We are grateful for your thorough review. For a comprehensive response, please refer to the attached link. All materials within the link are indexed starting with the letter ‘L’ (e.g., Fig. L1).

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A1. Practical Utility of the Method (Response to Weakness and Q1).

The pipeline is illustrated in Fig. L10 to aid comprehension of the proposed approach. In the actual industrial process of injection molding product development, the process follows the stages of design -> analysis -> mass production. According to this process, designers must verify the product's defect status through analysis before mass production. However, there is a bottleneck in the design -> analysis phase, and we aim to address this using T-EMNN.

The reason the design -> analysis process is a bottleneck is that the work is divided between the designer, who specializes in design, and the analyst, who specializes in numerical analysis. The designer does not have numerical analysis expertise and must rely on the expert analyst. This process requires iteration. When the designer modifies the design, they send it to the analyst for analysis. The analyst then reports the results to the designer, who evaluates whether there are any defects based on the analysis. If defects are suspected, the designer modifies the design and sends it again, creating a repeated cycle (design -> analysis -> design -> analysis). This iterative process typically takes about a week for a single product (based on the Basket dataset), leading to significant delays in mass production.

We expect that using the T-EMNN we developed will dramatically reduce this iterative process. First, T-EMNN allows designers to self-validate their designs without requiring any special analysis skills. Furthermore, we confirmed that with T-EMNN, it takes less than 3 minutes (based on the Basket dataset) to go from preprocessing to checking the analysis results for a single injection molding product. Thus, with T-EMNN, designers can quickly self-validate designs that are unlikely to have defects. This allows designers to filter out designs with low defect potential and send only those to the analyst, significantly reducing the iteration process. This approach will drastically shorten the time to mass production.

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A2. Generalizability of the Thickness Threshold (Response to Q2)

We evaluate our method on four additional settings (using 5 seeds for reliability):

• Deforming Plate (Node-level prediction - Position of Next timestep)
• Basket (Graph-level prediction - Max pressure)
• Circular Plate (Node-level prediction - Deflection of each node)
• SimJEB [1] (Node-level prediction - Magnitude of the displacement)

Regarding performance, as shown in Table L1, the thickness processor successfully handles even next-timestep prediction. In Table L2, our method demonstrates strong performance on both the Basket (graph-level prediction) and Circular Plate (node-level prediction) datasets. Regarding the SimJEB dataset, although our data-driven coordinate system does not perform well under the "In Dist." setting (where the original coordinate system is well-established but our system faces challenges in aligning coordinates between diverse shapes), it performs effectively in the "Out-of-Dist." setting. In this setting, our method successfully addresses the misalignment issue in the original coordinate system between shapes, ensuring E(3)-invariance.

To qualitatively demonstrate our model’s ability to process thickness, we include the learned valid thickness edges in Figs. L4, L5, and L6. As shown in these figures, our thickness processor effectively connects opposing surfaces (left), facilitating beneficial interactions, while filtering out opposing surfaces that negatively impact performance (right).

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A3. Discussion for Handling Non-watertight Surfaces (Response to Q3)

We believe that T-EMNN can be applied even in the case of non-watertight surfaces. However, when creating the thickness edge, we utilize a method where a ray is cast from the starting node in the opposite direction of the normal (i.e., into the interior of the object) until it reaches the opposing surface. If the object is not watertight, there may be cases where the ray does not touch the opposite surface. In such cases, if the ray does not touch a specific node, the thickness edge can simply be excluded. This should not pose any issues for training and prediction.

However, for our T-EMNN, the watertightness of the 3D shape data is not a critical factor. What matters is whether the physical properties between thickness pairs are similar. If they are similar, we expect that using T-EMNN will still yield good prediction results.

Thank you for your thoughtful review. For a comprehensive response, please refer to the attached link. All materials within the link are indexed starting with the letter ‘L’ (e.g., Fig. L1).

