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

Abstract

For artificial deep neural networks, we prove expression rates for analytic functions in the norm of where . Here denotes the Gaussian product probability measure on . We consider in particular and activations for integer . For , we show exponential convergence rates in . In case , under suitable smoothness and sparsity assumptions on , with denoting an infinite (Gaussian) product measure on , we prove dimension-independent expression rate bounds in the norm of . The rates only depend on quantified holomorphy of (an analytic continuation of) the map to a product of strips in (in for , respectively). As an application, we prove expression rate bounds of deep -NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.