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. Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfaces

    Apr 22, 2026Ganesh Sahadeo Meshram, Partha Pratim Chakrabarti, Suman ChakrabortyPDE Operator LearningPhysics-Informed Neural Operators

    Department of Mechanical Engineering, IIT Kharagpur, Kharagpur, India · Department of Computer Science & Engineering, IIT Kharagpur, Kharagpur, India

  2. Neural Shape Operator Surrogates -- Expression Rate Bounds

    Apr 20, 2026Helmut Harbrecht, Christoph SchwabPDE Operator LearningNeural Network Approximation Theory

  3. Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches

    Apr 18, 2026Yixuan WangPDE SolvingKolmogorov-Arnold Networks

    CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California · [math.NA]

  4. Late Fusion Neural Operators for Extrapolation Across Parameter Space in Partial Differential Equations

    Apr 17, 2026Eva van Tegelen, Taniya Kapoor, George A. K. van Voorn +2Surrogate ModelingPDE Surrogate Modeling

    Artificial Intelligence Group, Wageningen University and Research, Netherlands · Mathematical and Statistical Methods, Wageningen University and Research, Netherlands

  5. Operator Learning for Predicting Bulk Wave Parameters of Spectral Wave Models

    Apr 7, 2026Shukai Cai, Sourav Dutta, Mark Loveland +4PDE Surrogate ModelingPDE Operator Learning

    Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 E 24th St, Austin, 78712, TX, USA · Information Technology Laboratory, U.S. Army Engineer Research and Development Center, MS, USA · Department of Mechanical Engineering and Technology Management, Norwegian University of Life Sciences, Ås, Norway +1

  6. FNO∠θ^{\angle θ}: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems

    Apr 6, 2026Zhexian Li, Ketan SavlaFourier Neural OperatorPDE Operator Learning

    Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA. · Sonny Astani Department of Civil and Environmental Engineering, the Daniel Epstein Department of Industrial and Systems Engineering, and the Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA.

  7. Lotka-Sharpe Neural Operators for Control of Population PDEs

    Apr 4, 2026Miroslav Krstic, Iasson Karafyllis, Luke Bhan +1Feedback ControlPDE Operator Learning

    University of California, San Diego, USA · National Technical University of Athens, Greece · KTH Royal Institute Technology, Stockholm, Sweden

  8. Simple yet Effective: Low-Rank Spatial Attention for Neural Operators

    Apr 4, 2026Zherui Yang, Haiyang Xin, Tao Du +1Neural OperatorsPDE Operator Learning

    University of Science and Technology of China · Tsinghua University · Shanghai Qi Zhi Institute

  9. Learning Contractive Integral Operators with Fredholm Integral Neural Operators

    Apr 3, 2026Kyriakos C. Georgiou, Constantinos Siettos, Athanasios N. YannacopoulosNeural OperatorsPDE Operator Learning

    Department of Electrical Engineering and Information Technologies, University of Naples “Federico II”, Naples, Italy · Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, University of Naples “Federico II”, Naples, Italy

  10. PI-JEPA: Label-Free Surrogate Pretraining for Coupled Multiphysics Simulation via Operator-Split Latent Prediction

    Apr 1, 2026Brandon Yee, Pairie KohMultiphysics SimulationPDE Surrogate Modeling

    Physics Lab, Yee Collins Research Group · Graduate School of Business, Stanford University

  11. Lie Generator Networks for Nonlinear Partial Differential Equations

    Mar 31, 2026Shafayeth Jamil, Rehan KapadiaKoopman Operator LearningPDE Operator Learning

    Ming Hsieh Department of Electrical and Computer Engineering University of Southern California

  12. DSO: Dual-Scale Neural Operators for Stable Long-term Fluid Dynamics Forecasting

    Mar 26, 2026Huanshuo Dong, Hao Wu, Hong Wang +2PDE Operator LearningNeural Operators

    University of Science and Technology of China, Hefei, China · Tsinghua University, Beijing, China · Institute of Automation, Chinese Academy of Sciences, Beijing, China +1

  13. Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations

    Mar 25, 2026Heng Wu, Junjie Wang, Benzhuo LuPDE Operator LearningNeural Operators

    SKLMS, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

  14. Circuit-Inspired High-Order Neural Networks with Unified Neural Dynamics Modeling for PDE Solving and Visual Perception

    Mar 25, 2026Tongfei Chen, Jingying Yang, Linlin Yang +8Dynamic Neural NetworksPDE Operator Learning

    School of Artificial Intelligence (Institute of Artificial Intelligence), Beihang University, Beijing, China. · Communication University of China, Beijing, China. · School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. +3

