We develop a deep learning approach to learn the boundary-to-solution operator, i.e., to establish the boundary to steady solution mapping, in the Toth basin of arbitrary top and bottom topographies and two types of prescribed boundary conditions.
The machine-learned mapping is represented by a DeepOnet, which takes the geometrical data and boundary conditions as the inputs and produces the steady state solution as the output. In this approach, we approximate the top and bottom boundaries by either truncated Fourier series or piecewise linear representations. The DeepOnet maps directly the finite dimensional representations of the boundaries to the steady state solution of the ground water transport equation in the Toth basin. We present two different implementations of the DeepOnet: 1) the Toth basin is embedded in a rectangular computational domain, and 2) the Toth basin with arbitrary top and bottom boundaries is mapped into a rectangular computational domain via a nonlinear transformation. We 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. Both implementations yield the same results, showcasing a new deep learning approach to study ground water transport phenomena.