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

Mar 23, 2026

A Novel Method for Enforcing Exactly Dirichlet, Neumann and Robin Conditions on Curved Domain Boundaries for Physics Informed Machine Learning

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.

Suchuan Dong, Yuchuan Zhang

Center for Computational and Applied Mathematics Department of Mathematics Purdue University, USA

PDE SolvingPhysics-Informed ML
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Mar 23, 2026

Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach

We present a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solutions are sought in the function space spanned by physics-informed random projection neural networks (PI-RPNNs), and discretized via a collocation approach. These are shallow, 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 explicit generalized eigenvalue formulation governing the linear stability of stationary solutions, separating the physical domain interior dynamics from the algebraic constraints imposed by boundary conditions (BCs), 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, rigorously explaining the observed ill-conditioning.

Gianluca Fabiani, Michail E. Kavousanakis, Constantinos Siettos +1

Hopkins Extreme Materials Institute, Johns Hopkins University, Baltimore, 21218, MD, USA · Dept. of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, 21218, MD, USA · School of Chemical Engineering, National Technical University of Athens, 9 Iroon Polytechniou Street, Athens, 15772, Greece +2

Dynamical SystemsNeural PDE SolversPhysics-Informed ML
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Mar 22, 2026

Benchmarking Scientific Machine Learning Models for Air Quality Data

Accurate air quality index (AQI) forecasting is essential for the protecting public health in rapidly growing urban regions, and the practical model evaluation and selection are often challenged by the lack of rigorous, region-specific benchmarking on standardized datasets. Physics-guided machine learning and deep learning models could be a good and effective solution to resolve such issues with more accurate and efficient AQI forecasting. This research study presents an explainable and comprehensive benchmark that enables a guideline and proposed physics-guided best model by benchmarking classical time-series, machine-learning, and deep-learning approaches for multi-horizon AQI forecasting in North Texas (Dallas County). Using publicly available U.S. Environmental Protection Agency (EPA) daily observations of air quality data from 2022 to 2024, we curate city-level time series for PM2.5 and O3 by aggregating station measurements and constructing lag-wise forecasting datasets for LAG in {1,7,14,30} days. For benchmarking the best model, linear regression (LR), SARIMAX, multilayer perceptrons (MLP), and LSTM networks are evaluated with the proposed physics-guided variants (MLP+Physics and LSTM+Physics) that incorporate the EPA breakpoint-based AQI formulation as a consistency constraint through a weighted loss. Experiments using chronological train-test splits and error metrics MAE, RMSE showed that deep-learning models outperform simpler baselines, while physics guidance improves stability and yields physically consistent pollutant with AQI relationships, with the largest benefits observed for short-horizon prediction and for PM2.5 and O3. Overall, the results provide a practical reference for selecting AQI forecasting models in North Texas and clarify when lightweight physics constraints meaningfully improve predictive performance across pollutants and forecast horizons.

Khawja Imran Masud, Venkata Sai Rahul Unnam, Sahara Ali

University of North Texas, 76205 Texas, USA

Air Quality PredictionForecasting BenchmarksTime Series ForecastingPhysics-Informed ML
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Mar 21, 2026

Neutrino Oscillation Parameter Estimation Using Structured Hierarchical Transformers

Neutrino oscillations encode fundamental information about neutrino masses and mixing parameters, offering a unique window into physics beyond the Standard Model. Estimating these parameters from oscillation probability maps is, however, computationally challenging due to the maps' high dimensionality and nonlinear dependence on the underlying physics. Traditional inference methods, such as likelihood-based or Monte Carlo sampling approaches, require extensive simulations to explore the parameter space, creating major bottlenecks for large-scale analyses. In this work, we introduce a data-driven framework that reformulates atmospheric neutrino oscillation parameter inference as a supervised regression task over structured oscillation maps. We propose a hierarchical transformer architecture that explicitly models the two-dimensional structure of these maps, capturing angular dependencies at fixed energies and global correlations across the energy spectrum. To improve physical consistency, the model is trained using a surrogate simulation constraint that enforces agreement between the predicted parameters and the reconstructed oscillation patterns. Furthermore, we introduce a neural network-based uncertainty quantification mechanism that produces distribution-free prediction intervals with formal coverage guarantees. Experiments on simulated oscillation maps under Earth-matter conditions demonstrate that the proposed method is comparable to a Markov Chain Monte Carlo baseline in estimation accuracy, with substantial improvements in computational cost (around 240×\times fewer FLOPs and 33×\times faster in average processing time). Moreover, the conformally calibrated prediction intervals remain narrow while achieving the target nominal coverage of 90%, confirming both the reliability and efficiency of our approach.

