Elucidated Rolling Diffusion Models for Probabilistic Forecasting of Complex Dynamics

Thanks for your detailed review and insightful feedback. We're encouraged that you found our method to have strong empirical performance compared to the baselines.

• W1: "the contribution is somewhat limited, considering that is mainly consists of combining existing ideas for rolling diffusion with the EDM framework."

While we build upon established components, we respectfully argue that their successful integration into our ERDM framework is a non-trivial contribution tailored specifically for chaotic dynamical systems. The novelty lies in elucidating the specific adaptations required—which are not features of a standard EDM or RSDM—including a snapshot-dependent noise schedule, a bespoke temporal architecture, a specialized initialization strategy, and a tailored loss weighting.

As our ablation studies demonstrate (Appendix, Table 1), these carefully engineered components are crucial for the final performance, showing that a naive combination of existing ideas would not have been sufficient. This is complemented by strong empirical results.

• W2: "...sceptical about the word “Elucidating” in the title... I’m not convinced that this paper is “elucidating” how one should design diffusion model for forecasting in the best way... [it] promises more than what the paper delivers."

We thank the reviewer for this fair point regarding the paper's title and its perceived promise. The primary reason for the name is make the link to the Elucidated Diffusion Models (EDM) framework, from which our method is derived, explicit; an alternative such as "EDM Rolling Diffusion Models" would be redundant.

We agree that our paper is primarily methodological. Our use of "Elucidating" isn't meant to cover the entire design space, but to clarify the core principles—in architecture, initialization, and noise scheduling—needed to successfully adapt the EDM framework for dynamics forecasting.

• W3: "...the proposed temporal UNet model used for denoising seems to be a key component, and perhaps this deserves to be emphasized more..."

You're right, the temporal U-Net is essential to our method's success. We chose not to over-emphasize this specific architecture because our primary contribution is the more general ERDM framework, which is agnostic to the specific backbone.

Our temporal U-Net could be swapped with any other capable 3D or sequential architecture, which might even yield better performance (see also discussion with reviewer yaoq). The key architectural takeaway isn't our specific U-Net, but that a capable temporal backbone is necessary, as a naive, non-temporal model would fail. We don't want to tie the ERDM framework too closely to one architectural choice, but rather highlight the principles that make it work.

• Q1: "The proposed method has a higher computational cost (per NFE) than a standard EDM (...) does the observed improvement for ERDM primarily come from the fact that you effectively use W times the number of diffusion steps in ERDM...?"

This is a misunderstanding. ERDM actually needs less NFEs than EDM. However, each NFE is more costly for ERDM (due to the 3D vs. 2D denoiser architecture).

The key difference is:

• The standard autoregressive EDM must perform a full, expensive denoising schedule (with $N_{EDM} \approx 20$ steps) for every single time step it predicts.
• In contrast, ERDM's denoises in parallel. The first snapshots of the window start in a state that is already "almost denoised" and only needs a very small number of refinement steps ($N=2$ for our ERA5 experiments) to produce the next forecast before rolling forward.

To make this relationship perfectly clear and quantitative, we will add the following efficiency benchmark to our revised paper.

Legend: Efficiency metrics for ERDM vs. EDM. All measurements were performed on a single A100 GPU using mixed precision. Inference figures correspond to a 15-day (30 time steps), 5-member ensemble forecast with second-order Heun solver. Training uses a batch size of 1.

• Q2: "How are the forecasts produced with EDM, W=4? Does it also use rolling diffusion, or do you simply extract the first forecast in the window and throw away the rest?"

This baseline does not use rolling diffusion. It is a standard sequence-to-sequence EDM (using the same 3D denoiser arch. as ERDM) that operates autoregressively. Conditioned on a single frame (e.g., at time $t$), it jointly predicts a window of future frames ($t+1$ to $t+W$). Unlike ERDM, all frames in this window are denoised from the same initial noise level. To generate a long forecast, the last frame of the predicted window (at time $t+W$) is then used as the new condition for the next prediction step. We will clarify this distinction in the revised manuscript.

