mlsanityData to 14 Oct 2026

Neural and spectral operator surrogates: unified construction and expression rate bounds

arXiv 2207.04950 ↗First posted 11 Jul 2022math.NA cs.LG cs.NAcited by 21

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Abstract (arXiv)

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for Deep Neural Operator and Generalized Polynomial Chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and spectral operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity. We illustrate the abstract concepts by expression rate bounds for the coefficient-to-solution map for a linear elliptic PDE on the torus.

Key facts. Cited by 21 papers in the map (data through 14 Oct 2026). Builds on 15 of its 84 references that are in the map. First posted on arXiv 11 Jul 2022.

Builds on

15 of 84 references.

References · 15

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2021Choose a Transformer: Fourier or Galerkincited by 343
  3. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  4. 2022MIONet: Learning multiple-input operators via tensor productcited by 181
  5. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  6. 2017Solving ill-posed inverse problems using iterative deep neural networkscited by 145
  7. 2017Optimal Approximation with Sparsely Connected Deep Neural Networkscited by 91
  8. 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 83
  9. 2018Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantificationcited by 81
  10. 2021The Modern Mathematics of Deep Learningcited by 63
  11. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  12. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
3 more
  1. 2020Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularitiescited by 20
  2. 2021Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networkscited by 12
  3. 2022De Rham compatible Deep Neural Network FEMcited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 1; 2024: 5; 2025: 5; 2026: 9.

21 papers.

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  1. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  2. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  3. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  4. 2024Mixtures of Neural Operators Reduce Active Complexity in Operator Learningcited by 16
  5. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 11
  6. 2025A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theorycited by 9
  7. 2024Learning smooth functions in high dimensions: from sparse polynomials to deep neural networkscited by 7
  8. 2026Performance of Neural and Polynomial Operator Surrogatescited by 7
  9. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 5
  10. 2025Verification and Validation for Trustworthy Scientific Machine Learningcited by 4
  11. 2025Theory-to-Practice Gap for Neural Networks and Neural Operatorscited by 4
  12. 2026A short tour of operator learning theory: Convergence rates, statistical limits, and open questionscited by 4
9 more
  1. 2026Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learningcited by 4
  2. 2026Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networkscited by 3
  3. 2024Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEscited by 2
  4. 2026Neural Shape Operator Surrogates -- Expression Rate Boundscited by 1
  5. 2026Optimal Multiscale Learning of Linear Operatorscited by 1
  6. 2025Discretization-independent operator learning for partial differential equationscited by 0
  7. 2026Proximal-IMH: Proximal Posterior Proposals for Independent Metropolis-Hastings with Approximate Operatorscited by 0
  8. 2026Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operatorscited by 0
  9. 2026Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEscited by 0