We sincerely appreciate the reviewer for taking the time to review our paper and for providing valuable, in-depth questions and feedback. Below, we provide detailed responses to the questions and comments raised.

Q1: State contribution of the work extends beyond the clustering technique

We thank the reviewer for thoughtful review and for expressing concerns regarding the contributions of our paper. We appreciate the opportunity to clarify the distinctions between our method and EAGLE and to highlight the unique contributions our work brings to the field. While our approach shares some conceptual similarities with EAGLE, there are significant differences that constitute substantial contributions and lead to improved performance. Please see the summarized contributions of our work below:

1. Methodological Innovations

   (a) Novel Physics-Guided Segmentation Method:

   • [Physically Meaningful Segments]: Introduced a novel clustering algorithm that incorporates physics-informed features into the segmentation process. Unlike EAGLE's fixed-size, node-centric clustering, our method adaptively adjusts segment shapes and locations based on mesh topology and physical priors (e.g., distances to boundaries and obstacles), resulting in segments that align with the underlying dynamics.
   • [Superior Performance with Fewer Segments]: Achieved superior performance without the need for a large number of segments by leveraging physics-guided segmentation. Thoroughly analyzed the effect of segment number on performance, providing insights into how segmentation granularity impacts model accuracy and predicted mesh quality—an aspect not discussed by EAGLE.

   (b) Segment Feature Aggregation Mechanism:

   • [Simplified Aggregation with Average Pooling]: Reduced model complexity by using average pooling for segment feature aggregation, which is invariant to segment size, unlike EAGLE's GRU-based method.
   • [Avoidance of Sequence Dependency]: Avoided issues related to sequence dependency and inconsistencies due to random node ordering inherent in GRU-based methods, enhancing robustness and efficiency.

   (c) Meaningful Macro-Level Information Exchange:

   • [Physics-Informed Attention Mechanism]: Exchanged higher-level information through a physics-informed attention mechanism, as opposed to EAGLE's attention over numerous small clusters with less physical significance.
   • [Efficient Information Transfer]: Enabled more meaningful and targeted macro-level information exchange that closely reflects actual physical processes, leading to efficient information transfer and improved modeling of long-range interactions.

   (d) Extended Analysis and Enhancements:

   • [Incorporation of Positional Encoding (PE)]: Enhanced network expressivity by incorporating positional encoding and thoroughly analyzed its impact under various conditions.
   • [Investigation of Overlapping Segments]: Examined the benefits of overlapping segments, gaining insights into scenarios where overlaps enhance performance.

2. Applicability and Extension to New Domains

   (a) Application to Solid Mechanics and Lagrangian Systems:

   • [Extension Beyond Fluid Dynamics]: To the best of our knowledge, we are the first to apply clustering methods to solid mechanics problems, particularly those involving long-range interactions and Lagrangian systems with mesh deformation. This extends the applicability of hierarchical GNNs beyond fluid dynamics, which is the primary focus of EAGLE.
   • [Addressing Unique Challenges]: Developed a clustering/segmentation algorithm that integrates well into a hierarchical framework for deformable Lagrangian meshes, providing a method applicable to a wide range of dynamic systems, whether fluid or solid.

3. Robustness and Scalability

   (a) Handling Mesh Irregularity and Scalability:

   • [Flexibility in Segment Size]: Not constrained by equal-sized clusters or fixed nodes per segment, allowing better handling of varying mesh structures and irregular meshes than EAGLE.
   • [Adaptive to Mesh Density]: Mitigated issues in dense meshes where EAGLE's approach results in too many small clusters by adjusting segments based on mesh density and physical priors, enhancing generalization to larger and more complex systems.
   • [Generalizabiliy to Larger-scale Dataset]: Demonstrated ability to effectively generalize to larger-scale datasets, which is essential for real-world simulation applications."

(Response to Q1 to be continued)

(Continuation of the response to Q1)

4. Comprehensive Evaluation and Insights

   (a) Comprehensive Evaluation Across Metrics:

   • [Incorporation of Mesh Quality Metrics]: Considered multiple metrics to evaluate the quality of the predicted meshes. Mesh quality is crucial for Lagrangian systems because the mesh deforms along with the material; maintaining high-quality meshes is essential for accurately capturing dynamics and ensuring numerical stability over time.
   • [Balanced Performance]: Demonstrated that our method achieves an excellent balance across prediction accuracy, computational efficiency, and mesh quality—being the first to consider mesh quality in this context.

