Neural PDE Solvers

PDE: Partial Differential Equation

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Overview

Neural PDE solvers approximate fields, time evolution, or mappings from equation inputs to solutions, with potential benefits for repeated simulations, inverse problems, and high-dimensional settings. Physics-informed neural networks (PINNs) train on equation residuals, while weak-form and variational methods use integral or energy principles; neural operators learn mappings across families of PDE instances. Other approaches use randomized or separable features, and hybridize learned models with finite-element, finite-volume, or multigrid solvers. The central tension is achieving reliable accuracy, conservation, and long-term stability while reducing total computational cost; generalization beyond training conditions and rigorous error guarantees remain important challenges.

Latest papers

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272 papers47 papers in the last four weeks, up 74% on the four weeks before. 0.3% of all new papers.

Apr 7, 2026

Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies

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

Prashant Kumar, Rajesh Ranjan

Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016

Domain Decomposition MethodsNeural PDE SolversPhysics-Informed MLPDE SolvingScientific MLComputational Fluid Dynamics
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Apr 6, 2026

Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks

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

Anas Jnini, Elham Kiyani, Khemraj Shukla +5

Department of Information Engineering and Computer Science, University of Trento, Trento, Italy · Division of Applied Mathematics, Brown University, Providence, RI, USA · Departament de Física, Universitat d’Alacant, Spain +3

Physics-Informed MLNeural PDE SolversQuasi-Newton MethodsNatural Gradient Methods
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Apr 6, 2026

Learning the Stellar Structure Equations via Self-supervised Physics-Informed Neural Networks

Stellar astrophysics relies critically on accurate descriptions of the physical conditions inside stars. Traditional solvers such as MESA (Modules for Experiments in Stellar Astrophysics), which employ adaptive finite-difference methods, can become computationally expensive and challenging to scale for large stellar population synthesis (>109>10^9 stars). In this work, we present an self-supervised physics-informed neural network (PINN) framework that provides a mesh-free and fully differentiable approach to solving the stellar structure equations under hydrostatic and thermal equilibrium. The model takes as input the stellar boundary conditions (at the center and surface) together with the chemical composition, and learns continuous radial profiles for mass Mr(r)M_r(r), pressure P(r)P(r), density ρ(r)ρ(r), temperature T(r)T(r), and luminosity Lr(r)L_r(r) by enforcing the governing structure equations through physics-based loss terms. To incorporate realistic microphysics, we introduce auxiliary neural networks that approximate the equation of state and opacity tables as smooth, differentiable functions of the local thermodynamic state. These surrogates replace traditional tabulated inputs and enable end-to-end training. Once trained for a given star, the model produces continuous solutions across the entire radial domain without requiring discretization or interpolation. Validation against benchmark MESA models across a range of stellar masses yields a Mean Relative Absolute Error of 3.06%3.06\% and an average R2R^2 score of 99.98%99.98\%. To our knowledge, this is the first demonstration that the stellar structure equations can be solved in a fully self-supervised and data-free fashion employing PINNs. This work establishes a foundation for scalable, physics-informed emulation of stellar interiors and opens the door to future extensions toward time-dependent stellar evolution.

Manuel Ballester, Santiago Lopez-Tapia, Seth Gossage +9

SkAI Institute (NSF–Simons AI Institute for the Sky), Chicago, IL, USA · Department of Electrical and Computer Engineering, Northwestern University, Chicago, IL, USA · CIERA, Northwestern University, Chicago, IL, USA +4

AstrophysicsNeural PDE SolversPhysics-Informed ML
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Apr 4, 2026

A Theory-guided Weighted L2L^2 Loss for solving the BGK model via Physics-informed neural networks

While Physics-Informed Neural Networks offer a promising framework for solving partial differential equations, the standard L2L^2 loss formulation is fundamentally insufficient when applied to the Bhatnagar-Gross-Krook (BGK) model. Specifically, simply minimizing the standard loss does not guarantee accurate predictions of the macroscopic moments, causing the approximate solutions to fail in capturing the true physical solution. To overcome this limitation, we introduce a velocity-weighted L2L^2 loss function designed to effectively penalize errors in the high-velocity regions. By establishing a stability estimate for the proposed approach, we shows that minimizing the proposed weighted loss guarantees the convergence of the approximate solution. Also, numerical experiments demonstrate that employing this weighted PINN loss leads to superior accuracy and robustness across various benchmarks compared to the standard approach.

