We are pleased that the reviewer recognized the novelty of our physics-guided framework and the rigor of our experimental validation. We also sincerely appreciate the constructive feedback regarding the experimental demonstrations, which have been instrumental in improving the quality of our manuscript. The reviewers' comments are shown below in italicized font, and our responses are presented in regular font. We hope that these answers will meet with your approval.

Comment 1: The task operates under a multi-task learning framework. The network structure involves shared and task-specific components. How do changes in the ratio of shared-to-task-specific components (e.g., the degree of parameter sharing) affect model size and performance metrics?"

Response 1: Thank you for your valuable comments. In our work, to address various distribution shifts, we propose IDOL, which characterizes the distribution of feature embeddings to model both task-shared and task-specific invariant identity tokens. Unlike previous multi-task learning frameworks that typically separate shared and specific components from backbone features, our approach adopts physical priors to regulate the distributions of the features to learn physical invariance. Specifically, the original feature distributions are guided by physical priors and are then separately transformed into task-shared and task-specific identity tokens.

However, to explore how different degrees of parameter sharing affect model size and multi-task performance, we conducted supplementary experiments by varying the parameter ratios between task-shared and task-specific identity tokens. Specifically, we define the ratio based on the dimensional size of the corresponding identity token, with configurations including 1:1, 1:2, 1:3, 2:1, and 3:1. For example, a ratio of 1:2 means that the task-shared identity token has half the dimensionality of the task-specific identity token. The results are presented below.

Note: Wind, Inner-Core, and Outer-Core refer to the estimation tasks of wind speed (m/s), inner-core size (nmi), and outer-core size (nmi), respectively.

In summary, as shown in the table, under the given model design and experimental settings, varying the parameter ratio between task-shared and task-specific identity tokens has slight impact on model size and overall estimation performance. This is mainly because the proposed identity tokens, even with lightweight configurations, are sufficient to impose physical constraints on the feature distributions, thereby guiding the model to effectively capture the intrinsic characteristics required for multi-task learning.

Increasing the dimensionality of a specific identity token primarily enables more fine-grained representation learning, which may lead to slight variations in task-wise performance, but does not result in significant overall performance gains. This further confirms the effectiveness and efficiency of the proposed identity token design in ensuring consistent and robust multi-task learning.

Comment 2: Physical priors (Holland model) are central to IDOL but lack ablation on their necessity.

Response 2: Thank you for your thoughtful review. To further validate the contribution of the Holland model in modeling task-specific identities, we conducted supplementary experiments in which the priors is replaced with learnable linear layers. The corresponding results, which will be included in the revised manuscript, are presented as follows:

Note: Wind, Inner-Core, and Outer-Core refer to estimation tasks of Wind Speed (m/s), Inner-Core Size (nmi), and Outer-Core Size (nmi) errors, respectively. $\mathbf{id}_\mathtt{sp}$ refers to the proposed task-specific identitiy tokens.

In summary, the performance of the model with task-specific identities modeled by Holland is better than that of the model with task-specific identities learned by Linear layers. It’s because informing prior can help the model better capture the physical relationships among tasks and the development mechanism of tropical cyclones. Besides, from the table, we can also observe that the model with task-specific identities learned by Linear layers still performs better than the one without task-specific identity modeling, which demonstrates that even without prior knowledge, constraining the feature distribution to model task-specific identities is still effective for mitigating distribution shifts, thus improving the estimation accuracy of tropical cyclones.

Comment 3: Lacks formal proofs for invariance guarantees.

Response 3: Thank you for the valuable comment, which helps improve the theoretical completeness of our work. At this stage, the proposed IDOL is designed to learn invariant feature distributions in the tropical cyclone (TC) domain from a physical and causal modeling perspective. The learned invariance is empirically validated through comprehensive and rigorous experiments.

We are actively exploring a more principled theoretical framework to provide formal invariance guarantees. However, the representation of distribution shifts arising from the complex and evolving nature of TCs remains highly abstract and challenging to formulate explicitly. In the future, this limitation may be addressed by further integrating Bayesian Invariant Learning and Physics-Informed Learning, which together can model uncertainty while ensuring physical consistency across diverse environments.

