We sincerely thank you for your constructive suggestions. Below we will address your questions in detail.

$\bullet$ Response to weakness 1:

Thank you for your comments.

In the figure, edges of different colors represent connections between atoms of different types (see the lower left corner of Figure 3). Please note the red and green edges in Figure 3, which are highlighted with glowing effects for better visual distinction. It can be observed that applying the local frame leads to identical local structures (the red and green edges of two different atoms point in the same direction). In contrast, applying SPFrame preserves the differences in local structures (the red and green edges of the two atoms point in different directions). This structural identity in the local frame causes the node/atom features to become identical after message passing, resulting in information loss. For more details, please refer to our response to Reviewer 5W3r's weaknesses.

We hope our response addresses the reviewer’s question.

$\bullet$ Response to weakness 2:

We sincerely thank you for your insightful comments. Due to the highly symmetric nature of crystal structures, using the Gram-Schmidt orthogonalization-based rotation matrix construction method (detailed in Appendix A.4) may result in two invariant vectors being nearly identical, leading to redundancy and duplication of information. This can increase the difficulty of network training and hinder performance improvements. In contrast, quaternion-based methods do not suffer from this limitation.

In addition to Gram-Schmidt and quaternion, Ref. [1] proposes a rotation matrix construction method based on singular value decomposition (SVD). We conducted supplementary experiments using this method to construct $F_{\text{inv}}$ on the JARVIS dataset, and the results are shown below.

Table: Property prediction results on the JARVIS dataset

Compared to the quaternion-based approach, the SVD-based method did not yield significant improvements. In [1], the SVD-based method demonstrated better performance on scalar regression tasks (e.g., Experiment 4, where an MLP is used to predict scalar values). However, since our task is more complex—mapping from graph structures to scalar properties, this may account for the performance difference between the quaternion-based and SVD-based methods. A more detailed investigation will be left for future work.

We hope our response addresses the reviewer’s question.

[1] Learning with 3D rotations, a hitchhiker’s guide to SO(3). ICML2024.

$\bullet$ Response to weakness 3:

We sincerely thank you for your constructive comments.

Previously, DiffCSP++ [1] applied the frame method to crystal structure prediction and materials generation, where frames were used to construct invariant representations of crystals, thereby removing the requirement for equivariance in the subsequent denoising model. Our method can serve a similar purpose. Since our approach is learnable frames, a more recent concurrent work - RADM [2] (for molecular generation) offers an insightful application strategy. RADM employs frames to eliminate the need for equivariance in the denoising model, allowing the use of a more advanced non-equivariant diffusion transformer (DiT) as the denoising model to enhance generation quality. Additionally, it leverages an autoencoder to learn richer feature representations in the latent space.

Similarly, our SPFrame can be applied in crystal generation. During autoencoder training, SPFrame can preserve crystal symmetries and enable the learning of more expressive latent features. Moreover, by removing the equivariance constraint from the denoising model, SPFrame allows the adoption of powerful non-equivariant DiT to further improve generation performance.

We hope our response addresses the reviewer’s question.

[1] Space Group Constrained Crystal Generation. ICLR2024.

[2] Scalable Non-Equivariant 3D Molecule Generation via Rotational Alignment. ICML2025.

$\bullet$ Response to weakness 4:

We sincerely thank you for your constructive comments.

First, we discuss how these generation methods leverage crystal symmetry during the generation process. The core idea of these approaches is to simplify the data to be generated by exploiting the crystal symmetry (For example, by generating only the asymmetric unit and then reconstructing the full crystal structure using symmetry operations).

DiffCSP++ [1]: Due to crystal symmetry, the elements of the lattice matrix in crystals belonging to different space groups are subject to specific constraints, as described in Table 1 of [1]. Therefore, when generating the lattice matrix, DiffCSP++ only needs to generate the elements that are not restricted by the crystal symmetry, thereby simplifying the generation process. For the generation of atomic fractional coordinates and types, crystal symmetry allows further simplification. Since atoms at symmetrically equivalent positions can be derived from a single representative atom, it is sufficient to generate only one such atom, and the rest can be obtained through symmetry operations. DiffCSP++ separately and simultaneously generates the lattice, fractional coordinates, and atomic composition under the reduced form of the space group constraint. The generated crystals strictly satisfy the space group constraint.

