# Discretization Error of Fourier Neural Operators

arXiv: https://arxiv.org/abs/2405.02221 · HTML: https://mlsanity.com/paper/2405.02221/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 3 May 2024.

## Builds on (most cited first, 10 of 10)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Incremental Spatial and Spectral Learning of Neural Operators for Solving Large-Scale PDEs](https://mlsanity.com/paper/2211.15188/) (2022)
- [Learning Homogenization for Elliptic Operators](https://mlsanity.com/paper/2306.12006/) (2023)

## Built on by (most cited first, 22 of 22)

- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Performance of Neural and Polynomial Operator Surrogates](https://mlsanity.com/paper/2604.00689/) (2026)
- [Diagnosing Failure Modes of Neural Operators Across Diverse PDE Families](https://mlsanity.com/paper/2601.11428/) (2026)
- [Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators](https://mlsanity.com/paper/2408.09004/) (2024)
- [Multi-Level Monte Carlo Training of Neural Operators](https://mlsanity.com/paper/2505.12940/) (2025)
- [Split Conformal Prediction in the Function Space with Neural Operators](https://mlsanity.com/paper/2509.04623/) (2025)
- [Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems](https://mlsanity.com/paper/2409.03302/) (2024)
- [Can neural operators always be continuously discretized?](https://mlsanity.com/paper/2412.03393/) (2024)
- [RONOM: Reduced-Order Neural Operator Modeling](https://mlsanity.com/paper/2507.12814/) (2025)
- [Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictors](https://mlsanity.com/paper/2509.26443/) (2025)
- [Delay-adaptive Control of Nonlinear Systems with Approximate Neural Operator Predictors](https://mlsanity.com/paper/2508.20367/) (2025)
- [Can Data-Driven Dynamics Reveal Hidden Physics? There Is A Need for Interpretable Neural Operators](https://mlsanity.com/paper/2510.02683/) (2025)
- [Breaking Scale Anchoring: Frequency Representation Learning for Accurate High-Resolution Inference from Low-Resolution Training](https://mlsanity.com/paper/2512.05132/) (2025)
- [Predictor-Based Output-Feedback Control of Linear Systems with Time-Varying Input and Measurement Delays via Neural-Approximated Prediction Horizons](https://mlsanity.com/paper/2603.29117/) (2026)
- [QuadNorm: Resolution-Robust Normalization for Neural Operators](https://mlsanity.com/paper/2605.07375/) (2026)
- [Stability and Discretization Error of State Space Model Neural Operators](https://mlsanity.com/paper/2605.18905/) (2026)
- [Fourier Neural Operators with rank-1 lattice points and hyperbolic cross](https://mlsanity.com/paper/2606.08871/) (2026)
- [From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators](https://mlsanity.com/paper/2607.00320/) (2026)
- [A Variational Optimal Transport Operator on Incompressible Flow](https://mlsanity.com/paper/2609.13729/) (2026)

## Abstract (arXiv)

Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data. The Fourier Neural Operator (FNO) is one of the main model architectures used for operator learning. The FNO combines linear and nonlinear operations in physical space with linear operations in Fourier space, leading to a parameterized map acting between function spaces. Although in definition, FNOs are objects in continuous space and perform convolutions on a continuum, their implementation is a discretized object performing computations on a grid, allowing efficient implementation via the FFT. Thus, there is a discretization error between the continuum FNO definition and the discretized object used in practice that is separate from other previously analyzed sources of model error. We examine this discretization error here and obtain algebraic rates of convergence in terms of the grid resolution as a function of the input regularity. Numerical experiments that validate the theory and describe model stability are performed. In addition, an algorithm is presented that leverages the discretization error and model error decomposition to optimize computational training time.