Adaptive Flow Matching for Resolving Small-Scale Physics

We thank the reviewer for their remarks and for recognizing the extensive experimental validation of our work.

1. When we compare the visually generated images between SFM and CorrDiff, it seems CorrDiff is better. Can the authors provide more convincing explanations about the advantages of CorrDiff?

Response. While a single visual sample (e.g., Fig. 2) may suggest CorrDiff looks better, such examples can be misleading and not representative. A more robust comparison is provided in Fig. 3, where the spectral analysis shows that CorrDiff underperforms AFM at both low and high frequencies. Moreover, quantitative results in Table 2 demonstrate that AFM achieves better reconstruction accuracy and significantly improved uncertainty calibration. In the revised manuscript appendix, we have included another case in each of Figures 9 and 10 to shed more light in this comparison.

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2a. Though they obtain competitive performances and support theoretical proofs to demonstrate their method, it is not clear whether this method can be generalized or employed to other tasks, both physical science and nature image tasks. So this leaves a doubt about the wide application of this paper and I suggest the author can make some more demonstrations.

2b.I am also curious whether the proposed adaptive flow matching works for nature images, and I do expect to see such validated experiments to make this paper stronger.

2c.The applications to weather scenarios are limited, or we need to see large-scale validations to make this paper more convincing.

2d. How do the main contributions of adaptive flow matching from interpolating the latent from common flow-matching based nature image methods?

Response. Thank you for the suggestion. While our method was motivated by challenges in scientific data, it makes no domain-specific assumptions and is broadly applicable. To demonstrate generality, we include two diverse testbeds: (1) real-world weather downscaling using Taiwan’s CWA dataset, and (2) a synthetic PDE-based Kolmogorov flow dataset. Extending AFM to natural image tasks is a promising direction, but beyond the scope of this already extensive study. We have added this discussion to the conclusion of the revised manuscript.

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4a. Regarding Fig 2, I guess the SFM shall be the method proposed in this paper.

4b. SFM in Figures shall be well denoted, since we may think it shall be AFM.

Response. Thank you for noting this. We have fixed the AFM label in the figures of the revised manuscript.

We hope our responses and revisions have adequately addressed your remarks.

We thank the reviewer for their constructive comments. Below are our responses to your remarks.

1. In my opinion, VAE is a more natural choice ... (we shorten the questions due to the character limit)

Response. Thank you for the insightful question. As clarified in our Remark in Sec 4.2, while VAEs are a natural choice for stochastic encoding, we intentionally opt for a maximum-likelihood-based approach with a deterministic encoder and post-hoc noise injection. This choice is motivated by both simplicity and interpretability.

From a modeling standpoint, our approach yields a closed-form expression for the noise variance—simply the RMSE of the regression error—which enables straightforward tuning using validation data to control generalization. In contrast, VAEs require careful balancing of the KL term and reconstruction loss, which is known to be challenging and prone to issues such as posterior collapse and the prior hole.

From a physics perspective, particularly in applications like weather downscaling, large-scale structures are largely deterministic. A deterministic encoder is therefore well-suited to capture these coherent features, while the stochasticity in small scales is effectively modeled via calibrated noise.

While we agree that VAE-style encoders are a valid alternative, our design provides a principled and practical path that aligns well with physical intuition and avoids some known drawbacks of VAEs.

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2. The novelty of the paper is questionable...

Response. We respectfully disagree. While our method shares surface similarities with CorrDiff, AFM introduces key innovations that address fundamental limitations of CorrDiff and generalize it as a corner case.

Joint Training vs. Two-Stage: CorrDiff relies on a two-stage pipeline, first training a deterministic encoder, then fitting a diffusion model to the residuals. This separation causes overfitting in the encoder. In consequence, the residuals shrink and become uninformative, leaving the generative model. In contrast, AFM performs joint training of the encoder and the generative model, maintaining informative residuals throughout optimization.

** Uncertainty Modeling**: CorrDiff lacks an uncertainty-aware encoder. AFM introduces an adaptive noise injection mechanism that dynamically increases the encoder noise when the validation error rises, signaling overfitting. This prevents collapse and encourages the encoder to retain uncertainty.

** Channel-wise Adaptivity**: AFM performs per-variable noise scaling—crucial for scientific data where variables (e.g., temperature vs. radar reflectivity) exhibit different stochastic behaviors. CorrDiff applies uniform noise, which fails to capture such heterogeneity.

These design choices lead to significantly improved calibration and spread-skill performance (see Tables 2–3, 6, 13; Figures 5–6, 8).

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3. Lack of detailed explanation of noise scaling...

Response. Thank you for the remark. We have expanded our explanation of the noise scaling mechanism in Sec. 4.2, Appendix E, and Algorithm 1 of the revised manuscript.

Briefly, we estimate $\sigma_z$ using a maximum-likelihood criterion linked to the encoder's RMSE on validation data (see Eq. 8). As training progresses, if the encoder begins to overfit—reflected by rising validation error—our method increases $\sigma_z$, injecting more uncertainty into the latent space. This prevents the encoder from collapsing into a deterministic solution and ensures that the flow-matching model remains calibrated and robust to out-of-distribution inputs.

Unlike fixed or manually tuned noise levels, our approach adaptively adjusts $\sigma_z$ per variable (channel-wise), which is crucial for scientific data where different physical variables exhibit distinct noise scales and predictability (e.g., temperature vs. radar reflectivity, as shown in Fig. 5 and Fig. 8).

