# Bounding the Rademacher Complexity of Fourier neural operators

arXiv: https://arxiv.org/abs/2209.05150 · HTML: https://mlsanity.com/paper/2209.05150/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 31 references that are in the map.
- First posted on arXiv 12 Sep 2022.

## Builds on (most cited first, 20 of 20)

- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [Relative Flatness and Generalization](https://mlsanity.com/paper/2001.00939/) (2020)
- [Generalization bounds for deep convolutional neural networks](https://mlsanity.com/paper/1905.12600/) (2019)
- [Generalization Error in Deep Learning](https://mlsanity.com/paper/1808.01174/) (2018)
- [On Generalization Bounds of a Family of Recurrent Neural Networks](https://mlsanity.com/paper/1910.12947/) (2019)
- [Analyzing Finite Neural Networks: Can We Trust Neural Tangent Kernel Theory?](https://mlsanity.com/paper/2012.04477/) (2020)
- [On the Rademacher Complexity of Linear Hypothesis Sets](https://mlsanity.com/paper/2007.11045/) (2020)
- [Generalization bounds for graph convolutional neural networks via Rademacher complexity](https://mlsanity.com/paper/2102.10234/) (2021)
- [Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation](https://mlsanity.com/paper/2105.11632/) (2021)

## Built on by (most cited first, 16 of 16)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Data Complexity Estimates for Operator Learning](https://mlsanity.com/paper/2405.15992/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws](https://mlsanity.com/paper/2401.01783/) (2024)
- [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)
- [A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics](https://mlsanity.com/paper/2407.03945/) (2024)
- [Neural Operators Learn the Local Physics of Magnetohydrodynamics](https://mlsanity.com/paper/2404.16015/) (2024)
- [Learning Green's Function Efficiently Using Low-Rank Approximations](https://mlsanity.com/paper/2308.00350/) (2023)
- [Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operators](https://mlsanity.com/paper/2506.19396/) (2025)
- [Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators](https://mlsanity.com/paper/2603.00971/) (2026)
- [Analysis of Fourier Neural Operators via Effective Field Theory](https://mlsanity.com/paper/2507.21833/) (2025)
- [Dynamical Implicit Neural Representations](https://mlsanity.com/paper/2511.21787/) (2025)
- [Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs](https://mlsanity.com/paper/2605.10277/) (2026)
- [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026)

## Abstract (arXiv)

A Fourier neural operator (FNO) is one of the physics-inspired machine learning methods. In particular, it is a neural operator. In recent times, several types of neural operators have been developed, e.g., deep operator networks, Graph neural operator (GNO), and Multiwavelet-based operator (MWTO). Compared with other models, the FNO is computationally efficient and can learn nonlinear operators between function spaces independent of a certain finite basis. In this study, we investigated the bounding of the Rademacher complexity of the FNO based on specific group norms. Using capacity based on these norms, we bound the generalization error of the model. In addition, we investigated the correlation between the empirical generalization error and the proposed capacity of FNO. From the perspective of our result, we inferred that the type of group norms determines the information about the weights and architecture of the FNO model stored in the capacity. And then, we confirmed these inferences through experiments. Based on this fact, we gained insight into the impact of the number of modes used in the FNO model on the generalization error. And we got experimental results that followed our insights.