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A1. Problem Definition and Dynamic Analysis

The task is to predict node-wise 3D deformation for 28 unique shapes with thickness, each under 18 condition combinations. In Figure 7, we visualize the deformation by taking the L2-norm of each node's deformation, resulting in a scalar value per node. These shapes are tested under diverse conditions, which are defined as "static analysis," distinct from the dynamic analysis in MGN, which models material interactions. Static analysis focuses on how deformation changes under varying conditions (e.g., pressure), while dynamic analysis, such as the deforming plate in MGN, examines interactions over time (e.g., velocity).

Although they serve different purposes, we also perform next timestep deformation prediction with the Deforming Plate dataset to demonstrate how our thickness processor helps the model handle dynamic situations. Since our data-driven coordinate system is designed for static analysis to prevent misalignment from affecting the results, we use the original coordinate system for dynamic analysis. The same mesh edge and node features as in the MGN paper, along with an additional node feature—the shortest world distance from the actuator to the node—are used to model interactions with the external environment. For detailed results, please refer to Table L1 and Fig. L6, where our model successfully handles the deforming plate's thickness, improving performance.

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A2. Additional Datasets

Please refer to our rebuttal A2 to reviewer "pfVr" due to character limits.

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A3. Equivariance/Invariance Proof

Please refer to Sec. L1 for the detailed proof. Our model can be either invariant or equivariant, depending on whether the 'inverse transformation' is applied. We will make sure to be more rigorous in distinguishing between the terms "equivariance" and "invariance" in the revised version.

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A4. Thickness Node Pair Definition. Please refer to Fig. L7 and L8 in Sec. L3 for a comprehensive description. The dot product between the normal vectors of $v_i$ and $T(v_i)$ is included as an attribute of the thickness edge between $(v_i, T(v_i))$ to address the issue you raised. To better account for these non-perpendicular edges, we use the dot product to measure how well the normal vectors are aligned in the edge attribute.

A5. Ray Projection Distance ($d$). We use the ray.intersects_location function from the trimesh library to find intersections between the mesh and the ray. Since the ray has no distance limit, it may extend to the far opposing side, leading to what is more accurately described as width rather than thickness, as shown in Fig. 4 of the paper. To address this, we introduce the thickness edge feature as a distance measure and apply a learnable thickness threshold to filter out incorrect thickness edges, without requiring a distance limit $d$.

A6. Selection of $\tau$. The learnable threshold $\tau$ is essentially a thickness threshold that helps with the prediction. It defines the range of thickness edges where message passing is beneficial. As distance increases, the interaction or correlation between opposite nodes tends to decrease. This aligns with the design we intended, and the reason it is made learnable is to allow the model to dynamically adjust this threshold.

A7. Clarification of Node Features $g_i$ and $r_i$ Node features are described in Apx, lines 606-607. We will reference them in the main text for clarity.

A8. Thickness Activation Function. Please refer to Fig. L9 for details. The activation function in Eq. 15 includes $\alpha$, which controls transition sharpness. Increasing $\alpha$ sharpens the influence based on thickness, but the influence is gradually reduced, not abruptly cut off, to ensure stable training. For a detailed analysis of $\alpha$, see Apx. D. The term "exclude" was meant to indicate a gradual reduction, not a hard cutoff.

A9. Regarding MGN. While our task does not require dynamic modeling, we chose MGN as a baseline for its strong architecture, on which we build by adding a thickness processor and a data-driven coordinate system. As stated in line 142, our method shares the core MGN structure, making a fair comparison and emphasizing the impact of our additions.

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A10. Essential References Not Discussed

We would like to clarify that MGN is mentioned in line 141, where we note that our architecture is derived from the core MGN structure. However, for clarity, we will specifically reference it in Sec 4.2.2.

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A11. Regarding the gravity. Thank you for your insightful comment. We are indeed considering a coordinate system where equivariance operates beyond the direction of the gravitational field as part of our future work.

