  Modern data-driven surrogate models for weather forecasting provide accurate
short-term predictions but inaccurate and nonphysical long-term forecasts. This
paper investigates online weather prediction using machine learning surrogates
supplemented with partial and noisy observations. We empirically demonstrate
and theoretically justify that, despite the long-time instability of the
surrogates and the sparsity of the observations, filtering estimates can remain
accurate in the long-time horizon. As a case study, we integrate FourCastNet, a
state-of-the-art weather surrogate model, within a variational data
assimilation framework using partial, noisy ERA5 data. Our results show that
filtering estimates remain accurate over a year-long assimilation window and
provide effective initial conditions for forecasting tasks, including extreme
event prediction.
