27100 Pavia, Italy · Department of Civil Engineering and Architecture, University of Pavia, 27100 Pavia, Italy · Institut Camille Jordan, Lyon 1 Université, 69622 Villeurbanne, France +1

Neural PDE SolversGradient DescentPhysics-Informed ML
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May 6, 2025

Physics-Informed Sylvester Normalizing Flows for Bayesian Inference in Magnetic Resonance Spectroscopy

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.

Julian P. Merkofer, Dennis M. J. van de Sande, Alex A. Bhogal +1

Eindhoven University of Technology, Eindhoven, The Netherlands · University Medical Center Utrecht, Utrecht, The Netherlands

Normalizing FlowsBayesian InferencePhysics-Informed MLMagnetic Resonance Spectroscopy
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Mar 24, 2025

Residuals Are Not Enough: Limits of Physics-Informed Pre-Training for Scientific Foundation Models

Scientific foundation models (SciFMs) aim to learn generalizable representations of physical systems governed by partial differential equations (PDEs), enabling transfer across tasks and domains. While physics-informed methods, which leverage PDE residuals as supervisory signals, have shown promise in scientific machine learning (SciML) for improving accuracy and reducing data requirements, their potential in the context of SciFMs remains relatively unexplored. In this evaluation study, we investigate whether (and how) physics-informed pre-training improves the generalization, robustness, and data efficiency of SciFMs. We conduct systematic experiments across a diverse set of PDEs, ranging from simple problems with periodic boundary conditions to more challenging systems such as the Navier-Stokes equations and non-periodic geometries. Our results show that physics-informed pre-training provides clear benefits in nice,'' e.g., structured, well-aligned settings: it enhances generalization and reduces data dependence, compared to data-only pre-training. However, these advantages diminish significantly as the downstream tasks become harder,'' e.g., as they involve discontinuities or deviate from the pre-training distribution. In complex or structurally different problems, such as those involving new boundary conditions or PDE operators, physics-informed models may perform only on par with---or even worse---than data-driven baselines. While residual-based pre-training helps in idealized regimes, realizing broadly transferable SciFMs will likely require subtler spatiotemporal inductive biases and more principled integration of physical knowledge into model architectures.

Serge Kotchourko, Amin Totounferoush, Michael W. Mahoney +1

University of Stuttgart, Germany · ICSI, LBNL, and University of California, Berkeley, USA

PDE Surrogate ModelingPhysics-Informed ML
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Mar 3, 2025

Hyperspectral Image Restoration and Super-resolution with Physics-Aware Deep Learning for Biomedical Applications

Hyperspectral imaging is a powerful bioimaging tool which can uncover novel insights, thanks to its sensitivity to the intrinsic properties of materials. However, this enhanced contrast comes at the cost of system complexity, constrained by an inherent trade-off between spatial, spectral, and temporal resolution. To overcome this limitation, we present a self-supervised deep learning-based approach that restores and enhances pixel resolution post-acquisition without requiring external training data beyond the images to be restored. Fine-tuned using metrics aligned with the imaging model, our physics-aware method achieves a 16×\times pixel super-resolution enhancement and a 12×\times imaging speedup without the need of additional training data for transfer learning. Applied to both synthetic and experimental data from five different sample types, including healthy and diseased tissues, we demonstrate that the model preserves biological integrity, as we did not detect systematic loss of biological features or biologically consequential hallucinations in tested datasets. We also concretely demonstrate the model's ability to reveal disease-associated metabolic changes that would otherwise remain undetectable. Furthermore, we provide physical insights into the model's inner workings, paving the way for future refinements that could potentially reveal novel high resolution features in an explainable manner. All methods are available as open-source software on GitHub.

Yuchen Xiang, Zhaolu Liu, Monica Emili Garcia-Segura +12

Department of Metabolism, Digestion and Reproduction, Imperial College London, London, UK · Department of Mathematics, Imperial College London, London, UK · Department of Clinical Neurosciences and NIHR Biomedical Research Centre, University of Cambridge, Cambridge, UK +4

Hyperspectral ImagingPhysics-Guided Image RestorationHyperspectral Image Super-ResolutionSelf-Supervised LearningImage RestorationPhysics-Informed ML+1
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