Learning Molecular Symmetry Breaking via Symmetry-adapted Neural Networks

We thank the reviewer for the thoughtful review and valuable feedback. In what follows, we provide point-by-point responses to your comments.

W1: I think there is generally an issue with this paper in terms of claiming novelty for things that were proposed in previous works and correctly acknowledging previous contributions.

Response: Thank you for pointing out the potential issues with acknowledging previous contributions. We kindly refer the reviewer to the first paragraph of Section 3, the second paragraph of Section 1 (top of page 2), and the final paragraph of Section 3.1 where we attributed credit to Kaba and Ravanbakhsh 2023, Xie and Smidt 2024, and Wang et al., 2024 among others. In our revised paper, we have incorporated a discussion of Hofgard 2024, and emphasized the novelty of our contributions in comparison to these papers.

W2: The most important case of this is the loss function Eq.3, which was already proposed and discussed by Xie and Smidt 2024. This is not acknowledged and is claimed to be an original contribution of the paper. The paper should be updated to not claim that this is an original contribution, and to correctly acknowledge Xie and Smidt 2024.

Response: We thank the reviewer for pointing out that the loss function has been proposed by Xie and Smidt 2024. We did not notice this loss function appeared in the appendix of the Xie and Smidt 2024. We have removed this from our contribution statement and have revised the related statement from the entire paper.

W3: The paper "Symmetry breaking and equivariant neural networks" (Kaba and Ravanbakhsh 2023) introducing relaxed equivariance as a solution to the symmetry breaking problem is also not mentioned although related. It should be mentioned and discussed.

Response: We kindly direct the reviewer to the beginning of Section 3, where we have outlined the contributions of Kaba and Ravanbakhsh (2023) to the concept of relaxed equivariance. To clarify further, while Kaba and Ravanbakhsh (2023) introduced the potential application of relaxed equivariance to the spontaneous symmetry breaking (SSB) problem, they did not develop specific methods or models to address it. In our work, we first provide a clear explanation of why relaxed equivariance is well-suited for SSB. We then propose a framework that leverages this concept to effectively learn SSB.

W4: It also seems to me that there are instances in the paper where appropriate citations is not given to the right papers. For example, Baker et al. 2024 did not introduce either canonicalization or relaxed equivariance, yet the way in which the paper is cited seems to suggests that.

Response: In our revised paper, we clarified the contributions of Baker et al. 2024, which provides an algorithm for canonicalization.

W5: I generally suggest to expand the related works section to discuss what solutions were previously proposed to the symmetry breaking problem and how the proposed one differentiates itself from these methods. For example, it is said that "[other] approaches are not applicable to learning symmetry breaking". What does that mean and how is that so, given the proposed method is based on canonicalization which is one of these other approaches?

Response: Thank you for this suggestion. In our revised paper, we have both expanded the contributions section and added discussion emphasizing canonicalization, relaxed equivariance, and spontaneous symmetry breaking in the context of the related works.

W6: I found that the baselines specifications was incomplete. I did not see explained in the paper what SchNet + MLP or EGNN + MLP means in terms of implementation. The fact that these models struggle on all tasks seems to indicate that they are simply not appropriate baselines. I suggest that the authors instead compare with Hofgard 2024 et al. or Lawrence et al. 2024 instead.

Response: In SchNet+MLP and EGNN+MLP the output of each architecture has been fed through an additional two-layer MLP to break the symmetry enforcement. In the revision, we have emphasized this and provided a discussion that these architectures can perform well but do not always perform well as intended by the highlighting distinguishment.

We notice that there is ambiguity because the reviewer did not leave precise citations for [1] and [2]. We have provided citations below and we hope the reviewer can provide some clarification.

[1] Hofgard, Elyssa, Rui Wang, Robin Walters, and Tess Smidt. "Relaxed Equivariant Graph Neural Networks." arXiv, 30 July 2024, arXiv:2407.20471v1. https://arxiv.org/pdf/2407.20471.

[2] Lawrence, Hannah, Vasco Portilheiro, Yan Zhang, and Sékou-Oumar Kaba. "Improving Equivariant Networks with Probabilistic Symmetry Breaking." OpenReview, 2024, https://openreview.net/pdf?id=1VlRaXNMWO.

I thank the authors for their answers. They have addressed some of my concerns, but not all.

