Thank you very much for your positive feedback and thoughtful, detailed comments. We greatly appreciate your careful reading and constructive suggestions, which will help us improve both the clarity and impact of our paper. We summarize and respond to your questions as follows:

R1. Improve Presentation and Increase Background Knowledge

Thank you very much for your valuable comments. Your suggestions are both professional and elegant, and we greatly appreciate your insights. We will make the following revisions to the manuscript:

• (W1): We will refer to review [A] for additional background information on irreducible representations, Wigner-D matrices, and related concepts, and have incorporated these materials to enhance clarity for the reader.
• (W2): We will enhance the description of Fig. 1 for better understanding, and enlarge Fig. 3 to improve readability.
• (Q1): We will clarify that $C_H$ represents the number of channels in the node feature matrix $\boldsymbol{H}$.
• (MC): We will adopt your suggested rephrasing, as it is indeed more elegant. Thank you again.
• (L1): We will move the discussion of limitations in Appendix C to the main text.

R2 (Q2). Fairness of the Uniform Node and Edge Features

Thank you very much for your thoughtful question. The non-uniform case can be readily generalized and addressed within our framework. Our assumption of uniform node and edge features is primarily made to simplify the theoretical analysis, particularly in the context of Def. 3.3 on geometric isomorphism. We believe this assumption is reasonable for the purposes of our current work. Nevertheless, we agree that it is important to explicitly justify and clarify such assumptions:

• In practice, our analysis can be extended to non-uniform node/edge features by: (1) first determining geometric isomorphism based on graph structure, and (2) then ensuring that the mapping between nodes/edges also respects feature matching.
• We will revise the text to more explicitly state the scope and limitations of this assumption, and include supplementary comments in the appendix on extending results to the non-uniform case.

We also clarify that the purpose of Fig. 2 is to demonstrate the role of geometric structure (independent of feature information) in distinguishing points and facilitating the construction of valid virtual nodes.

R3 (W3). Additional Experiments on Large-scale Geometric Graphs

Thank you very much for your insightful comment and we decided to add an experiment on large-scale scenario. We acknowledge that our experiments do not cover ultra-large graph datasets (with 1K+ nodes). However, the main emphasis of our work is on establishing a theoretical framework for completeness, and the experimental results are intended primarily to provide supporting evidence for our theoretical contributions. It is worth emphasizing that, in the field of atomistic modeling, systems with more than one hundred nodes are widely recognized in literature as sufficiently challenging benchmarks. We believe that our results on such systems already provide a meaningful demonstration of our model's practical capabilities.

To further evaluate the performance of our model on larger datasets, we have conducted additional experiments on the more representative Water-3D (average >8K nodes/system) dataset in [B] as you suggested. Due to time limitations, we restrict the evaluation to a 1/5 subset called Water-3D-mini with 3,000/300/300 samples for train/valid/test from the original dataset (15,000/1,500/1,500), with a maximum training epoch of 1,000.

Table S1. Results on Water-3D-mini ($\times 10^{-4}$).

Our approach (EGNN$_\text{cpl}$-global/global) consistently outperforms the baselines at every depth, and achieves superior results with fewer layers. We believe this further demonstrates the efficiency and scalability of our approach, highlighting its potential for large-scale and real-world applications.

R4 (Q3). Discussion on Algo. 4 and Models

Thank you for raising this interesting and important question. The internal structure of the virtual node corresponds to $\vec{\boldsymbol{V}}^\top\vec{\boldsymbol{V}}$ in Algo. 4. According to our theoretical framework, this feature should indeed be included. However, we chose not to incorporate it mainly due to observed performance degradation in our empirical studies. We believe there are two possible explanations for this: (1) The virtual nodes in our model are learnable, and their parameters—i.e., the virtual node feature—involve in message passing, potentially providing sufficient informational capacity already; (2) In our experiments, we found that introducing this feature tended to reduce training stability. Notably, similar issues regarding training instability with inner product-like vector features have been reported in literature like [B].

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We sincerely appreciate your valuable suggestions, and we will include these clarifications and additional results in the revised manuscript.

