Fast Crystal Tensor Property Prediction: A General O(3)-Equivariant Framework Based on Polar Decomposition

Authors: Haowei Hua, Wanyu Lin, JINGWEN YANG
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: $O(3)$ group tensor equivariance, polar decomposition, tensor properties

Abstract:
Predicting tensor properties of the crystalline materials is a fundamental task in materials science. Unlike single-value property prediction, which is inherently invariant, tensor property prediction requires maintaining $O(3)$ group tensor equivariance. This equivariance constraint often introduces tremendous computational costs, necessitating specialized designs for effective and efficient predictions. 
To address this limitation, we propose a general $O(3)$-equivariant framework for fast crystal tensor prediction, called {\em GoeCTP}. 
Our framework is efficient as it does not need to impose equivalence constraints onto the network architecture. Instead, {\em GoeCTP} captures the tensor equivariance with a simple external rotation and reflection (R\&R) module based on the polar decomposition. The crafted external R\&R module can rotate and reflect the crystal into an invariant standardized crystal position in space without introducing extra computational cost. We show that {\em GoeCTP} is general as it is a plug-and-play module that can be smoothly integrated with any existing single-value property prediction network for predicting tensor properties. Experimental results indicate that the {\em GoeCTP} method achieves higher prediction performance and runs 13$\times$ faster compared to existing state-of-the-art models in elastic benchmarking datasets, underscoring its effectiveness and efficiency.

Decision: reject
Reviewer scores: 6, 5, 3, 6

Reviews and discussion:
<thread>
creation date:
1729178305784

thread_type:
official_review

<thread_object>
creator: reviewer_pk6i

<content>
summary:
The paper introduces a method for predicting how crystals react to forces applied from different directions, a challenge that requires maintaining consistency regardless of the crystal's orientation in space. 

The proposed method uses a sound mathematical technique (polar decomposition) to standardize crystal positions, enabling faster and more accurate predictions of these directional properties (known technically as tensor properties) while respecting the physics principle of orientation independence (technically called O(3) group equivariance). 

The proposal is significantly faster and more accurate than existing approaches, especially for predicting how materials deform under stress or respond to electric fields.

strengths:
1. The pre-processing step of standardizing crystal positions through polar decomposition is significant for multiple reasons. It ensures equivariance, simplifies model architecture, and preserves tensor properties across different orientations through an additional step.
2. The method achieves 13x speed improvement over prior methods in predicting tensor properties.
3. The paper effectively explains O(3)-equivariance in crystal tensor prediction through clear organisation, helpful diagrams, and accessible mathematical explanations, making sophisticated concepts understandable even to non-specialists.

weaknesses:
1. The paper focuses primarily on quantitative metrics (e.g. Frobenius norm and EwT percentages) to demonstrate the method's effectiveness. However, there is a lack of qualitative insights into how the method affects real-world predictions. Including case studies or qualitative analyses where the method's predictions are compared to known physical properties of materials would strengthen the practical significance.
2. Evaluation is carried out on two specific datasets for dielectric and elastic tensor prediction. While the results are promising, these datasets may not cover the full range of tensor property prediction challenges. Testing on a broader variety of materials, including more extreme cases, would strengthen the claim of generalisability.
3. As the authors discuss between lines 137 and 142, " the requirements for O(3) equivariance typically differ from the O(3)-equivariance
defined in the general molecular studies." Because of these specific requirements, the scope and the applicability of the proposed polar decomposition are limited.

questions:
1. Were there qualitative case studies where the proposed method's predictions were compared to known real-world material properties, such as elastic or dielectric responses of specific materials?
2. Were there plans to test the proposed method on other datasets, especially those involving more complex or extreme tensor property cases (e.g., materials with highly anisotropic properties or rare crystal structures)? 
3. Why was the dataset "Piezo" that was tested in a prior work [Yan et al.] not tested in this paper?
4. What explains the discrepancy in the number of samples in the "Elastic" dataset between the prior work [Yan et al.] and this paper? The prior work [Yan et al.] reports 14,220 samples in Table 1, while this paper reports 25,110 samples in Table 1 on line 364.
5. Could the specific requirements of O(3) equivariance for crystalline materials limit the use of polar decomposition? Additionally, how did these differences impact the potential generalisation of the proposed method to molecular systems, where O(3) equivariance is defined differently?

[Yan et al.] [A Space Group Symmetry Informed Network for O(3) Equivariant Crystal Tensor Prediction, In ICML 2024](https://openreview.net/forum?id=BOFjRnJ9mX).
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</thread>
<thread>
creation date:
1730128902923

thread_type:
official_review

<thread_object>
creator: reviewer_3tqt

<content>
summary:
This paper proposes a novel approach intended to achieve O(3) tensorial equivariance by transforming crystal structures into standardized positions, so that neural networks do not need to satisfy equivariance during prediction. The invariant predictions are then mapped back to equivariant outputs. While the goal of O(3) equivariant predictions is notable, prior work has already addressed this problem with established solutions, including ETGNN's vector outer product and GMTNet's equivariant networks. Additionally, existing techniques, such as frame averaging and minimal frame averaging, can be employed to achieve O(3) tensor equivariance effectively. Moreover, this work does not consider space group constraints, which are crucial for tensorial properties in crystallography. As such, the novelty and contribution of this work are limited and do not meet the standards for acceptance at current form.

strengths:
1. An alternative approach is proposed to achieve tensor O(3) equivariance. If used, it is faster than equivariant network based O(3) equivariant predictions like GMTNet.

weaknesses:
1. **Lack of Consideration for Space Group Constraints**

Space group constraints, which are fundamental in determining the tensor properties of crystals, are not accounted for in this work. No experimental results are provided to verify whether the proposed method can generate predictions that align with these constraints. Crystals exhibit unique tensor characteristics that are intrinsically tied to their crystal class or space group, and ignoring these symmetries is a significant oversight.

