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

arXiv: https://arxiv.org/abs/2305.19663 · HTML: https://mlsanity.com/paper/2305.19663/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 62 references that are in the map.
- First posted on arXiv 31 May 2023.

## Builds on (most cited first, 22 of 22)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities](https://mlsanity.com/paper/2210.01074/) (2022)
- [Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks](https://mlsanity.com/paper/2008.05730/) (2020)
- [Variable-Input Deep Operator Networks](https://mlsanity.com/paper/2205.11404/) (2022)
- [Multi-Scale Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2302.03580/) (2023)
- [cuFINUFFT: a load-balanced GPU library for general-purpose nonuniform FFTs](https://mlsanity.com/paper/2102.08463/) (2021)
- [Learning Incompressible Fluid Dynamics from Scratch -- Towards Fast, Differentiable Fluid Models that Generalize](https://mlsanity.com/paper/2006.08762/) (2020)

## Built on by (most cited first, 21 of 21)

- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space](https://mlsanity.com/paper/2410.11382/) (2024)
- [FUSE: Fast Unified Simulation and Estimation for PDEs](https://mlsanity.com/paper/2405.14558/) (2024)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators](https://mlsanity.com/paper/2505.22573/) (2025)
- [Simple yet Effective: Low-Rank Spatial Attention for Neural Operators](https://mlsanity.com/paper/2604.03582/) (2026)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems](https://mlsanity.com/paper/2409.03302/) (2024)
- [Invertible Koopman neural operator for data-driven modeling of partial differential equations](https://mlsanity.com/paper/2503.19717/) (2025)
- [Learning cardiac activation and repolarization times with operator learning](https://mlsanity.com/paper/2505.08631/) (2025)
- [GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains](https://mlsanity.com/paper/2605.03497/) (2026)
- [Adaptive Mamba Neural Operators](https://mlsanity.com/paper/2607.18043/) (2026)
- [SUPRA: Subspace Parameterized Attention for Neural Operator on General Domains](https://mlsanity.com/paper/2504.15897/) (2025)
- [Learning Mappings in Mesh-based Simulations](https://mlsanity.com/paper/2506.12652/) (2025)
- [Principled Operator Learning in Ocean Dynamics: The Role of Temporal Structure](https://mlsanity.com/paper/2510.09792/) (2025)
- [Deep Gaussian Processes for Functional Maps](https://mlsanity.com/paper/2510.22068/) (2025)
- [Late Fusion Neural Operators for Extrapolation Across Parameter Space in Partial Differential Equations](https://mlsanity.com/paper/2604.16721/) (2026)
- [Euclidean Fourier Neural Operators](https://mlsanity.com/paper/2608.28425/) (2026)
- [Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators](https://mlsanity.com/paper/2609.05860/) (2026)
- [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026)
- [METRO: Metric-Enhanced Token Routing Operator](https://mlsanity.com/paper/2610.05100/) (2026)

## 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.