• Q3: "...could you not at least run the proposed method for both 2020 and 2021 and carry out two separate evaluations depending on the test sets for the different baselines?"

We couldn't evaluate our model on the 2020 test set as it would be an unfair comparison; our model's training data includes the year 2020. Re-training our model with a different data split just for this comparison wasn't feasible due to the high computational cost of a full training run.

• Q4: "What’s the point of having two weighting functions $\lambda(\sigma)$ and $f(\sigma)$? ... Could you simply replace this product with some function $g(\sigma)$ that you tune based on empirical performance?"

Thank you for this insightful question. Our formulation using the product of $\lambda(\sigma)$ and $f(\sigma)$ was deliberately chosen to maintain consistency with the EDM framework, from which our method is derived. $\lambda(\sigma)$ is adopted directly from EDM, while our novel weighting $f(\sigma)$ is an adaptation of the EDM training noise distribution.

We agree with your assessment that these could be merged into a single tunable function, $g(\sigma)$, and we believe exploring such alternative weighting schemes is a valuable direction for future work. We will add this clarification to the manuscript.

• Q5: "...Is this really a correct interpretation? ...I got the feeling that your proposed model could accurately simulate from the multi-step forecasting distribution $p(y_{t+W}|y_t)$ ... but I don’t believe that this is the case..?"

Thank you for this excellent question. We agree that a complete denoising of the window to $t+W$ is not performed at each step. Our statement about "modeling progressive uncertainty" pertains to the effect of the noise schedule on the joint denoising process itself.

By escalating noise for distant future snapshots, the framework compels the denoiser to treat information from these frames as less reliable. This process implicitly teaches the model that uncertainty grows over the forecast horizon. Therefore, the principle of increasing uncertainty is embedded within the model's learning mechanism. We will clarify this distinction in the revised manuscript.

• Q6: "Why do you have “weather forecasting” in the title when the method is more general...?"

We agree that our method is more general. We chose to emphasize "Weather Forecasting" in the title because it's the primary application in our experiments and because it motivated our work. We view the Navier-Stokes experiment as a valuable but simpler proof-of-concept, as it's a smaller-scale problem that shares similar underlying chaotic dynamics with weather.

Thank you for your thorough and constructive review. We are grateful for your positive feedback, particularly on our comprehensive ablations, which you noted help to clearly identify the most impactful components of our model. We also appreciate you recognizing the paper as well-written with a clear narrative, and the integration of RDM and EDM as a novel, timely, and relevant contribution to the field.

• W1: "Variance in Loss Weighting vs. Noise Sampling - The authors replace sampling σ from a log-normal distribution [...] with an explicit loss-weighting scheme. [...] incorporating weights into the noise sampling distribution often reduces variance, so the chosen loss-weighting strategy may be suboptimal from a training perspective."

We thank the reviewer for this insightful technical point. The potential difference in gradient variance between weighting the loss versus importance-sampling the noise distribution is a valid consideration. While we acknowledge the theoretical benefits of noise sampling, we found our chosen explicit loss-weighting scheme to be highly effective in practice, yielding the strong empirical results reported. We will add a discussion of this trade-off, along with the suggested reference, to the section on limitations and future work.

• W2: "First-window initialisation - [...] ERDM trains a separate “initial” model for the first rollout window. However, this still entails training and tuning a second network. [...] on large-scale datasets [...], maintaining two models would probably be more demanding than tuning some additional hyperparameters."

This is a valid point regarding the practicality of our initialization strategy. In the context of our study, this approach was practical as a strong pretrained model (our EDM baseline) was already available. However, we acknowledge this may not be generalizable. We will explicitly state in our new limitations section that the reliance on a readily available, strong pretrained forecaster is a limitation of our current approach, particularly for novel problem domains.

• W3: "Missing Baseline: Original RDM formulation - The paper [...] does not compare directly to the original DDPM formulation of RSDM from [2]. Demonstrating gains over a DDPM-based rolling model [...] would more convincingly attribute performance improvements to the EDM design choices..."

