by Tobias Bischoff and Katherine Deck
Climate simulations play a crucial role in understanding and predicting climate change scenarios. However, the spatial resolution that simulations can be carried out with is often limited by computational resources to around ~50-250 km in the horizontal. This leads to a lack of high-resolution detail; moreover, since small-scale dynamical processes can influence behavior on larger scales, coarse resolution simulations can additionally be biased compared to a high-resolution “truth”. For example, simulations run at coarse resolutions fail to accurately capture important phenomena such as convective precipitation, tropical cyclone dynamics, and local effects from topography and land cover. Consequently, these simulations have a limited ability in making predictions on regional and sub-regional scales, especially for extreme temperatures and precipitation rates.
To overcome these challenges, downscaling techniques have been developed to enhance resolution and correct biases in fluid and climate simulations. Traditional downscaling approaches, like bias-correction spatial disaggregation, typically do not incorporate contextual information, like topographical datasets, and additionally lack multivariate capability. On the other hand, generative models developed by the machine learning community have demonstrated success at super-resolution and domain translation tasks involving multiple variables and can readily make use of contextual information. These models learn statistical relationships between datasets and are now being used for downscaling fluid simulations and climate data because they do not require paired datasets and can capture the correlations between state variables and spatial locations. In our work, we have pursued using diffusion models, a type of generative model, for the downscaling task [1, 2]. A distinct advantage of diffusion models relative to other generative models is that, once trained to generate climate data, they can be repurposed to generate new, realistic high resolution “data” using conditional sampling, providing flexibility in exploring specific features of interest.
We explore a novel approach for unsupervised downscaling of fluid simulations using diffusion bridges. We exploit the fact that coarsely resolved and highly resolved climate simulations share statistical similarities on large scales while differing on smaller scales. We employ a two-step process, where a prescribed forward diffusion process erases information on small scales, while a second learned reverse diffusion process allows us to reintroduce fine-scale features. This approach enables the generation of high-resolution information from low-resolution inputs without customization for specific translation tasks: during training time, paired samples of high- and low-resolution climate data fields are not necessary.
To test this algorithm, we first generated low- and high-resolution simulations of 2D turbulent flow, including a moisture variable. We additionally added to the equations a “context” term to mimic the role that topography plays in driving precipitation. We then trained a diffusion model on the high-resolution data, across all contextual “sites”, and then used the diffusion bridge to sample high-resolution flow fields conditional on the contextual information and the input low-resolution image (see  for additional details).
In Figure 1, the results of downscaling using diffusion models on the moisture field of the 2D fluid simulation are shown. As described above, initially noise is added to eliminate features except those on the largest scales (first four panels on the left), and then generated features are reintroduced on small and intermediate spatial scales (fourth panel to seventh panel). By not fully noising the initial snapshot, the resulting downscaled image contains large-scale features similar to the ones in the original snapshot. Additionally, this demonstrates how the downscaling procedure can be applied to the low-resolution supersaturation tracer field by conditioning on site-level contextual information (shown in the second to rightmost column). This means that known high-resolution contextual information can be introduced during the denoising step such that the generated snapshot exhibits a distinct signal in the fluid flow. For comparison, the rightmost column displays randomly chosen data samples at the same resolution as the generated sample.
To assess how well the diffusion bridge model works for downscaling with contextual information, we used several metrics. In the left panel of Figure 2 we check if small scale spectral information is captured in a statistically faithful way. We find that the diffusion bridge method can fill in small scale structures to statistically match the real high-resolution data. In the center panel of Figure 2 we compare the spatial mean statistics of the generated samples with the spatial mean statistics of the real high-resolution data. Indeed, we find that the distribution of spatial means is transformed using the diffusion bridge so that it is more consistent with the high-resolution data.
Finally, in the right panel of Figure 2, we show the distribution of an idealized condensation rate metric and focus on the right tail of the condensation rate distribution (the vertical line represents the 99th percentile); this allows us to gauge how well extreme condensation events are represented. We find that there is strong lift in the tails so that extreme event probabilities in the generated high-resolution data almost match the extreme event probabilities in the real high-resolution data (see  for additional details).
Diffusion models offer unique advantages compared to traditional downscaling methods, including the ability to generate samples, the ability to work with unpaired data, and the reusability of trained models. While the initial findings for using diffusion bridges for downscaling are promising, further validation of the methodology is necessary. In this context, it will be crucial to apply the method to a more realistic dataset, such as reanalysis or data from realistic atmospheric simulations. In doing so, we will be able to assess the performance in practical scenarios and ensure its applicability beyond simpler test cases. Additionally, it will be important to further investigate how well diffusion-based downscaling methods capture extreme events. Such events, characterized by their intensity and rarity, play a significant role in understanding climate dynamics and assessing climate risks. Better modeling them will contribute to understanding climate change impacts and enable better preparation and mitigation strategies for extreme events in the future.
 Bischoff & Deck, Unpaired Downscaling of Fluid Flows with Diffusion Bridges (2023; submitted) arXiv 2305.01822
 Deck & Bischoff, Easing Color Shifts in Score-Based Diffusion Models (2023; submitted) arXiv 2306.15832