SymmCD [2]: SymmCD only generates the asymmetric unit and, during the generation process, it produces only the unit parameters of the asymmetric unit rather than the full lattice matrix (as illustrated in Figure 1 of [2]). This approach differs from that of DiffCSP++, which focuses on generating the full lattice matrix and uses predefined structural templates to enforce space group constraints. SymmCD also provides experimental evidence suggesting that this approach in DiffCSP++ (using predefined structural templates) may limit the diversity and novelty of the generated samples.

Wyckoff Transformer [3]: Wyckoff Transformer is an autoregressive generation method, distinct from diffusion-based methods [1,2,4]. For atoms located at symmetric positions, it generates only discrete data such as the space group, atomic element, site symmetry, and enumeration, rather than the explicit atomic coordinates or the crystal lattice matrix. The final atomic structure is obtained by combining generated discrete information with an energy relaxation method.

WyckoffDiff [4]: WyckoffDiff, given a space group and Wyckoff positions, directly predicts the probability of the atom type occupying the corresponding position, as shown in Figure 1 of [4]. This approach bears some similarity to [5], where the probability of an element’s occurrence is predicted given the periodic table. However, WyckoffDiff incorporates symmetry constraints into the structure generation process.

Our method is applied to scalar property prediction of crystals, which differs from the use cases of the aforementioned generative methods. Our approach primarily addresses the limitation of local frames, which cause atoms at symmetry-equivalent positions to have identical local structures. This structural uniformity leads to identical node features and consequently results in information loss (see our response to Reviewer 5W3r's Weaknesses). Therefore, when designing the frame, we adopt an associative approach that combines an invariant local frame with an equivariant global frame shared across the atomic system. This design prevents the destruction of crystal symmetry and ensures that atoms at symmetry-equivalent positions retain their symmetric local environments, thereby enabling their node features to remain distinguishable.

[1] Space Group Constrained Crystal Generation. ICLR2024.

[2] SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models. ICLR2025.

[3] Wyckoff Transformer: Generation of Symmetric Crystals. ICML2025.

[4] WyckoffDiff -- A Generative Diffusion Model for Crystal Symmetry. ICML2025.

[5] Scalable Diffusion for Materials Generation. ICLR2024.

We will include a discussion above of these works in our paper. We hope our response addresses the reviewer’s question.

$\bullet$ Response to weakness 5:

We sincerely thank you for your valuable comments.

We conducted experiments using iComFormer [1] as the backbone, which outperforms eComFormer. Below, we provided the evaluation of JARVIS dataset. Notably, our solution outperforms or at least is on par with the baselines (Crystalframer). The preliminary results are as follows.

Table: Property prediction results on the JARVIS dataset

We hope our response addresses your concerns.

Thank you for your detailed answers and for running additional experiments to show improvements with iComFormer and the benefits of Quaternion-based SPFrame. Also, it would be good to add the explanation corresponding to Figure 3 in the paper. I retain my score and recommend acceptance.

We sincerely thank you for your useful comments. Below we will address your questions in detail.

$\bullet$ Response to weakness 1:

We sincerely thank you for your comments.

Frame-based methodologies are indeed a general geometric learning approach. Our contribution first lies in identifying and articulating the challenges that arise when directly applying general frame methods to crystalline materials (highly symmetric atomic systems). We then propose a targeted solution to address these challenges, which, to the best of our knowledge, has not been considered in prior work.

Moreover, we provide a theoretical justification demonstrating the superiority of SPFrame over conventional local frame methods (please refer to our response to Reviewer 5W3r’s weaknesses), which further strengthens the theoretical significance of our work.

Finally, SPFrame provides a new insight for both machine learning and materials science, namely, that general-purpose machine learning techniques must undergo specific adaptation to effectively address the unique characteristics of materials science problems.

We hope our response addresses your concerns.

$\bullet$ Response to weakness 2:

We sincerely thank you for your valuable comments. Our method primarily focuses on enhancing the performance of the backbone (As demonstrated by the performance gains achieved when applying SPFrame to eComFormer). Using a more powerful backbone is theoretically expected to yield even better results. To address the reviewer’s concerns, we conducted additional experiments by replacing the backbone to achieve more significant improvements over the baseline. Specifically, we used iComFormer [1], which outperforms eComFormer, as the new backbone. Below, we provided the evaluation of JARVIS dataset. Notably, our solution outperforms or at least is on par with the baselines (Crystalframer). The preliminary results are as follows.

Table: Property prediction results on the JARVIS dataset

We hope our response addresses your concerns.

[1] Complete and efficient graph transformers for crystal material property prediction. ICLR2024.

