mlsanityData to 7 Oct 2026

Data Complexity Estimates for Operator Learning

arXiv 2405.15992 ↗First posted 25 May 2024cs.LG cs.NA math.NAcited by 21

Not on the map. Why?

Abstract (arXiv)

Operator learning has emerged as a new paradigm for the data-driven approximation of nonlinear operators. Despite its empirical success, the theoretical underpinnings governing the conditions for efficient operator learning remain incomplete. The present work develops theory to study the data complexity of operator learning, complementing existing research on the parametric complexity. We investigate the fundamental question: How many input/output samples are needed in operator learning to achieve a desired accuracy ε? This question is addressed from the point of view of n-widths, and this work makes two key contributions. The first contribution is to derive lower bounds on n-widths for general classes of Lipschitz and Fr'echet differentiable operators. These bounds rigorously demonstrate a ``curse of data-complexity'', revealing that learning on such general classes requires a sample size exponential in the inverse of the desired accuracy ε. The second contribution of this work is to show that ``parametric efficiency'' implies ``data efficiency''; using the Fourier neural operator (FNO) as a case study, we show rigorously that on a narrower class of operators, efficiently approximated by FNO in terms of the number of tunable parameters, efficient operator learning is attainable in data complexity as well. Specifically, we show that if only an algebraically increasing number of tunable parameters is needed to reach a desired approximation accuracy, then an algebraically bounded number of data samples is also sufficient to achieve the same accuracy.

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). Builds on 14 of its 39 references that are in the map. First posted on arXiv 25 May 2024.

Builds on

14 of 39 references.

References · 14

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  3. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
  4. 2024Operator Learning: Algorithms and Analysiscited by 84
  5. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  6. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  7. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  8. 2021Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spacescited by 30
  9. 2021Two-layer neural networks with values in a Banach spacecited by 26
  10. 2023Deep Operator Network Approximation Rates for Lipschitz Operatorscited by 24
  11. 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
  12. 2022Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysiscited by 22
2 more
  1. 2022Bounding the Rademacher Complexity of Fourier neural operatorscited by 16
  2. 2022Learning ReLU networks to high uniform accuracy is intractablecited by 5

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 5; 2025: 9; 2026: 7.

21 papers.

Sort:

  1. 2025Operator Learning: A Statistical Perspectivecited by 14
  2. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  3. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 8
  4. 2024Optimal Convergence Rates for Neural Operatorscited by 7
  5. 2025KANO: Kolmogorov-Arnold Neural Operatorcited by 6
  6. 2026Performance of Neural and Polynomial Operator Surrogatescited by 5
  7. 2024Separable Operator Networkscited by 4
  8. 2024Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operatorscited by 4
  9. 2025Morephy-Net: An Evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed Neural Operator Learning Networkscited by 3
  10. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 3
  11. 2025Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictorscited by 2
  12. 2026A short tour of operator learning theory: Convergence rates, statistical limits, and open questionscited by 2
9 more
  1. 2025Learning where to learn: Training data distribution optimization for scientific machine learningcited by 1
  2. 2025Delay-adaptive Control of Nonlinear Systems with Approximate Neural Operator Predictorscited by 1
  3. 2024Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systemscited by 0
  4. 2025PIP2 Net: Physics-informed Partition Penalty Deep Operator Networkcited by 0
  5. 2026Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equationcited by 0
  6. 2026Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEscited by 0
  7. 2026Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalizationcited by 0
  8. 2026Learning solution operators of PDEs with sparse approximation methodscited by 0
  9. 2026Optimal Multiscale Learning of Linear Operatorscited by 0