A short tour of operator learning theory: Convergence rates, statistical limits, and open questions
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Abstract (arXiv)
This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and neural network approximations. Next, it illustrates fundamental performance limits in terms of sample size by adopting a minimax perspective and considering various notions of regularity beyond holomorphy. The paper ends with a discussion on the interplay between these two perspectives and related open questions.
Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 26 of its 45 references that are in the map. First posted on arXiv 28 Feb 2026.
Builds on
26 of 45 references.
References · 26
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- 2022Neural and spectral operator surrogates: unified construction and expression rate boundscited by 21
14 more
- 2024Data Complexity Estimates for Operator Learningcited by 21
- 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
- 2020Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularitiescited by 20
- 2025Operator Learning: A Statistical Perspectivecited by 14
- 2024Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Studycited by 13
- 2024Optimal deep learning of holomorphic operators between Banach spacescited by 12
- 2024A practical existence theorem for reduced order models based on convolutional autoencoderscited by 9
- 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 8
- 2025Operator learning meets inverse problems: A probabilistic perspectivecited by 8
- 2024Learning smooth functions in high dimensions: from sparse polynomials to deep neural networkscited by 7
- 2025Theory-to-Practice Gap for Neural Networks and Neural Operatorscited by 4
- 2025Optimal Recovery Meets Minimax Estimationcited by 3
- 2025Extension and neural operator approximation of the electrical impedance tomography inverse mapcited by 3
- 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 3
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Citing papers per year: 2026: 2.
2 papers.