Gyounghun Ko, Sung-Jun Son, Seung Yeon Cho +1

Neural PDE SolversPhysics-Informed ML
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Apr 2, 2026

General Explicit Network (GEN): A novel deep learning architecture for solving partial differential equations

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

Genwei Ma, Ting Luo, Ping Yang +1

National Center for Applied Mathematics Beijing, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China · The Academy for Multidisciplinary Studies, Capital Normal University, No. 105 West Third Ring Road North, Beijing, 100048, China · Academy of Information Network Security, People’s Public Security University of China, No.1 Muxidi Nanli, Beijing, 100038, China +2

Neural PDE SolversPDE SolvingPhysics-Informed ML
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Apr 2, 2026

Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing

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

Yilong Dai, Shengyu Chen, Xiaowei Jia +1

The University of Alabama, Tuscaloosa, Alabama, USA · University of Pittsburgh, Pittsburgh, Pennsylvania, USA

Neural PDE SolversPDE SolvingFlow-Based Generative Modeling
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Mar 31, 2026

Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNs

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

Zhangyong Liang, Huanhuan Gao, Manyu Xiao

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China · School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China

Neural PDE SolversPhysics-Informed ML
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Mar 30, 2026

Physics-Guided Transformer (PGT): Physics-Aware Attention Mechanism for PINNs

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

Ehsan Zeraatkar, Rodion Podorozhny, Jelena Tešić

Computer Science, Texas State University, San Marcos, Texas, US

Physics-Informed MLNeural PDE SolversTransformer AttentionPDE Inverse Problems
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Mar 26, 2026

A Neural Score-Based Particle Method for the Vlasov-Maxwell-Landau System

Plasma modeling is central to the design of nuclear fusion reactors, yet simulating collisional plasma kinetics from first principles remains a formidable computational challenge: the Vlasov-Maxwell-Landau (VML) system describes six-dimensional phase-space transport under self-consistent electromagnetic fields together with the nonlinear, nonlocal Landau collision operator. A recent deterministic particle method for the full VML system estimates the velocity score function via the blob method, a kernel-based approximation with O(n2)O(n^2) cost. In this work, we replace the blob score estimator with score-based transport modeling (SBTM), in which a neural network is trained on-the-fly via implicit score matching at O(n)O(n) cost. We prove that the approximated collision operator preserves momentum and kinetic energy, and dissipates an estimated entropy. We also characterize the unique global steady state of the VML system and its electrostatic reduction, providing the ground truth for numerical validation. On three canonical benchmarks -- Landau damping, two-stream instability, and Weibel instability -- SBTM is more accurate than the blob method, achieves correct long-time relaxation to Maxwellian equilibrium where the blob method fails, and delivers 50%50\% faster runtime with 4×4\times lower peak memory.

Vasily Ilin, Jingwei Hu

Department of Mathematics, University of Washington, Seattle, WA, USA. · Department of Applied Mathematics, University of Washington, Seattle, WA, USA.

Neural PDE Solvers
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Mar 26, 2026

System-Anchored Knee Estimation for Low-Cost Context Window Selection in PDE Forecasting

Autoregressive neural PDE simulators predict the evolution of physical fields one step at a time from a finite history, but low-cost context-window selection for such simulators remains an unformalized problem. Existing approaches to context-window selection in time-series forecasting include exhaustive validation, direct low-cost search, and system-theoretic memory estimation, but they are either expensive, brittle, or not directly aligned with downstream rollout performance. We formalize explicit context-window selection for fixed-window autoregressive neural PDE simulators as an independent low-cost algorithmic problem, and propose System-Anchored Knee Estimation (SAKE), a two-stage method that first identifies a small structured candidate set from physically interpretable system anchors and then performs knee-aware downstream selection within it. Across all eight PDEBench families evaluated under the shared L∈{1,…,16}L\in\{1,\dots,16\} protocol, SAKE is the strongest overall matched-budget low-cost selector among the evaluated methods, achieving 67.8% Exact, 91.7% Within-1, 6.1% mean regret@knee, and a cost ratio of 0.051 (94.9% normalized search-cost savings).

