Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs

We investigate non-adaptive methods of deep ReLU neural network approximation in Bochner spaces L2(U∞,X,μ) functions on U∞ taking values a separable Hilbert space X, where is either R∞ equipped with the standard Gaussian probability measure, or [−1,1]∞ Jacobi measure. Functions to be approximated are assumed satisfy certain weighted ℓ2-summability generalized chaos polynomial expansion coefficients respect measure μ. prove convergence rate this terms size approximating networks. These results then applied solution parametric elliptic PDEs random inputs for lognormal and affine cases.