Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in $L^2({\mathbb{R}}^d,{\gamma }_d)$

For artificial deep neural networks, we prove expression rates for analytic functions $f:{\mathbb{R}}^d\to \mathbb{R}$ in the norm of $L^2({\mathbb{R}}^d,{\gamma }_d)$ where $d\in \mathbb{N}\cup \{\mathrm{\infty }\}$. Here ${\gamma }_d$ denotes the Gaussian product probability measure on ${\mathbb{R}}^d$. We consider in particular ReLU and ReLU${}^k$ activations for integer $k\geq 2$\mathrm{.}For$d\in$\mathbb{N}$$,weshowexponentialconvergenceratesin$L^2($\mathbb{R}$^d,\gamma_d)$\mathrm{.}Incase$d=\infty$, under suitable smoothness and sparsity assumptions on $f:{\mathbb{R}}^{\mathbb{N}}\to \mathbb{R}$, with ${\gamma }_{\mathrm{\infty }}$ denoting an infinite (Gaussian) product measure on ${\mathbb{R}}^{\mathbb{N}}$, we prove dimension-independent expression rate bounds in the norm of $L^2({\mathbb{R}}^{\mathbb{N}},{\gamma }_{\mathrm{\infty }})$. The rates only depend on quantified holomorphy of (an analytic continuation of) the map $f$ to a product of strips in ${\mathbb{C}}^d$. As an application, we prove expression rate bounds of deep ReLU-NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.