Collocation approximation by deep neural ReLU networks for parametric elliptic PDEs with lognormal inputs

Abstract: We obtained convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ from the non-compact set $\mathbb{R}<sup>\infty$. The approximation error is measured in the norm of the Bochner space $L_2(\mathbb{R}^\infty, V, \gamma)$, where $\gamma$ is the infinite tensor product standard Gaussian probability measure on $\mathbb{R}<sup>\infty$ and $V$ is the energy space. We also obtained similar results for the case when the lognormal inputs are parametrized on $\mathbb{R}<sup>M$ with very large dimension $M$, and the approximation error is measured in the $\sqrt{g_M}$-weighted uniform norm of the Bochner space $$L_\infty^{{\sqrt{g}}(\mathbb{R}^M,} V)$$, where $g_M$ is the density function of the standard Gaussian probability measure on $\mathbb{R}<sup>M$.