Giorgio Morales, Gregory Lehaut, Antonin Vacheret +2

Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYC. Caen, France · Normandie Univ, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen. Caen, France

Parameter EstimationHigh-Energy PhysicsTransformerPhysics-Informed ML
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Mar 21, 2026

Interpretable Battery Aging without Extra Tests via Neural-Assisted Physics-based Modelling

State of health (SoH) is widely used for battery management, but it is a single scalar and offers limited interpretability. Two batteries with similar SoH can exhibit very different degradation behaviors and the lack of interpretability hinders optimal battery operation. In this paper, we propose IBAM for interpretable battery aging modelling with a neural-assisted physics-based framework. IBAM outputs a 2-D aging fingerprint without extra diagnostic tests and uses only routine logs from the battery management system. The fingerprint offers great interpretability by capturing a battery's curve-wide polarization voltage loss and the tail loss near the end-of-discharge. IBAM first creates a physics-based battery model based on a fractional-order equivalent circuit model, and then extracts per-cycle fingerprints from the model using a two-stage least-squares method. IBAM further anchors fingerprints on the SoH axis with physics-guided regression, where the per-cycle SoH is estimated via a bidirectional gated recurrent unit with customized multi-channel voltage features. Across batteries with short-, medium-, and long-lifespans, IBAM consistently yields the best physics model fidelity at different aging stages, and provides clear interpretations of degradation mechanisms and fingerprint patterns about batteries of different lifespans. The resulting fingerprints support interpretable battery health assessment and can inform battery control choices.

Yuan Qiu, Wei Li, Wei Zhang +4

Singapore Institute of Technology (SIT), Singapore, 828608 · Singapore University of Social Sciences (SUSS), Singapore, 599494 · Institute of Materials Research and Engineering (IMRE), Agency for Science, Technology and Research (A*STAR), 2 Fusionopolis Way, Innovis #08-03, Singapore 138634, Republic of Singapore

Lithium-Ion BatteriesPhysics-Informed MLInterpretable ML
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Mar 20, 2026

PhyUnfold-Net: Advancing Remote Sensing Change Detection with Physics-Guided Deep Unfolding

Bi-temporal change detection is highly sensitive to acquisition discrepancies, including illumination, season, and atmosphere, which often cause false alarms. We observe that genuine changes exhibit higher patch-wise singular-value entropy (SVE) than pseudo changes in the feature-difference space. Motivated by this physical prior, we propose PhyUnfold-Net, a physics-guided deep unfolding framework that formulates change detection as an explicit decomposition problem. The proposed Iterative Change Decomposition Module (ICDM) unrolls a multi-step solver to progressively separate mixed discrepancy features into a change component and a nuisance component. To stabilize this process, we introduce a staged Exploration-and-Constraint loss (S-SEC), which encourages component separation in early steps while constraining nuisance magnitude in later steps to avoid degenerate solutions. We further design a Wavelet Spectral Suppression Module (WSSM) to suppress acquisition-induced spectral mismatch before decomposition. Experiments on four benchmarks show improvements over state-of-the-art methods, with gains under challenging conditions.

Zelin Lei, Yaoxing Ren, Jiaming Chang

Xi’an Jiaotong University · Anhui University

Remote Sensing Change DetectionPhysics-Informed MLDeep Unfolding
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Mar 20, 2026

An Adaptive Machine Learning Framework for Fluid Flow in Dual-Network Porous Media

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.

V. S. Maduri, K. B. Nakshatrala

Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204, USA.

Neural PDE SolversPorous Media FlowPhysics-Informed MLPDE Inverse ProblemsComputational Fluid Dynamics
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Mar 20, 2026

Verifiable Error Bounds for Physics-Informed Neural Network Solutions of Lyapunov and Hamilton-Jacobi-Bellman Equations

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.

