# Non-equispaced Fourier Neural Solvers for PDEs

arXiv: https://arxiv.org/abs/2212.04689 · HTML: https://mlsanity.com/paper/2212.04689/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 52 references that are in the map.
- First posted on arXiv 9 Dec 2022.

## Builds on (most cited first, 30 of 38)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Twins: Revisiting the Design of Spatial Attention in Vision Transformers](https://mlsanity.com/paper/2104.13840/) (2021)
- [Adaptive Graph Convolutional Recurrent Network for Traffic Forecasting](https://mlsanity.com/paper/2007.02842/) (2020)
- [Intriguing Properties of Vision Transformers](https://mlsanity.com/paper/2105.10497/) (2021)
- [Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depth](https://mlsanity.com/paper/2103.03404/) (2021)
- [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)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [How Do Vision Transformers Work?](https://mlsanity.com/paper/2202.06709/) (2022)
- [Structured Sequence Modeling with Graph Convolutional Recurrent Networks](https://mlsanity.com/paper/1612.07659/) (2016)
- [Diffusion-Convolutional Neural Networks](https://mlsanity.com/paper/1511.02136/) (2015)
- [Global Filter Networks for Image Classification](https://mlsanity.com/paper/2107.00645/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Towards Robust Vision Transformer](https://mlsanity.com/paper/2105.07926/) (2021)
- [Are Convolutional Neural Networks or Transformers more like human vision?](https://mlsanity.com/paper/2105.07197/) (2021)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)

## Built on by (most cited first, 8 of 8)

- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning](https://mlsanity.com/paper/2405.09285/) (2024)
- [Re-Dock: Towards Flexible and Realistic Molecular Docking with Diffusion Bridge](https://mlsanity.com/paper/2402.11459/) (2024)
- [Protein 3D Graph Structure Learning for Robust Structure-based Protein Property Prediction](https://mlsanity.com/paper/2310.11466/) (2023)
- [Generating synthetic data for neural operators](https://mlsanity.com/paper/2401.02398/) (2024)
- [Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids](https://mlsanity.com/paper/2402.08187/) (2024)
- [Tokenizing Electron Cloud in Protein-Ligand Interaction Learning](https://mlsanity.com/paper/2505.19014/) (2025)

## Abstract (arXiv)

Solving partial differential equations is difficult. Recently proposed neural resolution-invariant models, despite their effectiveness and efficiency, usually require equispaced spatial points of data. However, sampling in spatial domain is sometimes inevitably non-equispaced in real-world systems, limiting their applicability. In this paper, we propose a Non-equispaced Fourier PDE Solver (textscNFS) with adaptive interpolation on resampled equispaced points and a variant of Fourier Neural Operators as its components. Experimental results on complex PDEs demonstrate its advantages in accuracy and efficiency. Compared with the spatially-equispaced benchmark methods, it achieves superior performance with 42.85% improvements on MAE, and is able to handle non-equispaced data with a tiny loss of accuracy. Besides, to our best knowledge, textscNFS is the first ML-based method with mesh invariant inference ability to successfully model turbulent flows in non-equispaced scenarios, with a minor deviation of the error on unseen spatial points.