In practice, to avoid abrupt changes, we update $\sigma_z$ using an exponential moving average (EMA) every $M=10{,}000$ training steps $\sigma_z \leftarrow (1{-}\beta)\, \sigma_z + \beta\, \sigma_z^{\mathrm{cur}}$ The final $\sigma_z$ is fixed for inference. An ablation in Sec. 5.3 (Table 4) demonstrates the effectiveness of this adaptive noise scaling in improving ensemble calibration and spread-skill alignment.

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4. Too many typos...

Response. Thank you for noticing the typos. We have corrected them in the revised manuscript.

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5.Where is definition of $\lambda$?..."

Response. Thank you for your comment. Currently $\lambda$ is introduced in Sec. 4.3. It balances the deterministic regression and stochastic diffusion terms. In the limit $\lambda \rightarrow \infty$, AFM reduces to a purely deterministic model. We have added this clarification to the main text in the revised manuscript.

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We hope our responses and revisions have adequately addressed your concerns.

We thank the reviewer for their comments and for finding our paper interesting.

1. In the abstract itself, it is claimed that AFM achieves SOTA on regional downscaling. However, this is not the case. According to Table 2 and Figure 2, improvements over CFM are arguable. However, on Kolmogorov-Flow, task performance seems to be much better. But overall, it feels marginally better.

Response. We thank the reviewer for their remark. Our claim of “SOTA” is based on the consistent gains observed across datasets, channels and metrics, particularly in those with greater stochastic variability. In Table2, for instance, radar reflectivity sees a clear improvement when using AFM over CorrDiff, and AFM’s ensemble calibration (e.g., SSR) is superior across all channels. Similarly in Table 13 AFM outperforms all other models in most metrics and channels and has better calibration by a large narging.
Also, in the Kolmogorov Flow experiments (Table 3), AFM clearly outperforms CFM and CorrDiff and the gap become wider as the misalignment increases (tau 3 -> 10). While the margins may appear moderate on more deterministic channels (e.g., temperature), these results overall indicate that AFM consistently delivers better calibration and uncertainty modeling, which is crucial in scientific tasks. We clarify in the revised abstract that our method excels especially under higher stochasticity and improves overall ensemble calibration.

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2. As flow matching maps the any to any distribution, there should be a baseline that just does that along with traditional CFM. Basically, direct $\mathcal{X} \to \mathcal{Y}$ mapping. Similarly, other methods, such as Diffusion Schrödinger bridge matching,' could also be added. Because of these prior works (and missing baselines), I am unsure if the claim for Section 4 Line 149-152 is valid.

Response. Thank you for the comment. In our setting, direct $\mathcal{X} \to \mathcal{Y}$ flow matching is not applicable due to both spatial and channel misalignments—$\mathcal{X}$ and $\mathcal{Y}$ often have different modalities (e.g., $\mathcal{X}$ includes forecast features, while $\mathcal{Y}$ includes precipitation and radar reflectivity). As such, intermediate alignment via an encoder is necessary. The closest viable baseline is Conditional Flow Matching (CFM), which we include. Methods like diffusion Schrödinger bridges (e.g., I2SB) also assume aligned domains and are similarly inapplicable here.

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3. Paper often misrepresents the Flow matching as a stochastic process (Abstract Line 25), which is misleading. Meanwhile, stochasticity comes from noise term in eq. (5).

Response. Thank you for pointing this out. You are correct—flow matching is inherently deterministic (ODE-based), and the stochasticity in our method arises from the noise injected into the encoder output. We have clarified this distinction in the abstract of the revised manuscript.

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4.Typos in Figures 2,3 & 4 where SFM is written instead of AFM. “scientific” type in conclusion

Response. Thank you for noticing the typos, that are now fixed in the revised manuscript.

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5. What is the final $\sigma_z$ value during the inference? Is it the one result at the end of training (Figure 5)?

Response. Yes, that is correct. During inference, we use the final converged value of $\sigma_z$ obtained at the end of training, as shown in Figure~5.

We hope our responses and revisions have adequately addressed the remaining of your concerns.

We thank the reviewer for their positive and constructive feedback, and for recognizing our clear motivation and rigorous methodology.

1. In figure 2, 4: there is a confusion between the acronyms: SFM -> AFM

Response. Thank you for noticing the typo. In the revised manuscript, we have used AFM consistently.

2. The authors could be more specific when describing the 'misalignment' between the input / output. How this mis-alignment could be characterized: is it global or local, subpixel or few pixels scale?

Response. Thank you for the suggestion. We have clarified the notion of misalignment in the revised manuscript. It arises in two ways: (1) channel-level mismatch, where input and output contain different modalities (e.g., forecasts vs. radar reflectivity); and (2) spatial misalignment due to differences in the underlying PDEs between coarse and fine models. Spatially, the misalignment can be both global and local. For instance, in Fig. 7, storm centers in wind fields are displaced by several grid points (large-scale shift), while in Fig. 8, smaller PDE discrepancies ($\tau{=}3$) lead to subpixel-to-few-pixel level local misalignment. In the revised manuscript we have added a section G.3 discussing this in further detail and also adapted Section 5 to more clearly define misalignment.

3. I understand that the input low resolution data is first rescaled by bilinear interpolation to the size of the output. Is it correct?

Response. Thank you for the question. Indeed, we apply bilinear upsampling to align dimensions, then let the encoder refine and align further. This is explained in Section 5.1 of the manuscript (and in more detail in G.1 of the Appendix). In the revised version we added further clarifications regarding this in the main text (Sec 5.1).

3. Could you envisage different fields of application? Could this approach be useful for other use cases?

Response. Certainly. Our AFM framework applies wherever partial or misaligned data requires super-resolution or channel synthesis, e.g., MRI subsampling in medical imaging. We highlight broader applicability in the Conclusion of the revised manuscript.

We hope our responses and revisions have adequately addressed your questions.