  Physics-informed neural networks (PINNs) are appealing data-driven tools for
solving and inferring solutions to nonlinear partial differential equations
(PDEs). Unlike traditional neural networks (NNs), which train only on solution
data, a PINN incorporates a PDE's residual into its loss function and trains to
minimize the said residual at a set of collocation points in the solution
domain. This paper explores the use of the Schwarz alternating method as a
means to couple PINNs with each other and with conventional numerical models
(i.e., full order models, or FOMs, obtained via the finite element, finite
difference or finite volume methods) following a decomposition of the physical
domain. It is well-known that training a PINN can be difficult when the PDE
solution has steep gradients. We investigate herein the use of domain
decomposition and the Schwarz alternating method as a means to accelerate the
PINN training phase. Within this context, we explore different approaches for
imposing Dirichlet boundary conditions within each subdomain PINN: weakly
through the loss and/or strongly through a solution transformation. As a
numerical example, we consider the one-dimensional steady state
advection-diffusion equation in the advection-dominated (high Peclet) regime.
Our results suggest that the convergence of the Schwarz method is strongly
linked to the choice of boundary condition implementation within the PINNs
being coupled. Surprisingly, strong enforcement of the Schwarz boundary
conditions does not always lead to a faster convergence of the method. While it
is not clear from our preliminary study that the PINN-PINN coupling via the
Schwarz alternating method accelerates PINN convergence in the
advection-dominated regime, it reveals that PINN training can be improved
substantially for Peclet numbers as high as 1e6 by performing a PINN-FOM
coupling.
