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Optimal Multiscale Learning of Linear Operators

arXiv 2606.16913 ↗First posted 15 Jun 2026math.ST cs.NA math.NAcited by 0

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Abstract (arXiv)

We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 28 of its 44 references that are in the map. First posted on arXiv 15 Jun 2026.

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References · 28

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  5. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  6. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  7. 2024Operator Learning: Algorithms and Analysiscited by 84
  8. 2023A Mathematical Guide to Operator Learningcited by 74
  9. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  10. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
  11. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 27
  12. 2023Elliptic PDE learning is provably data-efficientcited by 26
16 more
  1. 2022Learning Green's functions associated with time-dependent partial differential equationscited by 25
  2. 2023Deep Operator Network Approximation Rates for Lipschitz Operatorscited by 24
  3. 2022Neural and spectral operator surrogates: unified construction and expression rate boundscited by 21
  4. 2024Data Complexity Estimates for Operator Learningcited by 21
  5. 2025Operator Learning: A Statistical Perspectivecited by 14
  6. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  7. 2024Optimal deep learning of holomorphic operators between Banach spacescited by 12
  8. 2024Operator Learning of Lipschitz Operators: An Information-Theoretic Perspectivecited by 9
  9. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 8
  10. 2019Regularity and convergence analysis in Sobolev and H"older spaces for generalized Whittle-Mat'ern fieldscited by 3
  11. 2023Online Infinite-Dimensional Regression: Learning Linear Operatorscited by 3
  12. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 3
  13. 2026A short tour of operator learning theory: Convergence rates, statistical limits, and open questionscited by 2
  14. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
  15. 2026Convergence Rates for Learning Pseudo-Differential Operatorscited by 1
  16. 2026Hybrid operator learning of wave scattering maps in high-contrast mediacited by 1

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