Jun Liu

Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

HJB EquationsNeural PDE SolversPhysics-Informed MLOptimal ControlLyapunov Stability
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Mar 19, 2026

Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees

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.

Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1

Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

Neural Network GeneralizationNeural PDE SolversPDE SolvingPhysics-Informed MLGeneralization Bounds
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Mar 19, 2026

WarPGNN: A Parametric Thermal Warpage Analysis Framework with Physics-aware Graph Neural Network

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 prohibitively high computational costs, limiting their scalability for complex chiplet-package systems. In this paper, we present WarPGNN, an efficient 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 transferability across diverse package configurations. Our method first encodes 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 embeddings. 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 inference 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.

Haotian Lu, Jincong Lu, Sachin Sachdeva +1

Department of Electrical and Computer Engineering · University of California, Riverside · Riverside, CA, USA

Graph Neural NetworksPhysics-Informed ML
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Mar 18, 2026

A Family of Adaptive Activation Functions for Mitigating Failure Modes in Physics-Informed Neural Networks

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.

Krishna Murari

†Department of Mathematics, IIT Madras, Chennai 600036, India

Neural Network Activation FunctionsPhysics-Informed MLWavelet Methods
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Mar 18, 2026

Physics-Aware Machine Learning for Seismic and Volcanic Signal Interpretation

Modern seismic and volcanic monitoring is increasingly shaped by continuous, multi-sensor observations and by the need to extract actionable information from nonstationary, noisy wavefields. In this context, machine learning has moved from a research curiosity to a practical ingredient of processing chains for detection, phase picking, classification, denoising, and anomaly tracking. However, improved accuracy on a fixed dataset is not sufficient for operational use. Models must remain reliable under domain shift (new stations, changing noise, evolving volcanic activity), provide uncertainty that supports decision-making, and connect their outputs to physically meaningful constraints. This paper surveys and organizes recent ML approaches for seismic and volcanic signal analysis, highlighting where classical signal processing provides indispensable inductive bias, how self-supervision and generative modeling can reduce dependence on labels, and which evaluation protocols best reflect transfer across regions. We conclude with open challenges for robust, interpretable, and maintainable AI-assisted monitoring.

William Thorossian

Physics-Informed MLSeismology
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Mar 18, 2026

Verification and Validation of Physics-Informed Surrogate Component Models for Dynamic Power-System Simulation

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.

Petros Ellinas, Indrajit Chaudhuri, Johanna Vorwerk +1

Department of Wind and Energy Systems, Technical University of Denmark (DTU), Kgs. Lyngby, Denmark.

Surrogate ModelingPower SystemsPhysics-Informed MLNeural Surrogate Modeling
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Mar 18, 2026

Symmetry-Reduced Physics-Informed Learning of Tensegrity Dynamics

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.

Jing Qin, Muhao Chen

Department of Mathematics, University of Kentucky, Lexington, KY · Department of Mechanical and Aerospace Engineering, University of Houston, TX

Physics-Informed MLStructural MechanicsMultibody Dynamics
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Mar 18, 2026

Anisotropic Permeability Tensor Prediction from Porous Media Microstructure via Physics-Informed Progressive Transfer Learning with Hybrid CNN-Transformer

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.

Mohammad Nooraiepour

Faculty of Mathematics and Natural Sciences, University of Oslo, PO Box 1047 Blindern, 0316 Oslo, Norway

Materials Property PredictionPorous Media FlowPhysics-Informed MLTransfer Learning
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Mar 18, 2026

AirDDE: Multifactor Neural Delay Differential Equations for Air Quality Forecasting

Accurate air quality forecasting is essential for public health and environmental sustainability, but remains challenging due to the complex pollutant dynamics. Existing deep learning methods often model pollutant dynamics as an instantaneous process, overlooking the intrinsic delays in pollutant propagation. Thus, we propose AirDDE, the first neural delay differential equation framework in this task that integrates delay modeling into a continuous-time pollutant evolution under physical guidance. Specifically, two novel components are introduced: (1) a memory-augmented attention module that retrieves globally and locally historical features, which can adaptively capture delay effects modulated by multifactor data; and (2) a physics-guided delay evolving function, grounded in the diffusion-advection equation, that models diffusion, delayed advection, and source/sink terms, which can capture delay-aware pollutant accumulation patterns with physical plausibility. Extensive experiments on three real-world datasets demonstrate that AirDDE achieves the state-of-the-art forecasting performance with an average MAE reduction of 8.79% over the best baselines. The code is available at https://github.com/w2obin/airdde-aaai.

