Stochastic Deep-Ritz for Parametric Uncertainty Quantification

TL;DR

This paper introduces a deep learning-based numerical method that combines variational problem-solving with Monte Carlo sampling to effectively handle uncertainty in materials science applications.

Contribution

It presents a novel deep-learning approach for solving variational problems under uncertainty, integrating Monte Carlo sampling for improved accuracy.

Findings

Method effectively handles uncertainty in variational problems.

High accuracy demonstrated on multiple test cases.

Simple yet powerful approach for materials science applications.

Abstract

Scientific machine learning has become an increasingly important tool in materials science and engineering. It is particularly well suited to tackle material problems involving many variables or to allow rapid construction of surrogates of material models, to name just a few. Mathematically, many problems in materials science and engineering can be cast as variational problems. However, handling of uncertainty, ever present in materials, in the context of variational formulations remains challenging for scientific machine learning. In this article, we propose a deep-learning-based numerical method for solving variational problems under uncertainty. Our approach seamlessly combines deep-learning approximation with Monte-Carlo sampling. The resulting numerical method is powerful yet remarkably simple. We assess its performance and accuracy on a number of variational problems.