From the journal Digital Discovery Peer review history

13-Jan-2024

Dear Dr Wen:

Manuscript ID: DD-ART-12-2023-000233
TITLE: A Universal Equivariant Graph Neural Network for the Elasticity Tensors of Any Crystal System

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************

Reviewer 1

This paper reports a set new crystals. This paper seems related to some recent developments reported by DeepMind in Nature. The distinction and novelty is not clear and should be described in more detail especially when it comes to the method, and the application to materials discovery.

Interesting parts of the work include a focus on physically sound predictions that meet basic principles, a minimum requirement of such models. The reviewer is excited to see that the ML community is adopting physical principles in modeling. But there lies also the problem, in that how novel and unexpected the designs can be given that a lot of known physics is used to construct the algorithm?

Some additional comments:

-The claim that the model works for "any" crystal system is a bit broad and general, can the authors discuss this in a more nuanced way or tone down the claims?

-Novelty claims are problematic, sometimes, especially when it comes to crystalline structures that may exist already but have not been identified or experimentally assessed and compared against the analysis methods used here. The reviewer thinks that the work could be interesting but that the claims are overblown.

-Can the authors seek out cases where the model 'fails' to better understand limitations?

A nice paper overall! The reviewer hopes that the comments are helpful.

Reviewer 2

This is a very interesting paper, with research well performed and well reported, with good quality of the figures and readability of the text. I is a pleasure to read, and I strongly support its publication. Prediction of anisotropic properties, in particular mechanical ones, from microscopic characteristics is a difficult task. I do have a few comments to make below, starting with a very puzzling concern.

The main point I have is that I think there is an issue in the explanation of the elastic theory for crystalline systems, regarding the number of independent components of the elastic tensor for different symmetry classes. I have read the explanation by the authors that "In our opinion, there is still significant confusion on this topic." and note that the authors state that the following works are incorrect:

- Landau, Lev D.; Lifshitz, Evgeny M. (1970). Theory of Elasticity. Vol. 7 (2nd ed.). Pergamon Press. ISBN 978-0-08-006465-9.
- Nye, Physical properties of crystals

Despite being widely used, it would be possible. However, I failed to find in the references cited an explicit version of the authors' Figure 1.

Moreover, the authors argue that: "The tetragonal and trigonal systems are each divided into two symmetry classes, but the distinctions can be eliminated by a different choice of the coordinate system." However, the choice of the coordinate system is imposed by the crystallographic conventions: while the authors could decide on something different if they want, they are dealing with external data, in CIF files, represented according to those conventions of the international tables of crystallography.

Finally, if we simply open the Materials Project database, and look for crystals with the Laue classes in question (4 / m for tetragonal, or -3 for rhombohedral) that have elastic information available, we can clearly see the contradiction. To give an example, I searched for space group #87 (I4/m): https://next-gen.materialsproject.org/materials?has_props=elasticity&spacegroup_number=87

You find that most materials in this space group have nonzero C16 constants, like
mp-6304 https://next-gen.materialsproject.org/materials/mp-6304?has_props=elasticity&spacegroup_number=87#properties (to list just one example).

So I fail to reconcile the very assertive statements of the authors, and the concrete data available at hand. At the very least, I would argue that the presentation in Figure 1 and associated text is in contradiction with the actual data that is present in databases. But I wonder what the impact is on the authors' models, more broadly.

-----

Minor comments:

- I may have missed it, but are unit cell characteristics input in the GNN model? I do not think they are, but if so, this could be explicitly stated.

- The authors given MAE on different average moduli per crystal class. But I wonder at the inverse problem: given a cubic crystal as input, how "far" from a cubic-type tensor will the output of the model will be? In other terms, across all cubic crystals (which should have C11 = C22 = C33, C12 = C13 = C23, and C14 = 0 etc), how much do the predictions for elastic tensor deviate from this symmetry? Because, if I am correct, the symmetry is not itself input into the model? Unless it is, because the symmetry is retained in the atomic graphic? (And if so, it would be good to state it explicitly in the paper).

- About this sentence: "In addition, the theory of elasticity does permit negative bulk and Young’s moduli in general, though [62], and stable materials with negative moduli can be synthesized under certain conditions, such as in the form of thin films." I think it is strongly misleading, referring to materials that are not relaxed, infinite crystals (as studied by the authors here). That they have identified an acceptable limitation of their methodology is okay, but comparing it to a completely different situation is not necessary.

- The authors have tested the presence of negative bulk, shear and Young's moduli (averages). But a much more strict test (and very interesting!) would be to identify the presence of any negative eigenvalue in the elastic tensor (see Mouhat for example): those are mechanically unstable materials, and it will be more common to have one negative eigenvalue that a negative average modulus. It is also a common test performed to validate DFT calculations of elastic constants.

Reviewer 3

Decision of the reviewer: The reviewer recommends a major revision of the manuscript.

Description of the content of the paper
The paper proposes an equivariant graph neural network, called MatTen, to predict the elasticity tensor of inorganic compounds.
The training, validation, and testing data have been collected by running density functional theory (DFT) calculations using the Vienna Ab-Initio Simulation Package (VASP) and by using the stress tensor returned by VASP as labels for the atomic structures passed in input to the MatTen model. The VASP calculations to retrieve the strain tensor have been run on a portion of the atomic structures contained in the Materials Project dataset.
The authors performed an inverse design workflow that uses the trained MatTen model as surrogate for DFT to accelerate the exploration of new compounds. This AI-accelerated workflow has identified new atomic structures with improved strain properties with respect to the ones contained in the original data.

Although numerical results are promising from a computational standpoint, the narrative needs to be properly revised because it still lacks important physical aspects.

Technical assessment of the paper:

Major concerns:

• The authors use the term “universal” in the title. This adjective suggests that the approach is applicable to all the materials. However, the atomic structures used for this study are only taken from the Materials Project dataset, and these atomic structures have a Bravais lattice that is very small (less than one hundred atoms, most of them actually are even below 20 atoms in the structure). Because of the small size of the Bravais lattice that characterizes these structures, one can only model intermetallic (ordered) phases.

I suggest that the authors remove the adjective “universal” from the title and reword the related claims mentioned throughout the narrative.

