Neural Operators

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Neural operators learn maps from input fields, parameters, or geometries to output fields, often approximating the solution map of a partial differential equation. DeepONet uses branch and trunk networks, Fourier Neural Operators learn interactions in spectral space, and graph and transformer operators handle irregular meshes and complex geometries; physics-informed losses and structure-preserving designs add physical constraints. These models can make repeated simulation, forecasting, and inverse design much faster than running a numerical solver for every case. A central challenge is making predictions reliable across new resolutions, geometries, and physical regimes while preserving stability over long rollouts and matching classical solvers when high accuracy is required.

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313 papers52 papers in the last four weeks, up 24% on the four weeks before. 0.3% of all new papers.

Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs
May 31, 2026Lennon J. Shikhman, Shane GilbertiePDE Surrogate Modeling Scientific ML
College of Computing, Georgia Institute of Technology, Atlanta, 30332, GA, USA · Department of Aerospace, Physics and Space Sciences, Florida Institute of Technology, Melbourne, 32901, FL, USA
Limits of Resolution Equivariance in Fourier Neural Operators
May 30, 2026Alex Colagrande, Paul Caillon, Eva Feillet +1Fourier Neural Operator Equivariant Neural Networks
Miles Team, LAMSADE, Universit´e Paris Dauphine-PSL, Paris, France · Universit´e Paris-Saclay, CNRS, LISN, France · ESPCI PSL, Paris, France
Is Zero-Shot Super-Resolution Possible in Operator Learning?
May 29, 2026Unique Subedi, Ambuj TewariOperator Learning Statistical Learning Theory
Department of Statistics, University of Michigan
Functional Attention: From Pairwise Affinities to Functional Correspondences
May 29, 2026Jiefang Xiao, Maolin Gao, Simon Weber +2PDE Operator Learning Attention Mechanisms
Technical University of Munich, Germany · Munich Center for Machine Learning (MCML), Germany · PIXL, Department of Computer Science, University of Oxford, United Kingdom +1
LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition
May 29, 2026Jeongun Ha, Sanga Yoon, Donghun LeeDynamical Systems Neural PDE Solvers
Department of Mathematics, Korea University, 145 Anam-ro Seongbuk-gu Seoul, Republic of Korea.
Neural Operator-Based Surrogate Model for CFD:Helical Coil Steam Generator in Small Modular Reactor
May 28, 2026Minseo Lee, Seongmin Oh, Chaehyeon Song +5PDE Surrogate Modeling Reduced-Order Modeling
Division of Advanced Nuclear Engineering, POSTECH, 772026 Cheongam-Ro, Pohang-si, 37673, Gyeongbuk-do, Republic of Korea · Department of Quantum System Engineering, Jeonbuk National University, 567 Baekje-daero, Jeonju-si, 54896, Jeollabuk-do, Republic of Korea · Department of Nuclear Engineering, Hanyang University, Wangsimni-ro, Seongdong-gu, Seongdong-gu, 04763, Seoul, Republic of Korea +1
High-Fidelity Industrial Crash Dynamics Prediction via Geometry-Aware Operator Learning with Memory-Efficient Low-Rank Attention
May 26, 2026Deepak Akhare, Mohammad Amin Nabian, Corey Adams +2Low-Rank Attention Surrogate Modeling
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN · NVIDIA, San Jose, CA · General Motors, Detroit, MI
APEX: Amplitude Anchors and Phase Priors for Target-Scarce Higher-Frequency Wave Prediction
May 26, 2026Yifan Sun, Lei Cheng, Sijie Chen +3PDE Surrogate Modeling Neural Operators
College of Information Science and Electronic Engineering, Zhejiang University
Accelerating Bayesian inverse design in computational fluid dynamics using neural operators
May 25, 2026Bipin Tiwari, Omer SanNeural Surrogate Modeling Bayesian Inverse Problems
Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville,2026 TN 37996, USA
Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decomposition
May 24, 2026Anuj Kumar, Josiah Bjorgaard, Nikolaos Bouklas +2Neural Surrogate Modeling PDE Surrogate Modeling
Pasteur Labs, NY, NY, USA · Cornell University, Ithaca, NY, USA · Institute for Simulation Intelligence, NY, NY, USA
IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
May 24, 2026Shan Zhong, George BirosNeural PDE Solvers PDE Solving
Oden Institute for Computational Science and Engineering, The University of Texas at Austin, 201 E 24th St, Austin, 78712, TX, USA · Walker Department of Mechanical Engineering, The University of Texas at Austin, 3012026 E Dean Keeton St, Austin, 78712, TX, USA
WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations
May 23, 2026Muhammad Abid, Arth Sojitra, Omer SanPDE Solving Multi-Scale Feature Fusion
Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville Knoxville, TN 37996, USA.
Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds
May 23, 2026Zherui Yang, Tao Du, Ligang LiuPoint Cloud Learning Geometric Representation Learning
University of Science and Technology of China, China · Tsinghua University, China · Shanghai Qi Zhi Institute, China +1
Discontinuous Galerkin Neural Operator for Pathology Defocus Deblurring
May 22, 2026Shaoqing Duan, Haofei Song, Xintian Mao +2Computational Pathology Image Deblurring
Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, Shanghai, China.
Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation
May 21, 2026Xiaotian Liu, Shuyuan Shang, Xiaopeng Wang +2Neural PDE Solvers Spectral Bias
1Dartmouth College · 2CUHK Shenzhen · 3Lawrence Berkeley National Lab
Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations
May 21, 2026Shuang Chen, Juncai He, Xue-Cheng TaiNeural Network Approximation Theory Neural Operators
Qiuzhen College, Tsinghua University, Haidian District, Beijing 100084, China · Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing 100084, China · Norwegian Research Center, Fantoftvegen 38, 5072 Bergen, Norway
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)
Data-Efficient Neural Operator Training via Physics-Based Active Learning
May 20, 2026Alicja Polanska, Lorenzo Zanisi, Vignesh Gopakumar +1Active Learning Physics-Informed ML
University College London · United Kingdom Atomic Energy Authority
Nonlocal operator learning for fMRI encoding and decoding tasks
May 19, 2026Andreas Kramer, Saugat Acharya, Alice Giola +1Neural Decoding Neuroimaging
Department of Computer Science, Idaho State University 921 S. 8th Ave Mail Stop 8060, Pocatello, ID 83209-8023 · Department of Mathematics and Statistics, Idaho State University Physical Science Complex, 921 S. 8th Ave., Stop 8085, Pocatello, ID 83209
Neural Operators for Design-Space Surrogate Modeling of Tendon-Actuated Continuum Robots
May 18, 2026Branden Frieden, James M. Ferguson, Alan Kuntz +1Tendon-Driven Actuation Fourier Neural Operator
The Robotics Center and the Kahlert School of Computing at the University of Utah, Salt Lake City, UT 84112, USA. · The Departments of Computer Science and Electrical and Computer Engineering at Vanderbilt University, Nashville, TN, 37203, USA.
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
Function graph transformers universally approximate operators between function spaces
May 18, 2026Takashi Furuya, David Mis, Ivan Dokmanić +2Transformer Neural Operator Transformer Attention
Doshisha University, RIKEN AIP · Rice University · University of Basel +2
Stability and Discretization Error of State Space Model Neural Operators
May 17, 2026Abderrahim Bendahi, Adrien Fradin, Johan Peralez +2Fourier Neural Operator Convergence Analysis
École polytechnique Paris, France · Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007 Villeurbanne, France · CNRS, UniversitéJulieLyonDigne1, INSA Lyon, LIRIS Lyon, France +1
Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations
May 16, 2026Phuoc-Toan Huynh, Richard Archibald, Feng BaoUncertainty Quantification Stochastic PDEs
Department of Mathematics, Florida State University, Tallahassee, FL, USA · Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA
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
Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization
May 15, 2026Kai HidajatUncertainty Quantification Stochastic PDEs
Department of Applied Mathematics University of Washington Seattle, WA, USA
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
Discovering Physical Directions in Weight Space: Composing Neural PDE Experts
May 14, 2026Pengkai Wang, Pengwei Liu, Yuanyi Wang +7PDE Surrogate Modeling Model Merging
1The Hong Kong Polytechnic University · 2Zhejiang University · 3Tsinghua University +1
Topology-Preserving Neural Operator Learning via Hodge Decomposition
May 13, 2026Dongzhe Zheng, Tao Zhong, Christine Allen-BlanchetteHodge Decomposition PDE Operator Learning
Princeton University
U-HNO: A U-shaped Hybrid Neural Operator with Sparse-Point Adaptive Routing for Non-stationary PDE Dynamics
May 13, 2026Yingzhe Ma, Xiao Yang, Yuxin Xie +2Fourier Neural Operator PDE Surrogate Modeling
University of Electronic Science and Technology of China · Peking University

