Perceptually Constrained Precipitation Nowcasting Model

We appreciate the reviewer’s detailed feedback. We will address their concerns and eager to engage in a more detailed discussion with the reviewer.

Q1.

Thank you for the comment. In precipitation nowcasting, the high uncertainty in short-term evolution means a single historical observation can correspond to multiple future scenarios. ConvLSTM minimizes the mean squared error loss$\mathcal{L}_{pe}=\mathbb{E}\left[\|Y-\hat{Y}^*\|^2\right]$, causing predictions $\hat{Y}^*$ to converge to the conditional expectation $\mathbb{E}[Y|X]$. This produces a posterior mean sequence – a statistical average of all possible future precipitation outcomes under given input conditions.

Q2 & W1

In Section 3 and Appendix A, we outlined the gradient-stopping operation. To clarify: Due to diverging learning objectives between the precipitation estimator and the rectified flow model, gradient stopping is applied to prevent the rectified flow model from interfering with the estimator's acquisition of physical motion dynamics. This constraint forces the rectified flow model to actively learn physical consistency (e.g., motion continuity and distribution alignment) directly from input data. Meanwhile, end-to-end training improves model robustness and alleviates suboptimal solutions inherent in two-stage frameworks.

Regarding independence assumptions, our approach aligns with Freirich et al. (2021) in theoretical framework but diverges in the training methodology. Under the condition of stopping gradient propagation from $\hat{Y}^*$ to $\hat{Y}$, the generation of $\hat{Y}$ does not influence $\hat{Y}^*$. This establishes a Markov process: $\hat{Y} \leftarrow \hat{Y}^* \leftarrow X \rightarrow Y$. Consequently, given historical data $X$, $Y$ remains independent of both $\hat{Y}$ and $\hat{Y}^*$, while $\hat{Y}$ depends solely on $\hat{Y}^*$. This ensures compliance with the predictive independence assumption with theoretical justification, achieving causal decoupling between variables.

Q3 & W2.

Thank you for the comment. Our model employs mixed-precision training (FP16) on a single NVIDIA A100 80GB GPU, with supporting hardware including an Intel(R) Xeon(R) Platinum 8350C CPU @ 2.60GHz. Resource utilization metrics of our method compared to the ConvLSTM on the SEVIR dataset are summarized in the following Table.

Q4.

To understand the issue, we can see Figure 9 that short-term predictions are relatively easier due to closer alignment between predicted and real frame distributions, while long-term predictions suffer significant distribution drift - a discrepancy amplified by frame sampling strategies that prioritize learning long-term variations, thereby compromising short-term prediction accuracy. As shown in the Figure 9, the issue can be alleviated by using a moderate small $k$. However, a smaller $k$ may reduce long-term prediction accuracy. Hence, it is a trade-off.

Q5.

We present the experimental results of different LPIPS loss weight configurations in Table 4, Table 5, and Figure 6. These demonstrate that incorporating the LPIPS loss effectively suppresses checkerboard artifacts in the rectified flow model. Notably, within a reasonable range(0.5~1), the specific weighting configurations do not significantly affect the experimental outcomes. Detailed results for additional weight configurations will be supplemented as the table below.

---

Q6.

The experimental results and visualisations (Tables 4 and 6; Figures 5 and 7-9) show our exponential sampling strategy, where K determines the range of variation of the sampling probability. Figure 5 shows the k-probability relationship, while Table 4 evaluates different k settings. Figure 7 confirms larger k improves distant frame prediction accuracy. Figure 8 compares linear and exponential sampling. Experiments show that with more iterations, moderate increases in distant frame sampling probability improve long-term prediction. However, over-amplification undermines learning of other frames, causing performance drops.

We are grateful for the reviewer's acknowledgment of our work ​and their detailed feedback, which will help us refine our research.

Theoretical Claims.

Equation (2) can be solved through either Equation (20) or our proposed method, which has different error bounds. Freirich et al. established and showed that the theoretical optimal solution for Equation (20) is 2MMSE. Here we show the error bound of our method is smaller than 2MMSE by proving \\mathbb{E}\\left[\\left\\|\hat{Z}_1-\\hat{Y}^*\\right\\|^2\right] \\leq \\mathbb{E}\\left[\\left\\|Y-\\hat{Y}^*\\right\|^2\\right] in Appendix A. Since $\\hat{Z}_1$ is the final output ​under the independence assumptions, we can get the conclusion by using the following equation:

\\begin{aligned} \\mathbb{E}\\left[\\left\\|Y-\\hat{Z}_1\\right\\|^2\\right] & =\\mathbb{E}\\left[\\left\\|Y-\\hat{Y}^*\\right\\|^2\\right]+\\mathbb{E}\\left[\\left\|\\hat{Z}_1-\\hat{Y}^*\\right\|^2\\right] \\ & \\leq 2 \\mathbb{E}\\left[\\left\\|Y-\\hat{Y}^*\\right\\|^2\\right]=2MMSE\\end{aligned}

W1 & S3.

