mlsanityData to 7 Oct 2026

Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains

arXiv 2305.19663 ↗First posted 31 May 2023cs.LG cs.NA math.NAcited by 20

Not on the map. Why?

Abstract (arXiv)

The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. Leveraging the observation that a limited set of Fourier (Spectral) modes suffice to provide the required expressivity of a neural operator, we propose a simple method, based on the efficient direct evaluation of the underlying spectral transformation, to extend neural operators to arbitrary domains. An efficient implementation* of such direct spectral evaluations is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows us to extend neural operators to arbitrary point distributions with significant gains in training speed over baselines while retaining or improving the accuracy of Fourier neural operators (FNOs) and related neural operators.

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 22 of its 62 references that are in the map. First posted on arXiv 31 May 2023.

Builds on

22 of 62 references.

References · 22

  1. 2020An Image is Worth 16x16 Words: Transformers for Image Recognition at Scalecited by 29,915
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  4. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  5. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  6. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  7. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  8. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  9. 2022Message Passing Neural PDE Solverscited by 368
  10. 2021Choose a Transformer: Fourier or Galerkincited by 333
  11. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  12. 2021Factorized Fourier Neural Operatorscited by 237
10 more
  1. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  2. 2022Learning Operators with Coupled Attentioncited by 88
  3. 2023Solving High-Dimensional PDEs with Latent Spectral Modelscited by 84
  4. 2019Deep learning observables in computational fluid dynamicscited by 38
  5. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 29
  6. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 26
  7. 2022Variable-Input Deep Operator Networkscited by 24
  8. 2023Multi-Scale Message Passing Neural PDE Solverscited by 12
  9. 2021cuFINUFFT: a load-balanced GPU library for general-purpose nonuniform FFTscited by 11
  10. 2020Learning Incompressible Fluid Dynamics from Scratch -- Towards Fast, Differentiable Fluid Models that Generalizecited by 6

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 8; 2026: 9.

20 papers.

Sort:

By area:

  1. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  2. 2024Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Spacecited by 10
  3. 2024FUSE: Fast Unified Simulation and Estimation for PDEscited by 6
  4. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  5. 2026Simple yet Effective: Low-Rank Spatial Attention for Neural Operatorscited by 4
  6. 2025FNOPE: Simulation-based inference on function spaces with Fourier Neural Operatorscited by 3
  7. 2024Fourier Neural Operators for Learning Dynamics in Quantum Spin Systemscited by 2
  8. 2026Topological Neural Operatorscited by 2
  9. 2025Invertible Koopman neural operator for data-driven modeling of partial differential equationscited by 1
  10. 2025Learning cardiac activation and repolarization times with operator learningcited by 1
  11. 2026GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domainscited by 1
  12. 2025SUPRA: Subspace Parameterized Attention for Neural Operator on General Domainscited by 0
8 more
  1. 2025Learning Mappings in Mesh-based Simulationscited by 0
  2. 2025Principled Operator Learning in Ocean Dynamics: The Role of Temporal Structurecited by 0
  3. 2025Deep Gaussian Processes for Functional Mapscited by 0
  4. 2026Late Fusion Neural Operators for Extrapolation Across Parameter Space in Partial Differential Equationscited by 0
  5. 2026Adaptive Mamba Neural Operatorscited by 0
  6. 2026Euclidean Fourier Neural Operatorscited by 0
  7. 2026Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operatorscited by 0
  8. 2026Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solvercited by 0