  Physics-Informed Neural Networks (PINNs) are machine learning tools that
approximate the solution of general partial differential equations (PDEs) by
adding them in some form as terms of the loss/cost function of a Neural
Network. Most pieces of work in the area of PINNs tackle non-linear PDEs.
Nevertheless, many interesting problems involving linear PDEs may benefit from
PINNs; these include parametric studies, multi-query problems, and parabolic
(transient) PDEs. The purpose of this paper is to explore PINNs for linear PDEs
whose solutions may present one or more boundary layers. More specifically, we
analyze the steady-state reaction-advection-diffusion equation in regimes in
which the diffusive coefficient is small in comparison with the reactive or
advective coefficients. We show that adding information about these
coefficients as predictor variables in a PINN results in better prediction
models than in a PINN that only uses spatial information as predictor
variables. This finding may be instrumental in multiscale problems where the
coefficients of the PDEs present high variability in small spatiotemporal
regions of the domain, and therefore PINNs may be employed together with domain
decomposition techniques to efficiently approximate the PDEs locally at each
partition of the spatiotemporal domain, without resorting to different learned
PINN models at each of these partitions.
