A Generalized Schwarz-type Non-overlapping Domain Decomposition Method
  using Physics-constrained Neural Networks
Shamsulhaq Basir Inanc Senocak
We present a meshless Schwarz-type non-overlapping domain decomposition
method based on artificial neural networks for solving forward and inverse
problems involving partial differential equations (PDEs). To ensure the
consistency of solutions across neighboring subdomains, we adopt a generalized
Robin-type interface condition, assigning unique Robin parameters to each
subdomain. These subdomain-specific Robin parameters are learned to minimize
the mismatch on the Robin interface condition, facilitating efficient
information exchange during training. Our method is applicable to both the
Laplace's and Helmholtz equations. It 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. A key strength of our method lies in
its ability to learn a Robin parameter for each subdomain, thereby enhancing
information exchange with its neighboring subdomains. We observe 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.