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

Dinh Dũng

Sbornik. Mathematics, Tome 214 (2023) no. 4, pp. 479-515

Résumé

We find the convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ in the noncompact 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 infinite tensor-product standard Gaussian probability measure on ${\mathbb R}^\infty$ and $V$ is the energy space. We also obtain similar dimension-independent results in the case when the lognormal inputs are parametrized by ${\mathbb R}^M$ of 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 ${\mathbb R}^M$. Bibliography: 62 titles.

Détail

Keywords: high-dimensional approximation, collocation approximation, deep ReLU neural networks, parametric elliptic PDEs, lognormal inputs.