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.
Categories: math.NA cs.NA math.PR stat.ML