This paper addresses physics-informed deep learning schemes for satellite ocean remote sensing data. Such observation datasets are characterized by the irregular space-time sampling of the ocean surface due to sensors' characteristics and satellite orbits. With a focus on satellite altimetry, we show that end-to-end learning schemes based on variational formulations provide new means to explore and exploit such observation datasets. Through Observing System Simulation Experiments (OSSE) using numerical ocean simulations and real nadir and wide-swath altimeter sampling patterns, we demonstrate their relevance w.r.t. state-of-the-art and operational methods for space-time interpolation and short-term forecasting issues. We also stress and discuss how they could contribute to the design and calibration of ocean observing systems.