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Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs

Lennon J. Shikhman, Shane Gilbertie

College of Computing, Georgia Institute of Technology, Atlanta, 30332, GA, USA · Department of Aerospace, Physics and Space Sciences, Florida Institute of Technology, Melbourne, 32901, FL, USA

PDE Surrogate Modeling Scientific ML Magnetohydrodynamics Neural Operators Computational Fluid Dynamics

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Neural operators provide fast surrogate models for PDE simulations, but standard architectures often treat geometry and discretization as secondary to field data. Physical states are usually represented as grid-channel stacks, even when different quantities naturally belong on vertices, edges, faces, cells, boundaries, or interfaces and must satisfy compatibility constraints. We propose Cellular Sheaf Neural Operators, a discretization-aware framework for structure-preserving neural PDE surrogates. The method represents PDE states on oriented cell complexes, couples local feature spaces through learned restriction maps, and uses incidence/Hodge-informed message passing to follow computational geometry. Learned update heads pass through coboundary or flux maps, allowing selected constraints to arise from cell-complex structure rather than only from loss penalties. For magnetohydrodynamics, this yields face-based magnetic-flux updates driven by edge electromotive fields and finite-volume-style fluid updates driven by learned face fluxes and cell sources. On turbulent MHD and fusion-equilibrium surrogate tasks, the method improves structure-sensitive diagnostics, including rollout behavior, divergence control, spectral error, and equilibrium-regression accuracy. These results indicate that cellular-sheaf structure is a useful inductive bias for neural PDE surrogates in constrained multiphysics systems.