PDE Operator Learning

PDE: Partial Differential Equation

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24 papers in the last four weeks, up 140% on the four weeks before. 0.2% of all new papers.

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LLT: Local Linear Transformer for PDE Operator Learning
Jul 4, 2026Oded Ovadia, Eli TurkelPDE Surrogate Modeling Transformer Attention
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
LiNO: Lifting based multiresolution neural operator
Jul 2, 2026Himanshu Pandey, Subham Patel, Ratikanta BeheraWavelet Methods PDE Operator Learning
Department of Computational and Data Sciences, Indian Institute of Science, Bangalore 560012, India · IISc Mathematics Initiative, Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India
GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems
Jul 1, 2026Meenakshi Krishnan, Pranav Pulijala, Ke Chen +2PDE Solving Image Reconstruction
University of Maryland, College Park · University of Delaware, Newark
From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators
Jul 1, 2026Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan WaniorekDynamical Systems Fourier Neural Operator
University of Chicago
McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation
Jun 29, 2026Jiwei Jia, Xinliang Liu, Juntao Wang +1Neural PDE Solvers PDE Solving
School of Mathematics, Jilin University, Changchun 130012, China · Shenzhen Loop Area Institute, Shenzhen 518038, China · Ocean University of China, Qingdao 266100, China +1
Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantification
Jun 28, 2026Zirui Deng, Jingbo Sun, Deyu Meng +1Uncertainty Quantification PDE Surrogate Modeling
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China
Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps
Jun 28, 2026Chenhui Zhu, Fei WangFourier Neural Operator PDE Solving
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P. R. China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.
Higher-Order Fourier Neural Operator: Explicit Mode Mixer for Nonlinear PDEs
Jun 26, 2026Alex Colagrande, Paul Caillon, Eva Feillet +1Fourier Neural Operator Neural PDE Solvers
Miles Team, LAMSADE, Université Paris Dauphine-PSL, Paris, France · Université Paris-Saclay, CNRS, LISN, 91400, Orsay, France · ESPCI PSL, Paris, France
Operator Learning for Cubic Nonlinear Schrödinger Equation on Periodic Domains
Jun 25, 2026Emmanuel E. Oguadimma, Victory C. Obieke, Xueying YuFourier Neural Operator PDE Solving
Neural operator-based digital twins for modeling amyloid- $β$ and tau propagation and treatment optimization in Alzheimer's disease
Jun 23, 2026Xiaofeng Xu, Tingting Dan, Zifan Zhou +3Alzheimer's Disease PDE Operator Learning
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA · Department of Psychiatry, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA · School of Electrical Engineering and Computer Science, Pennsylvania State University, University Park, PA 16802, USA +3
Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment
Jun 23, 2026Jason Sulskis, Sathya RaviFourier Neural Operator PDE Operator Learning
Electronic Systems Laboratory, Applied Embedded Systems Division Georgia Tech Research Institute · Department of Computer Science University of Illinois at Chicago
Neural Operator Processes for Probabilistic Operator Learning under Partial Observations
Jun 22, 2026Jose Miguel Lara-Rangel, Serge GuillasNeural Processes PDE Operator Learning
University College London London, United Kingdom
Physics-Informed Neural Operator for Speech Production Analysis
Jun 21, 2026Kazuya Yokota, Xinmeng Luan, Debasish Ray Mohapatra +2Speech Processing PDE Operator Learning
Department of Mechanical Engineering, Nagaoka University of Technology, Japan · Schulich School of Music, McGill University, Canada · Department of Electrical and Computer Engineering, University of British Columbia, Canada
TF-SNO: Time-Frequency Gated Spectral Neural Operators for Learning Non-Stationary Partial Differential Equations
Jun 19, 2026Yitian Zhou, Chaoning Zhang, Zhenzhen Huang +8PDE Solving PDE Operator Learning
University of Electronic Science and Technology of China Chengdu, China · Institute of Software, Chinese Academy of Sciences Beijing, China · Harbin Institute of Technology Shenzhen, China
ELADO: Elliptic PDE Assessment Datasets for Operator Learning
Jun 18, 2026Frank Ehebrecht, Toni Scharle, Martin AtzmuellerSynthetic Benchmark Generation PDE Operator Learning
Osnabrück University, Semantic Information Systems Group, Osnabrück, Germany · ROSEN Technology and Research Center GmbH, Lingen, Germany · German Research Center for Artificial Intelligence (DFKI), Osnabrück, Germany
A fast direct solver based neural network for solving PDEs
Jun 18, 2026Jashwanth Reddy Kadaru, Vaishnavi GujjulaPDE Solving PDE Operator Learning
Department of Computer Science & Engineering, International Institute of Information Technology Bangalore (IIIT-B), India · Department of Data Science and Artificial Intelligence, International Institute of Information Technology Bangalore (IIIT-B), India
Hierarchical Attention via Domain Decomposition
Jun 16, 2026Stephan Köhler, Oliver RheinbachLow-Rank Attention PDE Operator Learning
Faculty of Mathematics and Computer Science Technische Universität Bergakademie Freiberg, 09596 Freiberg, Germany
Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems
Jun 16, 2026Kuilin Qin, Lianfang Wang, Xu Sun +4PDE Surrogate Modeling PDE Operator Learning
Factorized Neural Operators Decompose Dynamic and Persistent Responses
Jun 15, 2026Hao Tang, Yuechen Duan, Jiongyu Zhu +3Mechanistic Interpretability PDE Operator Learning
School of Medicine, University of Dundee, United Kingdom. · School of Data Science, Fudan University, China. · School of Mathematical Sciences, Fudan University, China. +3
MR-GVNO: A Geometry-Aware Variational Physics-Informed Neural Operator for Mindlin-Reissner Plates on Irregular Domains
Jun 15, 2026Siqi Wang, Daobo Sun, Yizheng Wang +4PDE Surrogate Modeling PDE Operator Learning
Institute of Computational Mechanics × AI & College of Intelligent Robotics and Advanced Manufacturing, Fudan University, Shanghai 200433, China · School of Aerospace Engineering, Xiamen University, Xiamen 361102, China · Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China +2
Towards Data-Efficient Cross-Device Generalization of Grad-Shafranov Equilibria via Transfer Learning Neural Operator
Jun 13, 2026Jay Phil Yoo, William Howes, Yashika Ghai +3Transfer Learning PDE Operator Learning
Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering Department, University of Illinois Urbana-Champaign, Urbana, IL, USA · Fusion Energy Division, Oak Ridge National Lab, Oak Ridge, TN, USA · National Center for Supercomputing Applications, Urbana, IL, USA +2
Zero-shot generalization of transformer neural operators to larger domains
Jun 12, 2026Armand de Villeroché, Sibo Cheng, Vincent Le Guen +5Local Attention PDE Operator Learning
CEREA, ENPC, EDF R&D, Institut Polytechnique de Paris, Île-de-France, France · SINCLAIR AI Laboratory, Saclay, Île-de-France, France · EDF R&D, Île-de-France, France +1
How Much Memory Do We Need? Adaptive Memory Gate for Neural Operators
Jun 11, 2026Jihyeon Hur, Yongseok Kwon, Min-Gi Jo +2Memory-Augmented Neural Networks Fourier Neural Operator
1KAIST, Republic of Korea.
Harness In-Context Operator Learning with Chain of Operators
Jun 10, 2026Minghui Yang, Ling Guo, Liu YangOOD Generalization PDE Operator Learning
Department of Mathematics, Shanghai Normal University, Shanghai, China · Department of Mathematics, National University of Singapore, Singapore
SirenFNO: Efficient and Full Frequency Learning of Fourier Neural Operators
Jun 9, 2026Pengqing Shi, Jie Yin, Stephen Tierney +1Fourier Neural Operator Spectral Bias
The University of Sydney, Australia
Topological Neural Operators
Jun 8, 2026Lennart Bastian, Samuel Leventhal, Mustafa Hajij +1PDE Operator Learning Neural Operators
Imperial College London, UK · University of San Francisco, USA
Operator learning for solving Fokker-Planck equations with various initial conditions
Jun 8, 2026Li Zeng, Xiaoliang Wan, Yaobin Wang +2PDE Solving PDE Operator Learning
School of Mathematics and Statistics, Fuzhou University, Fuzhou, China · Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge 70803, USA · Faculty of Science and Technology, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China. +2
Fourier Neural Operators with rank-1 lattice points and hyperbolic cross
Jun 7, 2026Jakob Dilen, Alexander Keller, Frances Y. Kuo +1Neural Network Generalization Fourier Neural Operator
On the training of physics-informed neural operators for solving parametric partial differential equations
Jun 4, 2026Nanxi Chen, Chuanjie Cui, Airong Chen +2Neural Network Optimization Gradient Interference
College of Civil Engineering, Tongji University, Shanghai 200092 · Department of Engineering Science, University of Oxford, Oxford OX1 3PJ · Institute for Foundations of Data Science, Yale University, New Haven, CT 06520
Learning solution operators of PDEs with sparse approximation methods
Jun 4, 2026Sebastian Neumayer, Daniel Potts, Fabian TaubertSparse Recovery PDE Operator Learning
Sebastian Neumayer · Daniel Potts · Fabian Taubert

