  Predictions of weather hazard require expensive km-scale simulations driven
by coarser global inputs. Here, a cost-effective stochastic downscaling model
is trained from a high-resolution 2-km weather model over Taiwan conditioned on
25-km ERA5 reanalysis. To address the multi-scale machine learning challenges
of weather data, we employ a two-step approach Corrector Diffusion
(\textit{CorrDiff}), where a UNet prediction of the mean is corrected by a
diffusion step. Akin to Reynolds decomposition in fluid dynamics, this isolates
generative learning to the stochastic scales. \textit{CorrDiff} exhibits
skillful RMSE and CRPS and faithfully recovers spectra and distributions even
for extremes. Case studies of coherent weather phenomena reveal appropriate
multivariate relationships reminiscent of learnt physics: the collocation of
intense rainfall and sharp gradients in fronts and extreme winds and rainfall
bands near the eyewall of typhoons. Downscaling global forecasts successfully
retains many of these benefits, foreshadowing the potential of end-to-end,
global-to-km-scales machine learning weather predictions.