• Experiment 6.1:
  • Is symmetry breaking required for this task? Why are the equivariant baselines performing well?
  • The model with best performance should be bolded in Table 1 even if it's not the proposed method to be consistent with the rest of the paper.
  • Ablation study regarding equivalence class scheme: I don't see a clear reference to this, what I see is an ablation study on the effect of the SBCD, which is a different point. I still don't see evidence that the clustering scheme based on equivalence classes is beneficial.
• I find the description in 6.2 confusing. Here are my issues :
  • It is not clear to me what exactly is predicted, is it the time evolution of the molecular configuration? That, and how is the prediction performed (what is given as input to the network) should be clarified. A sentence saying "in general, we use force regression to make predictions unless utilizing symmetry-breaking architectures." was added, but then what is used how are predictions made with symmetry-breaking architectures if not forces? Some clarification on this would be necessary.
  • I don't why this task exemplifies the symmetry breaking problem. I have not fully understood what the task is, but I assume it is to predict the motion of the molecule. For that position, and velocity are necessary. Even in the asymmetric stretch configuration, the molecule doesn't have the same symmetry as the stretch, but the input to the neural network (positions and velocities) will not have special symmetries. So an equivariant network is expected to be able to perform this task. Maybe this is the reason why some equivariant baselines perform as well or better than the proposed method.
  • The sentence "we see several expected results, in particular, that symmetry preserving and approximate symmetry methods are insufficient for resolving the broken symmetry." was added. This does not seem true, some equivariant perform fine and even outperform the proposed method, if I am not mistaken. The authors should be clear here and provide an explanation (see previous point). Assigning that to future work is not an option as this is related to the central point of the paper.
  • I don't understand why the proposed method is bolded in Table 2 if some baselines perform as well or better. Again, for consistency, the best method for each task should be bolded.
• The added experimental results have shown that either models relying on equivariant features (EGNN, PaiNN) or models with noise perform on par or better than the proposed method. In particular, some models are not supposed to solve the tasks properly due to equivariance and yet perform fine. The performance of the proposed method is therefore mitigated. Can the authors comment on that and suggest ways to update the paper accordingly?

Q5: I don't know why this task exemplifies the symmetry-breaking problem. I have not fully understood what the task is, but I assume it is to predict the motion of the molecule. For that position and velocity are necessary. Even in the asymmetric stretch configuration, the molecule doesn't have the same symmetry as the stretch, but the input to the neural network (positions and velocities) will not have special symmetries. So an equivariant network is expected to be able to perform this task. Maybe this is the reason why some equivariant baselines perform as well or better than the proposed method.

Response: The input to the model is the indicated structure of the H$\_2$​O molecule, and the objective is to predict the structure where the positions align with the endpoints of each indicated vector. Among the baseline models, PaiNN stands out as an outlier, performing well due to its ability to describe the normal modes using the eigenvalues of a higher-order tensor. To further support our conjecture about PaiNN, we have included results without the tensor-valued output in Appendix C.5. This task effectively tests symmetry breaking in vector-valued equivariant predictions.

Q6: The sentence "we see several expected results, in particular, that symmetry preserving and approximate symmetry methods are insufficient for resolving the broken symmetry." was added. This does not seem true, some equivariant perform fine and even outperform the proposed method, if I am not mistaken. The authors should be clear here and provide an explanation (see previous point). Assigning that to future work is not an option as this is related to the central point of the paper.

Response: We have revised the text for clarity. The only outlier in this task is PaiNN, which provides both a tensor-valued equivariant output and a vector-valued output. This highlights a critical limitation of this synthetic task for tensor-valued equivariant architectures. To address this, we demonstrate directly that PaiNN, when restricted to exclude the tensor-valued output, is unable to learn effectively. The corresponding results have been included in Appendix C.5.

Q7: I don't understand why the proposed method is bolded in Table 2 if some baselines perform as well or better. Again, for consistency, the best method for each task should be bolded.

Response: Thank you for pointing this out. We previously used bold text to indicate our method but have removed it for consistency. Please refer to our responses above for detailed explanations of why certain methods perform well.

Q8: The added experimental results have shown that either models relying on equivariant features (EGNN, PaiNN) or models with noise perform on par or better than the proposed method. In particular, some models are not supposed to solve the tasks properly due to equivariance and yet perform fine. The performance of the proposed method is therefore mitigated. Can the authors comment on that and suggest ways to update the paper accordingly?