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LLT: Local Linear Transformer for PDE Operator Learning

Oded Ovadia, Eli Turkel

School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

PDE Surrogate Modeling Transformer Attention Linear Attention PDE Operator Learning Neural Operators

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Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations. Transformer-based neural operators are of particular interest, since attention can learn long-range dependencies in the computational domain. However, standard attention has two major limitations when applied to PDEs: it scales quadratically with the number of computational nodes, and it lacks an explicit bias toward local interactions. To address these issues, we introduce Local Linear Transformer (LLT) for PDE operator learning. The architecture combines linear global attention with local spatial mixing, and incorporates coordinate and geometry information. We evaluate LLT on several PDE problems, including elasticity, plasticity, airfoil flow, pipe flow, and Darcy flow. The reference data for these problems span finite-element, finite-volume, and finite-difference discretizations on structured and unstructured meshes. Compared with other neural-operator and transformer baselines from prior studies, LLT achieves competitive or lower relative

L_2

error across these problems. On matched structured discretizations, wall-clock time per training iteration is reduced by factors of 1.8 to 2.5 relative to Transolver. We also scale the approach and apply it to a three-dimensional car aerodynamics dataset with 32,186 unstructured mesh points per sample. Together, these results indicate that LLT provides an accurate and computationally efficient operator for PDE problems across discretizations, mesh types, and problem settings.

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