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Operator Surrogates

by Ch. Schwab and J. Westermann and J. Zech

(Report number 2026-16)

Abstract
This chapter provides a mathematical introduction to operator surrogates, which are finite-parametric maps to approximate complex (typically nonlinear) operators between infinite-dimensional (function) spaces. A typical approach to constructing operator surrogates is the encoder-approximator-decoder paradigm, whereby infinite-dimensional inputs are mapped to a finite-dimensional latent space, processed, and then mapped back to the infinite-dimensional output space. We discuss various methods for creating efficient representation systems, including stable linear frameworks like Banach frames and data-driven techniques such as Principal Component Analysis (PCA), Active Subspaces, and nonlinear autoencoders. Subsequently, we survey key surrogate architectures, contrasting spectral methods, based on polynomial chaos expansions and related technologies, with more recent neural network approaches. We further discuss architectures that do not rely on representation systems, such as Fourier Neural Operators and transformers. We elaborate on their relationship and how they relate} to other neural network operator surrogates, particularly those obtained by so-called `algorithm-unrolling'. Additionally, we provide an introduction to the theoretical analysis of approximation and generalization errors. We explain how underlying low-dimensional structures, provided for example by holomorphy, are essential to overcome the so-called ``curse of dimensionality'': this refers to the fact that the complexity of numerical algorithms tends to increase exponentially in the problem dimensionality. Since operator surrogates deal with the approximation of mappings between infinite dimensional spaces, traditional approaches suitable for low-dimensional problems typically fail in these situations. We present convergence rate bounds for holomorphic operators that depend on the number of trainable parameters and the regularity of the input and output data. Throughout, we prioritize intuition over detailed technical proofs and provide precise pointers to the literature for detailed arguments.

Keywords: Deep Operator Networks, Neural Operator, Spectral Operator, Error bounds

@Techreport{SWZ26_1171,
  author = {Ch. Schwab and J. Westermann and J. Zech},
  title = {Operator Surrogates},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-16.pdf },
  year = {2026}
}

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