Dear Reviewer, we would like to confirm whether our rebuttal has sufficiently addressed your concerns or not. If there are any remaining questions or clarifications needed, we would be glad to discuss them further at this stage. Thank you in advance.

We are pleased that the reviewer recognized the well-designed architecture and comprehensive evaluations of our work. We also sincerely appreciate the constructive comments on the contribution demonstration, which have been highly valuable in helping us improve the manuscript. The reviewers' comments are shown below in italicized font, and our responses are presented in regular font. We hope that these answers will meet with your approval.

Comment 1: The trade-offs between slight performance over the new architecture are not explained. -It is unclear how much ground truth labels (which may not be available) are required for addressing domain shifts. -The paper can be strengthened with some considerations on simplification of the architecture to reduce the deployment since this has real word implications.

Response 1: Thank you for the valuable comments. As demonstrated in Table 1 of the main text and Table 4 in Appendix C.2, our proposed method achieves the best estimation performance without increasing model parameters or inference time. Since all multi-attribute labels for tropical cyclones are publicly available, our approach is implemented within a supervised learning framework. By jointly considering the short inference time and low computational complexity of stacked convolutions, IDOL achieves the balance between prediction accuracy and computational efficiency, making it readily deployable in real-world applications.

Moreover, to better understand the model complexity, we analyze the parameter size of the proposed identity modeling modules and the feature extraction backbone (two VGG13 networks) in IDOL. The parameter size of them are 9.78M and 266.11M, respectively. This indicates that the identity distribution-oriented learning module itself is lightweight. To further investigate the trade-off between accuracy and efficiency, we replaced the VGG13 backbone with a significantly simpler CNN architecture and retrained the model on the same dataset. The comparative results are presented below:

Note: Wind, Press, Inner-Core, and Outer-Core refer to the estimation tasks of wind speed (m/s), pressure (hpa), inner-core size (nmi), and outer-core size (nmi), respectively.

In summary, as shown in the table, replacing the backbone with a lightweight CNN architecture has slight impact on the performance on overall estimation performance. This demonstrates that the proposed identity distribution-oriented physical-invariant learning framework is not only effective and lightweight, but also robust to changes in the feature extraction backbone, making it highly adaptable and scalable for practical deployment.

Comment 2: Is there a way to show visually the split between invariant and variant components?

Response 2: Yes, it is possible to visualize the separation between invariant and variant components by first applying dimensionality reduction on them and then using kernel density estimation (KDE) to show visually their distributions across different time domains. However, as the rebuttal box does not support figure attachments, we do not present the result here, which will be added in the revised appendix.

Additionally, as shown in Figure 4(b) of the manuscript, we provide an intuitive illustration of the invariant and variant components across different temporal domains. In the context of invariant learning, features from different domains should remain close in distribution, which implies similar statistical variances across domains. Taking the components shown in Figure 4(b) as an example, the task-shared identity ($PID_{sh}$) and task-specific identities ($PID_v$, $PID_p$, $PID_{ri}$, and $PID_{ro}$) consistently exhibit low variance across time domains, and are therefore regarded as invariant components. In contrast, the features extracted by the baseline DeepTCNet demonstrate fluctuating variance across different temporal domains, which are regarded as variant components.

Comment 3: These phenomena change with changing atmospheric and other dynamics. Is there an efficient way to infuse new external variables?

Response 3: Yes, there is an efficient way to incorporate new external variables. Since the proposed Correlation-Aware Information Bridge consists of graph convolutions and learnable parameters of a mixed Gaussian distribution, we can effectively infuse new external environmental variables by increasing the number of nodes in the knowledge graph. This enables the model to better learn representations of input data and their latent correlations with output tasks, while applying these effects to constrain feature distributions.

Comment 4: How can this framework work in an unsupervised way?

