  The measurement of deep water gravity wave elevations using in-situ devices,
such as wave gauges, typically yields spatially sparse data. This sparsity
arises from the deployment of a limited number of gauges due to their
installation effort and high operational costs. The reconstruction of the
spatio-temporal extent of surface elevation poses an ill-posed data
assimilation problem, challenging to solve with conventional numerical
techniques. To address this issue, we propose the application of a
physics-informed neural network (PINN), aiming to reconstruct physically
consistent wave fields between two designated measurement locations several
meters apart.
  Our method ensures this physical consistency by integrating residuals of the
hydrodynamic nonlinear Schr\"{o}dinger equation (NLSE) into the PINN's loss
function. Using synthetic wave elevation time series from distinct locations
within a wave tank, we initially achieve successful reconstruction quality by
employing constant, predetermined NLSE coefficients. However, the
reconstruction quality is further improved by introducing NLSE coefficients as
additional identifiable variables during PINN training. The results not only
showcase a technically relevant application of the PINN method but also
represent a pioneering step towards improving the initialization of
deterministic wave prediction methods.