Reference

• [A] Han J, Cen J, Wu L, et al. A survey of geometric graph neural networks: Data structures, models and applications. Frontiers of Computer Science 2025.
• [B] Zhang Y, Cen J, Han J, et al. Improving equivariant graph neural networks on large geometric graphs via virtual nodes learning. ICML 2024.
• [C] Huang W, Han J, Rong Y, et al. Equivariant graph mechanics networks with constraints. ICLR 2022.

Thank you for your thorough and insightful review. We are grateful for your positive comments on both the theoretical contributions and the literature discussion, as well as for your helpful suggestions to further strengthen the manuscript.

R1 (W1). Additional Experiment on Atomic Dataset

Thanks for your suggestion. Since there are few equivariant tasks on atomic point clouds, we have now included results for the large-scale Water-3D dataset [A], which requires predicting future coordinates for fluid particles (average >8K nodes/system). Due to time constraints, we only evaluate a substantial subset (Water-3D-mini: 3,000/300/300 for train/val/test) using up to 1,000 training epochs.

Table S1. Results on Water-3D-mini ($\times 10^{-4}$).

Our approach (EGNN$_\text{cpl}$-global/local) consistently outperforms all baselines at every depth, and achieves superior results with fewer layers. These findings further support both the scalability and efficiency of our method of handling large, real-world systems.

R2 (W2 & MC1). Theoretical Discussion

Thank you very much for your thoughtful and insightful comments, which have greatly enriched our understanding of the theoretical landscape.

Connotation of Model Methodology: As you astutely observe, there are indeed two main perspectives on unifying scalarization-based and tensor-product-based methodologies. The first approach, which we employ in our work, relies on message passing consisting of message computation and aggregation and it achieves equivariant constraints by leveraging known equivariant variables. The second approach, based on group convolution and decomposition of the convolution kernel (representation matrix), as you have perceptively pointed out, proceeds from the opposite direction—deducing equivariant variables directly from established equivariant constraints. Historically, the second perspective predates the first, and can be traced back to Eqs. (10), (11), and (16) in the 3D Steerable CNN paper [A]. The two additional works you referenced are, in fact, further developments based on [A]. This method offers profound insight into the underlying algebraic structures; however, it is often less accessible to researchers without a strong background in group representation theory. As a consequence, the second approach is most prevalent in research concerning equivariance in physical domains, while the first approach tends to be favored within the literature on equivariant graph neural networks. We will include a dedicated section in the appendix of our revised manuscript to introduce this alternative methodology and will update the main text to include appropriate citations.

Spherical Harmonics and Bases: We apologize for the confusion. Our intention was to convey that "for functions on the sphere that are equivariant under $\mathrm{SO}(3)$, the spherical harmonics form a basis," rather than to imply that "the spherical harmonics are a basis for $\mathrm{SO}(3)$" itself. We will clarify this statement in the revised manuscript to prevent further misunderstanding.

From Sphere to Euclidean Space: This can be addressed by incorporating an RBF function, as described in Lines 141–143 of the original manuscript. This design is also inspired by [A] (see after Eq. (16) in Section 4.2). We sincerely appreciate your thoughtful suggestions, which have helped clarify these important distinctions.

Model Interpretation from Canonicalization: Very nice comment! As you have pointed out, the essence of Frame-Averaging [31] lies in eliminating the influence of group transformations on the input by constructing a suitable frame, thereby enabling compatibility with any downstream architecture. From this perspective, we agree that the canonical form utilized in our method is conceptually aligned with the principles underlying Frame-Averaging. However, we would like to note that our specific implementation differs from this general framework. While Frame-Averaging [31] proposes a versatile approach that can be applied to a broad range of groups, our method focuses on designing an efficient canonicalization procedure tailored specifically to geometric graphs. This enables us to achieve computational efficiency while maintaining the desired equivariant properties. We greatly appreciate your suggestion, and we will clarify this relationship more explicitly in the revised manuscript to better position our contribution within the broader context. Thank you again for your valuable feedback.