   (b) Comprehensive Comparison of Hierarchical Designs:

   • To the best of our knowledge, we are the first work to comprehensively compare hierarchical models for dynamic system prediction, including both multi-level structures (e.g., g-U-Net, BSMS-GNN) and cluster-based structures (e.g., EAGLE and our MMSGN). This comparison evaluates various metrics such as prediction accuracy, predicted mesh quality, and computational efficiency, offering valuable insights for future research.

5. Computational Efficiency and Real-World Applicability

   (a) Reduced Computational Burden:

   • By using fewer, physics-informed segments, we reduce computational burden compared to EAGLE, which requires many small segments, making our method more efficient and applicable to real-world problems.

   (b) Applicability to Real-World Problems:

   • Our method's efficiency and ability to handle complex, irregular meshes make it more applicable to real-world problems where computational resources and scalability are critical.

6. Newly Introduced DeformingBeam Dataset: (please refer to our response to Q10 for details)

   (a) Challenging Long-Range Interactions:

   • Our dataset includes complex long-range interactions, providing a rigorous test for model performance under scenarios susceptible to over-smoothing.

   (b) Diverse Mesh Structures and Interaction Scenarios:

   • It features a variety of mesh configurations and diverse interaction scenarios between beams and obstacles, enhancing the dataset's complexity and applicability.

   (c) Standard and Scaled-Up Versions:

   • To the best of our knowledge, this is the first dataset to offer both a standard and a scaled-up version, making it a valuable resource for directly evaluating model scalability.

Q2: Strengthen the paper with a better analysis of the results, including when and why the proposed model performs better.

We appreciate the reviewer’s suggestions and have incorporated several in-depth analyses of our method. Below is a high-level summary of the additions made to the paper:

• We added two additional metrics-Chamfer Distance and Aspect Ratio (details in Appendix D.1)-for a more comprehensive assessment of the predicted meshes, updating Tables 3 to 7 and using them to guide our ablation studies detailed in Appendix E.2.
• We introduced three additional metrics—Conductance, Edge Cut Ratio, and Silhouette Score (details in Appendix D.2)—to thoroughly evaluate the segmentation methods by assessing both inter-segment and intra-segment qualities.
• To understand how predicted mesh properties in each segment vary over time for different methods, we included additional visualizations in Figure 6 and analysis in Appendix D.3, where mesh nodes are colored based on the average prediction error within their segments.
• We analyzed the correlation between these segmentation metrics and overall dynamic system performance, including mesh quality and prediction error, as shown in Figure 9(a–c) and Appendix E.2.
• To comprehensively evaluate the effect of segment number and determine the optimal number of segments for a given dataset, we analyzed prediction accuracy across a wide range of segment counts during training (mainly on the DeformingBeam dataset), as shown in Figure 9(d) and Appendix E.2.
• We conducted a comprehensive analysis of the effects of position encoding and segment overlap on model performance under different segment counts across all three datasets, as shown in Figure 10 and Appendix E.2.

Thank you for your comprehensive reply. I appreciate the time you took to address my review but kindly suggest keeping replies as concise as possible. That said, I understand that long, negative reviews often call for detailed responses. Since some of your replies overlap, I will only address specific points.

Explicit list of contributions I appreciate that you clarified your contributions, particularly in comparison to Eagle. However, I still have concerns about the novelty, and some of the claims in your rebuttal are not well supported.

1. a) Your clustering method is indeed more relevant than the naive approach in Eagle and has a clear impact on performance. However, it is not physics-guided. The introduced features, while logical, are purely geometric (e.g., distance to obstacles/boundaries) and do not incorporate any explicit physical knowledge.
   b) Some of your claims are very misleading. Your attention mechanism is not physics-informed; it is neither constrained by nor aware of the underlying physics. For example, you assert that Eagle clusters have "less physical significance," but this is unsupported, and the fact that your model outputs better prediction is not a proof of the physical significance of the cluster. Additionally, you state that the information exchange is "more meaningful." What does this mean, and which experiment demonstrates the "meaningfulness" of this exchange?
2. Why would equal-sized clusters improve the handling of irregular meshes? You claim that your method "enhances generalization to larger and more complex systems," but where is the evidence to support this?

I acknowledge the other points of novelty mentioned in your response and agree with them, but these do not seem to constitute the core claims of the paper.

Added experiments on mesh quality and correlation with performance Thank you for including this impressive set of new results. I found these experiments extremely interesting in many aspects. Visual inspection of the clusters in Fig. 6 shows that they are less biased toward circular shapes, which makes a lot of sense for CylinderFlow. Furthermore, the results in Fig. 9 demonstrate a clear correlation between performance and geometric mesh quality metrics.

