Stochastic deep-Ritz for parametric uncertainty quantification

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.