论文标题
关于部分微分方程的基于深度学习的近似方法的概述
An overview on deep learning-based approximation methods for partial differential equations
论文作者
论文摘要
它是应用数学中最具挑战性的问题之一，可以近似求解高维偏微分方程（PDE）。最近，已经提出了几种用于攻击此问题的深度学习近似算法，并在数字上进行了许多高维PDE示例的测试。这引起了一个生动的研究领域，在该领域中，将基于深度学习的方法和相关的蒙特卡洛方法应用于高维PDE的近似值。在本文中，我们通过重新访问与PDE的深度学习近似方法相关的选定数学结果并回顾其证明的主要思想，从而对该研究领域进行了介绍。我们还简要概述了这一研究领域的最新文献。
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 by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.