R4. Improve Presentation and Correct Typos

Thank you very much for your valuable feedback. Your questions and suggestions are elegant and insightful. We will make the following changes to the manuscript:

• (W3 & MC5): We will include cross-references in the revised manuscript to improve readability.
• (MC2): We will provide the corresponding proof in a later section. Here is a brief explanation: we prioritize structural information to determine whether two graphs are geometrically isomorphic. For cases where node/edge features are not uniform, we can further determine this based on matching (please see our reply to MC4).
• (MC3): Sorry, this is a typo. EGNN/HEGNN and TFN both model based on edges, so they are 2-body interactions (number of neighbors $\nu=1$), while MACE corresponds to $\nu\geq 1$.
• (MC4): We will modify the statement of "Point Cloud Isomorphism" in Def. 3.3 as follows: "The two point clouds $\vec{\boldsymbol{X}}^{(\mathcal{G})}$ and $\vec{\boldsymbol{X}}^{(\mathcal{H})}$ are isomorphic, i.e. the matching set $\mathbb{M}(\mathcal{G}, \mathcal{H})=\\{(\sigma, \mathfrak{g})\in S\_N\times \mathrm{E}(3)\mid\forall i, \vec{\boldsymbol{x}}_i^{(\mathcal{G})}= \mathfrak{g}\cdot\vec{\boldsymbol{x}}\_{\sigma(i)}^{(\mathcal{H})}\\}$ is not empty.".

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Thank you again for your comprehensive review and high-level comments, which have significantly helped us to improve the clarity, rigor, and impact of our work. We will incorporate all clarifications, additional experiments, and references as discussed in the revised manuscript.

Reference

• [A] Weiler M, Geiger M, Welling M, et al. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. NIPS 2018.

Dear Reviewer,

We would like to express our sincere appreciation for your thorough and insightful review of our manuscript. Your positive feedback on both the theoretical contributions and the discussion of related literature, as well as your constructive suggestions, have been invaluable in shaping our revisions.

We would be very grateful if you could let us know whether our responses have satisfactorily addressed your comments, or if there are any further questions or points we could clarify.

Thank you again for your time, careful reading, and constructive feedback.

Best regards, The Authors

Thank you very much for your valuable and constructive feedback. We greatly appreciate your careful reading, positive remarks, and thoughtful questions. Please find our detailed responses below:

R1 (W1 & Q1). Clarification of Concepts

Thank you for highlighting areas where further clarification and improved definitions are needed. We acknowledge that terms such as "geometric graph" and "steerable function" may not be universally familiar to all readers, especially those who primarily work with topological graphs. We will address these points and strengthen the introductory section to include more context and definitions.

Geometric Graph and Equivariance: We agree that a clearer distinction between geometric and topological graphs would benefit readers. In our context, a "geometric graph" refers to a graph not only with nodes and edges, but with associated coordinates in Euclidean space. Equivariance for geometric graphs then refers to functions whose outputs are transformed in a specific way—consistent with group actions (e.g., $\mathrm{O}(3)$ rotations and reflections)—whenever the graph’s geometric data is transformed. You may refer to review [A] for a more detailed introduction.

Steerable Function: Indeed, a "steerable function" is synonymous with an "equivariant function" under the action of a group, particularly $\mathrm{O}(3)$ in our case. The term "steerable" is often used in signal processing and geometric deep learning communities to denote equivariant mappings of higher degree ($l>1$).

Clarification of why $\mathbb{F}^{(l)}(\mathcal{G})$ depends on $\mathcal{G}$ in Thm. 3.2: Here, $\mathbb{F}^{(l)}(\mathcal{G}) \subset \mathbb{R}^{2l+1}$ represents the set of all $l$th-degree equivariant function values on the geometric graph $\mathcal{G}$. Each such function outputs a vector in $\mathbb{R}^{2l+1}$, corresponding to the degrees of freedom for the representation of degree $l$ under $\mathrm{O}(3)$ transformations. Importantly, space $\mathbb{F}^{(l)}(\mathcal{G})$ may not span the entire $\mathbb{R}^{2l+1}$, depending on the structure of the graph $\mathcal{G}$ and its geometric realization. As discussed in [B] and outlined in Appendix A.1.3, in some configurations (i.e. symmetric graphs), the equivariant functions are restricted to a subspace. Thus, the expression and universality results must take these distinctions into account.