• The use of a small Bravais lattice in the dataset (the Materials Project) also questions the accuracy of the data itself. In fact, for alloys, one has to include at least 4-5 neighbor shells to ensure that all the most relevant long-range interactions are properly captured.

Do the authors think that the atomistic structures in the Materials Project dataset contain all the interactions needed to produce accurate predictions of a material? If the answer is ‘yes’, then the authors should justify why they do not need to include at least 4-5 neighbor shells, which is what the material science community traditionally recommends. Saying “running DFT calculations for larger Bravais lattices is too expensive” is not an acceptable excuse. If the answer is ‘no’, then the authors should honestly tone down the claims mentioned in the paper, acknowledging the serious modeling limitations of the atomistic structures contained in the Materials Project dataset.

• I assume that the VASP calculations have been performed only at the ground state (0 Kelvin). If this is the case, what is the value of the mechanical properties retrieved, considering that most of the materials will be eventually deployed for finite temperature far from 0 Kelvin? If the authors think that the mechanical properties computed at 0 Kelvin are a reasonable approximation of the mechanical behavior of the material also for non-zero temperatures, they should mention at what temperatures the material is going to be deployed, and provide references that support the legitimacy to extrapolate from 0 Kelvin to non-zero finite temperatures. Otherwise, if the authors think that there may be severe discrepancies in the mechanical properties across the temperature range of deployability, then this limitation should be explicitly mentioned.

• The new atomistic structures found have been obtained by performing an optimization problem that used the MatTen surrogate model to maximize the strain tensor properties. However, there is no guarantee that these atomic structures are chemically stable. In fact, the results do not show any check on the values of the formation energy, therefore they do not guarantee that the phase with optimal strength is manufacturable.

• The mechanical behavior of an alloy is strongly affected by the microstructure of the material (e.g., grain size, grain orientation, grain boundaries). However, the Materials Project dataset contains only ideal (e.g., single phases) crystal structures. This is a serious limitation, that the authors should mention.

• Given that the importance of equivariance is emphasized many times, this claim begs to be validated. Specifically, show numerical results where a non-equivariant GNN is compared with an equivariant GNN and illustrate the supposed benefits of equivariance.

Minor concerns:

• Among the previous works references in the scope of GNN for mechanical properties, I think that the authors should also mention the following works:
1. Hestroffer et al. (https://doi.org/10.1016/j.commatsci.2022.111894). In this work, the authors use GNN models to predict the mechanical behavior of polycrystals. Miscrostructural behavior of alloys (e.g., grain size, grain orientation, grain boundaries) are important aspects, and are neglected by the materials project dataset that contains only ideal (e.g., single phase) crystal structures.
2. Karimi et al. (https://doi.org/10.1016/j.scriptamat.2023.115559). In this work, the authors train a graph neural networks (GNN) model, with grain centers as graph nodes, to assess the predictability of micromechanical responses of nano-indented 310S steel surfaces, based on surface polycrystallinity.
3. M. Lupo Pasini et al. (https://doi.org/10.1016/j.commatsci.2023.112141). In this work, the authors use embedded atomic model (EAM) as prototypes of DFT to collect large volumes of data and assess the robustness of GNN predictions for disordered (solid solution) phases of alloys. The dataset is open-source and available at (https://www.osti.gov/biblio/1958172). Albeit EAM implemented in LAMMPS is of a lower fidelity than VASP, this dataset complements the Materials Project dataset, as indeed it tries to address the challenges raised by larger Bravais lattices.
• Formula (3) is not a definition; it is a tautology. In fact, F_i is defined in terms of F_j.
The formula should be rewritten so that quantities on the left-hand side do not depend on themselves on the right-hand side.
• Second line of page 6. Given that the readership may not be familiar with the term “intensive properties”, please briefly elaborate what you mean with this term.
• The fact that equivariance is helpful to preserve symmetries is repeated many times across the entire paper, and it is very redundant. State it clearly once, and that is enough.
• End of first paragraph on page 8. The authors claim that transferability is an important property of MatTen. In my opinion, transferability is an important property which is retained by any graph convolutional neural network (GCNN). In fact, GCNNs use convolutional layers to learn short range interactions between atoms, and then use this learnt convolutional kernel to transfer the learnt features across other neighborhoods. The authors should restate this claim, acknowledging that the transferability is not a unique property of MatTen, but it is a natural property of GCNN models in general.
• Please provide more technical details about the hyperparameter optimization algorithms used to tune the hyperparameters of the MatTen model. Moreover, provide more details about all the final hyperparameter configurations used to produce the results described in the paper.
• What precision is used to run the VASP calculations? The mechanical properties require “PREC=ACCURATE” in the input file. Otherwise, the mechanical properties are not converging to a sufficiently accurate value. Please, provide this detail in the dataset description.

Please see the uploaded response_letter.pdf.

This text has been copied from the PDF response to reviewers and does not include any figures, images or special characters:


January 21, 2024
Joshua Schrier
Editor, Digital Discovery
RE: Decision on submission to Digital Discovery - DD-ART-12-2023-000233
Dear Prof. Schrier:
We have received your email containing the review for our manuscript DD-ART-12-2023-000233 submitted to Digital Discovery. We would like to thank you and the reviewers for the assessment of the paper and the feedback! We have carefully read the comments and suggestions from the reviewers and have made changes to the manuscript accordingly. The reviewers’ comments and our responses, along with the changes made, are described in detail below. In addition, all the changes are highlighted in the revised manuscript and electronic supplementary information.
We believe that with the responses and the modifications to the manuscript, we have addressed the reviewers’ concerns and hope that our manuscript will be accepted for publication in Digital Discovery.
Sincerely,

