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

arXiv: https://arxiv.org/abs/2305.15618 · HTML: https://mlsanity.com/paper/2305.15618/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 40 of its 90 references that are in the map.
- First posted on arXiv 24 May 2023.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 40)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Contrastive Learning for Unpaired Image-to-Image Translation](https://mlsanity.com/paper/2007.15651/) (2020)
- [ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2108.02938/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Models](https://mlsanity.com/paper/2104.14951/) (2021)
- [Perception Prioritized Training of Diffusion Models](https://mlsanity.com/paper/2204.00227/) (2022)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Dual Diffusion Implicit Bridges for Image-to-Image Translation](https://mlsanity.com/paper/2203.08382/) (2022)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [EGSDE: Unpaired Image-to-Image Translation via Energy-Guided Stochastic Differential Equations](https://mlsanity.com/paper/2207.06635/) (2022)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [UNIT-DDPM: UNpaired Image Translation with Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2104.05358/) (2021)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)

## Built on by (most cited first, 30 of 33)

- [DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems](https://mlsanity.com/paper/2402.04467/) (2024)
- [Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation](https://mlsanity.com/paper/2311.07896/) (2023)
- [Fast, Scale-Adaptive, and Uncertainty-Aware Downscaling of Earth System Model Fields with Generative Machine Learning](https://mlsanity.com/paper/2403.02774/) (2024)
- [Generative Diffusion-based Downscaling for Climate](https://mlsanity.com/paper/2404.17752/) (2024)
- [Probabilistic Emulation of a Global Climate Model with Spherical DYffusion](https://mlsanity.com/paper/2406.14798/) (2024)
- [Thermalizer: Stable autoregressive neural emulation of spatiotemporal chaos](https://mlsanity.com/paper/2503.18731/) (2025)
- [Dynamical-generative downscaling of climate model ensembles](https://mlsanity.com/paper/2410.01776/) (2024)
- [Conditional diffusion models for downscaling and bias correction of Earth system model precipitation](https://mlsanity.com/paper/2404.14416/) (2024)
- [Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations](https://mlsanity.com/paper/2408.00157/) (2024)
- [Improving Flow Matching by Aligning Flow Divergence](https://mlsanity.com/paper/2602.00869/) (2026)
- [A probabilistic framework for learning non-intrusive corrections to long-time climate simulations from short-time training data](https://mlsanity.com/paper/2408.02688/) (2024)
- [RainShift: A Benchmark for Precipitation Downscaling Across Geographies](https://mlsanity.com/paper/2507.04930/) (2025)
- [Universal Diffusion-Based Probabilistic Downscaling](https://mlsanity.com/paper/2602.11893/) (2026)
- [Towards diffusion models for large-scale sea-ice modelling](https://mlsanity.com/paper/2406.18417/) (2024)
- [Towards Kriging-informed Conditional Diffusion for Regional Sea-Level Data Downscaling](https://mlsanity.com/paper/2410.15628/) (2024)
- [GEN2: A Generative Prediction-Correction Framework for Long-time Emulations of Spatially-Resolved Climate Extremes](https://mlsanity.com/paper/2508.15196/) (2025)
- [WIND: Weather Inverse Diffusion for Zero-Shot Atmospheric Modeling](https://mlsanity.com/paper/2602.03924/) (2026)
- [Multi-fidelity Reinforcement Learning Control for Complex Dynamical Systems](https://mlsanity.com/paper/2504.05588/) (2025)
- [Guided Unconditional and Conditional Generative Models for Super-Resolution and Inference of Quasi-Geostrophic Turbulence](https://mlsanity.com/paper/2507.00719/) (2025)
- [Controlling Ensemble Variance in Diffusion Models: An Application for Reanalyses Downscaling](https://mlsanity.com/paper/2501.14822/) (2025)
- [Wasserstein GAN-Based Precipitation Downscaling with Optimal Transport for Enhancing Perceptual Realism](https://mlsanity.com/paper/2507.17798/) (2025)
- [On Global Applicability and Location Transferability of Generative Deep Learning Models for Precipitation Downscaling](https://mlsanity.com/paper/2512.01400/) (2025)
- [Super-Resolving Coarse-Resolution Weather Forecasts With Flow Matching](https://mlsanity.com/paper/2604.00897/) (2026)
- [SwAIther-Precip: Lead-Time-Aware Bias Correction Enables Kilometer-Scale Downscaling of Global AI Precipitation Forecasts over Switzerland](https://mlsanity.com/paper/2605.16163/) (2026)
- [Generating time-consistent dynamics with discriminator-guided image diffusion models](https://mlsanity.com/paper/2505.09089/) (2025)
- [Generate the Forest before the Trees -- A Hierarchical Diffusion model for Climate Downscaling](https://mlsanity.com/paper/2506.19391/) (2025)
- [IPSL-AID: Generative Diffusion Models for Climate Downscaling from Global to Regional Scales](https://mlsanity.com/paper/2604.03275/) (2026)
- [Generative Unsupervised Downscaling of Climate Models via Domain Alignment: Application to Wind Fields](https://mlsanity.com/paper/2604.03341/) (2026)
- [Physical Fidelity Reconstruction via Improved Consistency-Distilled Flow Matching for Dynamical Systems](https://mlsanity.com/paper/2605.05975/) (2026)
- [Longwang: Zero-Shot Global Spatiotemporal Precipitation Downscaling with a Latent Generative Prior](https://mlsanity.com/paper/2605.17603/) (2026)

## Abstract (arXiv)

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.