Combining machine learning and data assimilation to learn dynamics from sparse and noisy observations

The recent introduction of machine learning techniques in the field of numerical geophysical prediction has expanded the scope so far assigned to data assimilation, in particular through efficient automatic differentiation, optimisation and nonlinear functional representations.  Data assimilation together with machine learning techniques, can not only help estimate the state vector but also the physical system dynamics or some of the model parametrisations. This addresses a major issue of numerical weather prediction: model error.
I will discuss from a theoretical perspective how to combine data assimilation and deep learning techniques to assimilate noisy and sparse observations with the goal to estimate both the state and dynamics, with, when possible, a proper estimation of residual model error. I will review several ways to accomplish this using for instance offline, variational algorithms and online, sequential filters. The skills of these solutions with be illustrated on low-order and intermediate chaotic dynamical systems, as well as data from meteorological models and real observations.

Examples will be taken from collaborations with J. Brajard, A. Carrassi, L. Bertino, A. Farchi, Q. Malartic, M. Bonavita, P. Laloyaux, and M. Chrust.