Wenshuo Wang, Fan Zhang

School of Future Technology, South China University of Technology, China · State Key Laboratory of Ocean Sensing & Ocean College, Zhejiang University, China · Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, USA

Neural PDE SolversPDE Surrogate Modeling
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Mar 25, 2026

Learning Mesh-Free Discrete Differential Operators with Self-Supervised Graph Neural Networks

Mesh-free numerical methods provide flexible discretisations for complex geometries; however, classical meshless discrete differential operators typically trade low computational cost for limited accuracy or high accuracy for substantial per-stencil computation. We introduce a parametrised framework for learning mesh-free discrete differential operators using a graph neural network trained via polynomial moment constraints derived from truncated Taylor expansions. The model maps local stencils relative positions directly to discrete operator weights. The current work demonstrates that neural networks can learn classical polynomial consistency while retaining robustness to irregular neighbourhood geometry. The learned operators depend only on local geometry, are resolution-agnostic, and can be reused across particle configurations and governing equations. We evaluate the framework using standard numerical analysis diagnostics, showing improved accuracy over Smoothed Particle Hydrodynamics, and a favourable accuracy-cost trade-off relative to a representative high-order consistent mesh-free method in the moderate-accuracy regime. Applicability is demonstrated by solving the weakly compressible Navier-Stokes equations using the learned operators.

Lucas Gerken Starepravo, Georgios Fourtakas, Steven Lind +3

School of Engineering, The University of Manchester, Manchester, UK · School of Engineering, Cardiff University, Cardiff, UK

Neural PDE SolversGraph Neural Networks
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Mar 25, 2026

Quantum Neural Physics: Solving Partial Differential Equations on Quantum Simulators using Quantum Convolutional Neural Networks

Neural Physics recasts local discretisations of partial differential equations (PDEs) as fixed convolutional operators, providing a physics-preserving alternative to data-driven surrogate modelling in scientific machine learning. However, existing realizations remain largely confined to classical AI hardware and do not directly connect to quantum structured operator design. To bridge this gap, we introduce a Quantum Neural Physics framework and develop a Hybrid Quantum-Classical CNN Multigrid Solver (HQC-CNNMG). The proposed method maps analytically prescribed stencil operators to local quantum convolutional primitives and embeds them within a classical multilevel W-cycle architecture, combining the operator-centric view of scientific ML with the numerical rigor of multigrid solvers. Using amplitude encoding together with the Linear Combination of Unitaries (LCU) and the Quantum Fourier Transform (QFT), the resulting local quantum operators admit logarithmic-depth implementation, with circuit depth scaling as O(log⁡K)\mathcal{O}(\log K) for an encoded block of size KK under the idealized parallel circuit model considered here. Numerical experiments on Poisson, transient diffusion, convection--diffusion, and incompressible Navier--Stokes problems demonstrate numerical consistency, stable multilevel behaviour, and workflow-level feasibility on noiseless simulators. Comparisons with representative quantum linear solver paradigms further show that the main strength of HQC-CNNMG lies in its balanced trade-off among local circuit depth, numerical robustness, and compatibility with PDE structure, rather than in fully quantum global inversion.

Jucai Zhai, Muhammad Abdullah, Boyang Chen +6

Resource Geophysics Academy, Imperial College London, London, SW7 2AZ UK · Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London, SW7 2AZ UK · Quantum AI Leadership (QUAIL) Ltd, London, WC2H 9JQ UK +2

Neural PDE SolversPDE SolvingQuantum Convolutional Neural NetworksHybrid Quantum-Classical ML
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Mar 24, 2026

Coordinate Encoding on Linear Grids for Physics-Informed Neural Networks

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

Tetsuro Tsuchino, Motoki Shiga

Graduate School of Engineering, Gifu University, Japan · International Center for Synchrotron Radiation Innovation Smart, Tohoku University, Japan · Unprecedented-scale Data Analytics Center, Tohoku University, Japan +3