Mingjian Wen
Assistant Professor, Presidential Frontier Faculty Fellow University of Houston mjwen@uh.edu
Reviewer 1
1. Comment: This paper reports a set new crystals. This paper seems related to some recent developments reported by DeepMind in Nature. The distinction and novelty is not clear and should be described in more detail especially when it comes to the method, and the application to materials discovery.
Response: We think the reviewer is referring to the GNoME paper by Merchant et al.: https:
//doi.org/10.1038/s41586-023-06735-9. Although both works heavily employ graph neural networks (GNNs), there are major differences:
• Method. Both GNoME and this work use equivarint GNNs and take atomic structures as input. The training target, however, is very different. GNoME trains on the scalars (atomic energy and interatomic potential energy), while this work trains on tensorial properties (elastic tensors). Predicting tensorial properties is far less explored and more difficult in general, given the symmetry requirements a model has to satisfy.
• Objective and Application. A major aim of GNoME is to discover new crystal structures that have never been reported before (in any experimental or computational database). New crystals are proposed, and their stability is assessed using the energy above the convex hull criterion. In contrast, this work focuses on building a structure–property model for elastic tensors. We do not propose new crystal structures; instead, we apply the model to screen existing crystals in the Materials Project database and identify those with improved mechanical properties.
We have updated the manuscript to be more explicit of the contributions of this work, in terms of method and application.
Changes: We have updated the Conclusion section with the following statement:
MatTen has several unique characteristics: 1). it learns the full elasticity tensor and automatically handles all symmetry requirements, without the need to build separate models for individual components of the tensor or for each crystal system; 2). any elastic properties such as the bulk, shear, and Young’s moduli can be computed from the predicted elasticity tensor, leading to a unified framework for modeling elasticity; and 3). it allows for the exploration of anisotropic elastic behaviors (not possible with existing machine learning models), demonstrated by screening for crystals with extreme directional Young’s modulus.
2. Comment: Interesting parts of the work include a focus on physically sound predictions that meet basic principles, a minimum requirement of such models. The reviewer is excited to see that the ML community is adopting physical principles in modeling. But there lies also the problem, in that how novel and unexpected the designs can be given that a lot of known physics is used to construct the algorithm?
Response: Geometric deep learning, particularly equivariant GNNs based on spherical harmonics and tensor products, has been well developed in the past two to three years. This work takes advantage of the advancement and shows how it can be applied to high-rank tensorial properties, which are far less explored than scalar properties.
Essentially, this work builds a structure–property model, consisting of three parts: the input, the backbone module, and the output module. Like any other structure–property model, it is standard to use crystal structure information (including, cell vectors, atomic coordinates, and atomic numbers) as the input. There is not much design needed here. For the backbone module, we adopt the equivariant GNN algorithm, which indeed already incorporates a lot of known physics as the reviewer mentioned. It is one of the main building blocks that guarantee the symmetry in the elastic tensors. This, however, is not enough.
In terms of the model architecture, the major contribution of this work comes from the output module. We designed the output module such that irreps tensors corresponding to the harmonic decomposition of the elastic tensors are properly collected and assembled into the output, satisfying all symmetry requirements.
So, to sum up, we did not create new algorithm for the backbone module, but designed the output module for tensorial properties. With both modules, the symmetry requirements are automatically guaranteed.
3. Comment: The claim that the model works for “any” crystal system is a bit broad and general, can the authors discuss this in a more nuanced way or tone down the claims?
Response: We thank the reviewer for pointing this out! By “any”, we intended to mean that the model works for all seven crystal systems, without the need to treat each of them separately.
We have made this more explicit to avoid any confusion. On a related note, we have removed the use of “universal” in the title and throughout the manuscript.
Changes: Multiple changes have been made throughout the manuscript, for example:
Title: An Equivariant Graph Neural Network for the Elasticity Tensors of All Seven Crystal System
Abstract: Consequently, it provides a unified treatment of elasticity tensors for all seven crystal systems across diverse chemical spaces, without the need to deal with each separately.
4. Comment: Novelty claims are problematic, sometimes, especially when it comes to crystalline structures that may exist already but have not been identified or experimentally assessed and compared against the analysis methods used here. The reviewer thinks that the work could be interesting but that the claims are overblown.
Response: We want to clarify that this work does not propose any new crystal structures. In the “Discovering new crystals with extreme properties” section, we performed a screening of the crystals already existing in the Materials Project database, and identified those with large Young’s modulus.
To avoid the connection between “discovery” and “crystals that do not exist before,” we have updated the manuscript to use “screening” to reflect what was conducted.
Changes:
• Using MatTen, we screened the Materials Project database for the identification of materials with a large maximum directional Young’s modulus.
• Screening of crystals with extreme properties
• Quantitatively, the mean of Edmax for the identified crystals is 606 GPa...
5. Comment: Can the authors seek out cases where the model ‘fails’ to better understand limitations?
Response: In the original manuscript, we briefly discussed that predicted bulk modulus, shear modulus, and Young’s modulus can exhibit negative values. This is one of the failure modes of the model, because the theory of elasticity does not permit negative moduli for stable crystals under the thermodynamic conditions (at a temperature of 0 K and an external pressure of 0 Pa) considered in this work.
To further investigate the failure modes of the model, we further analyzed the stability of the crystals in the test set by checking the positive definiteness of the predicted elastic tensors. Given that all the crystals in the dataset are stable, we expect the eigenvalues of the predicted elastic tensors in Voigt notation to be all positive. This is a more stringent test than the one we performed in the original manuscript, where we only examined the average moduli.
We have added a discussion of this failure mode and provided further analysis to categorize them in a new section “Failure analysis” in the supplementary.
Changes:
Main text:
The number of crystals with negative predicted moduli remains minimal, accounting for only 3, 2, and 2 out of the 1021 test data for bulk, shear, and Young’s moduli, respectively. The moduli alone, however, do not provide a comprehensive understanding. For a crystal to be elastically stable, the sufficient and necessary condition is that the Voigt matrix should be positive definite [8]. We checked this and found that 25 crystals in the test set do not satisfy this condition. The majority of them are due to the incorrect prediction of the relative magnitudes of the diagonal and off-diagonal components of the Voigt matrix. A breakdown of the errors is provided in the ESI. Nevertheless, this is not a concern in practical use; one can filter out the negative ones if desired.
SI:
We checked the positive definiteness of the predicted elasticity tensors for the crystal in the test set. The 25 cases with at least one negative eigenvalues are listed in the below table. For the cubic, tetragonal, and orthorhombic crystals, the failure happens all because of the incorrect prediction of the relative magnitude of the diagonal component and off-diagonal components. For example, for the orthorhombic Na4C4S4N4 crystal (mp-6633), the DFT elasticity tensor is:

(tensor)

while the model predicted is:

(tensor)

The predicted c11 is substantially smaller than the DFT value. For the more complex (in terms of the number of independent components) trigonal crystals, we did not observe any pattern. Nor for the two monoclinic crystals.

