Data assimilation (DA) aims at forecasting the state of a
dynamical system by combining a mathematical representation of the
system with noisy observations taking into account their uncertainties.
State of the art methods are based on the Gaussian error statistics and the linearization of the non-linear dynamics which may lead to sub-optimal methods. In this
respect, there are still open questions how to improve these methods.
In this paper, we propose a \textit{fully data driven deep learning architecture}
generalizing recurrent Elman networks and data assimilation algorithms which
approximate a sequence of prior and posterior densities conditioned on noisy observations. By construction our approach can be used for general nonlinear dynamics
and non-Gaussian densities.
On numerical experiments based on the well-known
Lorenz-95 system and with Gaussian error statistics, our architecture achieves
comparable performance to EnKF on both the analysis and the propagation of probability density functions of the system state at a given time without using any explicit regularization technique.