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

TL;DR

This paper establishes convergence rates for deep ReLU neural network approximations of solutions to elliptic PDEs with lognormal inputs, covering both infinite and large finite-dimensional parameter spaces.

Contribution

It provides the first rigorous convergence rate analysis for neural network collocation methods applied to elliptic PDEs with lognormal random inputs in infinite and high-dimensional settings.

Findings

Convergence rates are derived in the Bochner space $L_2( abla^ ext{infty}, V, ext{Gaussian})$.

Similar results are obtained for finite but large dimensions $M$ with weighted uniform norms.

The analysis extends neural network approximation theory to stochastic 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 $\mathbb{R}^\infty$. 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 obtained similar results for the case when the lognormal inputs are parametrized on $\mathbb{R}^M$ 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 $\mathbb{R}^M$.