Scalable two-level domain decomposition methods for the physics and equality constrained artificial neural networks

A family of scalable two-level methods, using both overlapping and non-overlapping domain decomposition, is proposed for the physics and equality constrained artificial neural networks (PECANN) to approximate solutions to partial differential equations. Various coarse components are constructed using a smaller neural network with carefully designed loss functions. By applying the coarse components to the domain decomposition-based PECANN algorithms, the number of outer iterations remains stable even as the number of subdomains increases. Extensive numerical experiments demonstrate the efficiency and scalability of the algorithms, with their performance especially notable when handling a large number of subdomains.