Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in L^2(ℝ^d,γ_d)
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For artificial deep neural networks, we prove expression rates for analytic functions f:ℝ^d→ℝ in the norm of L^2(ℝ^d,γ_d) where d∈ℕ∪{∞}. Here γ_d denotes the Gaussian product probability measure on ℝ^d. We consider in particular ReLU and ReLU^k activations for integer k≥ 2. For d∈ℕ, we show exponential convergence rates in L^2(ℝ^d,γ_d). In case d=∞, under suitable smoothness and sparsity assumptions on f:ℝ^ℕ→ℝ, with γ_∞ denoting an infinite (Gaussian) product measure on ℝ^ℕ, we prove dimension-independent expression rate bounds in the norm of L^2(ℝ^ℕ,γ_∞). The rates only depend on quantified holomorphy of (an analytic continuation of) the map f to a product of strips in 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.
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