  In this paper, the authors propose a new approach to solving the groundwater
flow equation in the Toth basin of arbitrary top and bottom topographies using
deep learning. Instead of using traditional numerical solvers, they use a
DeepONet to produce the boundary-to-solution mapping. This mapping takes the
geometry of the physical domain along with the boundary conditions as inputs to
output the steady state solution of the groundwater flow equation. To implement
the DeepONet, the authors approximate the top and bottom boundaries using
truncated Fourier series or piecewise linear representations. They present two
different implementations of the DeepONet: one where the Toth basin is embedded
in a rectangular computational domain, and another where the Toth basin with
arbitrary top and bottom boundaries is mapped into a rectangular computational
domain via a nonlinear transformation. They implement the DeepONet with respect
to the Dirichlet and Robin boundary condition at the top and the Neumann
boundary condition at the impervious bottom boundary, respectively. Using this
deep-learning enabled tool, the authors investigate the impact of surface
topography on the flow pattern by both the top surface and the bottom
impervious boundary with arbitrary geometries. They discover that the average
slope of the top surface promotes long-distance transport, while the local
curvature controls localized circulations. Additionally, they find that the
slope of the bottom impervious boundary can seriously impact the long-distance
transport of groundwater flows. Overall, this paper presents a new and
innovative approach to solving the groundwater flow equation using deep
learning, which allows for the investigation of the impact of surface
topography on groundwater flow patterns.
