Learning Interface Conditions in Domain Decomposition Solvers

Ali Taghibakhshi, Nicolas Nytko, Tareq Uz Zaman, Scott MacLachlan, Luke Olson, Matthew West

Keywords: Domain Decomposition, Optimized Schwarz Methods, Graph Neural Networks, Unsupervised Learning

TL;DR: We use graph neural networks and introduce an improved loss function in order to learn interface conditions for optimized Schwarz domain-decomposition algorithms, enabling their use on unstructured grids.

Abstract: Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the \textit{optimal} construction of these methods requires tedious analysis and is often available only in simplified, structured-grid settings, limiting their use for more complex problems. In this work, we generalize optimized Schwarz domain decomposition methods to unstructured-grid problems, using Graph Convolutional Neural Networks (GCNNs) and unsupervised learning to learn optimal modifications at subdomain interfaces. A key ingredient in our approach is an improved loss function, enabling effective training on relatively small problems, but robust performance on arbitrarily large problems, with computational cost linear in problem size. The performance of the learned linear solvers is compared with both classical and optimized domain decomposition algorithms, for both structured- and unstructured-grid problems.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2205.09833/code)