Deep Learning in High Dimension: Neural Network Expression Rates for Analytic Functions in \(\pmb{L^2(\mathbb{R}^d,\gamma_d)}\)

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 \(d\in \mathbb{N}\cup \{ \infty \}\) . Here \(\gamma_d\) denotes the Gaussian product probability measure on \(\mathbb{R}^d\) . We consider in particular \(\mathrm{ReLU}\) and \(\mathrm{ReLU}^k\) 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 \((\mathbb{R}^{\mathbb{N}}, \mathcal{B}(\mathbb{R}^{\mathbb{N}}))\) , 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 \(\mathbb{C}^d\) (in \(\mathbb{C}^{\mathbb{N}}\) for \(d=\infty\) , respectively). As an application, we prove expression rate bounds of deep \(\mathrm{ReLU}\) -NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.