PDE Operator Learning

PDE: Partial Differential Equation

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

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

Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach
May 22, 2026Qian Zhang, George Em KarniadakisPDE Operator Learning Operator Learning
Division of Applied Mathematics, Brown University
IKNO: Infinite-order Kernel Neural Operators
May 21, 2026Pengyuan Zhu, Ivor W. Tsang, Yueming LyuPDE Operator Learning Neural Operators
Nanyang Technological University · Centre for Frontier AI Research(CFAR), Agency for Science, Technology and Research (A*STAR)
Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions
May 19, 2026Ha Dang, Sebastian Schmidt, Juergen HesserPDE Operator Learning Deep Operator Networks
Mannheim Institute for Intelligent Systems in Medicine, Heidelberg University · Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University · Heidelberg Institute for Theoretical Studies (HITS), Heidelberg University +2
PACE-FNO: Physics-Aligned Canonical Equivariance for Fourier Neural Operators
May 18, 2026Jiaxiao Xu, Changhong Mou, Yeyu Zhang +1Fourier Neural Operator OOD Generalization
Shanghai University of Finance and Economics · Utah State University · University of Edinburgh
Symplectic Neural Operators for Learning Infinite Dimensional Hamiltonian Systems
May 15, 2026Yeang Makara, Yusuke Tanaka, Takashi Matsubara +1PDE Operator Learning Neural Operators
Graduate School of Science Kobe University 1-1 Rokkodai-cho, Nada-ku, Kobe, Japan · NTT Communication Science Laboratories 2-4, Hikaridai, Seika-cho, Soraku-gun, Kyoto, Japan · Faculty of Information Science and Technology Hokkaido University Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido, Japan +1
AOT-POT: Adaptive Operator Transformation for Large-Scale PDE Pre-training
May 15, 2026Qitan Lv, Hong Wang, Zhongkai Hao +5Transformer PDE Operator Learning
University of Science and Technology of China · 2Shanghai AI Laboratory · Department of Electronic Engineering, Tsinghua University, China
Eradicating Negative Transfer in Multi-Physics Foundation Models via Sparse Mixture-of-Experts Routing
May 14, 2026Ellwil Sharma, Arastu SharmaSparse Mixture-of-Experts PDE Operator Learning
Shodh AI
Topology-Preserving Neural Operator Learning via Hodge Decomposition
May 13, 2026Dongzhe Zheng, Tao Zhong, Christine Allen-BlanchetteHodge Decomposition PDE Operator Learning
Princeton University
Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation
May 13, 2026Runlong Xie, An LuoFourier Neural Operator PDE Solving
Independent Researcher · School of Statistics, University of Minnesota, MN, USA.
UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning
May 12, 2026Hanli Qiao, George Em Karniadakis, Muhammad MuniruzzamanCross-Domain Generalization PDE Operator Learning
Water and Mining Environment Unit, Geological Survey of Finland, Vuorimiehentie 5, Espoo, 02151, Finland · Division of Applied Mathematics, Brown University, 170 Hope Street, Providence, RI 02912, USA · Institute of Geosciences, University of Bonn, Kirschallee 1-3, Bonn, 53115, Germany
Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
May 12, 2026Takashi Furuya, Ryo Ozawa, Jenn-Nan WangReaction-Diffusion Systems PDE Operator Learning
Doshisha University, RIKEN AIP · Tohoku University · National Taiwan University
Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations
May 12, 2026Chanyoung Kim, Myeonghwan Seong, Yujin Kim +2Hybrid Quantum-Classical ML PDE Operator Learning
KAIST · Department of Statistics and Data Science, Yonsei University · Department of Applied Statistics, Yonsei University +2
Compositional Neural Operators for Multi-Dimensional Fluid Dynamics
May 12, 2026Hamda Hmida, Hsiu-Wen Chang, Youssef MesriPDE Operator Learning Physics-Informed Neural Operators
Mines Paris - PSL University, Centre for Material Forming (CEMEF) · Mines Paris - PSL University, Centre for Robotics (CAOR)
Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs
May 11, 2026Xuxiang Zhao, Angelica I. Aviles-RiveroFourier Neural Operator Representation Learning
Qiuzhen College, Tsinghua University · YMSC, Tsinghua University
Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs
May 11, 2026Koichi Taniguchi, Sho SonodaPDE Solving PDE Operator Learning
Department of Mathematical and Systems Engineering, Faculty of Engineering Shizuoka University 3-5-1 Johoku, Chuo-ku, Hamamatsu, 432-8561, Japan · RIKEN AIP / CyberAgent 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
jNO: A JAX Library for Neural Operator and Foundation Model Training
May 11, 2026Leon Armbruster, Rathan Ramesh, Georg Kruse +1PDE Foundation Models PDE Operator Learning
Fraunhofer Institute for Integrated Systems and Device Technology IISB, Germany
Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators
May 11, 2026Xiaoxiao Lu, Ye Yuan, Jiahao ShiLatent Dynamics Modeling PDE Surrogate Modeling
School of AIA, Huazhong University of Science and Technology
HS-FNO: History-Space Fourier Neural Operator for Non-Markovian Partial Differential Equations
May 10, 2026Lennon J. ShikhmanFourier Neural Operator PDE Solving
College of Computing, Georgia Institute of Technology Atlanta, GA, United States
CATO: Charted Attention for Neural PDE Operators
May 9, 2026Chun-Wun Cheng, Sifan Wang, Carola-Bibiane Schönlieb +1PDE Operator Learning Attention Mechanisms
DAMTP University of Cambridge · Institute for Foundations of Data Science Yale University · Yau Mathematical Sciences Center Tsinghua University
M $^3$ : Reframing Training Measures for Discretized Physical Simulations
May 9, 2026Yuan Mei, Xingyu Song, Xiaowen Song +1Neural Surrogate Modeling Adaptive Sampling
1Zhejiang University · *Visiting student at The University of Tokyo from Zhejiang University. · 2The University of Tokyo
When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains
May 8, 2026Brandon Yee, Pairie Koh, Jack Rodriguez +1Neural PDE Solvers PDE Solving
Physics Lab, Yee Collins Research Group
QuadNorm: Resolution-Robust Normalization for Neural Operators
May 8, 2026Bum Jun Kim, Makoto Kawano, Yusuke Iwasawa +1Numerical Integration PDE Operator Learning
The University of Tokyo, Japan
Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation
May 4, 2026Nicole HaoFourier Neural Operator PDE Solving
Department of Mathematics, Cornell University
Isotropic Fourier Neural Operators
May 4, 2026Michael F. StaddonFourier Neural Operator Neural PDE Solvers
Independent Researcher
HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs
May 1, 2026Jinpai Zhao, Nishant Panda, Yen Ting Lin +3Neural Surrogate Modeling Neural Network Interpretability
Oden Institute University of Texas at Austin Austin, TX, USA · Information Sciences, CAI-3 Los Alamos National Laboratory Los Alamos, NM 87545, USA · Norwegian University of Life Sciences Ås, Norway / Simula Research Lab. Oslo, Norway
Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries
May 1, 2026Rodolphe Barlogis, Ferhat Tamssaouet, Quentin Falcoz +1PDE Solving PDE Operator Learning
PROMES-CNRS, Université Perpignan Via Domitia (UPVD), Perpignan, France · LAAS-CNRS, Université de Toulouse (UT), Toulouse, France
PILIR: Physics-Informed Local Implicit Representation
May 1, 2026Jianfeng Li, Feng Wang, Ke TangNeural PDE Solvers Implicit Neural Representations
School of Computer Science, Wuhan University · School of Computer Science and Technology, Southern University of Science and Technology
RETO: A Rotary-Enhanced Transformer Operator for High-Fidelity Prediction of Automotive Aerodynamics
Apr 30, 2026Bojun Zhang, Huiyu Yang, Yunpeng Wang +4Rotary Positional Embeddings Automotive Aerodynamics
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, 518055, China · Shenzhen Key Laboratory of Complex Aerospace Flows, Southern University of Science and Technology,Shenzhen 518055, China · Ningbo Key Laboratory of Advanced Manufacturing Simulation, Eastern Institute of Technology, Ningbo, 315200, China +1
Shearlet Neural Operators for Anisotropic-Shock-Dominated and Multi-scale parametric partial differential equations
Apr 28, 2026Fabio Pereira dos Santos, Julio de Castro Vargas Fernandes, Adriano Mauricio de Almeida CortesPDE Solving PDE Operator Learning
Coordination of Mathematical and Computational Methods National Laboratory of Scientific Computing Av. Getúlio Vargas, 333 - Quitandinha, Petrópolis - RJ, 25651-075 · Systems and Computational Engineering Program, COPPE/UFRJ, Federal University of Rio de Janeiro, P.O. Box 68542, Rio de Janeiro, RJ, 21941-909, Brasil
Adaptive-Distribution Randomized Neural Networks for PDEs: A Low-Dimensional Distribution-Learning Framework
Apr 27, 2026You Yang, Fei WangNeural PDE Solvers PDE Operator Learning
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