Construction of $\tilde{\boldsymbol{V}}^{(l)}(\mathcal{G})$: The construction of the basis $\tilde{\boldsymbol{V}}^{(l)}(\mathcal{G})$ is two fold: 1. We formally prove (Thm. 3.8 and Appendix A.5) that such a basis always exists for the space of $l$th-degree equivariant functions for any $\mathcal{G}$; 2. In Section 3.5, we provide a concrete example: the set of coordinate differences $\\{\vec{\boldsymbol{x}}_{ij}\\}$ used in EGNNs can be leveraged to construct $\tilde{\boldsymbol{V}}^{(l)}(\mathcal{G})$ in practice.

We will update the main text to make the definitions, dependencies, and construction procedures more transparent, and will provide these explanations in the main manuscript rather than only in appendices.

R2 (Q2). Discussion of Performance Improvement Difference over Baselines

Thank you for your insightful question regarding the performance on the MD17 dataset. Indeed, we notice that the improvements of our dynamic method are less pronounced on MD17, with some metrics even underperforming compared to the baselines. We believe this is due to the following reasons:

• Baselines such as Equiformer and HEGNN directly utilize high-degree steerable features, which are closely related to spherical harmonics and can effectively represent complex patterns (such as electron cloud shapes) essential for molecular dynamics tasks. While their architectural advantages may contribute significantly to performance in this setting, it is important to note that computing high-degree steerable features also introduces substantial additional computational overhead.
• In contrast, our approach does not rely on the high-degree steerable features. Nonetheless, our method achieves superior performance on 5 out of 8 tasks and demonstrates better generalizability.

In addition, we supplement our work with a challenging experiment on the large-scale Water-3D dataset [C], which involves predicting future coordinates for fluid particles in systems averaging over 8,000 nodes each. Due to time constraints, our evaluation is conducted on a substantial subset (Water-3D-mini: 3,000/300/300 for train/validation/test) with training capped at 1,000 epochs.

Table S1. Results on Water-3D-mini ($\times 10^{-4}$).

Our approach (EGNN$_\text{cpl}$-global/local) consistently outperforms all baselines at every depth, and achieves superior results with fewer layers. These findings further support both the scalability and efficiency of our method of handling large, real-world systems.

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Thank you again for your generous recognition of our work; your encouragement is truly appreciated. We will address these considerations and limitations in the revised manuscript to provide a more balanced perspective on the strengths and applicable scenarios of our method.

Reference

• [A] Han J, Cen J, Wu L, et al. A survey of geometric graph neural networks: Data structures, models and applications. Frontiers of Computer Science 2025.
• [B] Cen J, Li A, Lin N, et al. Are high-degree representations really unnecessary in equivariant graph neural networks? NeurIPS 2024.
• [C] Zhang Y, Cen J, Han J, et al. Improving equivariant graph neural networks on large geometric graphs via virtual nodes learning. ICML 2024.

Dear Reviewer,

We would like to express our sincere gratitude for your careful review and positive evaluation of our paper. Your encouraging remarks and constructive suggestions have been truly valuable to us.

We would be most grateful if you could kindly let us know whether our responses have fully resolved your concerns, or if there are any remaining questions or points we could further clarify.

Thank you once again for your time and insightful feedback.

Best regards, The Authors

Thank you for your comments and questions, and we salute your hard work and time in the review process. We summarize and respond to your questions as follows:

R1 (W1). Discussion of Expressive Power

Thank you for highlighting this important aspect. Expressive power refers to a model's ability to approximate functions. In general, the more functions a model can accurately approximate, the greater its expressive power. Completeness, actually, denotes the strongest level of expressive power: it is the capacity of a model to approximate any equivariant function to arbitrary precision.