Table S2: Number of crystals with negative eigenvalues by crystal system

6. Comment: A nice paper overall! The reviewer hopes that the comments are helpful.
Response: We thank the reviewer for the high evaluation of the work! The comments do help a lot to improve the presentation of the work.
Reviewer 2
1. Comment: This is a very interesting paper, with research well performed and well reported, with good quality of the figures and readability of the text. It is a pleasure to read, and I strongly support its publication. Prediction of anisotropic properties, in particular mechanical ones, from microscopic characteristics is a difficult task. I do have a few comments to make below, starting with a very puzzling concern.
Response: We thank the reviewer for the positive evaluation of the work! We have revised the manuscript to address the concerns.
2. Comment: The main point I have is that I think there is an issue in the explanation of the elastic theory for crystalline systems, regarding the number of independent components of the elastic tensor for different symmetry classes. I have read the explanation by the authors that “In our opinion, there is still significant confusion on this topic.” and note that the authors state that the following works are incorrect:
- Landau, Lev D.; Lifshitz, Evgeny M. (1970). Theory of Elasticity. Vol. 7 (2nd ed.). Pergamon Press. ISBN 978-0-08-006465-9.
- Nye, Physical properties of crystals
Despite being widely used, it would be possible. However, I failed to find in the references cited an explicit version of the authors’ Figure 1.
Response: The classification of the elastic tensor for different symmetry classes was initially very puzzling to us as well. We find that a lot of recent papers (mentioned in the manuscript) are giving conflicting results.
This topic is explained very well in Ref. 44 that we cited in the manuscript: Ellad B Tadmor, Ronald E Miller, and Ryan S Elliott. Continuum mechanics and thermodynamics: from fundamental concepts to governing equations. Cambridge University Press, 2012.
In the original manuscript, we have the following statement: We refer to Ref. 44 for a historical note on the development of the categorization. However, we agree with the reviewer that this might not be sufficient, and thus have updated the manuscript to make the reference more explicit.
Changes:
Caption of Fig. 1: See Ref. 44 for a detailed treatment of this classification.
Main text:
• It turns out that there exists only eight distinct classes (Fig. 1) [44], proved via a purely algebraic approach by directly identifying the equivalence classes corresponding to Eq. (1)
[45].
• We refer to Section 6.5 of Ref. 44 for a historical note on the development of the categorization.
3. Comment: Moreover, the authors argue that: “The tetragonal and trigonal systems are each divided into two symmetry classes, but the distinctions can be eliminated by a different choice of the coordinate system.” However, the choice of the coordinate system is imposed by the crystallographic conventions: while the authors could decide on something different if they want, they are dealing with external data, in CIF files, represented according to those conventions of the international tables of crystallography.
Response: We agree with the review that, in practice, if we choose a coordinate system that is consistent with the crystallographic conventions, the tetragonal and trigonal systems will have “distinct” symmetry classes.
As the reviewer also pointed out, we do have the freedom to choose the coordinate system. It does not matter whether we are dealing with external data like in CIF files or not. For example, suppose we have two crystal tetragonal crystals of space group 4/m and 4mmm given in CIF files in the crystallographic conventions. The independent components of the elastic tensors should look different in this case. However, we can choose to read the crystals from the CIF files and rotate them (different rotations for 4/m and 4mmm). If we rotate the elastic tensors accordingly, the independent components of the two crystal systems will be the same. Of course, after rotation, it might not be possible to express the crystal structures as CIF files anymore. But CIF is just a format, and we do not have to stick to it to represent crystals.
4. Comment: Finally, if we simply open the Materials Project database, and look for crystals with the Laue classes in question (4/m for tetragonal, or -3 for rhombohedral) that have elastic information available, we can clearly see the contradiction. To give an example, I searched for space group # 87 (I4/m): https://next-gen.materialsproject.org/materials?has_pr ops=elasticity&spacegroup_number=87 You find that most materials in this space group have nonzero C16 constants, like mp-6304 https://next-gen.materialsproject.org/ma terials/mp-6304?has_props=elasticity&spacegroup_number=87*properties (to list just one example). So I fail to reconcile the very assertive statements of the authors, and the concrete data available at hand. At the very least, I would argue that the presentation in Figure 1 and associated text is in contradiction with the actual data that is present in databases. But I wonder what the impact is on the authors’ models, more broadly.
Response: We appreciate the reviewer’s effort to carefully examine this! The reason why the elastic tensors on the Materials Project database do not match the classification in Fig. 1 is they are presented in a coordinate system that is consistent with the crystallographic conventions. Specifically, it adopts the IEEE standard (IEEE standard on piezoelectricity. ANSI/IEEE Std 176-1987, 0–1 (1988).). In other words, they are more consistent with the Wallace and Nye classifications as in Fig. S1, but not the classification in Fig. 1. We are trying to update the data in the Materials Project database, and, hopefully, they will be out in the next Materials Project database release.
The model is agnostic to the coordinate system, so it does not matter how the elastic tensors are presented. The L2 norm is invariant to rotations, i.e. it is the same for the original and rotated tensors.
Let us take vectors as an example. Suppose we want to match a (column) vector v to its target v′. We minimize the MSE loss function, i.e., (v−v′)T(v−v′). In a different coordinate system where both vectors are rotated by the rotation matrix R, the loss function becomes (Rv−Rv′)T(Rv−Rv′) = (v−v′)TRTR(v−v′) = (v−v′)T(v−v′), where we have used the fact that RTR = I. We see that the loss functions are the same before and after rotation. For the fourth-rank elastic tensors, the same argument applies. It is just the rotation matrix R becomes four rotation matrices, one for each index of the tensor.
Therefore, the model is not affected by the choice of the coordinate system to present the elastic tensors.
5. Comment: I may have missed it, but are unit cell characteristics input in the GNN model? I do not think they are, but if so, this could be explicitly stated.
Response: The model takes the crystal structure as input, which includes the super cell vectors, atomic coordinates, and atomic numbers. The super cell vectors are used to consider periodic boundary conditions for the crystals.
