Three ways to solve partial differential equations with neural networks — A review. Issue 2 (28th May 2021)

• Record Type: Journal Article
• Authors: Blechschmidt, Jan; Ernst, Oliver G.
• Other Names: Benner Peter guestEditor.; Klawonn Axel guestEditor.; Stoll Martin guestEditor.
• Is Part Of: Mitteilungen der Gesellschaft für Angewandte Mathematik und Mechanik. Volume 44:Issue 2(2021)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2021-05-28
• Subjects: backward differential equation -- curse of dimensionality -- Feynman–Kac -- Hamilton–Jacobi–Bellman equations -- neural networks -- partial differential equation -- PINN -- stochastic process; Mathematics -- Periodicals; Mechanics, Applied -- Periodicals; 510.5
• DOI: 10.1002/gamm.202100006
• Languages: English
• ISSNs: 0936-7195

Abstract

Abstract: Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and their suitability for high‐dimensional problems: physics‐informed neural networks, methods based on the Feynman–Kac formula and methods based on the solution of backward stochastic differential equations. The article is accompanied by a suite of expository software in the form of Jupyter notebooks in which each basic methodology is explained step by step, allowing for a quick assimilation and experimentation. An extensive bibliography summarizes the state of the art.