  We investigate non-adaptive methods of deep ReLU neural network approximation
in Bochner spaces $L_2({\mathbb U}^\infty, X, \mu)$ of functions on ${\mathbb
U}^\infty$ taking values in a separable Hilbert space $X$, where ${\mathbb
U}^\infty$ is either ${\mathbb R}^\infty$ equipped with the standard Gaussian
probability measure, or ${\mathbb I}^\infty:= [-1,1]^\infty$ equipped with the
Jacobi probability measure. Functions to be approximated are assumed to satisfy
a certain weighted $\ell_2$-summability of the generalized chaos polynomial
expansion coefficients with respect to the measure $\mu$. We prove the
convergence rate of this approximation in terms of the size of approximating
deep ReLU neural networks. These results then are applied to approximation of
the solution to parametric elliptic PDEs with random inputs for the lognormal
and affine cases.
