  We propose a discretization-free approach based on the physics-informed
neural network (PINN) method for solving coupled advection-dispersion and Darcy
flow equations with space-dependent hydraulic conductivity. In this approach,
the hydraulic conductivity, hydraulic head, and concentration fields are
approximated with deep neural networks (DNNs). We assume that the conductivity
field is given by its values on a grid, and we use these values to train the
conductivity DNN. The head and concentration DNNs are trained by minimizing the
residuals of the flow equation and ADE and using the initial and boundary
conditions as additional constraints. The PINN method is applied to one- and
two-dimensional forward advection-dispersion equations (ADEs), where its
performance for various P\'{e}clet numbers ($Pe$) is compared with the
analytical and numerical solutions. We find that the PINN method is accurate
with errors of less than 1% and outperforms some conventional
discretization-based methods for $Pe$ larger than 100. Next, we demonstrate
that the PINN method remains accurate for the backward ADEs, with the relative
errors in most cases staying under 5% compared to the reference concentration
field. Finally, we show that when available, the concentration measurements can
be easily incorporated in the PINN method and significantly improve (by more
than 50% in the considered cases) the accuracy of the PINN solution of the
backward ADE.