Response: While several baselines equipped with symmetry-breaking sets demonstrate the ability to learn symmetry breaking, it is particularly noteworthy that our architecture can effectively handle equivariance despite not being inherently equivariant by design. We emphasize the importance of constructing the symmetry-breaking set and propose the SANN architecture as a compelling approach that warrants further exploration, especially in the context of architecture designs based on equivalence classes.

Thank you once again for considering our rebuttal. Please do not hesitate to reach out if the response and revisions do not fully address your concerns. We sincerely appreciate it if you could let us know if all your concerns about our paper have been cleared.

We sincerely thank the reviewer for their thoughtful and constructive feedback and your engagement in the discussion of our work. We acknowledge that there were errors in some of our synthetic tasks, which we have now thoroughly verified and corrected. Based on your further feedback, we have revised the paper accordingly, with the changes highlighted in red.

Overview: Upon further review, we identified an issue where the labels were incorrectly centered, leading to zero values. This has been corrected to ensure the reported results are accurate. The results reaffirm our theoretical findings that SANN with SBCD is capable of achieving SSB. Additionally, we observed that EGNN+Noise is capable of SSB in certain cases. However, all other architectures fail to surpass the threshold of 0.25 in 10-fold cross-validation.

We understand there is significant interest in identifying which architectures perform well on Task 6.2. To clarify, invariant architectures and approximate equivariant architectures do not perform well. In contrast, symmetry-breaking architectures perform well. Among the baseline models, PaiNN stands out as an outlier, performing well because the normal modes can be described by the eigenvalues of a higher-order tensor. Notably, introducing noise into PaiNN decreased its performance. As noted in the reviewer's earlier comments, symmetry-breaking combined with symmetry-breaking measures enables high performance. To address this further, we have added additional commentary and results to Appendix C.5 to support our conjecture regarding PaiNN.

In what follows, we provide point-by-point responses to your further comments.

Q1: Is symmetry breaking required for this task (experiment 6.1)? Why are the equivariant baselines performing well?

Response: Task 6.1, upon further review, we discovered that the labels were incorrectly centered, resulting in the entire label becoming zero. We have corrected this issue and updated our reported values accordingly. The corrected results confirm our theoretical findings that SANN with SBCD is capable of achieving SSB. Furthermore, we observe that EGNN+Noise demonstrates SSB capability in certain cases. However, all other architectures fail to exceed the threshold of 0.25 in 10-fold cross-validation.

Q2: The model with best performance should be bolded in Table 1 even if it's not the proposed method to be consistent with the rest of the paper.

Response: We have removed the boldface for consistency with the results in Table 2.

Q3: Ablation study regarding equivalence class scheme: I don't see a clear reference to this, what I see is an ablation study on the effect of the SBCD, which is a different point. I still don't see evidence that the clustering scheme based on equivalence classes is beneficial.

Response: In Task 6.1, upon further review, we discovered that the labels were incorrectly centered, which caused the entire label to become zero. We have corrected this issue and updated our reported values accordingly. These results affirm our theoretical findings that SANN with SBCD is capable of achieving SSB. Additionally, we observe that EGNN+Noise demonstrates SSB capability in certain cases. However, all other architectures fail to exceed the threshold of 0.25 in 10-fold cross-validation. Notably, the results from this task underscore that our clustering mechanism is more effective at achieving SSB than simple noise injection.

Q4: It is not clear to me what exactly is predicted, is it the time evolution of the molecular configuration? That, and how is the prediction performed (what is given as input to the network) should be clarified. A sentence saying "in general, we use force regression to make predictions unless utilizing symmetry-breaking architectures." was added, but then what is used how are predictions made with symmetry-breaking architectures if not forces? Some clarification on this would be necessary.

Response: The initial position is provided, and the position at the endpoint of each indicated vector is predicted. A vector-valued equivariant feature can be predicted by taking the derivative of a global invariant feature, similar to regressing energies to predict forces. We leverage this principle to train on equivariant features. In symmetry-breaking architectures, instead of making a global invariant prediction, this layer is replaced with an MLP, and the model is trained on the output of the MLP.

I acknowledge the authors' response, but some of my key concerns were not satisfyingly addressed.