May 22, 2026J/K move · Enter open · S save

Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach

Qian Zhang, George Em Karniadakis

Division of Applied Mathematics, Brown University

PDE Operator Learning Operator Learning Computational Fluid Dynamics

Open paper AskPDF arXiv

Reconstructing flow fields from sparse measurements is a fundamental problem in fluid mechanics with broad implications for modeling, control, and design. In this work, we propose a novel operator learning framework that leverages the architecture of language models to perform flow reconstruction in a mesh-free manner. We reformulate flow field reconstruction as a sequence-to-sequence learning task, where sparse measurements are treated as context and unobserved locations as queries. Our model learns to reconstruct the full flow field from sparse inputs, effectively capturing spatial correlations and long-range dependencies. We evaluate the proposed approach on four benchmark datasets: (1) two-dimensional vortex street simulations, (2) daily average temperature data across the contiguous United States, (3) three-dimensional blood flow simulations based on dissipative particle dynamics, and (4) three-dimensional turbulent jet flow measurements obtained via particle tracking velocimetry. Across all cases, our method demonstrates competitive reconstruction accuracy, even with highly incomplete data (less than 10% observed), and achieves efficient performance. The results highlight the potential of language models as robust and scalable tools for scientific data reconstruction, and suggest a promising direction toward the development of foundation models for scientific and engineering applications.