Neural and Spectral Operator Surrogates on Gaussian Spaces

Abstract

We prove expression rate bounds of finite-parametric, spectral and neural surrogates for holomorphic maps between separable Hilbert spaces. The surrogates have an encoder-approximator-decoder architecture, with Karhunen-Loéve encoders and frame decoders. We prove expression rate bounds for two classes of finite-parametric surrogates: i) spectral surrogates obtained by N-term truncations of Wiener polynomial chaos expansions and ii) neural surrogates obtained by approximation of parametric maps with deep feedforward neural networks, ReLU and RePU activation functions and uniformly bounded weights. We work under an algebraic decay assumption on the eigenvalues of the covariance of the Gaussian measure on the input space. We obtain convergence rates for mean-square errors, and additionally in first-order Gaussian Sobolev spaces, to account for errors in the approximation of gradients.