Response 4: Thank you for your insightful question. We believe our framework can be extended to an unsupervised setting in future work. One potential direction is to perform clustering on the TC convection structures to extract inherent patterns without label supervision. Another promising approach is to leverage contrastive learning between satellite images and descriptive text to align multi-modal features in an unsupervised manner. These adaptations would allow the model to learn meaningful representations without relying on explicit labels.

Dear Reviewer, we would like to confirm whether our rebuttal has sufficiently addressed your concerns or not. If there are any remaining questions or clarifications needed, we would be glad to discuss them further at this stage. Thank you in advance.

We are pleased that the reviewer recognized the effectiveness of our prior incorporation strategy and the promising experimental results. Moreover, we sincerely appreciate the valuable suggestions regarding experimental demonstrations, which have greatly helped us to revise and improve our paper. The reviewers' comments are shown below in italicized font, and our responses are presented in regular font. We hope that these answers will meet with your approval.

Comment 1: (weakness) It is not clear what happens to the performance if the prior models are approximate or not fully accurate. & (question) What happens if the model priors are erroneous and/or approximate?

Response 1: Thank you for the insightful comment. In our approach, we propose the Prior-aware Approximator (PriorAPP) to incorporate the Holland model as prior knowledge. Since Holland is a classical statistical model in tropical cyclone (TC) research, it is well known that it is not fully accurate. As shown in the first two rows of Table 2 in the manuscript, the task-specific identities ($\mathbf{id}_\mathtt{sp}$) are derived based on the Holland model. Therefore, the second row of Table 2 in the manuscript reflects the model’s estimation performance when the prior is not fully accurate. The superior performance of the model with Holland demonstrates both the effectiveness of incorporating physical priors and the robustness of our method to approximate priors.

As shown in the following table, to further validate the contribution of the prior model and the robustness of our design, we conducted additional experiments under intentionally perturbed prior conditions. Specifically, in the "w/ Noisy Prior" setting, we inject additional task correlations into the Holland model that are not physically validated in previous research, such as introducing an artificial dependency between inner-core size and outer-core size.

Note: Wind, Inner-Core, and Outer-Core refer to estimation tasks of Wind Speed (m/s), Inner-Core Size (nmi), and Outer-Core Size (nmi) errors, respectively.

These results clearly indicate that the model maintains stable and robust performance even under perturbed prior conditions. This suggests that the proposed Prior-aware Approximator, combined with the nonlinear representation power of deep networks, can effectively adapt to approximate and erroneous priors. These supplementary experimental results will be added to the appendix in the revised version.

Comment 2: For the cases of tropical cyclones, it is not clear how often OOD issues must be handled. The scenario does not see overly dynamic.

Response 2: Thank you for the insightful comment. To assess the extent of distributional shifts in tropical cyclone (TC) estimation, we analyzed the label distributions across all TCs in the training and test sets. Specifically, we compute the Jensen-Shannon Divergence (JSD) between each test TC’s label distribution and that of the training set. A higher JSD indicates a more significant distribution shift. Based on median statistics, we empirically set a threshold of 0.39 to define out-of-distribution (OOD) cases. Our results show that 44% of test samples exceed this threshold, confirming that OOD issues represent a significant challenge in tropical cyclone estimation that warrants further attention and dedicated handling.

Moreover, from a meteorological perspective, the development of tropical cyclone (TC) attributes is inherently dynamic and nonlinear. Their evolution is influenced by internal processes such as eyewall replacement cycles, convective bursts, and variations in structural features like intensity distribution and inner-core compactness [1]. These dynamic behaviors introduce significant variability in TC structure and intensity evolution. As a result, no training set can fully capture the entire spectrum of possible TC states, highlighting the necessity of building models with strong generalization ability under distribution shifts. This is precisely the motivation behind our proposed identity distribution-oriented learning framework.

• [1] Rojas, B., Didlake, A., & Zhang, J. (2024). Asymmetries During Eyewall Replacement Cycles of Hurricane Ivan (2004). Monthly Weather Review.

Comment 3: Can you show more dynamic use cases in your experiments? In other words, can you simulate for highly changing distribution shifts?