Binqing Wu, Zongjiang Shang, Shiyu Liu +3

State Key Laboratory of Blockchain and Data Security, Zhejiang University · College of Computer Science and Technology, Zhejiang University · State Key Laboratory of Clean Energy Utilization, Zhejiang University

Air Quality PredictionMultivariate Time Series ForecastingPhysics-Informed ML
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Mar 18, 2026

Physics-informed Deep Mixture-of-Koopmans Vehicle Dynamics Model with Dual-branch Encoder for Distributed Electric-drive Trucks

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.

Jinyu Miao, Pu Zhang, Rujun Yan +8

School of Vehicle and Mobility, and State Key Laboratory of Intelligent Green Vehicle and Mobility, Tsinghua University, Beijing, China · KargoBot Inc., Beijing, China

Vehicle DynamicsKoopman Operator LearningNonlinear System IdentificationPhysics-Informed ML
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Mar 17, 2026

Domain-informed explainable boosting machines for trustworthy lateral spread predictions

Explainable Boosting Machines (EBMs) provide transparent predictions through additive shape functions, enabling direct inspection of feature contributions. However, EBMs can learn non-physical relationships that reduce their reliability in natural hazard applications. This study presents a domain-informed framework to improve the physical consistency of EBMs for lateral spreading prediction. Our approach modifies learned shape functions based on domain knowledge. These modifications correct non-physical behavior while maintaining data-driven patterns. We apply the method to the 2011 Christchurch earthquake dataset and correct non-physical trends observed in the original EBM. The resulting model produces more physically consistent global and local explanations, with an acceptable tradeoff in accuracy (4--5%).

Cheng-Hsi Hsiao, Krishna Kumar, Ellen M. Rathje

Interpretable MLPhysics-Informed MLSeismology
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Mar 17, 2026

pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learning

Generalizing across disparate physical laws remains a fundamental challenge for artificial intelligence in science. Existing deep-learning solvers are largely confined to single-equation settings, limiting transfer across physical regimes and inference tasks. Here we introduce pADAM, a unified generative framework that learns a shared probabilistic prior across heterogeneous partial differential equation families. Through a learned joint distribution of system states and, where applicable, physical parameters, pADAM supports forward prediction and inverse inference within a single architecture without retraining. Across benchmarks ranging from scalar diffusion to nonlinear Navier--Stokes equations, pADAM achieves accurate inference even under sparse observations. Combined with conformal prediction, it also provides reliable uncertainty quantification with coverage guarantees. In addition, pADAM performs probabilistic model selection from only two sparse snapshots, identifying governing laws through its learned generative representation. These results highlight the potential of generative multi-physics modeling for unified and uncertainty-aware scientific inference.

Amirhossein Mollaali, Bongseok Kim, Christian Moya +1

School of Mechanical Engineering, Purdue University, West Lafayette, 47906, IN, USA. · Department of Mathematics, Purdue University, West Lafayette, 47906, IN, USA.

Physics-Informed Generative ModelingPDE Surrogate ModelingPartial Differential EquationsPhysics-Informed MLPDE Inverse ProblemsPDE Discovery
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Mar 16, 2026

Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Data

Identifying physical laws from noisy observational data is a central challenge in scientific machine learning. We present Minimum-Action Learning (MAL), a framework that selects symbolic force laws from a pre-specified basis library by minimizing a Triple-Action functional combining trajectory reconstruction, architectural sparsity, and energy-conservation enforcement. A wide-stencil acceleration-matching technique reduces noise variance by 10,000x, transforming an intractable problem (SNR ~0.02) into a learnable one (SNR ~1.6); this preprocessing is the critical enabler shared by all methods tested, including SINDy variants. On two benchmarks -- Kepler gravity and Hooke's law -- MAL recovers the correct force law with Kepler exponent p = 3.01 +/- 0.01 at ~0.07 kWh (40% reduction vs. prediction-error-only baselines). The raw correct-basis rate is 40% for Kepler and 90% for Hooke; an energy-conservation-based criterion discriminates the true force law in all cases, yielding 100% pipeline-level identification. Basis library sensitivity experiments show that near-confounders degrade selection (20% with added r^{-2.5} and r^{-1.5}), while distant additions are harmless, and the conservation diagnostic remains informative even when the correct basis is absent. Direct comparison with noise-robust SINDy variants, Hamiltonian Neural Networks, and Lagrangian Neural Networks confirms MAL's distinct niche: interpretable, energy-constrained model selection that combines symbolic basis identification with dynamical rollout validation.

