Domain decomposition methods (DDMs) have been extensively developed for fast solutions of algebraic systems obtained from classical finite element discretization of partial differential equations (PDEs).

   The problem domain is partitioned into overlapping or nonoverlapping subdomains and each local problem to each subdomain is solved iteratively with a suitable interface update to obtain convergent local solutions to the global solution. Each local problem can be solved in parallel and with the addition of a global coarse problem, the iteration convergence can become robust to the number of subdomains in the partition. Such an iteration convergence can provide both strong and weak scalability when the method is implemented on parallel processors.

   Recently, DDMs have been also applied to neural network approximation to PDEs. In the neural network approximation, the solution accuracy and training efficiency can be greatly improved by using DD methodologies. With that, major limitations of the neural network approximation, a long training time and a limited accuracy, can be addressed. In this talk, I will present the following three approaches, overlapping Schwarz, localized FETI, and Robin iterative methods with their convergence theory and parallel scalability results.