We have updated the manuscript to state more explicitly the input to the model.
Changes:
In the GNN model, the input crystal is represented as a graph G(V,E), with atoms as the nodes V and bonds as the edges E. The feature Fi ∈ V characterizes atom i, and the initial value of Fi is obtained by encoding the atomic number Zi using a one-hot scheme. A bond/edge between two atoms is created if the distance ∥r⃗ij∥ is smaller than a cutoff value, where r⃗ij denotes the distance vector between atoms i and j. Periodic boundary conditions are considered when constructing the bonds, using super cell vectors. The distance vector r⃗ij is separated into two parts: the unit vector rˆij from atom i to atom j and the scalar distance rij between them. The former is expanded on real spherical harmonics , and the latter is expanded on the Bessel radial basis functions [57]. In sum, these embedding modules extract structural information (coordinates of atoms, atomic numbers, and super cell vectors) from the crystal and provide them to the interaction blocks.
6. Comment: The authors given MAE on different average moduli per crystal class. But I wonder at the inverse problem: given a cubic crystal as input, how “far” from a cubic-type tensor will the output of the model will be? In other terms, across all cubic crystals (which should have C11 = C22 = C33, C12 = C13 = C23, and C14 = 0 etc), how much do the predictions for elastic tensor deviate from this symmetry? Because, if I am correct, the symmetry is not itself input into the model? Unless it is, because the symmetry is retained in the atomic graphic? (And if so, it would be good to state it explicitly in the paper).
Response: The output elastic tensor is guaranteed to exactly satisfy the symmetry. Yes, no symmetry is explicitly provided to the model; all such information comes from the crystal structure itself (atomic coordinates, atomic numbers, and super cell vectors). This is a reason why equivariant GNNs are a powerful tool for crystal structure–property modeling: it guarantees not only the transformation of the coordinate system but also the internal symmetry of the crystal structure.
We implied this in the original manuscript. For example, at the end of section 2.2, we have the following statement: The equivariance ensures that the model can produce an elasticity tensor C that respects the orientation of the input crystal structure. In other words, the choice of a specific coordinate system does not affect the model. This independence of the frame of reference characteristic is an indispensable property for models that predict tensors. In addition, any such model should also preserve the material symmetry of the crystal. By construction, MatTen guarantees the material symmetry reflected in the elasticity tensor. Concretely, if the predicted elasticity tensor is represented as a Voigt matrix, the symmetry and number of independent components in Fig. 1 are automatically maintained for any crystal system (proof in the ESI).
However, we agree with the reviewer that this kind of statement might be too abstract and elusive. We have updated the manuscript to reiterate this point in the Introduction section, and also provide a more concrete example in Section 2.2.
Thanks for the suggestion!
Changes:
Introduction:
he model satisfies two essential symmetry requirements for elasticity tensors: independence of the frame of reference, meaning that the choice of a specific coordinate system does not affect the model output, and preservation of material symmetry, meaning that the symmetry in a crystal is captured and reflected in the output elasticity tensor.
Section 2.2:
By construction, MatTen guarantees the material symmetry reflected in the elasticity tensor. Concretely, if the predicted elasticity tensor is represented as a Voigt matrix, the symmetry and number of independent components in Fig. 1 are automatically maintained for all seven crystal systems (proof in the ESI). For example, for a cubic crystal, the model guarantees that there are only three independent components c11 = c22 = c33, c12 = c13 = c23, and c44 = c55 = c66 and that all other components are zero.
7. Comment: About this sentence: ”In addition, the theory of elasticity does permit negative bulk and Young’s moduli in general, though [62], and stable materials with negative moduli can be synthesized under certain conditions, such as in the form of thin films.” I think it is strongly misleading, referring to materials that are not relaxed, infinite crystals (as studied by the authors here). That they have identified an acceptable limitation of their methodology is okay, but comparing it to a completely different situation is not necessary.
Response: We agree that this is misleading. To be more focused, we have removed this discussion of finite crystals and thin films.
8. Comment: The authors have tested the presence of negative bulk, shear and Young’s moduli (averages). But a much more strict test (and very interesting!) would be to identify the presence of any negative eigenvalue in the elastic tensor (see Mouhat for example): those are mechanically unstable materials, and it will be more common to have one negative eigenvalue that a negative average modulus. It is also a common test performed to validate DFT calculations of elastic constants.
Response: Great suggestion! We have performed this analysis. We found that 25 out of the 1021 crystals in the test data have at least one negative eigenvalue. In agreement with the reviewer’s guess: this is more common than having negative average moduli. We analyzed these failures and provided further discussion to categorize the failure modes of the model in a new section “Failure analysis” in the SI.
Changes:
Main text:
The number of crystals with negative predicted moduli remains minimal, accounting for only 3, 2, and 2 out of the 1021 test data for bulk, shear, and Young’s moduli, respectively. The moduli alone, however, do not provide a comprehensive understanding. For a crystal to be elastically stable, the sufficient and necessary condition is that the Voigt matrix should be positive definite [8]. We checked this and found that 25 crystals in the test set do not satisfy this condition. The majority of them are due to the incorrect prediction of the relative magnitudes of the diagonal and off-diagonal components of the Voigt matrix. A breakdown of the errors is provided in the ESI. Nevertheless, this is not a concern in practical use; one can filter out the negative ones if desired.
SI:
We checked the positive definiteness of the predicted elasticity tensors for the crystal in the test set. The 25 cases with at least one negative eigenvalues are listed in the below table. For the cubic, tetragonal, and orthorhombic crystals, the failure happens all because of the incorrect prediction of the relative magnitude of the diagonal component and off-diagonal components. For example, for the orthorhombic Na4C4S4N4 crystal (mp-6633), the DFT elasticity tensor is:
(tensor)

while the model predicted is:

(tensor)

The predicted c11 is substantially smaller than the DFT value. For the more complex (in terms of the number of independent components) trigonal crystals, we did not observe any pattern. Nor for the two monoclinic crystals.

