  Partitioned neural network functions are used to approximate the solution of
partial differential equations. The problem domain is partitioned into
non-overlapping subdomains and the partitioned neural network functions are
defined on the given non-overlapping subdomains. Each neural network function
then approximates the solution in each subdomain. To obtain the convergent
neural network solution, certain continuity conditions on the partitioned
neural network functions across the subdomain interface need to be included in
the loss function, that is used to train the parameters in the neural network
functions. In our work, by introducing suitable interface values, the loss
function is reformulated into a sum of localized loss functions and each
localized loss function is used to train the corresponding local neural network
parameters. In addition, to accelerate the neural network solution convergence,
the localized loss function is enriched with an augmented Lagrangian term,
where the interface condition and the boundary condition are enforced as
constraints on the local solutions by using Lagrange multipliers. The local
neural network parameters and Lagrange multipliers are then found by optimizing
the localized loss function. To take the advantage of the localized loss
function for the parallel computation, an iterative algorithm is also proposed.
For the proposed algorithms, their training performance and convergence are
numerically studied for various test examples.
