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.

  1. When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains

    May 8, 2026Brandon Yee, Pairie Koh, Jack Rodriguez +1Neural PDE SolversPDE Solving

    Physics Lab, Yee Collins Research Group

  2. Sparse Random-Feature Neural Networks with Krylov-Based SVD for Singularly Perturbed ODE

    May 8, 2026Kevin Kurian Thomas Vaidyan, Siddharth RoutNeural PDE SolversSingular Value Decomposition

    Institute of Applied Mathematics University of British Columbia Vancouver, BC V6T 1Z4

  3. Learned Lagrangian Models of PDEs via Euler-Lagrange Residual Minimization

    May 8, 2026Lyra Zhornyak, Eric Forgoston, M. Ani HsiehNeural PDE SolversPDE Surrogate Modeling

    University of Pennsylvania · Montclair State University

  4. Monotone Neural Policy Iteration for High-Dimensional First-Order Hamilton--Jacobi--Bellman Equations

    May 8, 2026Minseok Kim, Yeongjong Kim, Namkyeong Cho +1Neural PDE SolversStochastic Optimal Control

    Department of Applied Artificial Intelligence, Seoul National University of Science and Technology, Seoul, Republic of Korea. · Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea. · Department of Mathematical Finance and Big Data, Gachon University, Seongnam, Republic of Korea +1

  5. INEUS: Iterative Neural Solver for High-Dimensional PIDEs

    May 7, 2026Jean-Loup Dupret, Davide Gallon, Patrick CheriditoNeural PDE SolversPDE Solving

    Department of Mathematics and RiskLab, ETH Zurich, Switzerland · Applied Mathematics: Institute for Analysis and Numerics, University of Münster, Germany

  6. Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations

    May 6, 2026Zhangyong Liang, Huanhuan GaoNeural PDE SolversPDE Solving

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

  7. Random test functions, H−1H^{-1} norm equivalence, and stochastic variational physics-informed neural networks

    May 5, 2026Diego MarcondesNeural PDE SolversPDE Solving

  8. Posterior-First Neural PDE Simulation: Inferring What Downstream Tasks Need from a Single Field

    May 5, 2026Wenshuo Wang, Fan ZhangNeural PDE SolversPDE Surrogate Modeling

    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

  9. Isotropic Fourier Neural Operators

    May 4, 2026Michael F. StaddonFourier Neural OperatorNeural PDE Solvers

    Independent Researcher

  10. Mesh Based Simulations with Spatial and Temporal awareness

    May 2, 2026Paul Garnier, Vincent Lannelongue, Elie HachemGraph Neural NetworksNeural PDE Solvers

    1CEMEF - Mines Paris PSL

  11. PILIR: Physics-Informed Local Implicit Representation

    May 1, 2026Jianfeng Li, Feng Wang, Ke TangNeural PDE SolversImplicit Neural Representations

    School of Computer Science, Wuhan University · School of Computer Science and Technology, Southern University of Science and Technology

  12. A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions

    Apr 30, 2026Matteo Raviola, Benjamin PeherstorferNeural PDE SolversPDE Solving

    EPFL, Lausanne, Switzerland · Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

  13. Compositional Meta-Learning for Mitigating Task Heterogeneity in Physics-Informed Neural Networks

    Apr 29, 2026Beomchul Park, Minsu Koh, Heejo Kong +1Neural PDE SolversMeta-Learning

    Department of Artificial Intelligence, Korea University, Anam-dong, Seongbuk-gu, Seoul, 02841, Republic of Korea · Department of Brain and Cognitive Engineering, Korea University, Anam-dong, Seongbuk-gu, Seoul, 02841, Republic of Korea

  14. Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Regimes

    Apr 27, 2026Zhangyong LiangNeural PDE SolversGradient Interference

    National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China.

  15. Shared-kernel Wavelet Neural Networks for Poisson Image Reconstruction

    Apr 27, 2026Yuanhao Gong, Tan Tang, Qianyan LiuNeural PDE SolversImage Reconstruction

  16. Adaptive-Distribution Randomized Neural Networks for PDEs: A Low-Dimensional Distribution-Learning Framework

    Apr 27, 2026You Yang, Fei WangNeural PDE SolversPDE Operator Learning

    School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

  17. Collocation-based Robust Physics Informed Neural Networks for time-dependent simulations of pollution propagation under thermal inversion conditions on Spitsbergen

    Apr 24, 2026Leszek Siwik, Maciej Sikora, Natalia Leszczyńska +7Neural PDE SolversAir Quality Prediction