Neural PDE SolversPhysics-Informed MLPositional EncodingSpectral Bias
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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 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

Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control

We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by parameterizing solutions via Hilbert--Galerkin Neural Operators (HGNOs). We prove the first Universal Approximation Theorems (UATs) which are sufficiently powerful to address these problems, based on novel topologies for Hessian terms and corresponding novel continuity assumptions on the fully nonlinear operator. These topologies are non-sequential and non-metrizable, making the problem delicate. In particular, we prove UATs for functions on Hilbert spaces, together with their Fréchet derivatives up to second order, and for unbounded operators applied to the first derivative, ensuring that HGNOs are able to approximate all the PDE terms. For control problems, we further prove UATs for optimal feedback controls in terms of our approximating value function HGNO. We develop numerical training methods, which we call Deep Hilbert--Galerkin and Hilbert Actor-Critic (reinforcement learning) Methods, for these problems by minimizing the Lμ2(H)L^2_μ(H)-norm of the residual of the PDE on the whole Hilbert space, not just a projected PDE to finite dimensions. This is the first paper to propose such an approach. The models considered arise in many applied sciences, such as functional differential equations in physics and Kolmogorov and HJB PDEs related to controlled PDEs, SPDEs, path-dependent systems, partially observed stochastic systems, and mean-field SDEs. We numerically solve examples of Kolmogorov and HJB PDEs related to the optimal control of deterministic and stochastic heat and Burgers' equations, demonstrating the promise of our deep learning-based approach.

Samuel N. Cohen, Filippo de Feo, Jackson Hebner +1

Mathematical Institute, University of Oxford · Institut für Mathematik, Technische Universität Berlin

HJB EquationsPDE SolvingNeural PDE SolversPDE Operator LearningOptimal ControlNeural Network Approximation Theory+1
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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

Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Models

We propose a new Neural Galerkin Normalizing Flow framework to approximate the transition probability density function of a diffusion process by solving the corresponding Fokker-Planck equation with an atomic initial distribution, parametrically with respect to the location of the initial mass. By using Normalizing Flows, we look for the solution as a transformation of the transition probability density function of a reference stochastic process, ensuring that our approximation is structure-preserving and automatically satisfies positivity and mass conservation constraints. By extending Neural Galerkin schemes to the context of Normalizing Flows, we derive a system of ODEs for the time evolution of the Normalizing Flow's parameters. Adaptive sampling routines are used to evaluate the Fokker-Planck residual in meaningful locations, which is of vital importance to address high-dimensional PDEs. Numerical results show that this strategy captures key features of the true solution and enforces the causal relationship between the initial datum and the density function at subsequent times. After completing an offline training phase, online evaluation becomes significantly more cost-effective than solving the PDE from scratch. The proposed method serves as a promising surrogate model, which could be deployed in many-query problems associated with stochastic differential equations, like Bayesian inference, simulation, and diffusion bridge generation.

Riccardo Saporiti, Fabio Nobile

CSQI, École Polytechnique Féderale de Lausanne, Switzerland

Normalizing FlowsNeural PDE SolversPDE SolvingPDE Surrogate ModelingFokker–Planck Equation
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Mar 17, 2026

Physics-integrated neural differentiable modeling for immersed boundary systems

Accurately, efficiently, and stably computing complex fluid flows and their evolution near solid boundaries over long horizons remains challenging. Conventional numerical solvers require fine grids and small time steps to resolve near-wall dynamics, resulting in high computational costs, while purely data-driven surrogate models accumulate rollout errors and lack robustness under extrapolative conditions. To address these issues, this study extends existing neural PDE solvers by developing a physics-integrated differentiable framework for long-horizon prediction of immersed-boundary flows. A key design aspect of the framework includes an important improvement, namely the structural integration of physical principles into an end-to-end differentiable architecture incorporating a PDE-based intermediate velocity module and a multi-direct forcing immersed boundary module, both adhering to the pressure-projection procedure for incompressible flow computation. The computationally expensive pressure projection step is substituted with a learned implicit correction using ConvResNet blocks to reduce cost, and a sub-iteration strategy is introduced to separate the embedded physics module's stability requirement from the surrogate model's time step, enabling stable coarse-grid autoregressive rollouts with large effective time increments. The framework uses only single-step supervision for training, eliminating long-horizon backpropagation and reducing training time to under one hour on a single GPU. Evaluations on benchmark cases of flow past a stationary cylinder and a rotationally oscillating cylinder at Re=100 show the proposed model consistently outperforms purely data-driven, physics-loss-constrained, and coarse-grid numerical baselines in flow-field fidelity and long-horizon stability, while achieving an approximately 200-fold inference speedup over the high-resolution solver.

