  The idea of using machine learning (ML) methods to reconstruct the dynamics
of a system is the topic of recent studies in the geosciences, in which the key
output is a surrogate model meant to emulate the dynamical model. In order to
treat sparse and noisy observations in a rigorous way, ML can be combined to
data assimilation (DA). This yields a class of iterative methods in which, at
each iteration a DA step assimilates the observations, and alternates with a ML
step to learn the underlying dynamics of the DA analysis. In this article, we
propose to use this method to correct the error of an existent, knowledge-based
model. In practice, the resulting surrogate model is an hybrid model between
the original (knowledge-based) model and the ML model. We demonstrate
numerically the feasibility of the method using a two-layer, two-dimensional
quasi-geostrophic channel model. Model error is introduced by the means of
perturbed parameters. The DA step is performed using the strong-constraint
4D-Var algorithm, while the ML step is performed using deep learning tools. The
ML models are able to learn a substantial part of the model error and the
resulting hybrid surrogate models produce better short- to mid-range forecasts.
Furthermore, using the hybrid surrogate models for DA yields a significantly
better analysis than using the original model.