$\bullet$ Response to weakness 3:

We sincerely thank you for your valuable comments. We completed preliminary experiments on bulk and shear modulus properties in the JARVIS dataset, as shown below.

Table: Bulk and shear modulus in the JARVIS dataset prediction results

It can be observed from the table that SPFrame significantly improves the performance of the backbone method eComFormer, achieving the best prediction result for Shear Modulus (Gv).

[1] CrysGNN: Distilling pre-trained knowledge to enhance property prediction for crystalline materials. AAAI2023.

[2] A Diffusion-Based Pre-training Framework for Crystal Property Prediction. AAAI2024.

$\bullet$ Response to weakness 4:

Thank you for your careful review. We will revise these typos. Equation 1 was mistakenly written as an expression describing equivariance; we will correct it to properly describe invariance.

$\bullet$ Response to weakness 5:

We sincerely appreciate your useful comments. We will include a discussion below of these related works in our paper.

In crystal property prediction tasks, in addition to frame-based methods (which can also be regarded as a form of representation learning), pretraining [1,3,6] and representation learning [2,4,5,7] are two other important approaches for improving prediction accuracy.

Pretraining methods mainly focus on pretraining the backbone network architecture. CrysXPP [1] designs an autoencoder, CrysAE, to perform self-supervised pretraining, thereby capturing essential structural and chemical features from large amounts of unlabeled crystal graph data and reducing prediction errors. CrysGNN is a dedicated pretrained GNN framework that, in addition to common self-supervised strategies such as feature reconstruction used in CrysXPP, incorporates connectivity reconstruction and contrastive learning over different crystal systems, which further enhances prediction performance. CrysDiff [6], on the other hand, is a diffusion-based pretraining method. During the pretraining phase, CrysDiff performs a crystal structure reconstruction task based on a diffusion model to learn the underlying edge distribution of crystal structures. In the fine-tuning phase, the model generates the target property values under the guidance of the structural data.

Unlike pretraining, representation learning primarily focuses on constructing more effective representations of crystal structures. Beyond the commonly used bond angle information to encode directional features, ALIGNN-d [2] incorporates dihedral angles into the crystal representation. This simple extension leads to a memory-efficient graph representation that captures the complete geometry of atomic structures. CrysMMNet [4] leverages textual descriptions of materials to encode global structural information into the graph structure, enabling the learning of a more robust and enriched representation of crystalline materials. Geom3D [5] is a benchmarking study on geometric representations, systematically evaluating the effectiveness of various geometric encoding strategies such as spherical harmonics basis, frame-based basis, and angle-based features. CrysAtom [7] uses unlabeled crystal data in an unsupervised manner to learn a distributed representation of atoms, which significantly improves the performance of various property prediction models.

We hope our response addresses the reviewer’s concerns.

[1] CrysXPP: An explainable property predictor for crystalline materials. npj Computational Materials, 2022.

[2] Efficient and interpretable graph network representation for angle-dependent properties applied to optical spectroscopy. npj Computational Materials, 2022.

[3] CrysGNN: Distilling pre-trained knowledge to enhance property prediction for crystalline materials. AAAI2023.

[4] CrysMMNet: Multimodal Representation for Crystal Property Prediction. UAI2023.

[5] Symmetry-informed geometric representation for molecules, proteins, and crystalline materials. NeurIPS2023.

[6] A Diffusion-Based Pre-training Framework for Crystal Property Prediction. AAAI2024.

[7] CrysAtom: Distributed Representation of Atoms for Crystal Property Prediction. LoG2024.

We sincerely thank you for your thoughtful feedback. Below we will address your questions in detail.

$\bullet$ Response to weakness 1:

We sincerely thank you for your valuable comments.
First, it is important to clarify that atoms at the same Wyckoff position share similar local environments, but not identical ones. As illustrated in Figure 2 of our paper (page 4), they can be transformed into the same local environment through symmetry operations such as rotations. Below, we discuss our motivation both from an intuitive and theoretical perspective.

From an intuitive perspective: As mentioned in the Introduction of our paper, directional information between atoms is a crucial geometric feature. Leveraging this information can enhance network performance. For example, ComFormer [2] improves upon Matformer [1] by incorporating directional information into the message passing process. However, as shown in Figure 2 of our paper, applying a local frame to ComFormer causes the directional information of symmetry-equivalent atoms p and q to be lost. This results in the node features of atoms p and q becoming identical across all subsequent message passing layers of the network, effectively degrading ComFormer back to Matformer. In contrast, the original ComFormer preserves the directional distinction between p and q, ensuring that their node features remain different in all subsequent layers. SPFrame is designed to overcome this limitation of the local frame. It ensures that the node features of symmetry-equivalent atoms p and q remain distinct throughout all layers of message passing, just as in the original ComFormer, thereby preserving essential geometric information.