Table S2: Number of crystals with negative eigenvalues by crystal system

Reviewer 3
1. Comment: The paper proposes an equivariant graph neural network, called MatTen, to predict the elasticity tensor of inorganic compounds. The training, validation, and testing data have been collected by running density functional theory (DFT) calculations using the Vienna AbInitio Simulation Package (VASP) and by using the stress tensor returned by VASP as labels for the atomic structures passed in input to the MatTen model. The VASP calculations to retrieve the strain tensor have been run on a portion of the atomic structures contained in the Materials Project dataset. The authors performed an inverse design workflow that uses the trained MatTen model as surrogate for DFT to accelerate the exploration of new compounds. This AI-accelerated workflow has identified new atomic structures with improved strain properties with respect to the ones contained in the original data.
Although numerical results are promising from a computational standpoint, the narrative needs to be properly revised because it still lacks important physical aspects.
Response: We thank the reviewer for the evaluation of the work and the suggestions! We have revised the manuscript to provide more physical insights and make it clear the limitations of the model. We believe these updates have improved the narrative of the work and hope that they have addressed the concerns.
2. Comment: The authors use the term “universal” in the title. This adjective suggests that the approach is applicable to all the materials. However, the atomic structures used for this study are only taken from the Materials Project dataset, and these atomic structures have a Bravais lattice that is very small (less than one hundred atoms, most of them actually are even below 20 atoms in the structure). Because of the small size of the Bravais lattice that characterizes these structures, one can only model intermetallic (ordered) phases.
I suggest that the authors remove the adjective “universal” from the title and reword the related claims mentioned throughout the narrative.
Response: Following the suggestion, we have removed “universal” from the title and updated the entire manuscript.
Changes:
Title: An Equivariant Graph Neural Network for the Elasticity Tensors of All Seven Crystal System
3. Comment: The use of a small Bravais lattice in the dataset (the Materials Project) also questions the accuracy of the data itself. In fact, for alloys, one has to include at least 4-5 neighbor shells to ensure that all the most relevant long-range interactions are properly captured.
Do the authors think that the atomistic structures in the Materials Project dataset contain all the interactions needed to produce accurate predictions of a material? If the answer is ‘yes’, then the authors should justify why they do not need to include at least 4-5 neighbor shells, which is what the material science community traditionally recommends. Saying “running DFT calculations for larger Bravais lattices is too expensive” is not an acceptable excuse. If the answer is ‘no’, then the authors should honestly tone down the claims mentioned in the paper, acknowledging the serious modeling limitations of the atomistic structures contained in the Materials Project dataset.
Response: We thank the reviewer for pointing this out! We agree that the Materials Project data does not contain all the long-range interactions due to the use of small Bravais lattices. So, the answer is ‘no’.
The reason to use small Bravias lattices is indeed what the reviewer mentioned. It is totally OK to run DFT calculations using large Bravais lattices for a handful of crystals, but it is not feasible to do so for high-throughput calculations of such a large number of crystals as in the Materials Project. We do not provide this as an excuse, but a practical consideration.
A machine learning model can only be as good as the data it is trained on. So, apparently, our model inherits the limitations of the data. We have updated the manuscript to acknowledge the limitations.
Changes: The below is added:
A limitation of the trained model can come from the data. The data consists of DFT calculations of perfect single crystals with relatively small super cells at a temperature of 0 K. Given that the efficacy of the model is intrinsically tied to the scope of the training data, it is imperative to exercise caution when applying the model to scenarios that extend beyond these parameters.
4. Comment: I assume that the VASP calculations have been performed only at the ground state (0 Kelvin). If this is the case, what is the value of the mechanical properties retrieved, considering that most of the materials will be eventually deployed for finite temperature far from 0 Kelvin? If the authors think that the mechanical properties computed at 0 Kelvin are a reasonable approximation of the mechanical behavior of the material also for non-zero temperatures, they should mention at what temperatures the material is going to be deployed, and provide references that support the legitimacy to extrapolate from 0 Kelvin to non-zero finite temperatures. Otherwise, if the authors think that there may be severe discrepancies in the mechanical properties across the temperature range of deployability, then this limitation should be explicitly mentioned.
Response: Yes, the elastic tensors are computed at 0 K.
We understand that most materials will be deployed at finite temperatures, but mechanical properties at 0 K are important for several reasons. They provide a baseline understanding of the materials’ properties without the complexities introduced by thermal motion. In addition, the elastic tensors at 0 K can be used to predict the mechanical properties at finite temperatures, e.g., using the Debye model.
Whether the mechanical properties at 0 K are a reasonable approximation of the mechanical behavior of the material at finite temperatures depends on the material. For example, for ceramics, the mechanical properties at 0 K can be a reasonable approximation of those at finite temperatures; this, however, cannot be expected for metal alloys. This work is not targeted at a specific application of a specific material, so we cannot provide a general answer to at what temperatures a material is to be deployed.
We agree with the reviewer that the temperature effect is an important aspect to consider, and have updated the manuscript to point this out.
Changes: A limitation of the trained model can come from the data. The data consists of DFT calculations of perfect single crystals with relatively small super cells at a temperature of 0 K. Given that the efficacy of the model is intrinsically tied to the scope of the training data, it is imperative to exercise caution when applying the model to scenarios that extend beyond these parameters. For example, the model is not appropriate for crystals with defects, such as vacancies, dislocations, and grain boundaries. Additionally, it is not advisable to directly employ the model for estimating the mechanical properties at finite temperate, especially for those materials, like metallic alloys, which exhibit a pronounced temperature dependency.
5. Comment: The new atomistic structures found have been obtained by performing an optimization problem that used the MatTen surrogate model to maximize the strain tensor properties. However, there is no guarantee that these atomic structures are chemically stable. In fact, the results do not show any check on the values of the formation energy, therefore they do not guarantee that the phase with optimal strength is manufacturable.
Response: First, we want to clarify that we do not propose new crystals in this work. In the “Discovering new crystals with extreme properties” section, we performed a screening of the crystals already existing in the Materials Project database.
In fact, when doing the screening, our first step is to check the formation energy above the convex hull. As stated in the manuscript: We first filtered crystals from the Materials Project database based on their energy above the convex hull values, selecting those with a value of ≤ 50 meV/atom. This energy determines the thermodynamic stability of a crystal and has been shown to correlate with the synthesizability of crystals [66, 67].
Changes: We have renamed the section title to “Screening of crystals with extreme properties” to avoid any confusion.
6. Comment: The mechanical behavior of an alloy is strongly affected by the microstructure of the material (e.g., grain size, grain orientation, grain boundaries). However, the Materials Project dataset contains only ideal (e.g., single phases) crystal structures. This is a serious limitation, that the authors should mention.
Response: Yes, we agree that the microstructure of the material is important. We added a statement to clarify that this work focuses on single crystals.
Changes: A limitation of the trained model can come from the data. The data consists of DFT calculations of perfect single crystals with relatively small super cells at a temperature of 0 K. Given that the efficacy of the model is intrinsically tied to the scope of the training data, it is imperative to exercise caution when applying the model to scenarios that extend beyond these parameters. For example, the model is not appropriate for crystals with defects, such as vacancies, dislocations, and grain boundaries. Additionally, it is not advisable to directly employ the model for estimating the mechanical properties at finite temperate, especially for those materials, like metallic alloys, which exhibit a pronounced temperature dependency.
7. Comment: Given that the importance of equivariance is emphasized many times, this claim begs to be validated. Specifically, show numerical results where a non-equivariant GNN is compared with an equivariant GNN and illustrate the supposed benefits of equivariance.
Response: We want to clarify that the objective of this manuscript is not to argue that equivariant models are superior to non-equivariant ones in terms of accuracy and such.
The use of equivariant models do have major benefits, which need no numerical results to show. First, we demonstrate that the equivariant model can predict the full tensor while at the same time satisfying the symmetry requirements. This is impossible for non-equivariant models. Second, the equivariant model provides a unified framework to predict the elastic tensors for all seven crystal systems, without the need to treat each separately. This is impossible with invariant models, either. These two points apply no matter the employed invariant model is a GNN or not.
That being said, we do include numerical results comparing the proposed model with a strong invariant baseline model (AutoMatminer), although it is not a GNN model. As can be seen from Table 1, the proposed model has lower mean absolute errors in derived scalar properties (bulk, shear, and Young’s moduli).
In addition, in the “Training on tensor components” section (image copied below), we also include a comparison to show their accuracy in predicting individual tensor components of cubic, tetragonal, and orthorhombic crystals.