  15. Adversarial Vulnerabilities in Neural Operator Digital Twins: Gradient-Free Attacks on Nuclear Thermal-Hydraulic Surrogates

    Mar 23, 2026Samrendra Roy, Kazuma Kobayashi, Souvik Chakraborty +2Adversarial RobustnessNeural Operators

    Department of Nuclear, Plasma & Radiological Engineering, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA · Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016, India · National Center for Supercomputing Applications, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA

  16. Generalization Limits of In-Context Operator Networks for Higher-Order Partial Differential Equations

    Mar 23, 2026Jamie Mahowald, Tan Bui-ThanhPDE Operator LearningNeural Network Generalization

    Department of Mathematics, University of Texas at Austin, Austin, Texas · Department of Aerospace Engineering and Engineering Mechanics, Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, Texas

  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. UniFluids: Unified Neural Operator Learning with Conditional Flow-matching

    Mar 19, 2026Haosen Li, Qi Meng, Jiahao Li +4PDE Operator LearningPDE Surrogate Modeling

    Academy of Mathematics and Systems Science, Chinese Academy of Sciences · Microsoft Research Asia · Renmin University of China +1

  19. RHYME-XT: A Neural Operator for Spatiotemporal Control Systems

    Mar 18, 2026Marijn Ruiter, Miguel Aguiar, Jake Rap +2PDE Surrogate ModelingPDE Operator Learning

    Digital Futures and Division of Decision and Control Systems, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden. · Control Systems Group, EE Dept., Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

  20. Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model

    Mar 18, 2026Luca PellegriniFourier Neural OperatorPDE Operator Learning

    Department of Mathematics, University of Pavia, Pavia, Italy · Euler Institute, Faculty of Informatics, Universit`a della Svizzera italiana, Lugano, Switzerland

  21. Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Maps

    Mar 16, 2026Pengcheng ChengPDE Operator LearningNeural Operators

    School of Mathematics, Jilin University, Changchun, 130012, China

  22. Windowed Fourier Propagator: A Frequency-Local Neural Operator for Wave Equations in Inhomogeneous Media

    Mar 15, 2026Yiyang Cai, Zixuan Qiu, Yunlu Shu +4PDE Operator LearningNeural Operators

    School of Mathematical Sciences, Fudan University, Shanghai, 200433, China · Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200438, China · Center for Applied Mathematics, Fudan University, Shanghai, 200438, China

  23. Manifold-Orthogonal Dual-spectrum Extrapolation for Parameterized Physics-Informed Neural Networks

    Mar 14, 2026Zhangyong Liang, Huanhuan GaoPDE Operator LearningPhysics-Informed ML

  24. Out-of-distribution transfer of PDE foundation models to material dynamics under extreme loading

    Mar 4, 2026Mahindra Rautela, Alexander Most, Siddharth Mansingh +9PDE Foundation ModelsOOD Generalization

    AOT-IC Group, Los Alamos National Laboratory, Los Alamos, New Mexico, US, 87545 · CAI Division, Los Alamos National Laboratory, Los Alamos, New Mexico, US, 87545 · EES Divison, Los Alamos National Laboratory, Los Alamos, New Mexico, US, 87545 +2

  25. 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

  26. Extending Neural Operators: Robust Handling of Functions Beyond the Training Set

    Mar 4, 2026Blaine Quackenbush, Paul J. AtzbergerNeural OperatorsOOD Generalization

    Department of Mathematics, University of California Santa Barbara (UCSB). · Department of Mathematics, Department of Mechanical Engineering, University of California Santa Barbara (UCSB)

  27. Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization

    Mar 3, 2026Xindi Gong, Dingcheng Luo, Thomas O'Leary-Roseberry +2PDE Operator LearningNeural Operators

    Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas, USA · School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia · Centre for Data Science, Queensland University of Technology, Brisbane, Australia +3

  28. One Operator to Rule Them All? On Boundary-Indexed Operator Families in Neural PDE Solvers

    Mar 2, 2026Lennon J. ShikhmanPDE Operator LearningNeural Operators

    College of Computing, Georgia Institute of Technology · Department of Mathematics and Systems Engineering, Florida Institute of Technology

  29. Operator Learning Using Weak Supervision from Walk-on-Spheres

    Mar 1, 2026Hrishikesh Viswanath, Hong Chul Nam, Xi Deng +3PDE Operator LearningPhysics-Informed ML

    Purdue University, USA · ETH Zurich, Switzerland · California Institute of Technology, USA +1

  30. Quantum AS-DeepOnet: Quantum Attentive Stacked DeepONet for Solving 2D Evolution Equations

    Feb 28, 2026Hongquan Wang, Hanshu Chen, Ilia Marchevsky +1PDE Operator LearningDeep Operator Networks

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