From a theoretical perspective: Let a crystal structure be represented as a graph with $N$ atoms. Denote the node/atom feature corresponding to atom $i$ as $f_i$. We define the complete set of node/atom features under two different framing schemes as follows:

$A=\\{f_{local,i}|i=1,2,..,N\\}, B=\\{f_{sp,i}|i=1,2,..,N\\},$

where $A$ corresponds to node/atom features in the network using local frames, and $B$ corresponds to those in the network using the SPFrame approach.

We assume that the atoms with indices $p,q$ ($1<p,q<N$) are symmetry-equivalent. The use of the local frame results in identical local environments for atoms p and q (According to the illustration in Figure 2 in our paper, page 4). As a consequence, after message passing via Equation (3) (in our paper, page 4), we can have

$f_{local,p}=f_{local,q}.$

Consequently, the effective number of distinguishable node/atom representations is reduced. Denoting the cardinality as $|\cdot|$, we can have:

$|A|=N-1<N.$

In contrast, the SPFrame approach preserves the differences in the local environments of atoms p and q. After message passing via Equation (5) (in our paper, page 5), the node/atom representations satisfy:

$f_{local,p} \neq f_{local,p} \implies |B|=N.$

For scalar crystal property prediction, the final node features are first aggregated to get global graph-level representation. This graph-level representation is then passed through a regression head. We denote the prediction target as a variable $Y \in \mathcal{Y}$. The neural network induces the following mapping:

$h_{local}:A \mapsto \mathcal{Y}, \quad h_{sp}:B \mapsto \mathcal{Y}.$

Since $|B|>|A|$, there exists a surjective mapping $g_1:B \mapsto A$, such that $h_{local}\circ g_1=h_{sp}$. However, an injective mapping $g_2:A \mapsto B$ does not exist in general due to loss of distinguishability in $|A|$. Consequently, the information flow can be described via the following Markov chain: $Y \to B \to A.$

Applying the data processing inequality to this chain yields: $I(Y;B) \ge I(Y;A),$

with equality if and only if:

$I(B;Y | A) =0 \quad\text{and}\quad Y \to A \to B$

i.e., the chain $Y \to A \to B$ is also valid. However, the absence of a mapping $g_2:A \mapsto B$ implies that this reverse chain cannot be constructed. Therefore, the inequality is strict: $I(Y;B) \gt I(Y;A),$

This result implies that the node/atom representations obtained via SPFrame retain strictly higher mutual information with the target variable $Y$ than those obtained via the local frame method. In other words, SPFrame-based features preserve more task-relevant information.

[1] Periodic graph transformers for crystal material property prediction. NeurIPS2022.

[2] Complete and efficient graph transformers for crystal material property prediction. ICLR2024.

$\bullet$ Response to weakness 2 and question1:

We sincerely thank you for your constructive comments.

This work also ensures SE(3) invariance. We provide the proof below. Let atoms i and j be two neighboring atoms, with positions denoted by $p_i$ and $p_j$, respectively. In the message passing formulation of Equation (5) (in our paper, page 5), the edge scalar feature $e_{ij}$ can be expressed as $||p_i-p_j||$, and the directional vector $\widehat{ e}_{ij}$ can be expressed as $p_i-p_j$. After applying a global rotation $R$ and translation $t$, we have

