mlsanityData to 7 Oct 2026

Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

arXiv 2207.05209 ↗First posted 11 Jul 2022cs.LG cs.NA math.NAcited by 416

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Abstract (arXiv)

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is 105 times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

Key facts. Cited by 416 papers in the map (data through 7 Oct 2026). Builds on 15 of its 32 references that are in the map. First posted on arXiv 11 Jul 2022.

Builds on

15 of 32 references.

References · 15

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  4. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  5. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  6. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  7. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  8. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  9. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  10. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  11. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  12. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
3 more
  1. 2019Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networkscited by 144
  2. 2021Seismic wave propagation and inversion with Neural Operatorscited by 51
  3. 2022Physical Design using Differentiable Learned Simulatorscited by 37

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 8; 2023: 49; 2024: 91; 2025: 115; 2026: 153.

416 papers.

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  1. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 321
  2. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 299
  3. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 293
  4. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  5. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  6. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  7. 2024Poseidon: Efficient Foundation Models for PDEscited by 186
  8. 2023Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behaviorcited by 128
  9. 2022Clifford Neural Layers for PDE Modelingcited by 91
  10. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  11. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  12. 2023Solving High-Dimensional PDEs with Latent Spectral Modelscited by 84
38 more
  1. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 80
  2. 2023A Mathematical Guide to Operator Learningcited by 74
  3. 2022Continuous PDE Dynamics Forecasting with Implicit Neural Representationscited by 71
  4. 2023Separable Physics-Informed Neural Networkscited by 64
  5. 2023Improved Operator Learning by Orthogonal Attentioncited by 58
  6. 2023ACE: A fast, skillful learned global atmospheric model for climate predictioncited by 53
  7. 2024Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolationcited by 45
  8. 2022Efficient Learning of Mesh-Based Physical Simulation with BSMS-GNNcited by 43
  9. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  10. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 42
  11. 2023MgNO: Efficient Parameterization of Linear Operators via Multigridcited by 40
  12. 2023Geometric Clifford Algebra Networkscited by 37
  13. 2025CodePDE: An Inference Framework for LLM-driven PDE Solver Generationcited by 37
  14. 2023Domain Agnostic Fourier Neural Operatorscited by 36
  15. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  16. 2024PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamicscited by 33
  17. 2024Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solverscited by 31
  18. 2023Koopman neural operator as a mesh-free solver of non-linear partial differential equationscited by 29
  19. 2024Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networkscited by 28
  20. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  21. 2023Deep Operator Network Approximation Rates for Lipschitz Operatorscited by 24
  22. 2024HAMLET: Graph Transformer Neural Operator for Partial Differential Equationscited by 24
  23. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 24
  24. 2024BENO: Boundary-embedded Neural Operators for Elliptic PDEscited by 22
  25. 2025Erwin: A Tree-based Hierarchical Transformer for Large-scale Physical Systemscited by 21
  26. 2023Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domainscited by 20
  27. 2024Discretization Error of Fourier Neural Operatorscited by 20
  28. 2025GeoTransolver: Learning Physics on Irregular Domains Using Multi-scale Geometry Aware Physics Attention Transformercited by 20
  29. 2026Transolver-3: Scaling Up Transformer Solvers to Industrial-Scale Geometriescited by 20
  30. 2023Improved generalization with deep neural operators for engineering systems: Path towards digital twincited by 19
  31. 2023Deep Equilibrium Based Neural Operators for Steady-State PDEscited by 19
  32. 2024Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responsescited by 18
  33. 2024Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problemscited by 18
  34. 2024Physics-Informed Geometry-Aware Neural Operatorcited by 18
  35. 2024PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systemscited by 18
  36. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
  37. 2024Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solvingcited by 17
  38. 2023Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEscited by 16

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