Fig. S13: Mean absolute error (MAE) of the elasticity tensor in Voigt matrix. Multiple AutoMatminer models are trained for each crystal system, each model predicting a separate nonzero component of the Voigt matrix in Fig. 1 in the main text. For example, for the orthorhombic crystal system, nine AutoMatminer models are trained.
We hope these results are sufficient to show the benefits of equivariance in modeling tensors of crystals.
8. Comment: Among the previous works references in the scope of GNN for mechanical properties, I think that the authors should also mention the following works:
1. Hestroffer et al. (https://doi.org/10.1016/j.commatsci.2022.111894). In this work, the authors use GNN models to predict the mechanical behavior of polycrystals. Miscrostructural behavior of alloys (e.g., grain size, grain orientation, grain boundaries) are important aspects, and are neglected by the materials project dataset that contains only ideal (e.g., single phase) crystal structures.
2. Karimi et al. (https://doi.org/10.1016/j.scriptamat.2023.115559). In this work, the authors train a graph neural networks (GNN) model, with grain centers as graph nodes, to assess the predictability of micromechanical responses of nano-indented 310S steel surfaces, based on surface polycrystallinity.
3. M. Lupo Pasini et al. (https://doi.org/10.1016/j.commatsci.2023.112141). In this work, the authors use embedded atomic model (EAM) as prototypes of DFT to collect large volumes of data and assess the robustness of GNN predictions for disordered (solid solution) phases of alloys. The dataset is open-source and available at (https://www.osti.gov/biblio/ 1958172). Albeit EAM implemented in LAMMPS is of a lower fidelity than VASP, this dataset complements the Materials Project dataset, as indeed it tries to address the challenges raised by larger Bravais lattices.
Response: We thank the reviewer for providing these references, and giving explanation on what each of them is about! We have updated the manuscript to include these references.
Changes: In a nutshell, state-of-the-art ML models for elastic properties encode compositional information [19–21] and/or structural information [20–23] in a material as feature vectors and then map them to a target using some regression algorithms. This approach is adopted in many existing works for learning elastic properties of, e.g., alloys [24–27] and polycrystals [28, 29].
9. Comment: Formula (3) is not a definition; it is a tautology. In fact, Fi is defined in terms of Fj. The formula should be rewritten so that quantities on the left-hand side do not depend on themselves on the right-hand side.
Response: Fj on the right-hand side is the atom feature in an interaction block, and Fi on the left-hand side is the atom feature in the next interaction block. We have updated the formula to explicitly indicate this.
Changes: This is achieved via the tensor product convolution by updating the atom feature in the (k + 1)th interaction block from that in the kth interaction block:
Fik+1 = X R(rij)Y (rˆij) ⊗ Fjk, (1)
j∈Ni
10. Comment: Second line of page 6. Given that the readership may not be familiar with the term “intensive properties”, please briefly elaborate what you mean with this term.
Response: Updated.
Changes: For intensive properties such as the elasticity tensor, meaning that the property value does not depend on the size of the system, we adopt the mean pooling by averaging the features such that the representation of the crystal is independent of the number of atoms.
11. Comment: The fact that equivariance is helpful to preserve symmetries is repeated many times across the entire paper, and it is very redundant. State it clearly once, and that is enough.
Response: We have updated the manuscript to avoid redundancy.
Changes: Multiple changes have been made, e.g.,
• It takes a crystal structure as input, represents it as a three-dimensional crystal graph, performs feature update on the crystal graph, and finally outputs a tensor property.
• it learns the full elasticity tensor and automatically handles all symmetry requirements.
• Both MatTen and MatSca have smaller MAEs than AutoMatminer across all three moduli, owning to the effectiveness of the underlying neural networks in learning materials properties from structures.
12. Comment: End of first paragraph on page 8. The authors claim that transferability is an important property of MatTen. In my opinion, transferability is an important property which is retained by any graph convolutional neural network (GCNN). In fact, GCNNs use convolutional layers to learn short range interactions between atoms, and then use this learnt convolutional kernel to transfer the learnt features across other neighborhoods. The authors should restate this claim, acknowledging that the transferability is not a unique property of MatTen, but it is a natural property of GCNN models in general.
Response: There, we mentioned: Despite the slightly higher errors, it is notable that the model can still perform well for the crystal systems with a low presence in the training data, particularly for triclinic crystals. This is primarily because MatTen internally treats all crystals the same, enabling crystal systems with fewer data to leverage the abundant data from other crystal systems and acquire enhanced representations. We agree with the reviewer that in general for CGNN models, the transferability comes from the learned internal features.
Here, we are discussing a different type of transferability. One widely adopted approach to model elastic tensors is to treat each component of the tensor as a scalar and train a separate model for each component and for each type of crystal system (e.g., https://doi.org/10.1016/j.comm atsci.2021.110671 and https://doi.org/10.1021/acscentsci.8b00229). Because multiple separate models are trained, the model for a specific component of a specific crystal system does not benefit from the data of other components or other crystal systems. In contrast, with MatTen, one does not need to worry about the crystal system, and all components are trained together. As a result, the learned features of a specific crystal system can benefit from the data of other crystal systems.
In sum, the transferability is due to the learned features as the reviewer mentioned, but, in the manuscript, we meant the transferability across different crystal systems, which is a unique property of MatTen.
We have updated the manuscript to clear up the potential confusion.
Changes: This is primarily because MatTen internally treats all crystals the same, enabling crystal systems with fewer data to leverage the abundant data from other crystal systems and acquire enhanced representations. This type of transferability is not possible with models that are built separately for each crystal system.
13. Comment: Please provide more technical details about the hyperparameter optimization algorithms used to tune the hyperparameters of the MatTen model. Moreover, provide more details about all the final hyperparameter configurations used to produce the results described in the paper.
Response: In the original manuscript, we mentioned that the hyperparameters are obtained using a grid search, and provided the optimal hyperparameter values in Table S3 in the ESI.
We have updated Table S3 to include more info on the values of the hyperparameters that we searched over.
Changes:
Main text:
We performed a grid search to obtain model hyperparameters such as the rcut and c. Search ranges and their optimal values are listed in Table S4 in the ESI.
SI:
Table S4

