An overview on deep learning-based approximation methods for partial differential equations
This paper addresses the problem of solving high-dimensional PDEs for researchers and practitioners in applied mathematics and deep learning, providing a survey of existing methods rather than proposing new ones.
This paper reviews deep learning-based approximation methods for high-dimensional partial differential equations (PDEs), a challenging problem in applied mathematics. It revisits selected mathematical results and proofs related to these methods and provides an overview of recent literature in the field.
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.