Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review

Alexander Heinlein, Axel Klawonn, Martin Lanser, Janine Weber

Research output: Contribution to journalArticleScientificpeer-review

Abstract

Scientific machine learning (SciML), an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within SciML given by the combination of domain decomposition methods (DDMs) with machine learning techniques for the solution of partial differential equations. The aim of the present work is to make an attempt of providing a review of existing and also new approaches within this field as well as to present some known results in a unified framework; no claim of completeness is made. As a concrete example of machine learning enhanced DDMs, an approach is presented which uses neural networks to reduce the computational effort in adaptive DDMs while retaining their robustness. More precisely, deep neural networks are used to predict the geometric location of constraints which are needed to define a robust coarse space. Additionally, two recently published deep domain decomposition approaches are presented in a unified framework. Both approaches use physics-constrained neural networks to replace the discretization and solution of the subdomain problems of a given decomposition of the computational domain. Finally, a brief overview is given of several further approaches which combine machine learning with ideas from DDMs to either increase the performance of already existing algorithms or to create completely new methods.

Original languageEnglish
Article numbere202100001
JournalGAMM Mitteilungen
Volume44
Issue number1
DOIs
Publication statusPublished - 2021
Externally publishedYes

Keywords

  • adaptive coarse spaces
  • deep learning
  • Deep Ritz
  • domain decomposition methods
  • hybrid modeling
  • neural networks
  • PDEs
  • physics-informed neural networks
  • scientific machine learning

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