Response 3: Yes, we can show more dynamic use cases in our experiments. However, due to the inability to attach pictures in the rebuttal box, these cannot be displayed here. Nevertheless, as demonstrated in Figure 5 of the main text and Figures 11 to 13 in Appendix C.2, the presented cases already exhibit significant distribution shifts, effectively showing IDOL's capability to handle distribution shifts. Specifically, Figure 5(a) in the main text, Figure 11(a) and Figure 12 in Appendix C.2 highlight sample pairs with distinct TC shape organizations, e.g., a blurred center versus a clear eye shape, which indicates significant input data distribution shifts. Additionally, Figure 5(b) in the main text, Figure 11(b) and Figure 13 in Appendix C.2 illustrate notable label distribution shifts, reflected in entirely different distribution patterns.

Comment 4: Can the proposed approach handle domain shifts?

Response 4: Yes, we believe the proposed approach can handle domain shifts for the following reasons:

Firstly, researches on domain shifts often employ domain-invariant feature learning, using visualizations of stable feature distributions across different domains to demonstrate their methods' effectiveness [2, 3]. This suggests that domain shifts can be partially reflected through feature distribution shifts. Our approach leverages prior knowledge to constrain feature distributions with physical invariance, aligning the goal of modeling invariant task identities with extracting domain-invariant features. The key distinction of our IDOL lies in integrating physical priors into distribution constraints, which we argue enhances generalization learning capability.

Secondly, in invariant learning, features from different domains should be close to each other, which means they will have similar variances. Given that tropical cyclones across different years exhibit unique developmental traits and variable environments, temporal domain shifts are evident. If our model extracts physically invariant identity tokens across these time domains, their statistical variance should remain consistent. As shown in Figure 4(b) of the manuscript, our physical identity tokens display smaller variance across time domains compared to DeepTCNet features, highlighting greater robustness to domain shifts. This improvement supports that incorporating prior physics enables the model to better understand the internal TC dynamics and generalize to unseen, variable environments.

• [2] Chen, Yang, et al. "Achieving domain generalization for underwater object detection by domain mixup and contrastive learning." Neurocomputing 528 (2023): 20-34.
• [3] Chen, Haojie, et al. "Self-supervised domain feature mining for underwater domain generalization object detection." Expert Systems with Applications 265 (2025): 126023.

Comment 5: How does this approach compare to other invariant representation learning techniques?

Response 5: In Table 1 of the manuscript, we have already compared the proposed IDOL with other invariant representation learning methods, including classic approaches (e.g., IRM, V-Rex) and recent state-of-the-art (SOTA) methods (e.g., SADE, DirMixE). The results indicate that IDOL outperforms these general invariant learning methods, likely due to the integration of domain-specific knowledge. Moving forward, we plan to include additional invariant representation learning methods in the comparison to enhance the comprehensiveness of the experiments.

First of all, we are pleased to see that the reviewers have recognized the clarity of our presentation. We sincerely appreciate all the valuable suggestions, which have greatly helped us to improve the paper. The reviewer’s comments are shown below in italicized font, followed by our responses in regular font. We hope that our responses address the concerns raised and meet with your approval.

Comment 1: abstract language of a very complex pipeline may loose some applied ML scientists on reading, suggestion: add specific example with illustration in appendix and add a high-level description in layman's terms to main manuscript.

Response 1: Thank you for your valuable suggestion. Indeed, tropical cyclone (TC) estimation refers to regressing multiple TC attributes, such as maximum wind speed, pressure, and wind radii, from satellite-based multi-modal data. To better motivate our method and highlight the underlying challenges, Appendix A of the manuscript provides a detailed overview of the different types of distribution shifts present in TC estimation. In response to your suggestion, we will include a specific example with an illustration figure in the revised appendix to better demonstrate the full pipeline in an intuitive and step-by-step manner. Since figures cannot be shown in the rebuttal box, we do not present them here.

Comment 2: as acknowledged in Question 7, the model performance experiments do not ship with error bars.