This is a valid point. Our rationale for not including a direct comparison to the DDPM-based RSDM is that the EDM framework is a direct generalization of DDPM. As shown in the original EDM paper (c.f. Table 1), EDM recovers the DDPM formulation for specific choices of its noise schedule and preconditioning.

Given the demonstrated success of the specific design choices proposed in the EDM paper, we focused our efforts on adapting these superior settings to the rolling context. While our ERDM framework inherits this generality and could be configured to replicate a DDPM-style RDM, we concentrated on what we believe to be the most promising formulation. We agree, however, that a direct empirical comparison is valuable and will note this as an important direction.

• W4: "Navier-Stokes - Only five test trajectories are evaluated... Moreover, I think the authors could include one or two additional baselines... For example, PDE-Refiner [3]... Another related approach is DyDiff [4]."

We thank the reviewer for these suggestions regarding the Navier-Stokes evaluation. Our use of five test trajectories follows the protocol established in the NeurIPS benchmark paper that introduced this dataset. While we agree that including additional baselines would be valuable, implementing and carefully tuning & evaluating new complex models is unfortunately hard to fulfill within the rebuttal period. We hope that the comprehensive ERA5 experiments serve as, complementary, compelling empirical evidence.

• W5: "..., the IFS ENS comparison is flawed because IFS ENS is verified against operational analyses rather than the ERA5 reanalyses, further undermining its relevance These mismatches leave the autoregressive EDM baseline as the only truly comparable benchmark."

You're right that we verify our model against ERA5 reanalysis while IFS ENS is verified against operational analysis. However, this isn't a flaw that favors our model—it's actually the opposite. This evaluation setup is the recommended best practice by the established WeatherBench-2 (WB2) benchmark, precisely because it's known to be more favorable to the IFS ENS baseline.

An external check of the official WB2 CRPS scores for "IFS vs. Analysis" versus "IFS vs. ERA5" will confirm that the former are consistently lower (better). Therefore, our comparison is intentionally conservative. Our model's competitive performance is achieved against a baseline that is being evaluated under its most advantageous conditions. We will make sure to clarify this in the paper.

• W6: "Moreover, the results are fairly hard to interpret in the way they are currently presented. I personally think that a better way of visualising and summarising these results is through score cards."

That's a great suggestion, thank you. We agree that score cards are an excellent way to visualize and summarize results.

Based on your feedback, we have now created a score card comparing ERDM vs. our key EDM baseline, and we will add it to the paper. Since we're not allowed to share images here, this is a high-level summary of it: The score card is predominantly "blue", indicating that ERDM outperforms EDM on the vast majority of variables and lead times, with minor exceptions for the geopotential field at 12 hours and for the q50 variable.

• W7: "Memory Requirements - The paper does not quantify the additional computational and memory costs of ERDM versus baseline EDM."

Excellent suggestion. We will add the following efficiency benchmark to our revised paper.

Legend: Efficiency metrics for ERDM vs. EDM. All measurements were performed on a single A100 GPU using mixed precision. Inference figures correspond to a 15-day (30 time steps), 5-member ensemble forecast with second-order Heun solver. Training uses a batch size of 1.

• Q1: "How dataset-dependent are the most reasonable choices for these hyperparameters? For example, in the context of Navier-Stokes, would a different temporal resolution require re-tuning of these hyperparameters...?"

In our experiments, $\rho$ was kept constant for both the Navier-Stokes and ERA5 datasets, suggesting a degree of robustness. The noise boundaries, $\sigma_{min}$ and $\sigma_{max}$, are more data-dependent, which is consistent with the standard EDM framework. We used the same $\sigma_{min}$ (0.002) for both problems, but $\sigma_{max}$ was set higher for ERA5 (500 vs. 200) to account for the larger dynamic range of its variables (e.g., humidity).

