mlsanityData to 7 Oct 2026

Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators

arXiv 2408.09004 ↗First posted 16 Aug 2024stat.ML cs.LG cs.NAcited by 4

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Abstract (arXiv)

We study learning-theoretic foundations of operator learning, using the linear layer of the Fourier Neural Operator architecture as a model problem. First, we identify three main errors that occur during the learning process: statistical error due to finite sample size, truncation error from finite rank approximation of the operator, and discretization error from handling functional data on a finite grid of domain points. Finally, we analyze a Discrete Fourier Transform (DFT) based least squares estimator, establishing both upper and lower bounds on the aforementioned errors.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 14 of its 24 references that are in the map. First posted on arXiv 16 Aug 2024.

Builds on

14 of 24 references.

References · 14

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  3. 2017Size-Independent Sample Complexity of Neural Networkscited by 289
  4. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  5. 2024Operator Learning: Algorithms and Analysiscited by 84
  6. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  7. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  8. 2023Elliptic PDE learning is provably data-efficientcited by 26
  9. 2024Data Complexity Estimates for Operator Learningcited by 21
  10. 2024Discretization Error of Fourier Neural Operatorscited by 20
  11. 2022Bounding the Rademacher Complexity of Fourier neural operatorscited by 16
  12. 2024Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learningcited by 12
2 more
  1. 2019Learning Schatten--von Neumann Operatorscited by 7
  2. 2023Online Infinite-Dimensional Regression: Learning Linear Operatorscited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 1; 2026: 3.

4 papers.

  1. 2025Operator Learning: A Statistical Perspectivecited by 14
  2. 2026Diagnosing Failure Modes of Neural Operators Across Diverse PDE Familiescited by 5
  3. 2026Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regimecited by 1
  4. 2026QuadNorm: Resolution-Robust Normalization for Neural Operatorscited by 0