Response 2: Thank you for your valuable suggestion. We agree that including error bars is important to reflect the model's stability and reliability. We have now supplemented Table 1 in the manuscript with performance metrics including the mean absolute error (MAE), root mean squared error (RMSE), and standard deviation (STD) across multiple runs. The updated results are shown below:

Note: Wind, Inner-Core, and Outer-Core refer to estimation tasks of Wind Speed (m/s), Inner-Core Size (nmi), and Outer-Core Size (nmi) errors, respectively.

In summary, compared with traditional methods (i.e., ADT and MTCWA) and previous state-of-the-art approaches (e.g., DeepTCNet), the proposed IDOL demonstrates superior capability in handling distribution shifts, achieving both the best and most stable estimation performance.

Comment 3: About available code.

Response 3: As indicated in Question 5 of the checklist, the code is already publicly available at the location we cited, with the correct suffix "IODL-A370". It is possible that the cited text was wrapped in the manuscript and therefore cannot be accessed correctly. Furthermore, we plan to release the code on GitHub under the MIT license in the near future to ensure broader accessibility.

Comment 4: Does your approach compare to predicting TCs from state-of-the-art weather foundation models such as Google's GenCast, IBM's PrithviWxC, or Microsoft's Aurora?

Response 4: Thank you for the valuable review. The proposed IDOL is originally designed for tropical cyclone (TC) estimation. To further demonstrate its transferability and effectiveness under distribution shifts, we adopt TC-Diffuser, the current state-of-the-art (SOTA) model for end-to-end multi-task TC forecasting, as the backbone in a transfer setting. The corresponding results are presented in Table 6 of Appendix C.2 in our manuscript, which demonstrate IDOL’s strong ability to handle distribution shifts, a challenge that is also inevitable in TC forecasting.

Second, regarding TC forecasting, foundation weather models such as Google's GenCast do not adopt an end-to-end architecture for multi-task TC forecasting. Instead, they forecast TC trajectory and intensity by identifying the cyclone center based on Mean Sea Level Pressure (MSLP) in the predicted atmospheric fields from ERA5 datasets. Due to the massive scale of ERA5 datasets, which require substantial time for downloading and processing, we were unable to conduct direct performance comparisons with the mentioned foundation models on the same TC test cases within the limited rebuttal period.

Nevertheless, to further compare the performance of our approach and the foundation weather model, we have supplemented the performance of another publicly available weather foundation model, Pangu (as reported in paper [1]), on comparable TC forecasting tasks, as shown in the following table:

Note: Traj, Press, and Wind refer to forecasting tasks of Trajectory (km), Pressure (hPa), and Wind Speed (m/s) errors, respectively. Additional supplementary experiments comparing our approach with weather foundation models will be included in the revised manuscript.

In summary, our IDOL is efficient in handling distribution shifts, thus achieving better forecasting performance than the foundation model Pangu.

• [1] Zhang, Shiqi, et al. "TC-Diffuser: Bi-Condition Multi-Modal Diffusion for Tropical Cyclone Forecasting." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 39. No. 1. 2025.

Comment 5: Eq. (2): additional details on $g$ and $\mathcal{O}_\text{id}$ in supplementary material, please.

Response 5: In Eq. (2), the function $g$ refers to the proposed estimation heads based on the Correlation-Aware Attention mechanism, while $\mathcal{O}_\text{id}$ denotes the modules designed to model task identity tokens by leveraging internal invariant physical correlations to regulate feature distributions.

Specifically, $\mathcal{O}_\text{id}$ comprises the proposed Task Dependency Flow Learning and Correlation-Aware Information Bridge. Additional details will be provided in the supplementary material in future revisions.

Comment 6: In l246-l247 you state: "the Physical Dynamic TC datasets (PDTC) we constructed" which I read as you (partially) constructing a new dataset? If not, please clarify and adjust the main text.

Response 6: Yes, the Physical Dynamic TC Datasets (PDTC) is constructed by us. Detailed information and construction procedures of this dataset are provided in Appendix C.1 of the manuscript.

Comment 7: About paper formatting and language.

Response 7: Thank you for your valuable suggestions. We will carefully revise and improve the manuscript based on your comments to further enhance its clarity.