  We consider strongly-nonlinear and weakly-dispersive surface water waves
governed by equations of Boussinesq type, known as the Serre-Green-Naghdi
system; it describes future states of the free water surface and depth averaged
horizontal velocity, given their initial state. The lack of knowledge of the
velocity field as well as the initial states provided by measurements lead to
an ill-posed problem that cannot be solved by traditional techniques. To this
end, we employ physics-informed neural networks (PINNs) to generate solutions
to such ill-posed problems using only data of the free surface elevation and
depth of the water. PINNs can readily incorporate the physical laws and the
observational data, thereby enabling inference of the physical quantities of
interest. In the present study, both experimental and synthetic (generated by
numerical methods) training data are used to train PINNs. Furthermore,
multi-fidelity data are used to solve the inverse water wave problem by
leveraging both high- and low-fidelity data sets. The applicability of the PINN
methodology for the estimation of the impact of water waves onto solid
obstacles is demonstrated after deriving the corresponding equations. The
present methodology can be employed to efficiently design offshore structures
such as oil platforms, wind turbines, etc. by solving the corresponding
ill-posed inverse water waves problem.
