  This paper addresses variational data assimilation from a learning point of
view. Data assimilation aims to reconstruct the time evolution of some state
given a series of observations, possibly noisy and irregularly-sampled. Using
automatic differentiation tools embedded in deep learning frameworks, we
introduce end-to-end neural network architectures for data assimilation. It
comprises two key components: a variational model and a gradient-based solver
both implemented as neural networks. A key feature of the proposed end-to-end
learning architecture is that we may train the NN models using both supervised
and unsupervised strategies. Our numerical experiments on Lorenz-63 and
Lorenz-96 systems report significant gain w.r.t. a classic gradient-based
minimization of the variational cost both in terms of reconstruction
performance and optimization complexity. Intriguingly, we also show that the
variational models issued from the true Lorenz-63 and Lorenz-96 ODE
representations may not lead to the best reconstruction performance. We believe
these results may open new research avenues for the specification of
assimilation models in geoscience.
