Learning Interface Conditions in Domain Decomposition Solvers

TL;DR

This paper introduces a novel approach using Graph Convolutional Neural Networks to learn optimal interface conditions in domain decomposition methods, enabling efficient solutions for complex unstructured-grid problems with linear computational cost.

Contribution

It extends optimized Schwarz methods to unstructured grids by leveraging GCNNs and unsupervised learning, improving their applicability and performance on complex PDE problems.

Findings

Learned solvers outperform classical methods on unstructured grids.

Training with an improved loss function enables robustness on large problems.

Computational cost scales linearly with problem size.

Abstract

Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the 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.