Martin G. Frasch

Institute on Human Development and Disability, University of Washington, Seattle, WA 98195, USA · Health Stream Analytics, LLC, Seattle, WA, USA

Scientific DiscoveryModel SelectionPhysics-Informed MLNonlinear System IdentificationSymbolic Regression
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Mar 16, 2026

OMNIFLOW: A Physics-Grounded Multimodal Agent for Generalized Scientific Reasoning

Large Language Models (LLMs) have demonstrated exceptional logical reasoning capabilities but frequently struggle with the continuous spatiotemporal dynamics governed by Partial Differential Equations (PDEs), often resulting in non-physical hallucinations. Existing approaches typically resort to costly, domain-specific fine-tuning, which severely limits cross-domain generalization and interpretability. To bridge this gap, we propose OMNIFLOW, a neuro-symbolic architecture designed to ground frozen multimodal LLMs in fundamental physical laws without requiring domain-specific parameter updates. OMNIFLOW introduces a novel Semantic-Symbolic Alignment mechanism that projects high-dimensional flow tensors into topological linguistic descriptors, enabling the model to perceive physical structures rather than raw pixel values. Furthermore, we construct a Physics-Guided Chain-of-Thought (PG-CoT) workflow that orchestrates reasoning through dynamic constraint injection (e.g., mass conservation) and iterative reflexive verification. We evaluate OMNIFLOW on a comprehensive benchmark spanning microscopic turbulence, theoretical Navier-Stokes equations, and macroscopic global weather forecasting. Empirical results demonstrate that OMNIFLOW significantly outperforms traditional deep learning baselines in zero-shot generalization and few-shot adaptation tasks. Crucially, it offers transparent, physically consistent reasoning reports, marking a paradigm shift from black-box fitting to interpretable scientific reasoning.

Hao Wu, Yongheng Zhang, Yuan Gao +7

Tsinghua University · Tencent · Shenzhen Loop Area Institute +2

Scientific Reasoning in Language ModelsPhysics-Informed MLPDE SolvingMultimodal Reasoning
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Mar 16, 2026

Physics-Informed Neural Systems for the Simulation of EUV Electromagnetic Wave Diffraction from a Lithography Mask

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.

Vasiliy A. Es'kin, Egor V. Ivanov

Department of Radiophysics, University of Nizhny Novgorod 23 Gagarin Ave., Nizhny Novgorod 603022, Russia

Neural PDE SolversNeural OperatorsPhysics-Informed MLSemiconductor Manufacturing
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Mar 16, 2026

Building Trust in PINNs: Error Estimation through Finite Difference Methods

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.

Aleksander Krasowski, René P. Klausen, Aycan Celik +3

Department of Artificial Intelligence, Fraunhofer Heinrich Hertz Institute · Centre of eXplainable Artificial Intelligence, Technological University Dublin · Department of Electrical Engineering and Computer Science, Technische Universität Berlin +1

Physics-Informed MLPDE SolvingNeural Network Interpretability
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Mar 16, 2026

Physics-informed fine-tuning of foundation models for partial differential equations

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.

Vlad Medvedev, Leon Armbruster, Christopher Straub +2

Department Modeling and Artificial Intelligence Fraunhofer Institute for Integrated Systems and Device Technology IISB Schottkystrasse 10, 91058 Erlangen, Germany

PDE Foundation ModelsPDE Surrogate ModelingPhysics-Informed ML
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Mar 16, 2026

A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients

We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of nn-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.

