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. Improving the accuracy of physics-informed neural networks via last-layer retraining

    Mar 4, 2026Saad Qadeer, Panos StinisNeural PDE SolversPhysics-Informed ML

    Advanced Computing, Mathematics and Data Division, Pacific Northwest National Laboratory, Richland, WA

  2. Jacobian-Adaptive Weighting for Stability: Enhancing Long-term Rollout of Neural Partial Differential Equation Solvers via Spatially-Adaptive Regularization

    Mar 4, 2026Fengxiang Nie, Yasuhiro SuzukiNeural PDE SolversAlgorithmic Stability

    University of Hiroshima

  3. DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs

    Mar 2, 2026Yingjie Tan, Quanming Yao, Yaqing WangNeural PDE SolversPDE Surrogate Modeling

    Qiuzhen College, Tsinghua University · Department of Electronic Engineering, Tsinghua University · Beijing Institute of Mathematical Sciences and Applications

  4. Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces

    Mar 2, 2026Jingbo Sun, Fei WangNeural PDE SolversPDE Solving

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

  5. Structure-preserving Randomized Neural Networks for Incompressible Magnetohydrodynamics Equations

    Mar 1, 2026Yunlong Li, Fei Wang, Lingxiao LiStructure-Preserving Neural NetworksMagnetohydrodynamics

    School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. 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 Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, China. +1

  6. Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics

    Mar 1, 2026Haoning Dang, Shi Jin, Fei WangPDE SolvingNeural PDE Solvers

    School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Mathematical Sciences & Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

  7. A level-wise training scheme for learning neural multigrid smoothers with application to integral equations

    Mar 1, 2026Lingfeng Li, Yin King Chu, Raymond Chan +1Neural OperatorsNeural PDE Solvers

    Hetao Institute of Mathematics and Interdisciplinary Sciences, Shenzhen, Guangdong, China. · Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. · Department of Data Science, Department of Operations and Risk Management,Lingnan University, Tuen Mun, Hong Kong, China. +1

  8. Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder

    Mar 1, 2026Ryosuke YanoNeural PDE SolversPhysics-Informed ML

    Tokio Marine dR Co., Ltd., 1-5-1, Otemachi, Chiyoda-ku, Tokyo, 100-0004, Japan.

  9. TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions

    Feb 28, 2026Hongjie Jiang, Di LuoNeural PDE SolversPDE Solving

    School of Mathematical Sciences, Peking University, Beijing, China · Department of Physics, Tsinghua University, Beijing, China · Institute of Advanced Study, Tsinghua University, Beijing, China

  10. A Stability-Aware Frozen Euler Autoencoder for Physics-Informed Tracking in Continuum Mechanics (SAFE-PIT-CM)

    Feb 27, 2026Emil HovadPhysics-Informed MLPDE Inverse Problems

    Section for Statistics and Data Analysis, DTU Compute, Technical University of Denmark, Kgs. Lyngby, Denmark

  11. Deep Accurate Solver for the Geodesic Problem

    Feb 25, 2026Saar Huberman, Amit Bracha, Ron KimmelNeural PDE SolversRiemannian Geometry

    Technion - Israel Institute of Technology, Haifa, Israel

  12. Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correction

    Feb 23, 2026Pao-Hsiung Chiu, Jian Cheng Wong, Chin Chun Ooi +3Neural PDE SolversPhysics-Informed ML

    Institute of High Performance Computing, A*STAR, Singapore. · Centre for Frontier AI Research, A*STAR, Singapore. · School of Mechanical Engineering, Tianjin University, China. +1

  13. Neural-HSS: Hierarchical Semi-Separable Neural PDE Solver

    Feb 20, 2026Pietro Sittoni, Emanuele Zangrando, Angelo A. Casulli +2Neural PDE SolversPDE Operator Learning

    Gran Sasso Science Institute, L’Aquila, Italy · University of Edinburgh, UK

  14. Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations

    Feb 19, 2026Bongseok Kim, Jiahao Zhang, Guang LinNeural PDE SolversPDE Solving

  15. Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method

    Feb 19, 2026Yuhe Wang, Min WangNeural PDE SolversPhysics-Informed ML

    Institute for Scientific Computation, Texas A&M University, College Station, TX 77843, USA · Department of Mathematics, University of Houston, Houston, TX 77204, USA

