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

Summary

This paper proposed a new two-stage method for statistical downscaling by combining a coarse de-biasing step based on optimal transport and a conditional up-sampling step based on a diffusion model.

Strengths

Overall, the paper is very well written and clearly explains the proposed method. The proposed method is divided into two components in a straightforward manner and is justified clearly. Empirical results confirm the applicability of the proposed method and its superior performance compared to several baselines such as cycle-GAN and ViT based super-resolution.

Weaknesses

The main novelty of this work is the introduction of a debiasing step for downscaling applications due to the biased nature of the problem. But Figure 3 shows the debiased result can be quite different from LR data. Because the diffusion model is very dependent on the debiased result, this debiasing step is crucial for the success of accurate downscaling in my opinion. Comparatively, cycle-GAN seems to output a more similar result to the LR data. So it seems necessary to discuss further if this debiasing step is indeed correcting the bias in LR data or introducing additional error/dissimilarity.

Questions

1. Using diffusion models for super-resolution has been widely studied, so it would be helpful to clarify if Section 3.2 is related to any relevant works such as SNIPS and DDRM? If so then please cite them in the paper. 2. What is Constraint RMSE in Table 2? And why is cycle GAN not included in this comparison? From my above comment, such a comparison could be helpful.

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

4 excellent

Contribution

2 fair

Limitations

The authors have addressed limitations of the work adequately. Broader impacts are not applicable for this work.

Summary

The authors proposed a two-stage probabilistic framework for unpaired data. The problem is factorized into two steps, an optimal transport (OT) based mapping for debiasing and a diffusion-based model for up-sampling. The problem is demonstrated on fluid mechanics datasets representing difficult fluid and weather problems. The predicted results matched the statistics of the physical properties well.

Strengths

The current paper developed a statistical downscaling framework for unpaired data. Tackling this problem is very crucial in learning from multi-scale, multi-fidelity models, especially for large-scale applications like weather/climate modeling. Originality: The idea of correcting the low-frequency bias by an OT map is novel and interesting. Moreover, the OT map is also integrated into the SOTA diffusion model framework. Due to the gradient information in diffusion modeling, a posterior conditioning sampling can further improve the performance at inference time and satisfy the given constraints well. The two-stage factorization is novel since it doesn't require a cycle-consistency type of loss and allows computing the debiasing map in a lower dimensional space. Quality: The numerical results show the developed model has a superior performance compared to the baseline models. The developed model also has the ability to provide reasonable uncertainty estimation. Moreover, comprehensive ablation studies and training details are provided to help evaluate the model. Clarity: The paper is well-written and clearly guides the reviewer to understand it. Math is accurate, and adequate proof and derivation are provided. Significance: The current work tackles the statistical downscaling problem in the weather/climate model. The ability to improve the accuracy of high-resolution forecasts from low-fidelity data has a broader impact in practical applications, like real-time weather forecasting.

Weaknesses

Some additional details and explanations are needed. See the question part.

Questions

1. What is the Reynolds number of the NS equations. 2. Are there any difficulties applying it to real-world turbulence data set? 3. In Figure 2 (a), what does the true trajectory look like? (2) What does the corrected in Figure 2(b) mean? Does it mean OT correction only? Moreover, it is hard to distinguish the OT+cDfn, corrected, true, and UncondDfn. Therefore, it is hard to see the advantage of OT+cDfn from the figure. 4. What is the training cost?

Rating

8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.

Confidence

5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.

Soundness

4 excellent

Presentation

4 excellent

Contribution

4 excellent

Limitations

The computational cost of OT mapping can be further reduced, which leads to a future research direction.

Summary

The authors suggest a simple approach for the problem of statistical downsampling, which is the super-resolution of low-resolution weather grids. The approach involves first "debias-ing" the low-resolution grid via solving an optimal transport problem, then obtaining a high resolution image by solving an image super-resolution problem with a score-based diffusion model.

Strengths

**Novel application of diffusion models.** The authors appear to apply diffusion modeling to a novel problem, even though the application appears to be straightforward. **Promising results.** According to the metrics provided by the authors, the results are promising. (However, I am concerned about the validity of the metrics.)

Weaknesses

**Unclear motivations.** Since this is predominantly a climate modeling paper submitted to a machine learning conference, the posed problem is certainly unfamiliar to me, and probably unfamiliar to most readers. The authors need to clearly describe the motivation of the problem. How can statistical downscaling ever succeed in reproducing the high fidelity, high resolution outputs if the model is so chaotic that the initial conditions do not even matter (Lines 32-34, and Footnote 1, Page 6)? **Validity of "debiasing".** The authors propose to reduce the discrepancy between low-resolution and high-resolution weather grids via optimal transport. This assumes some kind of continuity of the grids, i.e. similar weather systems in the lower dimensional weather grids produce similar simulated trajectories, that also correspond to similar weather systems in the higher dimensional weather grids. But a central assumption that necessitates the practice of using "unpaired" data is the "discontinuity" of the problem. Similar weather systems do not produce similar trajectories (Lines 32-34, and Footnote 1, Page 6). Therefore, is debiasing even possible? Moreover, is optimal transport the correct approach? **Concerns with empirical evaluations.** This paper suggests that the proposed method obtains better performance according to their metrics, which appear to measure the deviation of the modeled statistics with the ground truth statistics. This comparison seems somewhat unfair, since the proposed super-resolution model is trained solely by modeling the statistics of the ground truth data. No other method except for the CycleGAN implementation even attempts to similarly model the ground truth distribution $\mathcal{X}$, and even CycleGAN has auxiliary (i.e., cycle consistency) losses. This is corroborated by Table 1: only CycleGAN approaches the performance of the proposed method. No competing deep learning-based approaches to statistical downsampling are used in the comparison. Moreover, the metrics used for evaluation differ greatly from [1], which appears to be a well-cited paper in deep learning-based statistical downsampling. **Clarity in writing and formatting.** There are multiple places where the writing quality affects the readability of the text. 5: "tandeming" Tandem is not a verb and feels awkward here, consider using a different word? 14-16: "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." What does this sentence mean?
 21-22: "Consequentially, accurate predictions ... need to be *downscaled* from *coarser lower-resolution* models' outputs." (*Emphasis* mine.) Since this paper is submitted to a machine learning conference, where topics in computer vision are much more common than those in climate modeling, the authors need to clearly delineate where *downscaling* corresponds to statistical downscaling in the climate modeling sense, versus downscaling in the computer vision sense, especially when their meanings are completely reversed. I spent a lot of time trying to parse this sentence. Figure 1 and its caption is unclear. Y' is mentioned here but it is not explained until Section 3. Citation links do not work. [1] DeepSD: Generating High Resolution Climate Change Projections through Single Image Super-Resolution. https://arxiv.org/pdf/1703.03126.pdf

