Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs

TL;DR

This paper explores the use of deep learning, specifically physics-informed neural networks and the Deep Ritz method, to efficiently approximate solutions of elliptic PDEs with stochastic coefficients, overcoming limitations of traditional methods.

Contribution

It introduces deep learning techniques for stochastic Galerkin approximations of elliptic PDEs, including the application of PINNs and the Deep Ritz method, with analysis of their efficiency and applicability.

Findings

Deep learning methods effectively approximate solutions to stochastic PDEs.

PINNs can be applied broadly but require strong residual minimization.

Deep Ritz method reduces training cost and guarantees solution uniqueness.

Abstract

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and infinity. A traditional numerical method for solving the resulting high-dimensional coupled system of PDEs is replaced by deep learning techniques. In order to achieve this, physics-informed neural networks (PINNs), which typically operate on the strong residual of the PDE and can therefore be applied in a wide range of settings, are considered. As a second approach, the Deep Ritz method, which is a neural network that minimizes the Ritz energy functional to find the weak solution, is employed. While the second approach only works in special cases, it overcomes the necessity of testing in variational problems while maintaining mathematical rigor and ensuring the existence of a unique solution. Furthermore, the residual is of a lower differentiation order, reducing the training cost considerably. The efficiency of the method is demonstrated on several model problems.