Overall, these results significantly strengthen the paper and, in my view, should have been central to its original message.

Comparison with Eagle dataset I understand your reasoning, but given that your paper is strongly inspired by Eagle while aiming to address its weaknesses, a comparison with the Eagle dataset seems mandatory. Relying on performance on other datasets to argue that your method would outperform Eagle on its dataset as well is not well supported.

I edited my review and updated my score and its motivation.

Q2: Your attention mechanism is not physics-informed and the fact that your model outputs better prediction is not a proof of the physical significance of the cluster.

• We apologize for any confusion our previous statement may have caused and appreciate the opportunity to clarify our points. The reviewer is correct that the attention mechanism itself is not inherently physics-informed and has no direct awareness of the underlying physics. What we intended to convey is that, in our method, the attention operates over mesh segments created through physics-guided segmentation. Consequently, the attention mechanism indirectly leverages the underlying physics by aggregating information across these physically significant regions.
• To demonstrate that a mesh clustering or segmentation method has physical significance, it is important to show that the clusters correspond to meaningful physical properties, behaviors, or phenomena within the system we are studying. We believe that our additional results (e.g., Figures 6 and 9) support this point by illustrating that our segmentation method not only enhances predictive accuracy but also produces clusters with greater physical significance. This is achieved by effectively capturing temporal and spatial dynamics that are aligned with the system's physics, thereby outperforming EAGLE in this regard.
• We sincerely value the reviewer's feedback and are open to any suggestions the reviewer may have regarding additional experiments or analyses that could further support this statement. We are more than willing to incorporate the reviewer's recommendations to strengthen our work.

Q3: You state that the information exchange is "more meaningful." What does this mean, and which experiment demonstrates the "meaningfulness" of this exchange?

When we say that the information exchange in our model is "more meaningful," we mean that it more effectively and accurately captures the essential physical interactions and dynamics of the system being modeled. This improvement arises because our model's information exchange is based on mesh segments that align with key physical regions and behaviors within the system. As a result, the model achieves better predictive accuracy and generalization by effectively capturing and leveraging critical physical interactions. Below are experiments that demonstrate the "meaningfulness" of the exchange:

• [Visualization of Segmentation Alignment]: (Figure 6) The segments produced by our physics-guided segmentation align with important physical regions in the system. This alignment allows the model to capture essential interactions more effectively, as it focuses on areas where significant physical processes occur.
• [Correlation Between Segmentation Quality and Model Performance]: (Figure 9) We observed that higher-quality segments, which better represent the physical structure of the system, lead to improved predictions. This correlation indicates that when the segmentation accurately reflects the system's physical properties, the information exchange becomes more effective, enhancing the model's performance.
• [Performance Improvements Over State-of-the-art Methods]: (Tables 1 and 3; Figure 3, 4, and 13-16) Our model outperforms existing methods in terms of prediction accuracy and other evaluation metrics. This superior performance suggests that the information exchange facilitated by our physics-guided segmentation leads to more accurate modeling of the system's dynamics, reinforcing the effectiveness of our approach.

We appreciate the reviewer's insightful question and are happy to provide further clarification. If the reviewer has any suggestions for additional experiments or analyses that could better demonstrate the meaningfulness of our information exchange, we are more than willing to consider them to strengthen our work.

Q4: Why would equal-sized clusters improve the handling of irregular meshes?

• We thank the reviewer for the question. However, there appears to be a misunderstanding and we are happy to clarify it. We did NOT claim that equal-sized clusters improve the handling of irregular meshes. In contrast, equal-sized clusters, like those used in EAGLE, are not effective for irregular meshes. They impose uniformity that does not adapt to varying node densities and complex geometries present in irregular meshes. In contrast, our method allows clusters to vary in size and shape, adapting to the mesh's inherent variability and aligns with the system's physical dynamics. This adaptability enhances the model's ability to handle irregular meshes and improves predictive accuracy.

We sincerely thank the reviewer for taking the time to review our paper and provide valuable feedback. Below, we address the questions and comments raised in detail:

Q1: While the hierarchical approach appears conceptually sound, MMSGN may be challenging to re-implement due to its multi-level structure.

• We acknowledge the concern regarding the potential complexity of re-implementing MMSGN due to its multi-level structure. In fact, the hierarchical design of MMSGN is modular and adaptable, making it straightforward to apply to various dynamic systems. To further alleviate any implementation challenges, as mentioned in the paper, we will make the complete code publicly available upon the paper’s acceptance. This accessibility will enable researchers to easily utilize and extend our framework for their specific applications.