Questions

See "Weaknesses" section.

Rating

3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

2 fair

Contribution

2 fair

Limitations

See "Weaknesses" section.

Summary

This work introduces a new framework to tackle statistical downscaling, a climate science equivalent of super-resolution, in two steps: The first step removes the bias while staying at low resolution with an optimal transport method and the second step increases the spatial resolution with a diffusion-based model. The performance of the method is shown on two fluid-dynamics datasets: 2d Navier-Stokes and 1d Kuramoto-Sivanshinski equation. The proposed method outperforms baselines such as CycleGAN and ViT on all suggested metrics.

Strengths

Originality: This work develops a new method to tackle statistical downscaling Quality: Well written and well setup experiments. Clarity: Good motivation and explanation for the two-step approach tackling this problem. Significance: This work addresses an important and impactful problem and has high practical societal relevance. Especially, large upsampling factors like used here make the problem hard to solve and motivate the need for advanced methods like the one proposed here. The work addresses a probabilistic formulation of statistical downscaling, which is still under-researched.

Weaknesses

This is a great paper, but some weaknesses are: - The metrics used are not common in statistical downscaling. Very common metric that could be included is e.g. Continuous Rank Probability Score, see e.g. https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2022MS003120 for more commonly used climate downscaling metrics - No comparison with existing work like ClimAlign (https://arxiv.org/pdf/2008.04679.pdf), BCSD - References to some relevant works are missing: https://arxiv.org/pdf/2211.16116.pdf (stat. downscaling using diffusion models) - The work is motivated by climate/weather modeling but doesn’t include real climate/weather datasets. There are datasets available that include to different simulations at lower and higher resolution. It would be great to see how this method performs on real-world data. Datasets that could be used are NorESM data at different resolution or WRF simulations. Or learning the mapping from ERA-interim (https://climatedataguide.ucar.edu/climate-data/era-interim) to WRF (https://rda.ucar.edu/datasets/ds612.0/#!) data, like done in the ClimAlign work.

Questions

- It would be great to motivate the selection of metrics more and explain why other, more common metric like CRPS are not used here - A discussion at the beginning of the paper talking about different kinds of statistical downscaling setups such as perfect prognogsis (see e.g. https://www.cambridge.org/core/books/statistical-downscaling-and-bias-correction-for-climate-research/4ED479BAA8309C7ECBE6136236E3960F) - To be applied in climate modeling it is also relevant to know the inference runtime (not only training time). References: I would suggest including more of existing climate super-res/statistical downscaling work. A extensive collection of DL for statistical downscaling literature can be found here: https://github.com/paulaharder/deep-downscaling-overview Minor: Why are don't the first two rows in Table 2 have the best scores in bold? Typos: Figure 1 caption: "an invertible" instead of "a invertible" Line 52: "an unknown" instead of "a unknown" Line 129: "a structured" instead of "an structured" Line 141: "an unconditional" instead of "a unconditional" Figure 2 caption: "and and" typo Line 309: "s" missing of "sample"

Rating

8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

4 excellent

Contribution

4 excellent

Limitations

I think the limitations of this work could be made clearer. Either in an additional limitations section or in the conclusion should be a mention that the method has only been applied to an idealized datasets and not a real climate model dataset. Also, it should be mentioned that this work tackles a specific subfield of statistical downscaling and other subfields, eg. perfect prognosis where predictors differ

Thank you and rebuttal feedback.

We would like to thank again the reviewers for their time and reviews. They have been helpful to improve and refine the manuscript. Given that the discussion period is finishing soon, we are eager to hear any further comments, including feedback on the rebuttal,

Reviewer response

Thanks for the author’s detailed response and additional baselines and metrics results. After reading other reviews, I find the concerns are addressed satisfactorily by the elaboration and the empirical evaluation. The motivation is elaborated, and I was convinced that dealing with unpaired data is practical and challenging in climate systems. And the OT is a good approach to achieve debiasing and map data between different manifolds. The OT+diffusion can improve the sample quality compared with diffusion only. The strong empirical evidence showed that the proposed method outperforms all the baselines for versatile metrics. I believe it is a solid work and can bring insights into the ML+climate science community. Overall, I’d be happy to recommend a strong acceptance.