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Three ways to solve partial differential equations with neural networks — A review
Author(s) -
Blechschmidt Jan,
Ernst Oliver G.
Publication year - 2021
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.202100006
Subject(s) - artificial neural network , simplicity , suite , partial differential equation , computer science , software , mathematics , differential equation , construct (python library) , algorithm , calculus (dental) , mathematical optimization , artificial intelligence , mathematical analysis , physics , medicine , archaeology , dentistry , quantum mechanics , history , programming language
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.