mlsanityData to 14 Oct 2026

Deep Operator Learning Lessens the Curse of Dimensionality for PDEs

arXiv 2301.12227 ↗First posted 28 Jan 2023cs.LGcited by 18

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Abstract (arXiv)

Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz operators over Banach spaces using DNNs with applications to various PDE solution operators. The goal is to specify DNN width, depth, and the number of training samples needed to guarantee a certain testing error. Under mild assumptions on data distributions or operator structures, our analysis shows that deep operator learning can have a relaxed dependence on the discretization resolution of PDEs and, hence, lessen the curse of dimensionality in many PDE-related problems including elliptic equations, parabolic equations, and Burgers equations. Our results are also applied to give insights about discretization-invariance in operator learning.

Key facts. Cited by 18 papers in the map (data through 14 Oct 2026). Builds on 27 of its 60 references that are in the map. First posted on arXiv 28 Jan 2023.

Builds on

27 of 60 references.

References · 27

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
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15 more
  1. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 7; 2024: 4; 2025: 5; 2026: 2.

18 papers.

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  1. 2024Operator Learning: Algorithms and Analysiscited by 85
  2. 2023Elliptic PDE learning is provably data-efficientcited by 26
  3. 2023Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problemscited by 13
  4. 2023Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific MLcited by 8
  5. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  6. 2023Operator learning for hyperbolic partial differential equationscited by 5
  7. 2024Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equationscited by 5
  8. 2024Micrometer: Micromechanics Transformer for Predicting Mechanical Responses of Heterogeneous Materialscited by 5
  9. 2025Deep Neural ODE Operator Networks for PDEscited by 3
  10. 2023On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networkscited by 2
  11. 2023Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equationscited by 1
  12. 2023Neural Network Approximation for Pessimistic Offline Reinforcement Learningcited by 1
6 more
  1. 2025Curse of Dimensionality in Neural Network Optimizationcited by 1
  2. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
  3. 2025Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domainscited by 0
  4. 2025Learning Biomolecular Motion: The Physics-Informed Machine Learning Paradigmcited by 0
  5. 2026Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEscited by 0
  6. 2026Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion Systemcited by 0