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

    Apr 7, 2026Prashant Kumar, Rajesh RanjanDomain Decomposition MethodsNeural PDE Solvers

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

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

    Apr 6, 2026Anas Jnini, Elham Kiyani, Khemraj Shukla +5Physics-Informed MLNeural PDE Solvers

    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

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

    Apr 6, 2026Manuel Ballester, Santiago Lopez-Tapia, Seth Gossage +9AstrophysicsNeural PDE Solvers

    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

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

    Apr 4, 2026Gyounghun Ko, Sung-Jun Son, Seung Yeon Cho +1Neural PDE SolversPhysics-Informed ML

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

    Apr 2, 2026Genwei Ma, Ting Luo, Ping Yang +1Neural PDE SolversPDE Solving

    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

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

    Apr 2, 2026Yilong Dai, Shengyu Chen, Xiaowei Jia +1Neural PDE SolversPDE Solving

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

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

    Mar 31, 2026Zhangyong Liang, Huanhuan Gao, Manyu XiaoNeural PDE SolversPhysics-Informed ML

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

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

    Mar 30, 2026Ehsan Zeraatkar, Rodion Podorozhny, Jelena TešićPhysics-Informed MLNeural PDE Solvers

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

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

    Mar 26, 2026Vasily Ilin, Jingwei HuNeural PDE Solvers

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

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

    Mar 26, 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

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

    Mar 25, 2026Lucas Gerken Starepravo, Georgios Fourtakas, Steven Lind +3Neural PDE SolversGraph Neural Networks

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

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

    Mar 25, 2026Jucai Zhai, Muhammad Abdullah, Boyang Chen +6Neural PDE SolversPDE Solving

    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

  13. Coordinate Encoding on Linear Grids for Physics-Informed Neural Networks

    Mar 24, 2026Tetsuro Tsuchino, Motoki ShigaNeural PDE SolversPhysics-Informed ML

    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

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

    Mar 23, 2026Gianluca Fabiani, Michail E. Kavousanakis, Constantinos Siettos +1Dynamical SystemsNeural PDE Solvers

    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

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

    Mar 20, 2026V. S. Maduri, K. B. NakshatralaNeural PDE SolversPorous Media Flow

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

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

    Mar 20, 2026Jun LiuHJB EquationsNeural PDE Solvers

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

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

    Mar 19, 2026Samuel N. Cohen, Filippo de Feo, Jackson Hebner +1HJB EquationsPDE Solving

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

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

    Mar 19, 2026Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1Neural Network GeneralizationNeural PDE Solvers

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

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

    Mar 19, 2026Riccardo Saporiti, Fabio NobileNormalizing FlowsNeural PDE Solvers

    CSQI, École Polytechnique Féderale de Lausanne, Switzerland

  20. Physics-integrated neural differentiable modeling for immersed boundary systems

    Mar 17, 2026Chenglin Li, Hang Xu, Jianting Chen +1PDE Surrogate ModelingComputational Fluid Dynamics

    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.

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

    Mar 17, 2026Andrew Qing He, Wei CaiFokker–Planck EquationPDE Solving

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

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

    Mar 16, 2026Vasiliy A. Es'kin, Egor V. IvanovNeural PDE SolversNeural Operators

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

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

    Mar 11, 2026V. S. Maduri, K. B. NakshatralaPorous Media FlowComputational Fluid Dynamics

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

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

    Mar 10, 2026Bradley H. Theilman, James B. AimoneNeuromorphic ComputingNeural Network Robustness

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

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

    Mar 9, 2026Weizheng Zhang, Xunjie Xie, Hao Pan +4Neural PDE SolversPDE Solving

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

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

    Mar 9, 2026Francesco Della Santa, Sandra Pieraccini, Maria StrazzulloPDE Surrogate ModelingNeural PDE Solvers

    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

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

    Mar 6, 2026Milad Shirani, Pete H. Gueldner, Murat Khidoyatov +2Structural MechanicsMaterials Science

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

  28. Frequency-Separable Hamiltonian Neural Network for Multi-Timescale Dynamics

    Mar 6, 2026Yaojun Li, Yulong Yang, Christine Allen-BlanchetteHamiltonian Neural NetworksDynamical Systems

    Princeton University, USA

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

    Mar 6, 2026Akshay Govind Srinivasan, Balaji SrinivasanNeural PDE SolversPDE Solving

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

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

    Mar 5, 2026Yuchen Xie, Jianyuan Yin, Lei ZhangNeural PDE Solvers

    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