An overview on deep learning-based approximation methods for partial differential equations

作     者：Beck, Christian Hutzenthaler, Martin Jentzen, Arnulf Kuckuck, Benno 

作者机构：Department of Mathematics ETH Zurich Zurich Switzerland Applied Mathematics: Institute for Analysis and Numerics Faculty of Mathematics and Computer Science University of Münster Münster Germany Faculty of Mathematics University of Duisburg-Essen Essen Germany School of Data Science Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen China 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2020年

核心收录：

主　　题：Partial differential equations 

摘      要：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 *** Codes 65M99 (Primary), 35-02, 65-02, 68T07 (Secondary) Copyright © 2020, The Authors. All rights reserved.