PDE Operator Learning

PDE: Partial Differential Equation

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Overview

PDE operator learning studies how to approximate solution maps that take functions or physical parameters—such as initial conditions, coefficients, and boundary data—and return PDE solution fields. It aims to make repeated simulation faster for tasks including forecasting, design, inverse problems, and control, using approaches such as Fourier neural operators, DeepONets, attention and graph models, and kernel methods. Researchers also use physics-informed losses, multiscale representations, and structure-preserving constraints to improve accuracy, efficiency, and physical consistency. A central challenge is reliable generalization across unseen geometries, resolutions, frequencies, and parameter regimes while maintaining stable long-term predictions; strong in-distribution accuracy alone does not guarantee it.

Latest papers

CardsList

231 papers36 papers in the last four weeks, up 44% on the four weeks before. 0.2% of all new papers.

  1. A Boundary Integral-based Neural Operator for Mesh Deformation

    Feb 27, 2026Zhengyu Wu, Jun Liu, Wei WangNeural OperatorsPDE Operator Learning

    School of Electronics and Information Engineering, Hangzhou Dianzi University, Hangzhou, China

  2. From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators

    Feb 25, 2026Zhihao Li, Yu Feng, Zhilu Lai +1PDE Operator LearningNeural Operators

    The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China · The Hong Kong University of Science and Technology, Hong Kong SAR, China

  3. Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelines

    Feb 22, 2026Siavash Khodakarami, Vivek Oommen, Nazanin Ahmadi Daryakenari +2Spectral BiasPhysics-Informed ML

    Division of Applied Mathematics, Brown University, Providence, RI 02912, USA · School of Engineering, Brown University, Providence, RI 02912, USA

  4. 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

  5. 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

  6. 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

  7. 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

  8. Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows

    Feb 17, 2026Ramansh Sharma, Matthew Lowery, Houman Owhadi +1PDE Operator LearningOperator Learning

    Kahlert School of Computing University of Utah UT, USA · Department of Computing and Mathematical Sciences California Institute of Technology CA, USA

  9. PDE foundation models are skillful AI weather emulators for the Martian atmosphere

    Feb 16, 2026Johannes Schmude, Sujit Roy, Liping Wang +10PDE Foundation ModelsWeather Forecasting

    IBM Research · NASA Marshall Space Flight Center, Huntsville, Al, USA · Earth System Science Center, University of Alabama in Huntsville, Al, USA +1

  10. 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.

  11. SFO: Learning PDE Operators via Spectral Filtering

    Jan 23, 2026Noam Koren, Rafael Moschopoulos, Kira Radinsky +1PDE Surrogate ModelingPDE Operator Learning

    Department of Computer Science Technion - Israel Institute of Technology · Department of Computer Science Princeton University

  12. 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

  13. Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation

    Dec 9, 2025Muhammad Abid, Omer SanPDE Surrogate ModelingPDE Operator Learning

    Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

  14. Solver-Integrated Adversarial Attacking and Training of Neural Operators

    Oct 21, 2025Yifei SunAdversarial TrainingAdversarial Attacks

  15. RheOFormer: A generative transformer model for simulation of complex fluids and flows

    Oct 1, 2025Maedeh Saberi, Amir Barati Farimani, Safa JamaliNeural Surrogate ModelingTransformer

    Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA · Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

  16. Discretization-independent operator learning for partial differential equations

    Jul 9, 2025Jacob Hauck, Yanzhi ZhangPDE Operator LearningNeural Network Approximation Theory

    Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409

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

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

  18. Latent Evolution Operator (LEO): State-to-State Forecasting of Parametric PDEs

    May 14, 2025Khalid Rafiq, Wenjing Liao, Aditya G. NairPDE Surrogate ModelingPDE Operator Learning

    Department of Mechanical Engineering, University of Nevada, Reno, Reno, USA · School of Mathematics, Georgia Institute of Technology, Atlanta, USA

  19. From Theory to Application: A Practical Introduction to Neural Operators in Scientific Computing

    Mar 7, 2025Prashant K. JhaNeural Surrogate ModelingFourier Neural Operator

    Department of Mechanical Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA.

  20. Active Learning with Bayesian Multi-Fidelity Laplace Neural Operators for Oscillatory Parametric PDEs

    Feb 1, 2025Bongseok Kim, Haoyang Zheng, Michael Penwarden +1Dynamical SystemsSurrogate Modeling

    School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, 47907, IN, USA. · Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, 47907, IN, USA.

  21. 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