Chenglin Li, Hang Xu, Jianting Chen +1

State Key Lab of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · State Key Laboratory of Maritime Technology and Safety, Shanghai Ship and Shipping Research Institute Co., Ltd, Shanghai 200135, China.

PDE Surrogate ModelingComputational Fluid DynamicsNeural PDE SolversDifferentiable Physics
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Mar 17, 2026

Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds

In this paper, we extend the Weak Adversarial Neural Pushforward Method to the Fokker--Planck equation on compact embedded Riemannian manifolds. The method represents the solution as a probability distribution via a neural pushforward map that is constrained to the manifold by a retraction layer, enforcing manifold membership and probability conservation by construction. Training is guided by a weak adversarial objective using ambient plane-wave test functions, whose intrinsic differential operators are derived in closed form from the geometry of the embedding, yielding a fully mesh-free and chart-free algorithm. Both steady-state and time-dependent formulations are developed, and numerical results on a double-well problem on the two-sphere demonstrate the capability of the method in capturing multimodal invariant distributions on curved spaces.

Andrew Qing He, Wei Cai

Department of Mathematics, Southern Methodist University, Dallas, TX 75275.

Fokker–Planck EquationPDE SolvingNeural PDE Solvers
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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 11, 2026

A Machine Learning-Enhanced Hopf-Cole Formulation for Nonlinear Gas Flow in Porous Media

Accurate modeling of gas flow through porous media is critical for many technological applications, including reservoir performance prediction, carbon capture and sequestration, and fuel cells and batteries. However, such modeling remains challenging due to strong nonlinear behavior and uncertainty in model parameters. In particular, gas slippage effects described by the Klinkenberg model introduce pressure-dependent permeability, which complicates numerical simulation and obscures deviations from classical Darcy flow behavior. To address these challenges, we present an integrated modeling framework for gas transport in porous media that combines a Klinkenberg-enhanced constitutive relation, Hopf-Cole-transformed mixed-form linear governing equations, a shared-trunk neural network architecture, and a Deep Least-Squares (DeepLS) solver. The Hopf-Cole transformation reformulates the original nonlinear flow equations into an equivalent linear system closely related to the Darcy model, while the mixed formulation, together with a shared-trunk neural architecture, enables simultaneous and accurate prediction of both pressure and velocity fields. A rigorous convergence analysis is performed both theoretically and numerically, establishing the stability and convergence properties of the proposed solver. Importantly, the proposed framework also naturally facilitates inverse modeling of pressure-dependent permeability and slippage parameters from limited or indirect observations, enabling efficient estimation of flow properties that are difficult to measure experimentally. Numerical results demonstrate accurate recovery of flow dynamics and parameters across a wide range of pressure regimes, highlighting the framework's robustness, accuracy, and computational efficiency for gas transport modeling and inversion in tight formations.

V. S. Maduri, K. B. Nakshatrala

Graduate Student, Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204. · Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204.

Porous Media FlowComputational Fluid DynamicsNeural PDE SolversPDE Inverse Problems
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Mar 10, 2026

Intrinsic Numerical Robustness and Fault Tolerance in a Neuromorphic Algorithm for Scientific Computing

The potential for neuromorphic computing to provide intrinsic fault tolerance has long been speculated, but the brain's robustness in neuromorphic applications has yet to be demonstrated. Here, we show that a previously described, natively spiking neuromorphic algorithm for solving partial differential equations is intrinsically tolerant to structural perturbations in the form of ablated neurons and dropped spikes. The tolerance band for these perturbations is large: we find that as many as 32 percent of the neurons and up to 90 percent of the spikes may be entirely dropped before a significant degradation in the accuracy results. Furthermore, this robustness is tunable through structural hyperparameters. This work demonstrates that the specific brain-like inspiration behind the algorithm contributes to a significant degree of robustness expected from brain-like neuromorphic algorithms.