Based on our results across these heterogeneous datasets, our current settings provide a robust baseline. While exploring the precise dependency of the noise schedule on data characteristics is a valuable direction for future work, and further tuning could yield marginal improvements, we found these settings to generalize well.

• Q2: "Could the authors comment on why they haven't included an RDM-based baseline too (with the DDPM formulation)?"

See response to W3. Besides, the rolling diffusion model paper [2] doesn't provide open-source code.

• Q3: "Could you clarify how the sampling procedure looks like when N is not an integer?"

$N$ doesn't represent a literal number of discrete solver steps. Instead, it controls the amount of denoising per step, defined as $\Delta t = 1/N$. Our sampler takes discrete steps until an internal 'global diffusion time' accumulator crosses 1, at which point the window is rolled forward.

For example, when $N=1.25$, then $\Delta t = 0.8$. If the global time starts at 0.0, it takes two steps to reach 1.6, crossing the integer threshold and triggering a roll. The new global time becomes 0.6, etc. Thus, the number of actual sampling steps would always be 1 or 2 here.

• Q4: "How much of a limitation are the increased memory and computational complexity, with the vision that such a model could be used to model, for example, higher resolution weather data (0.25° or even 0.1° ERA5)?"

You're right to point out that the increased memory complexity from our 3D architecture is the primary bottleneck for scaling to higher-resolution data.

We are hopeful: While standard techniques like gradient checkpointing can trade speed for memory, we believe a more powerful solution is to integrate ERDM with latent space diffusion. Performing the diffusion process on a compressed representation would drastically reduce the memory and compute costs of the 3D backbone, making the framework much more scalable without sacrificing the benefits of the rolling formulation.

• Q5: "The SSR shows that ERDM goes from a region of underdispersion [...] to overdispersion [...] followed by underdispersion again. [...] Can the authors comment on this---what are the potential causes of overdispersion at medium timesteps, and whether the SSR could be stabilised through other hyperparameter choices?"

Great observation! Note that for ERA5 (Fig. 11 in the appendix) the SSR curves are qualitatively different and this behavior doesn't happen for EDM any more. Generally, we've found that the SSR can be "improved" (for ERDM, EDM and other models) simply by stopping the training earlier. This comes at a cost of accuracy, however. Because we consider the CRPS as the key metric, we avoid doing this in this paper.

Thank you again for your constructive feedback. We will certainly include these additional comments and results in our revised paper, as we believe that they significantly strengthen it.

W1 and W4: Navier-Stokes additional baselines

Upon your suggestion, we've trained two new baselines on the Navier-Stokes dataset: 1) ERDM VP, corresponding to our method, but using the VP schedule and preconditioning as in [45]; 2) PDE-Refiner [29]. PDE-Refiner excels for the first few timesteps of the rollout, but underpeforms ERDM in later stages of the rollout (being $\approx 2\times$ worse than ERDM at timestep $t=64$). We hope that these new baselines address your concerns about missing baselines. Please find a summary results table below (the first four rows correspond to a tabular version of Fig. 4 (left) in our paper).

Legend: CRPS results for five example lead times of the test rollout (best in bold, second best italized; $\times 100$ for clarity). New baselines: 1) ERDM VP: Replaces the EDM-based noise schedule and preconditioning in ERDM with the corresponding variance-preserving (VP) choices (see Table 1 in [23]), which is used by the RSDM from [45]; 2) PDE-Refiner [29] is a next-step diffusion-derived forecaster. The reported scores are attained after tuning of the two key HPs of PDE-Refiner, with its final values being K=6,\sigma_\min=0.002. The recommended values from [29], K=4,\sigma_\min=2e\text{-}7, did not work as well. We use the same denoiser architecture as for EDM. PDE-Refiner performs extremely well for the first few timesteps of the rollout, but rapidly diverges after that (potentially because as a next-step forecaster, it cannot model temporal interactions).

W4: We agree and will add this footnote, as requested.

W7 and Q4: This is a very valid concern. While strong latent spaces are developed, simple techniques like gradient checkpointing can be used to mitigate memory concerns in the meantime (at the cost of speed). As promised, we will include a new limitations paragraph that explicitly states all limitations of our method, including the ones we discussed with you. Please find our current draft for this new paragraph below.

