Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs

TL;DR

This paper studies how deep ReLU neural networks can effectively approximate functions in infinite-dimensional Bochner spaces, with applications to solving parametric PDEs with random inputs, establishing convergence rates based on function properties.

Contribution

It introduces non-adaptive deep ReLU neural network approximation methods in Bochner spaces and derives convergence rates, applying these results to parametric elliptic PDEs with random inputs.

Findings

Established convergence rates for neural network approximation in Bochner spaces.

Applied approximation results to parametric PDEs with random inputs.

Demonstrated effectiveness for lognormal and affine cases.

Abstract

We investigate non-adaptive methods of deep ReLU neural network approximation in Bochner spaces $L_2({\mathbb U}^\infty, X, \mu)$ of functions on ${\mathbb U}^\infty$ taking values in a separable Hilbert space $X$, where ${\mathbb U}^\infty$ is either ${\mathbb R}^\infty$ equipped with the standard Gaussian probability measure, or ${\mathbb I}^\infty:= [-1,1]^\infty$ equipped with the Jacobi probability measure. Functions to be approximated are assumed to satisfy a certain weighted $\ell_2$-summability of the generalized chaos polynomial expansion coefficients with respect to the measure $\mu$. We prove the convergence rate of this approximation in terms of the size of approximating deep ReLU neural networks. These results then are applied to approximation of the solution to parametric elliptic PDEs with random inputs for the lognormal and affine cases.