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

arXiv: https://arxiv.org/abs/2207.04950 · HTML: https://mlsanity.com/paper/2207.04950/

## 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 (most cited first, 15 of 15)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Optimal Approximation with Sparsely Connected Deep Neural Networks](https://mlsanity.com/paper/1705.01714/) (2017)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [The Modern Mathematics of Deep Learning](https://mlsanity.com/paper/2105.04026/) (2021)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities](https://mlsanity.com/paper/2010.12217/) (2020)
- [Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks](https://mlsanity.com/paper/2109.11354/) (2021)
- [De Rham compatible Deep Neural Network FEM](https://mlsanity.com/paper/2201.05395/) (2022)

## Built on by (most cited first, 21 of 21)

- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (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)
- [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)
- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)
- [Verification and Validation for Trustworthy Scientific Machine Learning](https://mlsanity.com/paper/2502.15496/) (2025)
- [Theory-to-Practice Gap for Neural Networks and Neural Operators](https://mlsanity.com/paper/2503.18219/) (2025)
- [A short tour of operator learning theory: Convergence rates, statistical limits, and open questions](https://mlsanity.com/paper/2603.00819/) (2026)
- [Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning](https://mlsanity.com/paper/2605.22724/) (2026)
- [Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks](https://mlsanity.com/paper/2604.01961/) (2026)
- [Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs](https://mlsanity.com/paper/2410.14788/) (2024)
- [Neural Shape Operator Surrogates -- Expression Rate Bounds](https://mlsanity.com/paper/2604.18012/) (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)
- [Proximal-IMH: Proximal Posterior Proposals for Independent Metropolis-Hastings with Approximate Operators](https://mlsanity.com/paper/2602.21426/) (2026)
- [Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operators](https://mlsanity.com/paper/2605.04164/) (2026)
- [Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs](https://mlsanity.com/paper/2605.10277/) (2026)

## 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.