Limitations. (new paragraph to be placed in the conclusion section of the main text)

While ERDM demonstrates strong performance, its practical application is constrained by several key limitations. Its hybrid 3D denoiser architecture is computationally demanding, making each forward pass more costly in memory and speed than the 2D networks used by models like EDM. This is offset by requiring a lower number of total neural function evaluations, such that the overall sampling time is comparable to next-step EDM forecasting. The high memory complexity presents a significant barrier to scaling to higher resolutions, but could be addressed with gradient checkpointing or, potentially, latent-space diffusion. This computational cost is coupled with a notable performance weakness on ERA5 in short-range forecasts compared to state-of-the-art external baselines like IFS ENS. ERDM depends on an external forecaster for initialization, a constraint that limits its applicability in domains where a suitable model is unavailable. Lastly, ERDM relies on an explicit loss weighting to compensate for the lack of training-time noise sampling found in EDM; while this design is critical for the model's performance, it may be suboptimal in practice (Dieleman, S. (2024). Noise schedules considered harmful).

Title: Thank you for bringing this up. Given the similar suggestion by reviewer jthi, and the generality of ERDM, we are planning to rename our paper to "Elucidated Rolling Diffusion Models for Probabilistic Forecasting of Complex Dynamics", unless another reviewer or the AC object.

We really appreciate all your feedback, please let us know of any remaining concerns or suggestions.

Thank you for your review and constructive feedback. We are encouraged that you found our core approach to be "compelling". We also appreciate you recognizing that our method is validated by a comprehensive series of experiments and that the paper is well-organized, clearly written, and easy for future work to build upon.

• W1: "The motivation of the paper appears to be unclear. It is not evident what limitations in existing methods the proposed techniques...are intended to address. The contributions seem more like engineering refinements rather than principled innovations."

Thank you for this feedback, which helps us clarify the core motivation and principled nature of our work. Our approach is designed to address key limitations of existing generative methods when applied to chaotic dynamical systems, particularly in data-scarce (non Internet-scale) scenarios. Standard video diffusion models often struggle with limited data, while next-step autoregressive diffusion models may not adequately capture non-stationarity and non-Markovianity. Our ERDM framework is proposed as a principled middle ground, using a rolling window approach to better model the dynamics.

Regarding our contributions, components like the weighted loss and initialization strategy are not mere engineering refinements but crucial, principled innovations required to make ERDM effective. For example, the weighted loss is essential for (1) correctly adapting the EDM training noise distribution to a sequential rolling setting and (2) is empirically vital for the model's performance, as validated by our ablation studies (Appendix, Table 1). The novelty lies in the synergistic integration of these carefully engineered components (including the noise schedule, initialization, and architecture), which are indispensable for adapting diffusion models to this challenging problem space.

• W2: "...there is a lack of objective quantitative tables to evaluate the performance of the proposed method compared to baseline approaches."

Thank you for this suggestion. While we believe our charts offer a more comprehensive view across all lead times—an approach common in the AI weather forecasting literature—we agree that a table serves as a valuable and objective reference. Therefore, for completeness, we provide the requested table below and will add it to the Appendix of our revised manuscript.

Tabular CRPS for ERA5

Legend: CRPS (lower is better) for 10m u-component wind (u10m) and 500 hPa geopotential (z500) at lead times of 3, 7, and 14 days.

• W3: a) "The paper lacks an analysis of its limitations." b) " Moreover, the experiments should include measurements of inference time and computational overhead..."

a) Thank you for this valuable feedback. We agree that a dedicated discussion of limitations and computational overhead will significantly strengthen the paper.

