  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 $d\in {\mathbb{N}}\cup\{ \infty \}$. Here
$\gamma_d$ denotes the Gaussian product probability measure on $\mathbb{R}^d$.
We consider in particular ReLU and ReLU${}^k$ activations for integer $k\geq
2$. For $d\in\mathbb{N}$, we show exponential convergence rates in
$L^2(\mathbb{R}^d,\gamma_d)$. In case $d=\infty$, under suitable smoothness and
sparsity assumptions on $f:\mathbb{R}^{\mathbb{N}}\to\mathbb{R}$, with
$\gamma_\infty$ denoting an infinite (Gaussian) product measure on
$\mathbb{R}^{\mathbb{N}}$, we prove dimension-independent expression rate
bounds in the norm of $L^2(\mathbb{R}^{\mathbb{N}},\gamma_\infty)$. 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.