Thank you for your thoughtful review. For a comprehensive response, please refer to the attached link. All materials within the link are indexed starting with the letter ‘L’ (e.g., Fig. L1).

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A1. Novelty of Our Work.

We would like to organize our contributions into two key aspects: 1) Thickness-Aware Framework and 2) Data-driven Coordinates, which effectively achieve E(3)-equivariance. Given your concern about the novelty of the GNN architecture, we will focus on emphasizing the Thickness-Aware Framework in this rebuttal, rather than the data-driven coordinates.

Perhaps our architecture may appear simple, but we would like to emphasize that the input meshes do not inherently contain any information regarding the "thickness." As a result, a basic GNN without any thickness processing faces challenges in interacting between opposing surfaces, despite their high correlation, as shown in Fig. 1 in the introduction.

Our novelty lies in how the GNN model adaptively addresses the "thickness," even though this information is not provided in the input data. To achieve this, we define the "thickness edge" from scratch and address potential issues, such as the negative correlation between opposing surfaces when the thickness is too large to share the correlative reaction. To overcome this challenge, we propose a learnable thickness threshold that adaptively identifies the optimal thickness edges. As a result, our method demonstrates strong performance, particularly on the $R^2$ metric, showcasing its ability to predict more realistic behavior.

We would like to emphasize that our method is not distinguished from others simply by utilizing several conditional embeddings through concatenation. In fact, the architecture of the baseline methods was modified to allow for the use of exactly the same conditional embeddings, including both condition and coordinate embeddings, as in our method, ensuring a fair comparison. Our distinct contribution lies in whether the model can handle opposing surfaces with high correlation, and our model successfully achieves this by defining and processing the thickness in a learnable way.

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A2. Evaluation on Four Additional Datasets.

We evaluate our method on four additional datasets (using 5 seeds for reliability):

• Basket (Graph-level prediction - Max pressure)
• Circular Plate (Node-level prediction - Deflection of each node)
• SimJEB [1] (Node-level prediction - Magnitude of the displacement)

The dataset and results are presented in Table L2 and Figures L1-L5. In Table L2, our method demonstrates strong performance on both the Basket (graph-level prediction) and Circular Plate (node-level prediction) datasets. Regarding the SimJEB dataset, although our data-driven coordinate system does not perform well under the "In Dist." setting (where the original coordinate system is well-established but our system faces challenges in aligning coordinates between diverse shapes), it performs effectively in the "Out-of-Dist." setting. In this setting, our method successfully addresses the misalignment issue in the original coordinate system between shapes, ensuring E(3)-invariance.

To qualitatively demonstrate our model’s ability to process thickness, we include the learned valid thickness edges in Fig. L4 and L5. As shown in these figures, our thickness processor effectively connects opposing surfaces (left), facilitating beneficial interactions, while filtering out opposing surfaces that negatively impact performance (right).

• Deforming Plate (Node-level prediction - Position of Next timestep)

Moreover, to evaluate the dynamic capabilities of our framework—specifically the thickness processor—we perform next-timestep deformation prediction using the Deforming Plate dataset (as described in the MGN paper) to demonstrate how our thickness processor helps the model handle dynamic scenarios. Since our data-driven coordinate system is designed for static analysis to prevent misalignment from affecting the results, we use the original coordinate system for dynamic analysis. The same mesh edge and node features as in the MGN paper, along with an additional node feature—the shortest world distance from the actuator to the node—are used to model interactions with the external environment. For detailed results, please refer to Table L1 and Fig. L6, where our model successfully accounts for the deforming plate’s thickness, leading to improved performance.

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A3. Reorganization of related work. Thank you for pointing this out. We will ensure that the related work regarding methods related to E(n)-equivariance/invariance is clearly addressed and reorganized separately.

[1] Whalen et al. "SimJEB: simulated jet engine bracket dataset." Computer Graphics Forum. Vol. 40. No. 5. 2021.