In our revision, we will add a new Limitations section to the conclusion. This section will consolidate and expand upon several points. We will reiterate the key limitation already noted in our conclusion—the higher memory footprint and cost per function evaluation of our 3D architecture compared to 2D alternatives. Furthermore, we will explicitly discuss other limitations of the current work, including:

1. The scope of our ERA5 experiments, which are not performed at the highest 0.25° resolution.
2. The framework's reliance on a pretrained forecaster for initialization.

b) Excellent point. To address the need for concrete efficiency metrics, we'll include the following efficiency benchmark in our revised paper.

Legend: Efficiency metrics for ERDM vs. EDM. All measurements were performed on a single A100 GPU using mixed precision. Inference figures correspond to a 15-day (30 time steps), 5-member ensemble forecast with second-order Heun solver. Training uses a batch size of 1.

• Q1: "The authors should provide a clearer explanation of the rationale and applicability of diffusion models for this task."

The rationale for using diffusion models is that they often represent the state-of-the-art for probabilistic forecasting, which is essential for two primary reasons in the context of chaotic dynamical systems. First, their ability to generate sharp, diverse ensembles is crucial for robust uncertainty quantification. Second, they produce high-fidelity samples that are more physically consistent than the blurry, averaged-out predictions often generated by deterministic models trained with mean-squared error. We will add this clarification to the revised manuscript. See our Diffusion models for spatiotemporal data related work section for relevant references.

• Q2: "In the Rolling EDM noise schedule, the authors identify the setting of the $\rho$ as particularly important. However, they only provide a simple empirical value without discussing its optimality or offering a thorough analysis."

The insight guiding our approach is that while progressive noise is necessary, an excessively large $\rho$ destroys the temporal information that is crucial for the sequence model to learn dependencies between frames (see Fig. 2). As detailed in our ablation studies (Appendix, Table 1), the model's performance is robust across a wide range of small $\rho$ values (e.g., -5 to -30). Note that we use the same $\rho$ for Navier-Stokes and ERA5 experiments. We will add this detailed analysis to the revised manuscript.

• L1: "The authors could provide a more thorough discussion of the societal limitations and potential impact of their work."

We discuss the broader impact of our work in Appendix A. Is there anything specific that the reviewer is missing there?

Thank you for your thoughtful and constructive review, and for recognizing our model's improved performance over strong baselines, as well as its ability to effectively capture long-range dependencies through an efficient sampling process.

• W1: The short-range performance on ERA5 data is inferior to competitors, and the provided explanation "needs further experiments to validate."

Thank you for this excellent point. We agree that our short-range performance on ERA5 data merits further discussion and that suboptimal initialization is likely only one contributing factor.

We believe a more significant factor is the denoiser architecture. For both the Navier-Stokes and the ERA5 experiments, we used the same general-purpose 3D U-Net. Our method performs extremely well on the simpler Navier-Stokes problem, where designing a bespoke architecture is not as critical. However, for a complex system like ERA5, state-of-the-art performance is increasingly tied to specialized architectures (e.g., graph networks or transformers) that carefully model the spherical structure of the Earth, and its various short- and long-range dependencies. The focus of our paper was on introducing the ERDM framework for sequential probabilistic forecasting, rather than designing a new SOTA architecture for weather.

We are confident that future work extending current SOTA architectures like GraphCast, Aurora, or Pangu-Weather to ingest temporal data, for use within our proposed ERDM framework, would significantly advance weather forecasting performance. We will clarify this distinction in the revised manuscript to better contextualize our results.

• W2: It is "unclear how the window size $W$ affect the performance," and the choices for $\sigma_{min}$ and $\sigma_{max}$ require more discussion.

Thank you for raising this important point. We have performed an ablation study on the window size $W$, $\sigma_{min}$, and $\sigma_{max}$, with the results presented in Table 1 of the Appendix. We agree that these findings should be highlighted more clearly.

The ablation shows no significant sensitivity to these hyperparameters within a reasonable range. We only observed a degradation in performance for overly large window sizes, which makes the current backbone architecture less efficient for feature extraction. In our revision, we will add a sentence to the main text to explicitly summarize this and direct the reader to the detailed ablation study in the Appendix.