    AGH University of Krakow, Faculty of Computer Science, Al. Mickiewicza 30, Kraków, 30-059, Poland · Medical University of Silesia-Katowice, Faculty of Medical Sciences, ul. Poniatowskiego 15, Katowice, 40-055, Poland · The University Centre in Svalbard, Longyearbyen, Box 156 N-9171, Longyearbyen, Norway +4

  18. A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization

    Apr 23, 2026Mohammad Mahdi Abedi, David Pardo, Tariq AlkhalifahNeural PDE SolversPDE Solving

    University of the Basque Country (UPV/EHU), Department of Mathematics, Spain · Basque Center for Applied Mathematics (BCAM), Bilbao, Spain · Ikerbasque, Basque Foundation for Science, Bilbao, Spain +1

  19. CO2_2 sequestration hybrid solver using isogeometric alternating-directions and collocation-based robust variational physics informed neural networks (IGA-ADS-CRVPINN)

    Apr 22, 2026Askold Vilkha, Tomasz Służalec, Marcin Łoś +1Neural PDE SolversPorous Media Flow

    AGH University of Krakow, Faculty of Computer Science, Al. Mickiewicza 2026 30, Kraków, 30-059, Poland

  20. Virtual boundary integral neural network for three-dimensional exterior acoustic problems

    Apr 19, 2026Jiahao Li, Qiang Xi, Ilia Marchevskiy +1Neural PDE SolversPDE Solving

    College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China · Department of Applied Mathematics, Bauman Moscow State Technical University, Moscow 105005, Russia

  21. FlowRefiner: Flow Matching-Based Iterative Refinement for 3D Turbulent Flow Simulation

    Apr 18, 2026Yilong Dai, Yiming Sun, Yiheng Chen +3Flow MatchingNeural PDE Solvers

    University of Alabama · University of Pittsburgh

  22. PINNACLE: An Open-Source Computational Framework for Classical and Quantum PINNs

    Apr 17, 2026Shimon Pisnoy, Hemanth Chandravamsi, Ziv Chen +4Neural PDE SolversPDE Surrogate Modeling

    †Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Haifa, Israel. · ‡Corresponding author. · §The Andrew and Erna Viterbi Faculty of Electrical & Computer Engineering, Technion Israel Institute of Technology, Haifa, Israel. +2

  23. A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws

    Apr 17, 2026Jiamin Jiang, Shanglin Lv, Jingrun ChenGraph Neural NetworksNeural PDE Solvers

    School of Artificial Intelligence and Data Science and Suzhou Institute for Advanced Research, University of Science and Technology of China, and Suzhou Big Data & AI Research and Engineering Center, China · School of Mathematical Sciences, University of Science and Technology of China, China · School of Mathematical Sciences and Suzhou Institute for Advanced Research, University of Science and Technology of China, and Suzhou Big Data & AI Research and Engineering Center, China

  24. G-PARC: Graph-Physics Aware Recurrent Convolutional Neural Networks for Spatiotemporal Dynamics on Unstructured Meshes

    Apr 16, 2026Jack T. Beerman, Tyler J. Abele, Mehdi Taghizadeh +6Neural PDE SolversPDE Surrogate Modeling

    School of Data Science, University of Virginia, Charlottesville, VA 22903, United States · Department of Engineering Systems and Environment, University of Virginia, Charlottesville, VA 22903, United States · Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22903, United States +1

  25. Auxiliary Finite-Difference Residual-Gradient Regularization for PINNs

    Apr 15, 2026Stavros KassinosNeural PDE SolversPhysics-Informed ML

    Computational Sciences Laboratory, Department of Mechanical Engineering, University of Cyprus, 1 University Avenue, Aglantzia, 2109, Nicosia, Cyprus

  26. Randomized Neural Networks for Integro-Differential Equations with Application to Neutron Transport

    Apr 15, 2026Haoning Dang, Fei Wang, Yifan Chen +3Neural PDE Solvers

    School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. +1

  27. Physics-Informed Neural Networks for Solving Derivative-Constrained PDEs

    Apr 15, 2026Kentaro Hoshisashi, Carolyn E Phelan, Paolo BaruccaNeural PDE SolversPDE Solving

    Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK

  28. Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains

    Apr 10, 2026Zhangyong LiangNeural PDE SolversPDE Solving

    National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China.

  29. Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation

    Apr 9, 2026Andrew Qing He, Wei Cai, Sihong ShaoNeural PDE Solvers

    *Department of Mathematics, Southern Methodist University, Dallas, TX, USA. · †School of Mathematical Sciences, Peking University, Beijing, China.

  30. A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributions

    Apr 7, 2026Xiaolong Wang, Jing Feng, Qi Liu +3PDE SolvingNeural PDE Solvers

    School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, China · School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China · MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, Xi’an, 710072, China +2