  Accurate short-term predictions of phase-resolved water wave conditions are
crucial for decision-making in ocean engineering. However, the initialization
of remote-sensing-based wave prediction models first requires a reconstruction
of wave surfaces from sparse measurements like radar. Existing reconstruction
methods either rely on computationally intensive optimization procedures or
simplistic modelling assumptions that compromise the real-time capability or
accuracy of the subsequent prediction process. We therefore address these
issues by proposing a novel approach for phase-resolved wave surface
reconstruction using neural networks based on the U-Net and Fourier neural
operator (FNO) architectures. Our approach utilizes synthetic yet highly
realistic training data on uniform one-dimensional grids, that is generated by
the high-order spectral method for wave simulation and a geometric radar
modelling approach. The investigation reveals that both models deliver accurate
wave reconstruction results and show good generalization for different sea
states when trained with spatio-temporal radar data containing multiple
historic radar snapshots in each input. Notably, the FNO demonstrates superior
performance in handling the data structure imposed by wave physics due to its
global approach to learn the mapping between input and output in Fourier space.
