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.

翻译：科学机器学习已成为材料科学与工程领域日益重要的工具。它特别适合处理涉及多变量的材料问题，或快速构建材料模型的代理模型，仅举几例。从数学角度看，材料科学与工程中的许多问题可表述为变分问题。然而，在变分框架下处理材料中普遍存在的不确定性，对科学机器学习仍是挑战。本文提出一种基于深度学习的数值方法，用于求解含不确定性变分问题。该方法巧妙融合深度学习近似与蒙特卡洛采样，所得数值方法既强大又异常简洁。我们通过一系列变分问题评估了其性能与精度。