Q2: The paper briefly mentions comparisons with other state-of-the-art methods but would benefit from a more in-depth analysis to further explore the capabilities and limitations of MMSGN.

We appreciate the reviewer’s suggestions and have incorporated several in-depth analyses of our method. Below is a high-level summary of the additions made to the paper:

• We added two additional metrics-Chamfer Distance and Aspect Ratio (details in Appendix D.1)-for a more comprehensive assessment of the predicted meshes, updating Tables 3 to 7 and using them to guide our ablation studies detailed in Appendix E.2.
• We introduced three additional metrics—Conductance, Edge Cut Ratio, and Silhouette Score (details in Appendix D.2)—to thoroughly evaluate the segmentation methods by assessing both inter-segment and intra-segment qualities.
• To understand how predicted mesh properties in each segment vary over time for different methods, we included additional visualizations in Figure 6 and analysis in Appendix D.3, where mesh nodes are colored based on the average prediction error within their segments.
• We analyzed the correlation between these segmentation metrics and overall dynamic system performance, including mesh quality and prediction error, as shown in Figure 9(a–c) and Appendix E.2.
• To comprehensively evaluate the effect of segment number and determine the optimal number of segments for a given dataset, we analyzed prediction accuracy across a wide range of segment counts during training (mainly on the DeformingBeam dataset), as shown in Figure 9(d) and Appendix E.2.
• We conducted a comprehensive analysis of the effects of position encoding and segment overlap on model performance under different segment counts across all three datasets, as shown in Figure 10 and Appendix E.2.

Q3: How does MMSGN handle complex boundary conditions?

• MMSGN is fully data-driven and learns the dynamics directly from the simulations, including the imposed boundary conditions. Boundary conditions are incorporated by encoding the node type in the input features, allowing it to distinguish boundary mesh nodes from others. Among the datasets we investigated, CylinderFlow adopts fixed values for boundary nodes; DeformingPlate and DeformingBeam main fixed positions for boundary handle nodes. During rollout, MMSGN explicitly enforces these boundary conditions by assigning the fixed values to the relevant boundary nodes, ensuring consistency with the physical setup.

Q4: Could it effectively simulate systems where boundaries have non-standard behaviors or physical constraints?

• MMSGN is capable of learning system dynamics even when non-standard boundary behaviors are involved, as it directly learns the evolution of state variables from ground truth simulations that inherently capture boundary condition effects. However, the current model does not explicitly enforce complex boundary conditions. To address this, future work could integrate strategies from existing studies that focus on incorporating boundary conditions and physical constraints in model design. Notable examples include:
  [1] Saad, Nadim, et al. "Guiding continuous operator learning through physics-based boundary constraints." arXiv preprint arXiv:2212.07477 (2022).
  [2] Yang, Shuqi, Xingzhe He, and Bo Zhu. "Learning physical constraints with neural projections." Advances in Neural Information Processing Systems 33 (2020): 5178-5189.

We sincerely thank the reviewer for their time, valuable feedback, and thoughtful suggestions. Below, we provide detailed responses to the questions and comments raised:

Q1: Mesh Continuity

• We thank the reviewer for valuable feedback regarding the inclusion of additional metrics for mesh evaluation. Following the recommendation, we have incorporated Aspect Ratio and Chamfer Distance as additional metrics in the paper (details in Appendix D.1). These metrics provide a broader and more standardized assessment of mesh quality and geometric fidelity. We have updated all relevant tables (Table 3 to Table 7) to include these two metrics for completeness. Additionally, these metrics are utilized to guide the evaluation of the various ablation studies added, as detailed in Appendix E.2.

Q2: Underwhelming PE Impact: Could the authors elaborate on the observed impact of PE?

We’ve done thorough analysis on the effect of PE and the details can be found in Appendix E.2. Below are our observed findings regarding PE:

• [Effectiveness of PE Relates to Segment Counts]: In both the CylinderFlow and DeformingPlate datasets, incorporating PE with fewer segments enhances performance across multiple metrics by reducing positional ambiguity. When segment counts are low, each segment encompasses larger and more diverse areas, limiting the model's spatial detail and understanding of segment relationships. PE provides explicit positional information, enabling the model to differentiate between distinct regions within the same segment and better comprehend how segments neighbor and interact with each other. In contrast, as the number of segments increases, the inherent spatial resolution improves, and the benefits of PE decrease. In some cases, PE may introduce unnecessary complexity that hinders performance.
• [Dataset-Specific Factors]: The Deforming Beam dataset, with its complex geometry and deformation, did not benefit from PE. This suggests that the effectiveness of PE is not only dependent on segment count but also on how well the PE implementation aligns with the dataset's specific characteristics.
• [Need for Tailored PE Approaches]: For complex systems, a generic PE may not suffice. Customized positional encoding strategies that consider the unique geometry and deformation patterns might be necessary to realize large performance gains.
• [Conclusion]: While PE unquestionably enhances the performance of graph-based networks, supported by strong theoretical foundations and an increasing volume of research, further mathematical advancements are crucial. Identifying optimal encoding strategies is essential for achieving consistent improvements, particularly when applying PE to mesh segments in a wide range of dynamic systems.

