Machine learning and domain decomposition methods -- a survey

TL;DR

This survey reviews how machine learning techniques are integrated with domain decomposition methods to improve computational efficiency and convergence in scientific computing and engineering applications.

Contribution

It categorizes and summarizes existing research on combining ML with DDMs, highlighting new approaches and future research directions.

Findings

ML enhances convergence of DDMs

ML accelerates training of physics-aware neural networks

ML as a discretization method for PDEs

Abstract

Hybrid algorithms, which combine black-box machine learning methods with experience from traditional numerical methods and domain expertise from diverse application areas, are progressively gaining importance in scientific machine learning and various industrial domains, especially in computational science and engineering. In the present survey, several promising avenues of research will be examined which focus on the combination of machine learning (ML) and domain decomposition methods (DDMs). The aim of this survey is to provide an overview of existing work within this field and to structure it into domain decomposition for machine learning and machine learning-enhanced domain decomposition, including: domain decomposition for classical machine learning, domain decomposition to accelerate the training of physics-aware neural networks, machine learning to enhance the convergence properties or computational efficiency of DDMs, and machine learning as a discretization method in a DDM for the solution of PDEs. In each of these fields, we summarize existing work and key advances within a common framework and, finally, disuss ongoing challenges and opportunities for future research.