# Deep Operator Network Approximation Rates for Lipschitz Operators

arXiv: https://arxiv.org/abs/2307.09835 · HTML: https://mlsanity.com/paper/2307.09835/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 69 references that are in the map.
- First posted on arXiv 19 Jul 2023.

## Builds on (most cited first, 21 of 21)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [A Probabilistic Graphical Model Foundation for Enabling Predictive Digital Twins at Scale](https://mlsanity.com/paper/2012.05841/) (2020)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [Elementary superexpressive activations](https://mlsanity.com/paper/2102.10911/) (2021)
- [The Parametric Complexity of Operator Learning](https://mlsanity.com/paper/2306.15924/) (2023)
- [Neural Network Architecture Beyond Width and Depth](https://mlsanity.com/paper/2205.09459/) (2022)
- [Quantitative approximation results for complex-valued neural networks](https://mlsanity.com/paper/2102.13092/) (2021)
- [De Rham compatible Deep Neural Network FEM](https://mlsanity.com/paper/2201.05395/) (2022)

## Built on by (most cited first, 24 of 24)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Data Complexity Estimates for Operator Learning](https://mlsanity.com/paper/2405.15992/) (2024)
- [Operator Learning of Lipschitz Operators: An Information-Theoretic Perspective](https://mlsanity.com/paper/2406.18794/) (2024)
- [The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures](https://mlsanity.com/paper/2410.23440/) (2024)
- [A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory](https://mlsanity.com/paper/2510.25379/) (2025)
- [Operator learning without the adjoint](https://mlsanity.com/paper/2401.17739/) (2024)
- [Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks](https://mlsanity.com/paper/2404.03761/) (2024)
- [Performance of Neural and Polynomial Operator Surrogates](https://mlsanity.com/paper/2604.00689/) (2026)
- [An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning](https://mlsanity.com/paper/2408.13222/) (2024)
- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)
- [New universal operator approximation theorem for encoder-decoder architectures](https://mlsanity.com/paper/2503.24092/) (2025)
- [One model to solve them all: 2BSDE families via neural operators](https://mlsanity.com/paper/2511.01125/) (2025)
- [Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning](https://mlsanity.com/paper/2605.22724/) (2026)
- [Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees](https://mlsanity.com/paper/2503.08642/) (2025)
- [Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks](https://mlsanity.com/paper/2604.01961/) (2026)
- [DiGRAF: Diffeomorphic Graph-Adaptive Activation Function](https://mlsanity.com/paper/2407.02013/) (2024)
- [Upper Approximation Bounds for Neural Oscillators](https://mlsanity.com/paper/2512.01015/) (2025)
- [Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalization](https://mlsanity.com/paper/2606.01244/) (2026)
- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)
- [Discretization-independent operator learning for partial differential equations](https://mlsanity.com/paper/2507.07292/) (2025)
- [The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositions](https://mlsanity.com/paper/2602.21910/) (2026)
- [Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs](https://mlsanity.com/paper/2605.10277/) (2026)
- [NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes](https://mlsanity.com/paper/2607.14361/) (2026)
- [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026)

## 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.