Thank you for the advice. We incorporate additional experiments with CasCast and reported the pooled CSI prediction results as shown in the following table. The experimental results further validate the effectiveness of our method. The results will be updated in the revised manuscript.

W2 & S2.

Thanks for your careful review. We have reviewed Appendix C and corrected the experimental results in Tables 5–6, which will be updated as follows:

S1.

Thank you for identifying this issue. The precipitation estimator ​reconstructs the input 13 frames and ​predicts 36 future frames, resulting in a total sequence length of 49. To eliminate ambiguity, we will revise Table 1 as follows:

Thanks for the reviewer's valuable suggestions. We will try to address the reviewer's concerns and are eager to engage in a more detailed discussion with the reviewer.

Theoretical Claims.

Thank you for pointing out this issue. We perform ​normalization (not binarization) to rescale images to [0, 1]: SEVIR (0–255) is divided by 255, and MeteoNet (0–70) by 70. We will correct the 'binarized' with 'normalized' in the revised manuscript.

Experimental Designs Or Analyses 1

We would like to clarify that unlike diffusion models reconstructing precipitation predictions via conditional integration, our method employs end-to-end learning to directly optimize the posterior mean sequence distribution from the precipitation estimator. To enhance this framework, we introduce two key components: ​temperature-weighted scaling and ​LPIPS perceptual loss. Comprehensive ablation studies demonstrate: (1) LPIPS regularization successfully suppresses checkerboard artifacts in rectified flow, enhancing visual coherence(Tables 4 & 5); (2) Temperature weighting significantly improves long-term frame prediction accuracy(Tables 4 & 6); (3) The rectified flow module achieves exceptional modeling of data distributions, generating meteorologically plausible precipitation patterns that effectively address issues such as high echo attenuation and missing details(Tables 5, Figure 4 & 6).

To further compare the performance of diffusion models and Rectified Flow in precipitation prediction, we conducted experiments by replacing the Rectified Flow module with a diffusion model. Specifically, due to the instability caused by adapting end-to-end training to diffusion models, we first constructed a pre-trained precipitation estimator. While keeping other configurations unchanged, we then utilized noise and predicted frame as inputs for diffusion modeling during the frame sampling process. Additionally, we employed CasCast as a baseline comparison. CasCast is a non-end-to-end precipitation prediction framework where its first stage originally uses a Vision Transformer (ViT) for precipitation estimation, followed by a second stage that applies diffusion models for distribution refinement. In our implementation, we replaced CasCast's ViT-based precipitation estimator with a ConvLSTM model. The experimental results, presented in the following Table, further validate the effectiveness of the rectified flow model. The results will be supplemented in the revised manuscript.

Experimental Designs Or Analyses 2

These materials will be supplemented in the revised manuscript.

W

Our experiments have included SOTA methods DiffCast (CVPR 2024), Earthfarseer (AAAI 2024). Following your suggestion, we also compare our method with CasCast (ICML 2024) as shown in above tables and the results will be updated in the revised manuscript. This ensures necessary comparisons with 2024 conference benchmarks.

[1] Yu, D., Li, X., Ye, Y., Zhang, B., Luo, C., Dai, K., Wang, R.,and Chen, X. Diffcast: A unified framework via residual diffusion for precipitation nowcasting. In The IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024

[2] Wu, H., Liang, Y., Xiong, W., Zhou, Z., Huang, W., Wang,S., and Wang, K. Earthfarsser: Versatile spatio-temporal dynamical systems modeling in one model. In Proceed-ings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 15906–15914, 2024.

[3] Gong, J., Bai, L., Ye, P., Xu, W., Liu, N., Dai, J., Yang,X., and Ouyang, W. Cascast: Skillful high-resolutionprecipitation nowcasting via cascaded modelling. In International Conference on Machine Learning, pp. 15809–15822. PMLR, 2024

S

Thank you for identifying this inconsistency. We will correct the description from "blue box" to "black box" in the revised manuscript.

We thank the reviewer for recognizing our ideas and theory.

W1

Thank you for your question. Due to the introduction of perceptual constraint, our model has the advantage of accurately preserving high-value part in prediction image, which indicates extreme weather storms. As shown in Figures 4 and 13, our model accurately predicts the evolution of the heavy precipitation band (above 160) and gives reliable intensity estimates.In spite of the advantage, our model may also fail to predict sudden convective storms that develop precipitation abruptly where no storm signals appear at the beginning. Improving such predictions requires incorporating atmospheric variables including temperature, humidity, and wind patterns during precipitation formation, which is a key objective for our subsequent research. We will add necessary discussions in our final version.

W2

We have elaborated on the impact of weight configurations for loss functions and distance sampling (Tables 4-6) in both the experiments and appendices. For other hyperparameters (e.g., learning rates), we identified appropriate values within the range of 1e-3 to 1e-5 and documented them in the main text(Section 5.1 Implementation Details). All hyperparameters in the model have been thoroughly specified, and the experimental code will be made publicly available on a community platform shortly as well.