Domain decomposition based physics and equality constrained neural networks for the Helmholtz equation

Overlapping and non-overlapping domain decomposition methods are proposed for the physics and equality constrained neural networks (PECANN) to solve the Helmholtz equation. A coarse component is introduced using a small neural network defined on the whole domain with carefully designed loss functions. This coarse component supplies both function values and normal derivatives on the subdomain interfaces to the local solvers, ensuring the scalability of the algorithms; the number of outer iterations remains nearly constant as the number of subdomains increases. Numerical experiments conducted with a wide range of wavenumbers over both standard and enlarged domains demonstrate the efficiency and scalability of the proposed algorithms.