Seungwan Han, Kwanghyuk Park, Jiaxi Gu +1

Reaction-Diffusion SystemsPDE SolvingPhysics-Informed MLDynamical Systems
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Mar 16, 2026

Machine learning cannot decarbonise the subsurface until it can be audited

Geological CO2 storage, geothermal energy and underground hydrogen storage depend on irreversible or high-cost decisions made with sparse data and incomplete physical understanding. Machine learning (ML) increasingly informs these decisions, yet methods are usually judged on predictive accuracy rather than on whether their outputs can be trusted, traced and acted upon. We argue that subsurface ML should be designed as decision infrastructure with shared systems that carry calibrated uncertainty and provenance from observation to operating decision. We propose a decision-readiness checklist, an audit-trail workflow for monitoring updates, and benchmark designs that score models by the decisions they support. We identify validation needs shared across the three technologies, and the data-access and governance steps needed for ML to support regulatory and investment decisions.

Hannah P. Menke, Ahmed H. Elsheikh, Lingli Wei +2

Institute of GeoEnergy Engineering, Heriot-Watt University, Edinburgh, UK · Subsurface Energy Transition and Innovation Centre, Heriot-Watt University, Edinburgh, UK · SubsurfaceLabs, Abu Dhabi, UAE +1

Scientific MLPhysics-Informed MLUncertainty Quantification
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Mar 16, 2026

PASTE: Physics-Aware Scattering Topology Embedding Framework for SAR Object Detection

Current deep learning-based object detection for Synthetic Aperture Radar (SAR) imagery mainly adopts optical image methods, treating targets as texture patches while ignoring inherent electromagnetic scattering mechanisms. Though scattering points have been studied to boost detection performance, most methods still rely on amplitude-based statistical models. Some approaches introduce frequency-domain information for scattering center extraction, but they suffer from high computation cost and poor compatibility with diverse datasets. Thus, effectively embedding scattering topological information into modern detection frameworks remains challenging. To solve these problems, this paper proposes the Physics-Aware Scattering Topology Embedding Framework (PASTE), a novel closed-loop architecture for comprehensive scattering prior integration. By building the full pipeline from topology generation, injection to joint supervision, PASTE elegantly integrates scattering physics into modern SAR detectors. Specifically, it designs a scattering keypoint generation and automatic annotation scheme based on the Attributed Scattering Center (ASC) model to produce scalable and physically consistent priors. A scattering topology injection module guides multi-scale feature learning, and a scattering prior supervision strategy constrains network optimization by aligning predictions with scattering center distributions. Experiments on real datasets show that PASTE is compatible with various detectors and brings relative mAP gains of 2.9% to 11.3% over baselines with acceptable computation overhead. Visualization of scattering maps verifies that PASTE successfully embeds scattering topological priors into feature space, clearly distinguishing target and background scattering regions, thus providing strong interpretability for results.

Jiacheng Chen, Yuxuan Xiong, Haipeng Wang

Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University, Shanghai 200433, China

Object DetectionSynthetic Aperture RadarPhysics-Informed ML
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Mar 15, 2026

Predicting Stress-strain Behaviors of Additively Manufactured Materials via Loss-based and Activation-based Physics-informed Machine Learning

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.

Chenglong Duan, Dazhong Wu

Department of Mechanical and Aerospace Engineering, College of Engineering and Computer Science, University of Central Florida, Orlando, FL 32816, USA

Materials Property PredictionPhysics-Informed MLAdditive Manufacturing
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Mar 14, 2026

Manifold-Orthogonal Dual-spectrum Extrapolation for Parameterized Physics-Informed Neural Networks

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 (P2^2INNs) 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 P2^2INNs 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.

Zhangyong Liang, Huanhuan Gao

PDE Operator LearningPhysics-Informed MLOOD GeneralizationParameter-Efficient Fine-Tuning
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Mar 13, 2026

Assessing the Utility of Volumetric Motion Fields for Radar-based Precipitation Nowcasting with Physics-informed Deep Learning

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.

Peter Pavlík, Anna Bou Ezzeddine, Viera Rozinajová

Kempelen Institute of Intelligent Technologies, Sky Park Offices, Bottova 7939/2A, Bratislava 811 09, Slovakia · Faculty of Information Technology, Brno University of Technology, Božetěchova 2, Brno 612 00, Czech Republic

Precipitation NowcastingPrecipitation ForecastingPhysics-Informed ML
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