1. The authors propose an architecture based on two components, first canonicalization and second "symmetry-adapted linear combinations". It was already known that canonicalization is sufficient to achieve symmetry breaking. The authors do not provide any (experimental) evidence that the symmetry adapted linear combinations help, as opposed to standard message passing. To assess that, an ablation study using a canonicalization scheme and a standard message passing GNN is expected.
2. Key details are missing from the description of experiment 6.2. What is the data generation process, how many samples were the networks trained on, what are the inputs and targets, etc. Basically, this experiment is far from being reproducible with the information provided in the paper. The same is true to a lesser extent with 6.1.
3. I am not satisfied with the explanation provided for the performance of PaiNN. PainNN is strictly equivariant and it was proved that equivariant networks cannot break symmetries. But PaiNN still performs well on this task. Two explanations are logically possible, either PaiNN is not strictly equivariant or the task does not require symmetry breaking. The explanation provided by the authors does not seem to fall in one of these categories and therefore does not convince me.
4. All the experiments are toy and it is not clear that the proposed method will be of interest in practical applications. Some experiments on a real dataset that is not hand-curated for the purpose of this paper are necessary.

All in all, I think the paper would benefit from some more work on the experimental part, discussion of previous work and mention of limitations. I therefore maintain my score.

We thank the reviewer for the thoughtful review and valuable feedback. In what follows, we provide point-by-point responses to your comments.

Q1: The significance of symmetry breaking in related fields, while highlighted, remains somewhat unclear. Both the number and scale of real-world experiments are limited. The authors attribute this to the lack of accessible datasets, but further evidence demonstrating the importance of symmetry breaking in relevant domains would strengthen the argument that this is not merely an edge case. Could the authors kindly provide more concrete evidence where symmetry breaking plays a critical role in molecular modeling tasks?

Response: We appreciate the reviewer’s invaluable suggestions. We have added a discussion in Appendix C.4 to stress the importance of learning symmetry-breaking while ensuring the model’s equivariance guarantees. We summarize the crucial points in the following:

• For a foundational reference on symmetry breaking in the mechanics of molecular structures, we refer the reviewer to [1]. This text covers order-disorder and structural transitions as well as the study of crystalline solids, liquid crystals, ferromagnetism, superfluids, and superconductors.

• Properties: The Hirshfeld dipole moment represents the charge separation within the molecule or material, offering a more nuanced understanding of molecular polarization. The Hirschfeld approach has been used to design semiconductors and dielectric materials where precise control of polarization is necessary [2]. In lead optimization, Hirshfeld dipole analysis has identified regions of polarity that need adjustment to enhance binding affinity or solubility [3].

• Trajectories: MD simulations introduce asymmetry by capturing thermal fluctuations as structures minimize energy over time. These processes perturb the symmetric starting structure. Understanding these perturbations is critical to drug design [4]. The MD17 dataset contains snapshots of structures during relaxation to predict forces and energies from each snapshot. This equivariant task is marginal compared to predicting the relaxed structure from the initial structure, which requires symmetry breaking.

• Interactions: The open catalyst project contains interactions between adsorbates and catalysts. The adsorbates are small molecules that are ideally suited to our architecture. However, the catalysts are crystalline structures with permutation group symmetries that are not directly addressed by our approach. Therefore, we leave this to future work.

[1] Kivelson, S. A., J. M. Jiang, and J. Chang. Statistical Mechanics of Phases and Phase Transitions. Princeton University Press, 2024. Google Books.

[2] Spackman, Mark A., and Patrick G. Byrom. "A Novel Definition of a Molecule in a Crystal." Chemical Physics Letters, vol. 267, no. 3–4, 1997, pp. 215–220. ScienceDirect, https://doi.org/10.1016/S0009-2614(97)00100-0.

[3] Bultinck, Patrick, et al. "Critical Analysis and Extension of the Hirshfeld Atoms in Molecules." Journal of Computational Chemistry, vol. 23, no. 1, 2002, pp. 822-834. Wiley Online Library, https://onlinelibrary.wiley.com/doi/full/10.1002/jcc.10241.

[4] De Vivo, Marco, Matteo Masetti, Giovanni Bottegoni, and Andrea Cavalli. "Role of Molecular Dynamics and Related Methods in Drug Discovery." Journal of Medicinal Chemistry, vol. 59, no. 9, 2016, pp. 4035-4061. American Chemical Society, https://doi.org/10.1021/acs.jmedchem.5b01684.

Thank you for considering our rebuttal. We appreciate your feedback and are happy to address further questions on our paper.