Thank you for taking the time to review our work. For a comprehensive response, please refer to the attached link. All materials within the link are indexed starting with the letter ‘L’ (e.g., Fig. L1).

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A1. Additional Datasets for Challenging Meshes (Response to Methods And Evaluation Criteria and W3)

We evaluate our method on four additional datasets (using 5 seeds for reliability):

• Deforming Plate (Node-level prediction - Position of Next timestep)
• Basket (Graph-level prediction - Max pressure)
• Circular Plate (Node-level prediction - Deflection of each node)
• SimJEB [1] (Node-level prediction - Magnitude of the displacement)

To visualize the thickness edges on other challenging datasets, we include the learned valid thickness edges in Figs. L4, L5, and L6. As shown in these figures, our thickness processor effectively connects opposing surfaces (left), facilitating beneficial interactions, while filtering out opposing surfaces that negatively impact performance (right).

Regarding performance, as shown In Table L2, our method demonstrates strong performance on both the Basket (graph-level prediction) and Circular Plate (node-level prediction) datasets. Regarding the SimJEB dataset, although our data-driven coordinate system does not perform well under the "In Dist." setting (where the original coordinate system is well-established but our system faces challenges in aligning coordinates between diverse shapes), it performs effectively in the "Out-of-Dist." setting. In this setting, our method successfully addresses the misalignment issue in the original coordinate system between shapes, ensuring E(3)-invariance.

Moreover, to evaluate the dynamic capabilities of our framework—specifically the thickness processor—we perform next-timestep deformation prediction using the Deforming Plate dataset (as described in the MGN paper) to demonstrate how our thickness processor helps the model handle dynamic scenarios. Since our data-driven coordinate system is designed for static analysis to prevent misalignment from affecting the results, we use the original coordinate system for dynamic analysis. The same mesh edge and node features as in the MGN paper, along with an additional node feature—the shortest world distance from the actuator to the node—are used to model interactions with the external environment. For detailed results, please refer to Table L1, where our model successfully accounts for the deforming plate’s thickness, leading to improved performance.

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A2. Discussion on Limitations in Scenarios Where Thickness is Not Dominant for Prediction (Response to Claims And Evidence)

Since not all objects have thickness, our method may not be very helpful for such shapes. However, in these cases, we adjust the threshold where thickness influences performance in a learnable manner. In such situations, the thickness threshold will adapt so that no thickness edge contributes to the prediction. If thickness is not important, the thickness edge will be ignored, and the model will ultimately be equivalent to the MGN with our proposed data-driven coordinate architecture.

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A3. Equivariance/Invariance Proof (Response to Theoretical Claims, W2 and Q2)

Please refer to Sec. L1 for the detailed proof. Our model can be either invariant or equivariant, depending on whether the 'inverse transformation' is applied.

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A4. Surface Mesh vs. Volume Mesh (Response to W1 and Q1)

Please refer to Appendix G for a comparison of GPU memory usage, performance, and inference time between volume-based and surface-based methods. We use the same model architecture (MGN) for a fair comparison, with the distinction that the input data is either volume mesh or surface mesh. The surface mesh utilizes only 11% of the GPU memory and takes 19% of the inference time compared to the volume mesh, while also delivering better performance. We argue that the performance degradation of the volume mesh arises from the over-smoothing or over-squashing problem that occurs when the mesh density is too high in GNN-based methods. For more details, please refer to Appendix G.

Regarding the DetailRecon, which was published in January 2025, we did not recognize the paper, but we will address it in the related work section of the revised version. Moreover, it is worth noting that FEM is not a neural network-based method, but rather a traditional numerical technique used for solving partial differential equations. While FEM remains a cornerstone in engineering simulations, it differs fundamentally from neural network-based approaches like ours, which aim to learn patterns from data rather than relying solely on predefined physical models.