A Schwarz-type domain decomposition method for physics-constrained neural networks

We present a Schwarz-type, non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). We adopt a generalized Robin-type interface condition, which is a convex combination of Dirichlet and Neumann conditions with a unique Robin parameter assigned to each subdomain. These subdomain-specific Robin parameters are learned to minimize the mismatch at the subdomain interfaces, facilitating efficient information exchange during training. Our method is applicable to both the Laplace's and Helmholtz equations. Our overall meshless solution method represents local solutions by an independent neural network model which is trained to minimize the loss on the governing PDE while strictly enforcing boundary and interface conditions through an augmented Lagrangian formalism. Our results show that the learned Robin parameters adapt to the local behavior of the solution, domain partitioning and subdomain location relative to the overall domain. Extensive experiments on forward and inverse problems, including one-way and two-way decompositions with crosspoints, demonstrate the versatility and performance of our proposed approach.