Dear Reviewer, Do you mind letting the authors know if their rebuttal has addressed your concerns and questions? Thanks! -AC

Thanks for the response. However, it does not fully address my concerns regarding the significance of symmetry breaking and the overall contribution of the work. Convincing and concrete evidence of the importance of symmetry breaking and the paper's contributions would be demonstrated by improved performance on real-world tasks where most current state-of-the-art models struggle. However, the experimental evaluation on the real-world dataset, QM7-X, includes only Schnet and EGNN as baselines, omitting comparisons with many recent state-of-the-art equivariant models. This significantly weakens the empirical support for the contribution.

In addition, the weakness highlighted by my fellow reviewers raised my concern regarding the novelty of the work. E.g., the authors claimed that "we propose a new symmetry breaking measure (SBM)" and even listed this as the first point of the contribution in their initial submission. But Reviewer SA1v pointed out that this has already been proposed by previous work and wasn’t acknowledged. While this was corrected in the revised version, the novelty of the work is thus limited.

I’m unable to raise my rating given these issues.

We thank the reviewer for the thoughtful review and valuable feedback. In what follows, we provide point-by-point responses to your comments.

W1: Stronger baselines should be considered. For instance, many approximate equivariant models are now available and may perform significantly better on SBB tasks compared to strictly equivariant counterparts.

Response: We have included additional numerical results on testing state-of-the-art (SOTA) equivariant and approximate equivariant models in the revised paper. In particular, we have incorporated DimeNet++ and PaiNN as SOTA invariant and equivariant architectures, respectively. We incorporate approximate equivariant architectures by injecting noise into the positional data in the forward training pass as discussed in [1]. In the revised version, we have provided a study in Section 6.1. We observe that approximate equivariance is not capable of strict symmetry breaking.

[1] Lawrence, Hannah, Vasco Portilheiro, Yan Zhang, and Sékou-Oumar Kaba. "Improving Equivariant Networks with Probabilistic Symmetry Breaking." OpenReview, 2024, https://openreview.net/pdf?id=1VlRaXNMWO.

W2: Depending on the symmetry group, calculating the symmetry-breaking measure in Equation 3 might not be straightforward by simply evaluating the metric (m(x,y) ) for each element in the group. How would this measure be computed in such cases? Or does this paper only consider finite groups? This needs to be clarified in the discussion.

Response: You are correct that the construction of the symmetry-breaking measure is only applicable to finite groups. Thank you for pointing this out. We have included terminology in the paper to reflect this and expanded the discussion of our algorithm for generating the stabilizers.

Notice that the set of molecular point groups that are infinite is $C_{\infty v}, D_{\infty h}, K_h$ [1]. $K_h$ represents perfect spherical symmetry which is not observed in practice. $C_{\infty v}, D_{\infty h}$ are linear molecules with or without a center of inversion, respectively. Because the data is aligned, the stabilizers are fixed to the $z$-axis and thus the relaxed metric only penalizes for discrepancies from the $z$-axis, and hence can be implemented as a finite summation.

[1] Harris, D. C., & Bertolucci, M. D. (1989). Symmetry and spectroscopy : an introduction to vibrational and electronic spectroscopy. Dover Publications.

W3: Although the paper is well-structured and provides essential foundational context, it lacks an explanation of why the proposed message-passing framework can address spontaneous symmetry breaking. This point requires further discussion in the paper.

Response: Thank you for your invaluable feedback. As outlined in Proposition 4.1, we demonstrate that the symmetry of the entire molecule is determined by the symmetries of the equivalence classes of its atoms. Building on this, Corollary 4.2 explains that when symmetry breaking (SB) occurs, it must manifest within at least one of these equivalence classes. We have included this explanation in the main text to provide the necessary context. Based on this insight, we propose a message-passing framework that first learns features at the level of equivalence classes, rather than directly from neighboring nodes in the graph. This approach effectively captures SB, as it is directly tied to SB within a class of equivalent atoms. We then aggregate the learned features across equivalence classes to retain the overall symmetry of the molecule, as inspired by the intersection described in Proposition 4.1. This framework enables a more targeted and efficient detection of SB within the molecular structure.

W4: The figures illustrating symmetry-breaking concepts are excellent. An additional illustration that explains the proposed message-passing mechanism would also be very helpful.

