mlsanityData to 7 Oct 2026

Deep Operator Network Approximation Rates for Lipschitz Operators

arXiv 2307.09835 ↗First posted 19 Jul 2023math.NA cs.LG cs.NAcited by 24

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Abstract (arXiv)

We establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or H"older) continuous maps mathcal G:mathcal X→mathcal Y between (subsets of) separable Hilbert spaces mathcal X, mathcal Y. The DON architecture considered uses linear encoders mathcal E and decoders mathcal D via (biorthogonal) Riesz bases of mathcal X, mathcal Y, and an approximator network of an infinite-dimensional, parametric coordinate map that is Lipschitz continuous on the sequence space ℓ2(mathbb N). Unlike previous works ([Herrmann, Schwab and Zech: Neural and Spectral operator surrogates: construction and expression rate bounds, SAM Report, 2022], [Marcati and Schwab: Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations, SAM Report, 2022]), which required for example mathcal G to be holomorphic, the present expression rate results require mere Lipschitz (or H"older) continuity of mathcal G. Key in the proof of the present expression rate bounds is the use of either super-expressive activations (e.g. [Yarotski: Elementary superexpressive activations, Int. Conf. on ML, 2021], [Shen, Yang and Zhang: Neural network approximation: Three hidden layers are enough, Neural Networks, 2021], and the references there) which are inspired by the Kolmogorov superposition theorem, or of nonstandard NN architectures with standard (ReLU) activations as recently proposed in [Zhang, Shen and Yang: Neural Network Architecture Beyond Width and Depth, Adv. in Neural Inf. Proc. Sys., 2022]. We illustrate the abstract results by approximation rate bounds for emulation of a) solution operators for parametric elliptic variational inequalities, and b) Lipschitz maps of Hilbert-Schmidt operators.

Key facts. Cited by 24 papers in the map (data through 7 Oct 2026). Builds on 21 of its 69 references that are in the map. First posted on arXiv 19 Jul 2023.

Builds on

21 of 69 references.

References · 21

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9 more
  1. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  2. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 8; 2025: 7; 2026: 9.

24 papers.

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  1. 2024Operator Learning: Algorithms and Analysiscited by 84
  2. 2024Data Complexity Estimates for Operator Learningcited by 21
  3. 2024Operator Learning of Lipschitz Operators: An Information-Theoretic Perspectivecited by 9
  4. 2024Operator learning without the adjointcited by 8
  5. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 8
  6. 2024Learning smooth functions in high dimensions: from sparse polynomials to deep neural networkscited by 7
  7. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  8. 2025A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theorycited by 6
  9. 2026Performance of Neural and Polynomial Operator Surrogatescited by 5
  10. 2025Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guaranteescited by 3
  11. 2025New universal operator approximation theorem for encoder-decoder architecturescited by 3
  12. 2025One model to solve them all: 2BSDE families via neural operatorscited by 3
12 more
  1. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 3
  2. 2024DiGRAF: Diffeomorphic Graph-Adaptive Activation Functioncited by 2
  3. 2025Upper Approximation Bounds for Neural Oscillatorscited by 1
  4. 2026Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networkscited by 1
  5. 2026Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learningcited by 1
  6. 2025Discretization-independent operator learning for partial differential equationscited by 0
  7. 2026The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositionscited by 0
  8. 2026Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEscited by 0
  9. 2026Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalizationcited by 0
  10. 2026Optimal Multiscale Learning of Linear Operatorscited by 0
  11. 2026NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processescited by 0
  12. 2026Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solvercited by 0