Bradley H. Theilman, James B. Aimone

Neural Exploration & Research Laboratory, Sandia National Laboratories, Albuquerque, NM

Neuromorphic ComputingNeural Network RobustnessNeural PDE SolversSpiking Neural Networks
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Mar 9, 2026

MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries

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

Weizheng Zhang, Xunjie Xie, Hao Pan +4

Shandong University, Qingdao, China · Tsinghua University, Beijing, China · Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, China

Neural PDE SolversPDE SolvingPhysics-Informed MLComputational Fluid Dynamics
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Mar 9, 2026

Graph-Instructed Neural Networks for parametric problems with varying boundary conditions

This work addresses the accurate and efficient simulation of physical phenomena governed by parametric Partial Differential Equations (PDEs) characterized by varying boundary conditions, where parametric instances modify not only the physics of the problem but also the imposition of boundary constraints on the computational domain. In such scenarios, classical Galerkin projection-based reduced order techniques encounter a fundamental bottleneck. Parametric boundaries typically necessitate a re-formulation of the discrete problem for each new configuration, and often, these approaches are unsuitable for real-time applications. To overcome these limitations, we propose a novel methodology based on Graph-Instructed Neural Networks (GINNs). The GINN framework effectively learns the mapping between the parametric description of the computational domain and the corresponding PDE solution. Our results demonstrate that the proposed GINN-based models, can efficiently represent highly complex parametric PDEs, serving as a robust and scalable asset for several applied-oriented settings when compared with fully connected architectures.

Francesco Della Santa, Sandra Pieraccini, Maria Strazzullo

Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy · Gruppo Nazionale per il Calcolo Scientifico INdAM, Piazzale Aldo Moro 5, 00185, Rome, Italy

PDE Surrogate ModelingNeural PDE SolversGraph Neural Networks
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Mar 6, 2026

Physics-Consistent Neural Networks for Learning Deformation and Director Fields in Microstructured Media with Loss-Based Validation Criteria

In this work, we study the mechanical behavior of solids with microstructure using the framework of Cosserat elasticity with a single unit director. This formulation captures the coupling between deformation and orientational fields that arises in many structured materials. To compute equilibrium configurations of such media, we develop two complementary computational approaches: a finite element formulation based on variational principles and a neural network-based solver that directly minimizes the total potential energy. The neural architecture is constructed to respect the fundamental kinematic structure of the theory. In particular, it enforces frame invariance of the energy, satisfies the unit-length constraint on the director field, and represents deformation and director fields through separate networks to preserve their kinematic independence in the variational setting. Beyond satisfying balance laws, however, physically admissible solutions must also correspond to stable energy minimizers. To assess this requirement, we derive the quasiconvexity condition, rank-one convexity condition, and the Legendre-Hadamard inequalities for the Cosserat model and formulate them in a manner suitable for evaluating neural network predictions. These necessary stability conditions provide a physics-based validation framework: network outputs that violate these necessary conditions cannot correspond to stable energy minimizers and can therefore be rejected. In this way, we integrate classical variational stability theory with modern machine-learning solvers, establishing a computational workflow in which equilibrium solutions are not only learned but also assessed for energetic consistency.

Milad Shirani, Pete H. Gueldner, Murat Khidoyatov +2

Department of Biomedical Engineering, Yale University, New Haven, CT, USA · Department of Mechanical Engineering, Yale University, New Haven, CT, USA

Structural MechanicsMaterials ScienceNeural PDE Solvers
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Mar 6, 2026