2. **Limited Improvement in Performance**

The integration of the proposed module does not enhance eComformer’s performance beyond achieving O(3) equivariance, as shown in Table 4. Other alternatives, such as ETGNN, frame averaging, and minimal frame averaging, also achieve O(3) equivariance but were not discussed or compared in this work. A more thorough discussion and comparative analysis of technical contributions and novelty would be beneficial.

3. **Absence of Experiments on Piezoelectric Tensors**

The work lacks experiments on piezoelectric tensors, which are especially sensitive to space group constraints. Including these experiments would strengthen the evaluation of the proposed approach’s applicability across tensorial properties with varying sensitivity to symmetry constraints.

4. **Performance on Elastic Tensors**

The performance of the proposed method on elastic tensors is significantly lower than GMTNet's original results. This suggests potential limitations in the approach’s effectiveness.

5. **Efficiency Gains Not Attributable to Proposed Method**

The efficiency gains claimed largely derive from the use of the lightweight eComformer, not the proposed approach. Similar speed-ups could be achieved by combining eComformer with other O(3) equivariant methods such as ETGNN’s vector outer-product approach or minimal frame averaging.

questions:
As listed above in Weaknesses.
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</thread>
<thread>
creation date:
1730451389587

thread_type:
official_review

<thread_object>
creator: reviewer_rvfl

<content>
summary:
The authors propose an O(3)-equivariant framework, GoeCTP, for crystal tensor prediction. GoeCTP utilizes polar decomposition to rotate and reflect the crystal into a standardized invariant position in space. The orthogonal matrix obtained from the polar decomposition is used to achieve equivariant tensor property predictions. The GoeCTP method achieves higher quality prediction results and runs more than 13× faster on the elastic benchmarking dataset.

strengths:
1. GoeCTP is plug-and-play as it can be readily integrated with any existing single-value property prediction network for predicting tensor properties. 
2. GoeCTP does not introduce excessive computational overhead.

weaknesses:
1. The article has limited contributions in terms of methodological innovation, as the methods and main structure used by the authors are derived from DiffCSP++[1] and Comformer[2]. For detail, the polar decomposition method used may have been inspired by DiffCSP++, while the code implementation adopts the structure of Comformer.   
2. The article does not clearly explain why Equation 2 needs to be satisfied. It is suggested that the authors provide more background or explanation regarding the physical or mathematical significance of Equation 2 in relation to tensor property prediction. This would help readers better understand the importance of this equation within the proposed framework.   
3. There are some citation issues in lines 339-340 of the article.


References:   
[1] Rui Jiao, Wenbing Huang, Yu Liu, Deli Zhao, and Yang Liu. Space group constrained crystal generation. In The Twelfth International Conference on Learning Representations, 2024.  
[2] Keqiang Yan, Cong Fu, Xiaofeng Qian, Xiaoning Qian, and Shuiwang Ji. Complete and efficient graph transformers for crystal material property prediction. In The Twelfth International Conference on Learning Representations, 2024.

questions:
1. Is it the case that all tensor properties need to satisfy Equation 2, or only certain tensor properties? Why？Please provide specific examples of tensor properties, indicating which properties need to satisfy Equation 2 and which do not, along with an explanation of the underlying reasons for this distinction. This would help deepen the understanding of the method's applicability and limitations.
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</thread>
<thread>
creation date:
1731005983158

thread_type:
official_review

<thread_object>
creator: reviewer_q6uc

<content>
summary:
This paper presents GoeCTP, a novel O(3)-equivariant framework for predicting tensor properties of crystalline materials. The key innovation is using polar decomposition to handle tensor equivariance through an external rotation and reflection (R&R) module, rather than building it into the network architecture.

strengths:
1. Novel use of polar decomposition for handling tensor equivariance
2. Strong theoretical foundation with clear mathematical proofs
3. Clear illustrations and explanations of complex concepts

weaknesses:
1. Limited discussion of potential limitations or failure cases
2. Only two datasets are used.1.

questions:
1. What are the limitations of using polar decomposition for this application? Are there edge cases where it might not work well?

2. How does the method perform on different types of crystal structures beyond those tested?
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<thread>
creation date:
1732285733512

thread_type:
official_comment

<thread_object>
creator: authors

<content>
comment:
We sincerely thank all four reviewers for their thoughtful feedback and insightful comments. 
We are particularly encouraged by the reviewers’ feedback. We have made a heavy revision to our paper according to the reviewer's constructive suggestions. Below, we summarize some key modifications in the updated PDF document:

---------------------------------

*Main text*

$\bullet$  (Section 5.1) Add an introduction for new piezoelectric tensor
dataset

$\bullet$ (Section 5.2) Add comparison of prediction performance 
on the piezoelectric dataset


$\bullet$ (Section 5.2) Add ablation study for verifying the $O(3)$ equivariance with piezoelectric dataset

$\bullet$ (Section 5.2) Add efficiency comparison on the piezoelectric dataset

---------------------------------

*Appendix*

$\bullet$ (Appendix A.2)
Add an introduction for $O(3)$ equivariance for different crystal tensor properties

$\bullet$ (Appendix A.3)
Add an introduction for the number of independent components in the piezoelectric tensor

$\bullet$ (Appendix A.5)
Add detailed discussion for limitations and extensibility of our method

$\bullet$ (Appendix A.6)
Add discussion for how to further utilize the tensor properties symmetry

---------------------------------

If further consideration remains, please kindly let us know. We are very happy to make a further revision in light of your great suggestions. We will address comments by each of the reviewers individually.
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Metareview:
metareview:
This paper received very weak support from the reviewers, and some of them raised major concerns on this paper.