PDE Solving

PDE: Partial Differential Equation

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Overview

PDE solving seeks approximate solutions to equations describing processes such as heat flow, fluids, waves, and diffusion, where exact solutions are often unavailable. Classical approaches include finite difference, finite element, finite volume, spectral, and domain-decomposition methods; learning-based approaches include physics-informed neural networks, neural operators, and randomized or basis-based solvers. Current work also combines these methods, using learned models for acceleration, adaptive refinement, or surrogate prediction while retaining numerical constraints such as conservation and boundary conditions. A central challenge is balancing speed and flexibility with accuracy, stability, reliable generalization, and guarantees that residual errors reflect true solution errors.

Latest papers

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255 papers30 papers in the last four weeks, up 20% on the four weeks before. 0.2% of all new papers.

  1. Adaptively trained Physics-informed Radial Basis Function Neural Networks for Solving Multi-asset Option Pricing Problems

    Jan 19, 2026Yan Ma, Yumeng Ren, Elisabeth LarssonQuantitative FinancePDE Solving

    aCollege of Mathematics and Statistics, Northwest Normal University, Lanzhou, China · bDepartment of Mathematics, City University of Hong Kong, Hong Kong SAR, China · cDepartment of Information Technology, Uppsala University, Sweden

  2. 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

  3. Quantum-Classical Physics-Informed Neural Networks for Solving Reservoir Seepage Equations

    Dec 3, 2025Xiang Rao, Yina Liu, Yuxuan ShenPDE SolvingPhysics-Informed ML

    School of Petroleum Engineering, Yangtze University, Wuhan 430100, China · College of Future Technology, Yangtze University, Wuhan 430100, China · State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University), Wuhan 430100, China +2

  4. Hybrid coupling with operator inference and the overlapping Schwarz alternating method

    Nov 20, 2025Irina Tezaur, Eric Parish, Anthony Gruber +3PDE SolvingReduced-Order Modeling

    Sandia National Laboratories, Livermore, CA and Albuquerque, NM, USA · Rice University, Houston, TX, USA

  5. 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

  6. 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

  7. A Greedy PDE Router for Blending Neural Operators and Classical Methods

    Sep 29, 2025Sahana Rayan, Yash Patel, Ambuj TewariPDE Solving

    Department of Statistics University of Michigan Ann Arbor, MI 48104, USA

  8. A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulations

    Sep 19, 2025Simon Klaes, Axel Klawonn, Natalie Kubicki +4PDE SolvingPDE Surrogate Modeling

    Department of Mathematics and Computer Science, University of Cologne, Germany · Center for Data and Simulation Science, University of Cologne, Germany · Institute for Advanced Simulation (IAS-2), Forschungszentrum Jülich, Germany +1

  9. 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

  10. DPG loss functions for learning parameter-to-solution maps by neural networks

    Jun 23, 2025Pablo Cortés Castillo, Wolfgang Dahmen, Jay GopalakrishnanPDE SolvingPDE Surrogate Modeling

  11. 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

  12. Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs

    Oct 18, 2024Takashi Furuya, Anastasis KratsiosStochastic Differential EquationsPDE Solving

    Doshisha University, Japan · Department of Mathematics McMaster University and Vector Institute, Canada

  13. FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation

    Nov 29, 2022Adarsh Ganeshram, Valentin Duruisseaux, Haydn Maust +6Physics-Informed Neural OperatorsPDE Solving

    Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

  14. Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

    Dec 15, 2021Carlo Marcati, Christoph SchwabPDE SolvingPDE Operator Learning

    Seminar for Applied Mathematics, ETH Zurich, Switzerland

  15. Deep learning based numerical approximation algorithms for stochastic partial differential equations

    Dec 2, 2020Christian Beck, Sebastian Becker, Patrick Cheridito +2Stochastic Differential EquationsNeural PDE Solvers

    Department of Mathematics, ETH Zurich · School of Data Science and School of Artificial Intelligence,2025 The Chinese University of Hong Kong, Shenzhen (CUHK-Shenhen) · Applied Mathematics: Institute for Analysis and Numerics, University of Münster +1