Frequency-Separable Hamiltonian Neural Network for Multi-Timescale Dynamics

While Hamiltonian mechanics provides a powerful inductive bias for neural networks modeling dynamical systems, Hamiltonian Neural Networks and their variants often fail to capture complex temporal dynamics spanning multiple timescales. This limitation is commonly linked to the spectral bias of deep neural networks, which favors learning low-frequency, slow-varying dynamics. Prior approaches have sought to address this issue through symplectic integration schemes that enforce energy conservation or by incorporating geometric constraints to impose structure on the configuration-space. However, such methods either remain limited in their ability to fully capture multiscale dynamics or require substantial domain specific assumptions. In this work, we exploit the observation that Hamiltonian functions admit decompositions into explicit fast and slow modes and can be reconstructed from these components. We introduce the Frequency-Separable Hamiltonian Neural Network (FS-HNN), which parameterizes the system Hamiltonian using multiple networks, each governed by Hamiltonian dynamics and trained on data sampled at distinct timescales. We further extend this framework to partial differential equations by learning a state- and boundary-conditioned symplectic operators. Empirically, we show that FS-HNN improves long-horizon extrapolation performance on challenging dynamical systems and generalizes across a broad range of ODE and PDE problems.

Yaojun Li, Yulong Yang, Christine Allen-Blanchette

Princeton University, USA

Hamiltonian Neural NetworksDynamical SystemsNeural PDE Solvers
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Mar 6, 2026

Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs

Modeling stiff partial differential equations (PDEs) with sharp gradients remains a significant challenge for scientific machine learning. While Physics-Informed Neural Networks (PINNs) struggle with spectral bias and slow training times, Physics-Informed Extreme Learning Machines (PIELMs) offer a rapid, closed-form linear solution but are fundamentally limited by physics-agnostic, random initialization. We introduce the Gaussian Mixture Model Adaptive PIELM (GMM-PIELM), a probabilistic framework that learns a probability density function representing the ``location of physics'' for adaptively sampling kernels of PIELMs. By employing a weighted Expectation-Maximization (EM) algorithm, GMM-PIELM autonomously concentrates radial basis function centers in regions of high numerical error, such as shock fronts and boundary layers. This approach dynamically improves the conditioning of the hidden layer without the expensive gradient-based optimization(of PINNs) or Bayesian search. We evaluate our methodology on 1D singularly perturbed convection-diffusion equations with diffusion coefficients ν=10−4ν=10^{-4}. Our method achieves L2L_2 errors up to 77 orders of magnitude lower than baseline RBF-PIELMs, successfully resolving exponentially thin boundary layers while retaining the orders-of-magnitude speed advantage of the ELM architecture.

Akshay Govind Srinivasan, Balaji Srinivasan

Indian Institute of Technology, Madras Chennai, TN 600107, India

Neural PDE SolversPDE SolvingPhysics-Informed MLAdaptive Sampling
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Mar 5, 2026

A Geometry-Adaptive Deep Variational Framework for Phase Discovery in the Landau-Brazovskii Model

The discovery of ordered structures in pattern-forming systems, such as the Landau-Brazovskii (LB) model, is often limited by the sensitivity of numerical solvers to the prescribed computational domain size. Incompatible domains induce artificial stress, frequently trapping the system in high-energy metastable configurations. To resolve this issue, we propose a Geometry-Adaptive Deep Variational Framework (GeoDVF) that jointly optimizes the infinite-dimensional order parameter, which is parameterized by a neural network, and the finite-dimensional geometric parameters of the computational domain. By explicitly treating the domain size as trainable variables within the variational formulation, GeoDVF naturally eliminates artificial stress during training. To escape the attraction basin of the disordered phase under small initializations, we introduce a warmup penalty mechanism, which effectively destabilizes the disordered phase, enabling the spontaneous nucleation of complex three-dimensional ordered phases from random initializations. Furthermore, we design a guided initialization protocol to resolve topologically intricate phases associated with narrow basins of attraction. Extensive numerical experiments show that GeoDVF provides a robust and geometry-consistent variational solver capable of identifying both stable and metastable states without prior knowledge.

Yuchen Xie, Jianyuan Yin, Lei Zhang

School of Mathematical Sciences, Peking University, Beijing, 100871, China · School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, 100875, China · School of Mathematical Sciences, Beijing International Center for Mathematical Research, Center for Quantitative Biology, Center for Machine Learning Research, Peking University, Beijing, 100871, China

Neural PDE Solvers
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