$

\boldsymbol{f}_i^{(k)} &= \psi^{(k)} ( \boldsymbol{f}_i^{(k-1)}, \sum_{j \in \mathcal{N}(i)} \phi^{(k)} (\boldsymbol{f}_i^{(k-1)}, \boldsymbol{f}_j^{(k-1)},e_{ij}, \widehat{e}_{ij} \mathbf{F}^{\top}_{\text{global}} \mathbf{F}^{\top}_{\text{INV},i})) \\ &= \psi^{(k)} ( \boldsymbol{f}_i^{(k-1)}, \sum_{j \in \mathcal{N}(i)} \phi^{(k)} (\boldsymbol{f}_i^{(k-1)}, \boldsymbol{f}_j^{(k-1)},||p_i-p_j||, (p_i-p_j) \mathbf{F}^{\top}_{\text{global}} \mathbf{F}^{\top}_{\text{INV},i})) \\ &\text{Apply R and t:}\\ &= \psi^{(k)} ( \boldsymbol{f}_i^{(k-1)}, \sum_{j \in \mathcal{N}(i)} \phi^{(k)} (\boldsymbol{f}_i^{(k-1)}, \boldsymbol{f}_j^{(k-1)},||(p_iR-t)-(p_jR-t)||, ((p_iR-t)-(p_jR-t)) (\mathbf{F}_{\text{global}} R)^{\top}\mathbf{F}^{\top}_{\text{INV},i})) \\ &= \psi^{(k)} ( \boldsymbol{f}_i^{(k-1)}, \sum_{j \in \mathcal{N}(i)} \phi^{(k)} (\boldsymbol{f}_i^{(k-1)}, \boldsymbol{f}_j^{(k-1)},||p_i-p_j||, (p_i-p_j)RR^{\top} \mathbf{F}^{\top}_{\text{global}} \mathbf{F}^{\top}_{\text{INV},i})) \\ &= \psi^{(k)} ( \boldsymbol{f}_i^{(k-1)}, \sum_{j \in \mathcal{N}(i)} \phi^{(k)} (\boldsymbol{f}_i^{(k-1)}, \boldsymbol{f}_j^{(k-1)},e_{ij}, \widehat{e}_{ij} \mathbf{F}^{\top}_{\text{global}} \mathbf{F}^{\top}_{\text{INV},i})) \\

$

Applying a rotation $R$ and translation $t$ does not change the expression in Equation (5). Therefore, Equation (5) is unaffected by rotation and translation, indicating that it is SE(3)-invariant.

We hope our response addresses your concerns.

$\bullet$ Response to weakness 4 and question 2:

We sincerely thank you for your valuable comments.

We conducted experiments using iComFormer [1] as the backbone, which outperforms eComFormer. Below, we provided the evaluation of JARVIS dataset. Notably, our solution outperforms or at least is on par with the baselines. The preliminary results are as follows.

Table: Property prediction results on the JARVIS dataset

The performance improvement of SPFrame on another backbone (iComFormer) demonstrates its general effectiveness.

We hope our response addresses your concerns.

[1] Complete and efficient graph transformers for crystal material property prediction. ICLR2024.

$\bullet$ Response to weakness 3 and question 3:

We sincerely thank you for your comments. For fair comparisons, we just follow the experimental settings in the literature [1,2,3,4]. We agree that statistical results can be more rigorous. To reflect the statistical results, we evaluate various backbones for addressing the significance of our solution. Specifically, we replaced the backbone with a more powerful model to achieve a more significant improvement over the baseline. The detailed results are as iComFormer results (in Response to weakness 4 and question 2).

We hope our response addresses your concerns.

[1] Periodic graph transformers for crystal material property prediction. NeurIPS2022.

[2] Complete and efficient graph transformers for crystal material property prediction. ICLR2024.

[3] Crystalformer: Infinitely Connected Attention for Periodic Structure Encoding. ICLR2024.

[4] Rethinking the role of frames for SE (3)-invariant crystal structure modeling. ICLR2025.

$\bullet$ Response to question 4:
Thank you for your comment. In fact, $\widehat{\boldsymbol{e}}_{ij}$ refers to the edge vector between atoms $i$ and $j$.

$\bullet$ Response to Typos:

We sincerely thank you for your careful review. We will revise these typos. We will remove the "rightarrow" notation on Line 98 and retain only the equation $(\mathbf{A},\mathbf{Q}_g\mathbf{X}+\mathbf{t}_g,\mathbf{Q}_g\mathbf{L})=(\mathbf{A},\mathbf{X},\mathbf{L})$.

We sincerely thank you for your constructive suggestions. Below we will address your questions in detail.

$\bullet$ Response to weakness 1 (formal analysis of the expressive power):

We sincerely thank you for your constructive suggestions.

We provide a mutual information-based proof to explain why the SPFrame approach yields more informative node/atom representations than the local frame method.

Let a crystal structure be represented as a graph with $N$ atoms. Denote the node/atom feature corresponding to atom $i$ as $f_i$. We define the complete set of node/atom features under two different framing schemes as follows:

$A=\\{f_{local,i}|i=1,2,..,N\\}, B=\\{f_{sp,i}|i=1,2,..,N\\},$

where $A$ corresponds to node/atom features in the network using local frames, and $B$ corresponds to those in the network using the SPFrame approach.