We sincerely thank the reviewer for their constructive feedback, which offers valuable guidance for future research. We also appreciate their recognition of the strengths of our paper. We recognize that adding physical constraints at contact points and enforcing boundary consistency, especially for systems with diverse material properties, are important areas for improvement. These enhancements are part of our planned future work to further refine our method. We believe addressing these points will enhance the robustness, flexibility, and applicability of our method in more complex scenarios.

Q1: Achieving optimal performance requires careful tuning of segmentation and message-passing parameters, which may limit ease of application in new scenarios.

• Regarding hyperparameter selection, we acknowledge the challenge it poses and have added a section to the paper providing a guideline for selecting optimal segment numbers (please refer to Appendix E.2). This guideline aims to assist users in achieving better performance with reduced trial-and-error efforts.

Q2: The paper does not discuss run time for the different configurations. It would be good to include some measure of the computational performance.

• The run times for training and testing across different models and datasets are reported in Table 8. Additionally, Section 4.2.2 includes a discussion of computational efficiency, where we compare different models in terms of accuracy and efficiency. If more detailed information is needed, such as run times for specific model hyperparameters or architectures, we would be glad to provide additional analysis or supplementary materials.

We sincerely thank the reviewer for taking the time to review our paper and provide valuable feedback. We also appreciate the reviewer’s recognition of our work. Below, we address the questions and comments raised in detail:

Q1: In Mesh Segment Feature Dispatch part, how does the extracted micro-level information from MeshGraphNet integrates with the macro-level features extracted from the mesh segments?

• For mesh node $i$, the integration is performed by concatenating its micro-level feature $\mathbf{h}_i$, obtained from the Encoder-Process-Decode (EPD) structure, with the macro-level feature ${{\mathbf{h}}_S}_i$, where $S_i$ is the segment that $i$ is assigned to. We found this straightforward integration efficient and effective for dynamic system prediction.

Q2: How the "prior information I" mentioned in line 249 calculated? How does these information utilized in the Physics-guided Segmentation part?

• In our work, the prior information $I$ for the three datasets includes boundary information (e.g., walls in the Cylinder Flow dataset, edges of deformable materials) and obstacle information. These are important because they define critical physical boundaries and obstacles that significantly influence the system's dynamics, affecting how forces and flows propagate through the mesh. We incorporate this prior information by defining functions to calculate prior-related features, as described in Eq (4) of our paper. Specifically, we use various distance measures—such as distances to obstacle nodes and boundary nodes—to compute these features, which are then utilized by our Physics-guided Segmentation module to guide the segmentation process.
• While we employ distance measures in this study, other features can also be used to represent prior information, such as material property variations, stress or strain distributions, or external force fields. These alternatives can provide additional insights into the physical behavior of the system and further inform the segmentation. In our datasets, we focused on single-material deformable objects, so material-related prior information was not included. However, our method can easily incorporate material-related features when simulating multi-material systems with different properties and behaviors. In such cases, the segmentation method would adjust based on material properties—for example, segmenting hyperelastic materials by regions of high elastic deformation, and plastic materials by areas undergoing permanent deformation due to yielding.
• In conclusion, our method is versatile and capable of incorporating various priors and user-defined features to guide segment selection. We plan to explore the integration of additional priors, such as material properties and stress distributions, in future work to enhance the segmentation's physical relevance.

Q3: It mentions that "predicted mesh in Lagrangian systems", just wondering what the predicted mesh indicates? Does it mean the method also output a mesh in addition to the current states on each nodes of a mesh?

• By "predicted mesh," we refered to the predicted states of the mesh nodes, rather than the generation of a separate or additional mesh. In Lagrangian systems, the positions of mesh nodes evolve over time. To evaluate the quality of these predictions, we introduced metrics that assess the mesh quality based on the predicted node positions. To improve the clarity, we have modified "predicted mesh" to "mesh with the predicted node positions" in our manuscript.