Deep ReLU neural network approximation of parametric and stochastic elliptic PDEs with lognormal inputs

We investigate non-adaptive methods of deep ReLU neural network approximation of the solution $u$ to parametric and stochastic elliptic PDEs with lognormal inputs on non-compact set $\mathbb{R}^\infty$. The approximation error is measured in the norm of the Bochner space $L_2(\mathbb{R}^\infty, V, \gamma)$, where $\gamma$ is the tensor product standard Gaussian probability on $\mathbb{R}^\infty$ and $V$ is the energy space. The approximation is based on an $m$-term truncation of the Hermite generalized polynomial chaos expansion (gpc) of $u$. Under a certain assumption on $\ell_q$-summability condition for lognormal inputs ($0< q <\infty$), we proved that for every integer $n > 1$, one can construct a non-adaptive compactly supported deep ReLU neural network $\boldsymbol{\phi}_n$ of size not greater than $n$ on $\mathbb{R}^m$ with $m = \mathcal{O} (n/\log n)$, having $m$ outputs so that the summation constituted by replacing polynomials in the $m$-term truncation of Hermite gpc expansion by these $m$ outputs approximates $u$ with an error bound $\mathcal{O}\left(\left(n/\log n\right)^{-1/q}\right)$. This error bound is comparable to the error bound of the best approximation of $u$ by $n$-term truncations of Hermite gpc expansion which is $\mathcal{O}(n^{-1/q})$. We also obtained some results on similar problems for parametric and stochastic elliptic PDEs with affine inputs, based on the Jacobi and Taylor gpc expansions.