Debias Coarsely, Sample Conditionally: Statistical Downscaling through Optimal Transport and Probabilistic Diffusion Models

We introduce a two-stage probabilistic framework for statistical downscaling between unpaired data. Statistical downscaling seeks a probabilistic map to transform low-resolution data from a (possibly biased) coarse-grained numerical scheme to high-resolution data that is consistent with a high-fidelity scheme. Our framework tackles the problem by tandeming two transformations: a de-biasing step that is performed by an optimal transport map, and a super-resolution step that is achieved via a probabilistic diffusion model with a posteriori conditional sampling. This approach characterizes a conditional distribution without the need for 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; they 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, the procedure is faithful to the correct statistics of physical quantities, even when the low-frequency energy profiles of the inputs and the desired outputs do not match, a crucial but difficult-to-satisfy assumption by current state-of-the-art alternatives.