  Variational data assimilation optimizes for an initial state of a dynamical
system such that its evolution fits observational data. The physical model can
subsequently be evolved into the future to make predictions. This principle is
a cornerstone of large scale forecasting applications such as numerical weather
prediction. As such, it is implemented in current operational systems of
weather forecasting agencies across the globe. However, finding a good initial
state poses a difficult optimization problem in part due to the non-invertible
relationship between physical states and their corresponding observations. We
learn a mapping from observational data to physical states and show how it can
be used to improve optimizability. We employ this mapping in two ways: to
better initialize the non-convex optimization problem, and to reformulate the
objective function in better behaved physics space instead of observation
space. Our experimental results for the Lorenz96 model and a two-dimensional
turbulent fluid flow demonstrate that this procedure significantly improves
forecast quality for chaotic systems.
