PDE Operator Learning

PDE: Partial Differential Equation

Follow this topic Ask about this topic

Momentum

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

Jul 13Week of Sep 28

Latest papers 148

All topics

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 +1PDE Operator Learning Neural Operators
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 PDE Operator Learning
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.
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 Learning Physics-Informed Neural Operators
Department of Mechanical Engineering, IIT Kharagpur, Kharagpur, India · Department of Computer Science & Engineering, IIT Kharagpur, Kharagpur, India
Neural Shape Operator Surrogates -- Expression Rate Bounds
Apr 20, 2026Helmut Harbrecht, Christoph SchwabPDE Operator Learning Neural Network Approximation Theory
Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches
Apr 18, 2026Yixuan WangPDE Solving Kolmogorov-Arnold Networks
CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California · [math.NA]
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 Modeling PDE Surrogate Modeling
Artificial Intelligence Group, Wageningen University and Research, Netherlands · Mathematical and Statistical Methods, Wageningen University and Research, Netherlands
Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model
Mar 18, 2026Luca PellegriniFourier Neural Operator PDE Operator Learning
Department of Mathematics, University of Pavia, Pavia, Italy · Euler Institute, Faculty of Informatics, Universit`a della Svizzera italiana, Lugano, Switzerland
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 Solvers Algorithmic Stability
University of Hiroshima
Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows
Feb 17, 2026Ramansh Sharma, Matthew Lowery, Houman Owhadi +1PDE Operator Learning Operator Learning
Kahlert School of Computing University of Utah UT, USA · Department of Computing and Mathematical Sciences California Institute of Technology CA, USA
Transolver-3: Scaling Up Transformer Solvers to Industrial-Scale Geometries
Feb 4, 2026Hang Zhou, Haixu Wu, Haonan Shangguan +4Efficient Neural Network Inference Neural PDE Solvers
School of Software, BNRist, Tsinghua University, China.
SFO: Learning PDE Operators via Spectral Filtering
Jan 23, 2026Noam Koren, Rafael Moschopoulos, Kira Radinsky +1PDE Surrogate Modeling PDE Operator Learning
Department of Computer Science Technion - Israel Institute of Technology · Department of Computer Science Princeton University
PGOT: A Physics-Geometry Operator Transformer for Complex PDEs
Dec 29, 2025Zhuo Zhang, Xi Yang, Ying Miao +5Neural PDE Solvers PDE Surrogate Modeling
National University of Defense Technology · Peking University · National University of Singapore +2
Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation
Dec 9, 2025Muhammad Abid, Omer SanPDE Surrogate Modeling PDE Operator Learning
Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.
Solver-Integrated Adversarial Attacking and Training of Neural Operators
Oct 21, 2025Yifei SunAdversarial Training Adversarial Attacks
RheOFormer: A generative transformer model for simulation of complex fluids and flows
Oct 1, 2025Maedeh Saberi, Amir Barati Farimani, Safa JamaliNeural Surrogate Modeling Transformer
Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA · Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Discretization-independent operator learning for partial differential equations
Jul 9, 2025Jacob Hauck, Yanzhi ZhangPDE Operator Learning Neural Network Approximation Theory
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409
DPG loss functions for learning parameter-to-solution maps by neural networks
Jun 23, 2025Pablo Cortés Castillo, Wolfgang Dahmen, Jay GopalakrishnanPDE Solving PDE Surrogate Modeling
From Theory to Application: A Practical Introduction to Neural Operators in Scientific Computing
Mar 7, 2025Prashant K. JhaNeural Surrogate Modeling Fourier Neural Operator
Department of Mechanical Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA.
Active Learning with Bayesian Multi-Fidelity Laplace Neural Operators for Oscillatory Parametric PDEs
Feb 1, 2025Bongseok Kim, Haoyang Zheng, Michael Penwarden +1Dynamical Systems Surrogate 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.
Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations
Dec 15, 2021Carlo Marcati, Christoph SchwabPDE Solving PDE Operator Learning
Seminar for Applied Mathematics, ETH Zurich, Switzerland

May 8, 2026, cs.LGJ/K move · Enter open · S save

When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains

Brandon Yee, Pairie Koh, Jack Rodriguez, Mihir Tekal

Physics Lab, Yee Collins Research Group

Neural PDE Solvers PDE Solving PDE Operator Learning Physics-Informed ML

Open paper AskPDF arXiv

We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-domain neural operators. We introduce the \textbf{Multi-Scale Attention Transformer} (\msat{}), a deep learning architecture that encodes spatiotemporal solution histories as token sequences and trains end-to-end via a composite supervised objective with optional physics-informed regularization terms. We conduct a comprehensive empirical evaluation against nine baselines -- including physics-informed neural networks (PINNs), neural operators (FNO, DeepONet, GNOT), and state-space models (Mamba-NO) -- across five benchmark problems from the PINNacle suite, using identical train/test splits and reference data for all methods. \msat{} achieves state-of-the-art generalization on complex geometry problems (

L^2_\mathrm{rel} = 0.0101

on Heat2D-CG, a

3.7\times

improvement over FNO) at

34\,\mathrm{s}

total inference vs.\

120{,}812\,\mathrm{s}

for Mamba-NO. Ablation studies over the physics regularization component reveal a precise inductive bias tradeoff: physics priors reduce test error on diffusion-dominated problems but degrade generalization on chaotic and recirculating-flow regimes, directly characterizing the prior misspecification boundary. Approximation error bounds as a function of domain boundary complexity

κ

provide a theoretical basis for these empirical findings and a principled rule for architecture selection.

Momentum

Related topics

Latest papers 148