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

It is one of the most challenging problems in applied mathematics to\napproximatively solve high-dimensional partial differential equations (PDEs).\nRecently, several deep learning-based approximation algorithms for attacking\nthis problem have been proposed and tested numerically on a number of examples\nof high-dimensional PDEs. This has given rise to a lively field of research in\nwhich deep learning-based methods and related Monte Carlo methods are applied\nto the approximation of high-dimensional PDEs. In this article we offer an\nintroduction to this field of research by revisiting selected mathematical\nresults related to deep learning approximation methods for PDEs and reviewing\nthe main ideas of their proofs. We also provide a short overview of the recent\nliterature in this area of research.\n