Uncertainty quantification in variational data assimilation with deep learning

The spatio-temporal reconstruction of a dynamical process from some observationaldata is at the core of a wide range of applications in geosciences. This is particularly true for weather forecasting, operational oceanography and climate studies. However, the re35 construction of a given dynamic and the prediction of future states must take into ac36 count the uncertainties that affect the system. Thus, the available observational measurements are only provided with a limited accuracy. Besides, the encoded physical equa38 tions that model the evolution of the system do not capture the full complexity of the real system. Finally, the numerical approximation generates a non-negligible error. For these reasons, it seems relevant to calculate a probability distribution of the state system rather than the most probable state. Using recent advances in machine learning techniques for inverse problems, we propose an algorithm that jointly learns a parametric distribution of the state, the dynamics governing the evolution of the parameters, and a solver. Experiments conducted on synthetic reference datasets, as well as on datasets describing environmental systems, validate our approach.