We assume that the atoms with indices $p,q$ ($1<p,q<N$) are symmetry-equivalent. The use of the local frame results in identical local environments for atoms p and q (According to the illustration in Figure 2 in our paper, page 4). As a consequence, after message passing via Equation (3) (in our paper, page 4), we can have

$f_{local,p}=f_{local,q}.$

Consequently, the effective number of distinguishable node/atom representations is reduced. Denoting the cardinality as $|\cdot|$, we can have:

$|A|=N-1<N.$

In contrast, the SPFrame approach preserves the differences in the local environments of atoms p and q. After message passing via Equation (5) (in our paper, page 5), the node/atom representations satisfy:

$f_{local,p} \neq f_{local,p} \implies |B|=N.$

For scalar crystal property prediction, the final node features are first aggregated to get a global graph-level representation. This graph-level representation is then passed through a regression head. We denote the prediction target as a variable $Y \in \mathcal{Y}$. The neural network induces the following mapping:

$h_{local}:A \mapsto \mathcal{Y}, \quad h_{sp}:B \mapsto \mathcal{Y}.$

Since $|B|>|A|$, there exists a surjective mapping $g_1:B \mapsto A$, such that $h_{local}\circ g_1=h_{sp}$. However, an injective mapping $g_2:A \mapsto B$ does not exist in general due to loss of distinguishability in $|A|$. Consequently, the information flow can be described via the following Markov chain: $Y \to B \to A.$

Applying the data processing inequality to this chain yields: $I(Y;B) \ge I(Y;A),$

with equality if and only if:

$I(B;Y | A) =0 \quad\text{and}\quad Y \to A \to B$

i.e., the chain $Y \to A \to B$ is also valid. However, the absence of a mapping $g_2:A \mapsto B$ implies that this reverse chain cannot be constructed. Therefore, the inequality is strict: $I(Y;B) \gt I(Y;A),$

This result implies that the node/atom representations obtained via SPFrame retain strictly higher mutual information with the target variable $Y$ than those obtained via the local frame method. In other words, SPFrame-based features preserve more task-relevant information.

We hope our response can address your concerns.

$\bullet$ Response to weakness 2 (smooth) and questions 1: We sincerely thank you for your insightful comments. In practical applications, our method can be made smooth. Take the Quaternion-based SPFrame as an example. In our implementation, for a quaternion (a,b,c,d), we only predict the values of b,c, and d, while a is fixed to 1 by default (see our Anonymous GitHub repository, file ./models/transformer.py, line 229, function quaternion_1ijk_to_rotation_matrix(q)). This design prevents the network from encountering the discontinuity issue when a = b = c = d = 0 during the normalization step in rotation matrix construction.

Moreover, this paper focuses primarily on scalar property prediction for crystals, which does not impose smoothness requirements. If one aims to extend the method to tasks such as force prediction, smooth rotation matrix construction methods can be adopted to inherently avoid discontinuities, such as the SVD-based rotation projection proposed in [1].

We hope our response addresses your concerns.

[1] Learning with 3D rotations, a hitchhiker’s guide to SO(3). ICML2024.

$\bullet$ Response to questions 2:

Thank you for your question. This work primarily investigates the application of frame-based methods in the prediction of scalar properties of crystals, where invariance is required. Therefore, the features f in equations 2 and 3 are invariant with respect to rotations. In contrast, methods such as NequIP focus on predicting interatomic potentials, where equivariance is necessary for accurate force prediction. As a result, these methods incorporate spherical harmonics and tensor products in their network architecture to construct equivariant representations. We hope our response addresses the reviewer’s question.

$\bullet$ Response to questions 3:

We sincerely thank you for your comments. Frame-based methods have already been applied to force prediction tasks [1,2], as they decouple the network from the strict requirement of equivariance. This allows them to avoid the high computational costs associated with architectures that rely on spherical harmonics and tensor products to achieve equivariance. Instead, frame construction methods offer lower computational overhead compared to tensor products, enabling more efficient models. This work primarily focuses on the application of frame-based methods for scalar property prediction in crystals, where the goal is to achieve invariance and to improve predictive accuracy. If our method is extended to force prediction tasks, it can theoretically achieve higher efficiency in the same manner as shown in [1,2].

[1] Graph neural network with local frame for molecular potential energy surface. LoG2022.

[2] A new perspective on building efficient and expressive 3D equivariant graph neural networks. NeurIPS2023.