Learning Interface Conditions in Domain Decomposition Solvers
This work addresses a bottleneck in computational science for researchers and engineers dealing with complex PDEs on unstructured grids, representing a novel method for a known limitation rather than an incremental improvement.
The authors tackled the problem of constructing optimal domain decomposition methods for unstructured-grid partial differential equations, which traditionally requires tedious analysis and is limited to simplified settings. They developed a method using Graph Convolutional Neural Networks and unsupervised learning to learn optimal interface modifications, achieving computational cost linear in problem size and robust performance on large problems.
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.