The objective of this work is to investigate the possibility of using the PINNs to approximate the Shallow Water Equations, a system of hyperbolic PDEs that simulates free-surface flow problems. Specifically, parametric cases are intentionally built upon selected benchmark problems, for the purpose of testing the degree to which the PINN is a convenient tool to be used when dealing with \emph{many-query} problems. The PINN seems to be a promising tool to deal with this class of problems for one fundamental reason: following one training session, the PINN is able to instantly provide simulations for every requested parameter value. Consequently, the computational cost of the complete parameter space analysis is, basically, the one of a single training session.