Near-term prediction of the structured spatio-temporal processes driving our climate is of profound importance to the safety and well-being of millions, but the prounced nonlinear convection of these processes make a complete mechanistic description even of the short-term dynamics challenging. However, convective transport provides not only a principled physical description of the problem, but is also indicative of the transport in time of informative features which has lead to the recent successful development of ``physics free'' approaches. In this work we demonstrate that their remains an important role to be played by physically informed models, which can successfully leverage deep learning (DL) to project the process onto a lower dimensional space on which a minimal dynamical description holds. Our approach synthesises the feature extraction capabilities of DL with physically motivated dynamics to outperform existing model free approaches, as well as state of the art hybrid approaches, on complex real world datasets including sea surface temperature and precipitation.