  This paper is a contribution in the context of variational data assimilation
combined with statistical learning. The framework of data assimilation
traditionally uses data collected at sensor locations in order to bring
corrections to a numerical model designed using knowledge of the physical
system of interest. However, some applications do not have available data at
all times, but only during an initial training phase. Hence, we suggest to
combine data assimilation with statistical learning methods; namely, deep
learning. More precisely, for time steps at which data is unavailable, a
surrogate deep learning model runs predictions of the `true' data which is then
assimilated by the new model. In this paper, we also derive a priori error
estimates on this SVDA approximation. Finally, we assess the method by
numerical test cases.