• W3: Latency that limits "immediate operational adoption," despite the model's relevance to other scientific domains.

Thank you for this very relevant comment on the practical implications of sampling latency. We agree with your assessment. If minimal inference latency is the single most important criterion, our model—like other iterative diffusion models such as EDM—may not be the preferred choice.

In the context of operational weather forecasting, our model (like most other AI-based models) is orders of magnitude faster than incumbent physics-based systems like IFS ENS. Thus, for many applications in this domain, inference speed is not the primary limiting factor for adoption.

Furthermore, this sampling speed issue could be alleviated via established techniques, such as diffusion in a compressed latent space (common in video generation), to significantly reduce the inference cost in future work.

• W4: The conceptual novelty is "incremental" as the model uses existing building blocks.

While we build upon established components, we respectfully argue that their successful integration into our ERDM framework is a non-trivial contribution tailored specifically for chaotic dynamical systems. The novelty lies in elucidating the specific adaptations required—which are not features of a standard EDM or RSDM—including a snapshot-dependent noise schedule, a bespoke temporal architecture, a specialized initialization strategy, and a tailored loss weighting.

As our ablation studies demonstrate (Appendix, Table 1), these carefully engineered components are crucial for the final performance, showing that a naive combination of the building blocks would not have been sufficient.

• Q1: "Since the authors mentioned the inferior performance of the ERDM is due to the EDM prediction initialization, could the authors replace the current initialization with IFS ENS or deterministic model (e.g. FourCastNet) and repeat the experiment? to close the gap"

Thank you for this excellent suggestion for a follow-up experiment.

It would certainly be possible to initialize our ERDM framework with predictions from a different model. However, we intentionally chose to use our next-step EDM baseline for initialization to ensure a carefully controlled and fair comparison. Our goal was to isolate and benchmark the specific advantages of the proposed ERDM framework against a baseline trained with the exact same setup. Using a more powerful, external model for initialization would confound the analysis, making it difficult to determine whether performance gains came from our framework or the superior initialization.

As we discussed in our response to W1, we believe that another significant factor for the short-range performance gap is the general-purpose 3D U-Net architecture, which is not as specialized for complex weather data as SOTA architectures.

• Q2: a) "The noise level for the first snapshot within the window is small, does this mean it takes fewer solves to fully denoise the first snapshot?" b) "Does this have any implications of the number of NFEs compared to other models?" c) "have you profiled whether the 3 D attention layers or memory bandwidth dominate? An ablation with a lighter temporal backbone would clarify where to optimize."

a) Yes, the first snapshot of the first window is emitted after $N$ sampling steps (recall, $N=2$ for ERA5).

b) Indeed, our ERDM framework is significantly more efficient in terms of the NFEs compared to the autoregressive EDM baseline. To generate a 30-step (15-day) forecast for our ERA5 experiments with a Euler solver, EDM requires 20 solver steps for each forecast, totaling $20 \times 30 = 600$ NFEs. In contrast, ERDM has a cost of only $N=2$ solver steps per forecast, resulting in just $2 \times 30 = 60$ NFEs (excluding a one-time initialization cost). These NFE counts double when using a second-order Heun solver, but the 10x relative efficiency of ERDM is maintained.

c) We experimented with a lighter 3D U-Net architecture by removing temporal attention from the shallower blocks. However, this modification resulted in negligible memory savings.

• Q3: "How does the choice of window size and accordingly σmin and σmax affect the performance?"

See response to W2 above.

• Q4: "What is the energy cost per 15 day ensemble forecast relative to IFS ENS, measured on the same hardware? This would strengthen the efficiency claim and inform practitioners."

A direct energy comparison on the same hardware is impractical, as IFS ENS runs on specialized supercomputers, not GPUs. Nevertheless, the trend of AI models being vastly more energy-efficient is well-established; ECMWF's AI model, for example, is cited as being 1,000x more efficient than IFS ENS. Our model aligns with this trend, generating a 15-day, 5-member ensemble forecast in just 209 seconds on a single A100 GPU.