Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in L2(ℝd,γd)

For artificial deep neural networks, we prove expression rates for analytic functions f:ℝd→ℝ in the norm of L2(ℝd,γd) where d∈ mathbbNcup ∞ . Here γd denotes the Gaussian product probability measure on ℝd. We consider in particular ReLU and ReLUk activations for integer k≥ 2. For d∈mathbbN, we show exponential convergence rates in L2(ℝd,γd). In case d=∞, under suitable smoothness and sparsity assumptions on f:ℝ^mathbbN→ℝ, with γ_∞ denoting an infinite (Gaussian) product measure on ℝ^mathbbN, we prove dimension-independent expression rate bounds in the norm of L2(ℝ^mathbbN,γ_∞). The rates only depend on quantified holomorphy of (an analytic continuation of) the map f to a product of strips in mathbbCd. As an application, we prove expression rate bounds of deep ReLU-NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.