14. Comment: What precision is used to run the VASP calculations? The mechanical properties require “PREC=ACCURATE” in the input file. Otherwise, the mechanical properties are not converging to a sufficiently accurate value. Please, provide this detail in the dataset description.
Response: Yes, we used “PREC=ACCURATE” in VASP input. The full computational workflow with all VASP input information is available in the atomate package. To make it easier for readers, we have updated the “Data Generation” section of the manuscript to provide more details on the VASP input information.
Changes:
The elasticity tensors were computed by a liner fitting of the stresses and strains obtained from DFT calculations using the Vienna Ab Initio Simulation Package (VASP) [75]. The calculations follow the same procedures discussed in [10], using PREC=Accurate, a tight convergence criterion of EDIFF=1e-6, an energy cutoff of ENCUT=700 eV, and a k-points density of 64 ˚A−3 in the reciprocal space to sample the Brillouin zone. Two additional improvements are made. First, to get more precise stresses for calculating the elasticity tensor, the projection operators in VASP are evaluated in the reciprocal space, that is, the setting LREAL=False was adopted. Second, to reduce numerical error in the calculations, the stresses are symmetrized according to the crystal symmetry.

01-Feb-2024

Dear Dr Wen:

Manuscript ID: DD-ART-12-2023-000233.R1
TITLE: An Equivariant Graph Neural Network for the Elasticity Tensors of All Seven Crystal Systems

Thank you for your submission to Digital Discovery, published by the Royal Society of Chemistry. I sent your manuscript to reviewers and I have now received their reports which are copied below.

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Reviewer 3

I thank the authors for having addressed my comments and properly revised the manuscript accordingly.

Reviewer 2

I think the authors have addressed the questions raised in my review.

Reviewer 1

The reviewer thanks the authors for the careful explanation and revisions. The approach is a lot more clear now.

Given the focus of the paper on using GNNs to predict tensorial properties, this paper should be cited (Yang et al., https://www.nature.com/articles/s41524-022-00879-4). This work is directly relevant since it also predicts atomic level properties like stresses, albeit focused on larger scales. Still, it should be discussed in this work.

Nice work!

Please see the attached response letter.

This text has been copied from the PDF response to reviewers and does not include any figures, images or special characters:

February 1, 2024
Joshua Schrier
Editor, Digital Discovery

RE: Decision on submission to Digital Discovery - DD-ART-12-2023-000233.R1
Dear Prof. Schrier:
Addressing the comments by reviewer 1, we have added a reference to the manuscript and discussed what it is about. We thank you and the reviewers for the comments and suggestions for helping us to improve the manuscript!
Sincerely,

Mingjian Wen
Assistant Professor, Presidential Frontier Faculty Fellow University of Houston

1
Reviewer 1
1. Comment: The reviewer thanks the authors for the careful explanation and revisions. The approach is a lot more clear now.
Given the focus of the paper on using GNNs to predict tensorial properties, this paper should be cited (Yang et al., https://www.nature.com/articles/s41524-022-00879-4). This work is directly relevant since it also predicts atomic level properties like stresses, albeit focused on larger scales.
Still, it should be discussed in this work.
Nice work!

Response: We thank the reviewer for providing this work that we are not aware of before! We have added a reference to it in the manuscript.
Changes: This approach is adopted in many existing works for learning elastic properties (e.g., for alloys [24–27] and polycrystals [28, 29]) and related atomic properties like stress and energy fields [30].

Reviewer 2
1. Comment: I think the authors have addressed the questions raised in my review.

Response: Thanks!

Reviewer 3
1. Comment: I thank the authors for having addressed my comments and properly revised the manuscript accordingly.

Response: Thanks!

02-Feb-2024

Dear Dr Wen:

Manuscript ID: DD-ART-12-2023-000233.R2
TITLE: An Equivariant Graph Neural Network for the Elasticity Tensors of All Seven Crystal Systems

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