mlsanityData to 14 Oct 2026

Group Equivariant Fourier Neural Operators for Partial Differential Equations

arXiv 2306.05697 ↗First posted 9 Jun 2023cs.LG cs.NA math.NAcited by 45

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Abstract (arXiv)

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting G-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Key facts. Cited by 45 papers in the map (data through 14 Oct 2026). Builds on 37 of its 49 references that are in the map. First posted on arXiv 9 Jun 2023.

Builds on

37 of 49 references.

References · 37

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  5. 2016Group Equivariant Convolutional Networkscited by 1,165
  6. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 776
  7. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 746
  8. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  9. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  10. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  11. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  12. 2021Machine learning accelerated computational fluid dynamicscited by 471
25 more
  1. 2018Spherical CNNscited by 456
  2. 2019General E(2)-Equivariant Steerable CNNscited by 446
  3. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
  4. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
  5. 2022Message Passing Neural PDE Solverscited by 386
  6. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 354
  7. 2016Steerable CNNscited by 305
  8. 2016Harmonic Networks: Deep Translation and Rotation Equivariancecited by 295
  9. 2021Factorized Fourier Neural Operatorscited by 249
  10. 2021Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformerscited by 248
  11. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 205
  12. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 204
  13. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 197
  14. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 126
  15. 2019Deep Scale-spaces: Equivariance Over Scalecited by 114
  16. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 97
  17. 2022AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutionscited by 82
  18. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 77
  19. 2020A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applicationscited by 53
  20. 2021Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamicscited by 47
  21. 2021Learning Dissipative Dynamics in Chaotic Systemscited by 42
  22. 2021Solving PDE-constrained Control Problems Using Operator Learningcited by 42
  23. 2022MAgNet: Mesh Agnostic Neural PDE Solvercited by 28
  24. 2022Transform Once: Efficient Operator Learning in Frequency Domaincited by 20
  25. 2021Rotation Equivariant Operators for Machine Learning on Scalar and Vector Fieldscited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 5; 2024: 16; 2025: 11; 2026: 13.

45 papers.

Sort:

  1. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 28
  2. 2024Clifford-Steerable Convolutional Neural Networkscited by 25
  3. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 25
  4. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 24
  5. 2024BENO: Boundary-embedded Neural Operators for Elliptic PDEscited by 24
  6. 2023Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolutioncited by 20
  7. 2024Equivariance via Minimal Frame Averaging for More Symmetries and Efficiencycited by 19
  8. 2024Space-Time Continuous PDE Forecasting using Equivariant Neural Fieldscited by 18
  9. 2023BubbleML: A Multi-Physics Dataset and Benchmarks for Machine Learningcited by 17
  10. 2024Diffeomorphism Neural Operator for various domains and parameters of partial differential equationscited by 15
  11. 2024SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equationscited by 14
  12. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
33 more
  1. 2023Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspectivecited by 11
  2. 2025Enhancing Fourier Neural Operators with Local Spatial Featurescited by 10
  3. 2024Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular gridscited by 7
  4. 2025Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogatescited by 6
  5. 2025EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scalecited by 6
  6. 2025FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulationcited by 5
  7. 2024Spectral Convolutional Conditional Neural Processescited by 4
  8. 2025Equivariant Neural Networks for General Linear Symmetries on Lie Algebrascited by 4
  9. 2024Solving Differential Equations with Constrained Learningcited by 3
  10. 2025A Two-Phase Deep Learning Framework for Adaptive Time-Stepping in High-Speed Flow Modelingcited by 3
  11. 2026Discovering Physical Directions in Weight Space: Composing Neural PDE Expertscited by 3
  12. 2024Fourier Neural Operators for Learning Dynamics in Quantum Spin Systemscited by 2
  13. 2024Current Symmetry Group Equivariant Convolution Frameworks for Representation Learningcited by 2
  14. 2024A Geometry-Aware Message Passing Neural Network for Modeling Aerodynamics over Airfoilscited by 2
  15. 2025Hierarchical-embedding autoencoder with a predictor (HEAP) as efficient architecture for learning long-term evolution of complex multi-scale physical systemscited by 2
  16. 2024G-RepsNet: A Fast and General Construction of Equivariant Networks for Arbitrary Matrix Groupscited by 1
  17. 2024Physics-embedded Fourier Neural Network for Partial Differential Equationscited by 1
  18. 2025Can Data-Driven Dynamics Reveal Hidden Physics? There Is A Need for Interpretable Neural Operatorscited by 1
  19. 2026Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistencycited by 1
  20. 2025Fourier-Invertible Neural Encoder (FINE) for Homogeneous Flowscited by 0
  21. 2025Implicit Augmentation from Distributional Symmetry in Turbulence Super-Resolutioncited by 0
  22. 2025Conditional Clifford-Steerable CNNs for PDE Modelingcited by 0
  23. 2026Linear-Nonlinear Fusion Neural Operator for Partial Differential Equationscited by 0
  24. 2026Isotropic Fourier Neural Operatorscited by 0
  25. 2026Deep Wave Network for Modeling Multi-Scale Physical Dynamicscited by 0
  26. 2026PACE-FNO: Physics-Aligned Canonical Equivariance for Fourier Neural Operatorscited by 0
  27. 2026EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEscited by 0
  28. 2026Distillation of Foundation Models for Time-dependent PDEscited by 0
  29. 2026Where to Compute and How to Interact: Operator-Readable Adaptation with Gauge-Aware Transportcited by 0
  30. 2026Scale-Split Neural Operator for Memory- and Data-Efficient 3D Turbulence Predictioncited by 0
  31. 2026Green-Routed Neural Operators: Physics Determines Where the Network Readscited by 0
  32. 2026Poisson-GENERIC Neural Operators: Exact Metriplectic Structure in Function Space via Casimir Entropiescited by 0
  33. 2026A Bregman Proximal Viewpoint on Neural Operatorscited by 0