mlsanityData to 14 Oct 2026

Bounding the Rademacher Complexity of Fourier neural operators

arXiv 2209.05150 ↗First posted 12 Sep 2022cs.LG stat.MLcited by 16

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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.

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

20 of 31 references.

References · 20

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 776
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  5. 2017Spectrally-normalized margin bounds for neural networkscited by 686
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  7. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  8. 2015Norm-Based Capacity Control in Neural Networkscited by 299
  9. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 297
  10. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 234
  11. 2016A vector-contraction inequality for Rademacher complexitiescited by 142
  12. 2017Fisher-Rao Metric, Geometry, and Complexity of Neural Networkscited by 100
8 more
  1. 2020Relative Flatness and Generalizationcited by 88
  2. 2019Generalization bounds for deep convolutional neural networkscited by 59
  3. 2018Generalization Error in Deep Learningcited by 47
  4. 2019On Generalization Bounds of a Family of Recurrent Neural Networkscited by 42
  5. 2020Analyzing Finite Neural Networks: Can We Trust Neural Tangent Kernel Theory?cited by 33
  6. 2020On the Rademacher Complexity of Linear Hypothesis Setscited by 17
  7. 2021Generalization bounds for graph convolutional neural networks via Rademacher complexitycited by 17
  8. 2021Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equationcited by 16

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 6; 2025: 4; 2026: 4.

16 papers.

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  1. 2024Operator Learning: Algorithms and Analysiscited by 85
  2. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  3. 2024Data Complexity Estimates for Operator Learningcited by 26
  4. 2025Operator Learning: A Statistical Perspectivecited by 17
  5. 2024Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Lawscited by 11
  6. 2026Diagnosing Failure Modes of Neural Operators Across Diverse PDE Familiescited by 6
  7. 2024Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operatorscited by 5
  8. 2024A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamicscited by 4
  9. 2024Neural Operators Learn the Local Physics of Magnetohydrodynamicscited by 2
  10. 2023Learning Green's Function Efficiently Using Low-Rank Approximationscited by 1
  11. 2025Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operatorscited by 1
  12. 2026Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operatorscited by 1
4 more
  1. 2025Analysis of Fourier Neural Operators via Effective Field Theorycited by 0
  2. 2025Dynamical Implicit Neural Representationscited by 0
  3. 2026Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEscited by 0
  4. 2026Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solvercited by 0