It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research, we review some of the main ideas of deep learning-based approximation methods for PDEs, we revisit one of the central mathematical results for deep neural network approximations for PDEs, and we provide an overview of the recent literature in this area of research.
翻译:这是应用数学解决高维部分差异方程式(PDEs)方面最具有挑战性的问题之一。最近,为解决这一问题,提出了若干基于学习的深度近似算法,并在数字上根据一些高维PDE的例子进行了测试。这产生了一个活跃的研究领域,在这个领域,深学习方法和相关的蒙特卡洛方法被用于高维PDEs的近似近似法。在本文中,我们介绍了这一研究领域,我们审视了以深学习为基础的PDEs近似法的一些主要想法,我们重新审视了PDEs深神经网络近似法的中央数学结果之一,我们概述了这一研究领域最近的文献。