The monitoring of waves at the ocean surface is critical for both operational needs (e.g., maritime traffic) and scientific studies (e.g., air-sea interactions). Synthetic aperture radar (SAR) Satellites provide one of the only remote sensing observations to retrieve ocean wave information on a global scale. However state-of-the-art SAR processing schemes often lead to poor inversion performance due to overly-simplistic assumptions. Here we leverage deep learning schemes to address these shortcomings. We state the targeted measurement of the ocean wave spectrum at sea surface as a neural mapping from SAR satellite observations. We exploit supervised deep learning schemes trained from a large-scale collocation dataset between real SAR observations and Wavewatch III model data. Our results emphasize for the first time how deep learning schemes can outperform the state-of-the-art analytical SAR-based inversion with an improvement in terms of mean square error greater than 65%. We analyse and discuss further the key features of the trained neural processing.