  We introduce a two-stage probabilistic framework for statistical downscaling
using unpaired data. Statistical downscaling seeks a probabilistic map to
transform low-resolution data from a biased coarse-grained numerical scheme to
high-resolution data that is consistent with a high-fidelity scheme. Our
framework tackles the problem by composing two transformations: (i) a debiasing
step via an optimal transport map, and (ii) an upsampling step achieved by a
probabilistic diffusion model with a posteriori conditional sampling. This
approach characterizes a conditional distribution without needing paired data,
and faithfully recovers relevant physical statistics from biased samples. We
demonstrate the utility of the proposed approach on one- and two-dimensional
fluid flow problems, which are representative of the core difficulties present
in numerical simulations of weather and climate. Our method produces realistic
high-resolution outputs from low-resolution inputs, by upsampling resolutions
of 8x and 16x. Moreover, our procedure correctly matches the statistics of
physical quantities, even when the low-frequency content of the inputs and
outputs do not match, a crucial but difficult-to-satisfy assumption needed by
current state-of-the-art alternatives. Code for this work is available at:
https://github.com/google-research/swirl-dynamics/tree/main/swirl_dynamics/projects/probabilistic_diffusion.