  16. Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

    Feb 18, 2026Jiaquan Zhang, Fachrina Dewi Puspitasari, Songbo Zhang +7Neural PDE SolversPDE Operator Learning

    School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, China · Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China · Institute of Software Chinese Academy of Sciences, Beijing, China +2

  17. Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings

    Feb 18, 2026Zhenzhen Huang, Haoyu Bian, Jiaquan Zhang +6Neural PDE SolversPhysics-Informed ML

    School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, China · Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China · Institute of Software Chinese Academy of Sciences, Beijing, China +1

  18. Flowers: A Warp Drive for Neural PDE Solvers

    Feb 17, 2026Till Muser, Alexandra Spitzer, Matti Lassas +2Neural PDE SolversPDE Surrogate Modeling

    Department of Mathematics and Computer Science, University of Basel, Basel, Switzerland · Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland · Simons Chair in Computational and Applied Mathematics and Earth Science, Rice University, Houston, TX, USA

  19. BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations

    Feb 16, 2026Jonathan Gorard, Ammar Hakim, James JunoNeural PDE SolversPartial Differential Equations

    Princeton University Princeton, NJ 08544, USA · Princeton Plasma Physics Laboratory Princeton, NJ 08540, USA

  20. Parameter-Minimal Neural DE Solvers via Horner Polynomials

    Feb 16, 2026T. Matulić, D. SeršićNeural PDE SolversPolynomial Neural Networks

    Dept. of Electronic Systems and Information Processing, Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia

  21. A Unified Physics-Informed Neural Network for Modeling Coupled Electro- and Elastodynamic Wave Propagation Using Three-Stage Loss Optimization

    Feb 14, 2026Suhas Suresh Bharadwaj, Reuben Thomas ThovelilNeural PDE SolversPDE Solving

    Department of Electrical and Electronics Engineering, BITS Pilani, Dubai Campus, Dubai, United Arab Emirates

  22. Transolver-3: Scaling Up Transformer Solvers to Industrial-Scale Geometries

    Feb 4, 2026Hang Zhou, Haixu Wu, Haonan Shangguan +4Efficient Neural Network InferenceNeural PDE Solvers

    School of Software, BNRist, Tsinghua University, China.

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

    Jan 1, 2026Abhisek Ganguly, Santosh Ansumali, Sauro SucciDynamical SystemsNeural PDE Solvers

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

  24. PGOT: A Physics-Geometry Operator Transformer for Complex PDEs

    Dec 29, 2025Zhuo Zhang, Xi Yang, Ying Miao +5Neural PDE SolversPDE Surrogate Modeling

    National University of Defense Technology · Peking University · National University of Singapore +2

  25. Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations

    Dec 22, 2025Xiaolong Wu, Qifeng LiaoNeural PDE SolversPDE Solving

    School of Information Science and Technology, ShanghaiTech University, Shanghai, 201210, China

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

    Nov 19, 2025Victorita Dolean, Daria Hrebenshchykova, Stéphane Lanteri +1Neural PDE SolversPDE Solving

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

  27. LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries

    Oct 29, 2025René P. Klausen, Ivan Timofeev, Jonas Naujoks +4Neural PDE SolversPDE Solving

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

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

    Aug 29, 2025Bangti Jin, Longjun WuNeural PDE SolversPDE Solving

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

  29. Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra

    May 16, 2025Ivan Bioli, Carlo Marcati, Giancarlo SangalliNeural PDE SolversGradient Descent

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

  30. Gabor-Enhanced Physics-Informed Neural Networks for Fast Simulations of Acoustic Wavefields

    Feb 24, 2025Mohammad Mahdi Abedi, David Pardo, Tariq AlkhalifahNeural PDE SolversPDE Solving

    Basque Center for Applied Mathematics, Bilbao, Spain · University of the Basque Country, Department of Mathematics, Spain · Ikerbasque, Basque Foundation for Science, Bilbao, Spain +1