Both of Thm. 3.2 in our paper and Thm. 2 in D-spanning [1] ultimately establish completeness, although their derivation processes differ substantially. Specifically, whereas D-spanning [1] relies on constructing equivariant polynomials via sufficient layers of tensor-product operations, our approach requires a set of complete invariant scalar functions along with a full-rank steerable basis set.

Furthermore, our canonical form can substitute for complex and high-order equivariant functions in practical applications (not just simple and low-order equivariant functions), as our porposed model is a complete model of arbitrary degree $l$.

R2 (W2). Comparison with Existing Invariant Representations

We are grateful for your insightful suggestion regarding comparisons with common invariant representations such as FPFH and GotenNet. In response, we have conducted additional experiments as follows:

1. Inv$_\text{FPFH}$: We construct additional node features using the FPFH descriptor and introduce them into the EGNN framework.
2. Inv$_\text{GotenNet}$: We similarly inject node features generated by the embedding layer of GotenNet.

All methods are evaluated on the N-body dataset, with the results presented below (Table S1):

Table S1. Results on the N-body Dataset ($\times 10^{-2}$).

We observe that, FPFH-derived features can enhance EGNN performance, and GotenNet embeddings only outperform EGNN at shallower depths. In contrast, our model consistently achieves leading or comparable results, especially as depth increases. We will discuss these insights more thoroughly in the revised manuscript.

R3 (W3). Additional experiments on Complex Real-world Systems

Thank you for encouraging us to investigate more challenging datasets. As QM9 is primarily an invariant prediction task, but our work aims to advance the performance of equivariant functions, we instead select the large-scale Water-3D dataset from [A] for evaluation. This dataset involves an equivariant task to predict future coordinates for fluid particles (average >8K nodes/system). Due to time constraints, we only evaluate a substantial subset (Water-3D-mini: 3,000/300/300 for train/val/test) using up to 1,000 training epochs.

Table S2. Results on Water-3D-mini ($\times 10^{-4}$).

Our approach (EGNN$_\text{cpl}$-global/local) consistently outperforms the baselines at every depth, and achieves superior results with fewer layers. These findings further support both the scalability and efficiency of our method of handling large, real-world systems.

R4 (W4). Improved the Presentation of Original Thm. 3.5

We apologize for our previous confusing presentation. Our intention was to present a theorem regarding the time complexity of Algo. 3. We will clearly state this theorem and provide a detailed proof to ensure the correctness and completeness of our claims in the revised manuscript.

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Thank you once again for your constructive feedback, which has directly contributed to improving the clarity, empirical evaluation, and theoretical discussion of our work. We will incorporate all clarifications and additional results into the revised version of the manuscript. We sincerely hope that our responses and revisions will address your concerns, and we would be grateful if you might consider these improvements in your evaluation.

Reference

• [A] Zhang Y, Cen J, Han J, et al. Improving equivariant graph neural networks on large geometric graphs via virtual nodes learning. ICML 2024.

Dear Reviewer,

Thank you very much for your thoughtful and valuable feedback, and for the considerable time and effort you devoted to reviewing our work.

If you have any further questions or concerns regarding our submission, please do not hesitate to let us know. We would greatly appreciate any additional opportunity for discussion or clarification that could further improve our work. We also sincerely hope that, in light of the revisions and clarifications we have provided, you might kindly reconsider your evaluation of our manuscript.

Thank you once again for your valuable input and attention. We look forward to your reply.

Best regards, The Authors

Thank you for your rebuttal. Most concerns addressed and score raised.

However, one point needs to be clarified: although QM9 predicts invariant features, it heavily relies on higher-degree equivariant features, such as energy-prediction methods. Invariant features are essentially a special case of equivariant features (with $l=0$). Many works based on spherical harmonic representations employ higher-degree equivariant representations to learn atomic interactions during the message-passing process, ultimately compressing these higher-degree equivariant features ($l>0$) into invariant ones. Therefore, I believe experiments on datasets like QM9, and even larger ones such as Molecule3D and OC20, are necessary, as they can accurately reflect whether such invariant representations are truly effective. I hope the authors can include related case studies in future work.