  Data assimilation is a central problem in many geophysical applications, such
as weather forecasting. It aims to estimate the state of a potentially large
system, such as the atmosphere, from sparse observations, supplemented by prior
physical knowledge. The size of the systems involved and the complexity of the
underlying physical equations make it a challenging task from a computational
point of view. Neural networks represent a promising method of emulating the
physics at low cost, and therefore have the potential to considerably improve
and accelerate data assimilation. In this work, we introduce a deep learning
approach where the physical system is modeled as a sequence of coarse-to-fine
Gaussian prior distributions parametrized by a neural network. This allows us
to define an assimilation operator, which is trained in an end-to-end fashion
to minimize the reconstruction error on a dataset with different observation
processes. We illustrate our approach on chaotic dynamical physical systems
with sparse observations, and compare it to traditional variational data
assimilation methods.