Response: Thank you for the suggestion. Due to the limited space for the illustration in the main text, we have incorporated it in Appendix C.1 of the revised version.

W5: Define the Earth Mover’s Distance, as it may be unfamiliar to some in the ML community.

Response: Thank you for bringing this to our attention. In practice, we use the Chamfer distance, which is often used as a proxy for the more computationally demanding EMD. The Chamfer distance measures the similarity between two point sets by averaging the squared distances from each point in one set to its nearest neighbor in the other.

We have included a broader discussion about why this is a useful metric for comparing point clouds in the revision. It is a simple and efficient metric widely used in 3D shape analysis and computer vision.

Dear Reviewer, Do you mind letting the authors know if their rebuttal has addressed your concerns and questions? Thanks! -AC

We thank the reviewer for the thoughtful review and valuable feedback. In what follows, we provide point-by-point responses to your comments.

W1: The paper lacks evaluations and comparisons to a broader range of baseline models. While it compares its results against EGNN and SchNet, multiple equivariant models in the literature are not considered.

Response: In the revised paper, we have expanded the comparative analysis to include additional state-of-the-art models. Specifically, we have incorporated DimeNet++ and PaiNN for each task, broadening the range of invariant and equivariant architectures evaluated. We believe this addition provides a more comprehensive comparison and addresses your concerns regarding the scope of baseline models.

W2: The experimental setup and model evaluation could be expanded to include more benchmarks beyond QM7-X to fully assess the generalizability of the proposed approach.

Response: We have further studied approximate equivariant architectures by injecting noise into the positional data during training following paper [1]. We summarize the limitations of existing datasets below.

Properties: The Hirshfeld dipole moment represents the charge separation within the molecule or material, offering a more nuanced understanding of molecular polarization. The Hirschfeld approach has been used to design semiconductors and dielectric materials where precise control of polarization is necessary [2]. In lead optimization, Hirshfeld dipole analysis has identified regions of polarity that need adjustment to enhance binding affinity or solubility [3]. For this reason, we feel that our property prediction is sufficient to assess the proposed approach.

Trajectories: MD17 contains snapshots of structures during relaxation to predict forces and energies from each snapshot. This equivariant task is marginal compared to predicting the relaxed structure from the initial structure, which requires symmetry breaking. Relaxed structure prediction is critical in drug design [4].

Interactions: The open catalyst project contains interactions between adsorbates and catalysts. The adsorbates are small molecules that are ideally suited to our architecture. However, the catalysts are crystalline structures with permutation group symmetries that are not directly addressed by our approach. Therefore, we leave this to future work.

[1] Lawrence, Hannah, Vasco Portilheiro, Yan Zhang, and Sékou-Oumar Kaba. "Improving Equivariant Networks with Probabilistic Symmetry Breaking." OpenReview, 2024, https://openreview.net/pdf?id=1VlRaXNMWO.

[2] Spackman, Mark A., and Patrick G. Byrom. "A Novel Definition of a Molecule in a Crystal." Chemical Physics Letters, vol. 267, no. 3–4, 1997, pp. 215–220. ScienceDirect, https://doi.org/10.1016/S0009-2614(97)00100-0.

[3] Bultinck, Patrick, et al. "Critical Analysis and Extension of the Hirshfeld Atoms in Molecules." Journal of Computational Chemistry, vol. 23, no. 1, 2002, pp. 822-834. Wiley Online Library, https://onlinelibrary.wiley.com/doi/full/10.1002/jcc.10241.

[4] De Vivo, Marco, Matteo Masetti, Giovanni Bottegoni, and Andrea Cavalli. "Role of Molecular Dynamics and Related Methods in Drug Discovery." Journal of Medicinal Chemistry, vol. 59, no. 9, 2016, pp. 4035-4061. American Chemical Society, https://doi.org/10.1021/acs.jmedchem.5b01684.

We sincerely thank the reviewer for their engagement in the discussion of our work. We acknowledge that there were small errors in some of our synthetic tasks (see details in our further response to Reviewer SA1v), which we have now thoroughly verified and corrected. Based on further reviewer feedback, we have revised the paper accordingly, with the changes highlighted in red.

Response: We have revised this sentence to emphasize the novelty of our approach.

We hope this addresses the reviewer's concerns. We would greatly appreciate it if the reviewer could provide more specific details regarding any remaining concerns with our paper. This will help us address them more effectively and ensure that our revisions meet the expectations of the review.