Equivariant Networks for Crystal Structures
Abstract
Supervised learning with deep models has tremendous potential for applications in materials science.
Recently, graph neural networks have been used in this context, drawing direct inspiration from models for molecules.
However, materials are typically much more structured than molecules, which is a feature that these models do not leverage.
In this work, we introduce a class of models that are equivariant with respect to crystalline symmetry groups.
We do this by deﬁning a generalization of the message passing operations that can be used with more general permutation groups, or that can alternatively be seen as deﬁning an expressive convolution operation on the crystal graph.
Empirically, these models achieve competitive results with state-of-the-art on property prediction tasks.
Introduction
Deep learning has seen remarkable applications in computational chemistry, both on the side of molecular property prediction and molecule generation.
For small molecules, these methods are close to a level of precision that would make them suitable for practical applications [48].
However, less attention has been put to the problem of designing neural network architectures for the larger assemblies of atoms that constitute materials.
This is a crucial problem, as deep learning could advantageously complement computationally demanding ab initio simulation methods like DFT [39, 28].
Random forests and kernel methods have already yielded promising results for concrete applications in materials science, thanks to a large amount of materials data now available.
Examples include determination of stable structures [76], photovoltaic cells design [72, 35] and discovery of magnetic and superconducting materials [42, 62, 17].
In this paper we address the problem of designing equivariant layers and use them for supervised learning on materials.
We focus on crystals, materials characterized by the ordered arrangement of their atoms in lattices.
Crystalline materials are largely present around us and essential in technological applications, as they include a large number of metals, ceramics, and salts, among others [3].
They can be described as the regular repetition of a set of atoms, called the unit cell , in all directions of space.
These patterns are analogous to wallpaper patterns, for which the repeated structure is an image.
Crystals are characterized by a high degree of symmetry which is fundamental in understanding their physical properties.
Consequently, we suggest that equivariant deep learning provides an appropriate framework to design function approximators for crystals.
Graph Neural Networks (GNNs) have recently been proposed for supervised prediction tasks on crystals [56, 74, 12].
We hypothesize here that these models may fall short in exploiting crystal symmetry and that structural information is lost when mapping a crystal structure to a graph.
In particular, GNNs are equivariant to the permutation of atoms given as input to the model.
This condition is overly restrictive and amounts to forgetting about the ordered nature of a crystal.
We propose to use models equivariant to a product of groups G Λ × S C , where G Λ acts at the level of the Bravais lattice , the underlying periodic grid of the crystal, and S C on the unit cells.
We show that our proposed equivariant architecture is more expressive than GNNs on this data structure.
By using the crystal structure, our approach amounts to deﬁning a group-equivariant convolution kernel on the crystal in a way that is completely analogous to convolutional neural networks (CNNs).
This convolution is deﬁned on a graph associated with a crystal structure.
Our contributions are the following: 1) We derive different equivariant models based on the group-theoretical properties of crystals.
We show some of the limitations of GNNs for crystal data and propose an alternative data structure to be used with our architecture.
We perform a rigorous analysis, cleaning, and processing of the Materials Project database [33] and share the resulting processed dataset to serve as a benchmark for materials applications.
We perform experimental tests of our models on the
Materials Project database and report results comparable to or better than baselines.
Related works
Equivariant neural networks It is well known that neural networks have to incorporate inductive biases to be useful in practice [70, 7].
Using models that are invariant or equivariant to the symmetry of the data has proven to be a particularly important inductive bias to promote generalization [8].
The ﬁrst notable application of this idea was the CNN, for which each convolution layer is equivariant to translation [44].
An alternative parameter-sharing view also appears in early works [59]; while more recent equivariant networks have used both convolutional [13, 41, 15, 20], and parametersharing view [53, 25].
A few notable symmetries considered in recent years are rotation in image and volumetric data [13, 71, 68], permutation symmetry in sets [77, 52] and graphs [40, 46], as well as Euclidean [63, 10], and rotational symmetry [14, 2, 22, 57, 24].
Deep learning for materials The increasing availability of large materials datasets from highthroughput calculations [19, 33, 37, 49, 31, 11], makes deep learning more and more relevant for materials science.
Following the successes of GNNs on molecular data, similar models have been proposed for materials as alternatives to methods based on feature engineering.
Note that in what follows, we refer to GNNs in a general sense that includes message passing neural networks (MPNNs)
[27].Many variants of GNNs exist [36, 65, 75, 46], the underlying idea being for each node to aggregate features of neighboring nodes in a permutation invariant, or in the case of [40] in an equivariant way.
The CGCNN [74] and MEGNet models [12] rely on mapping crystal structures to graphs and applying GNNs to obtain a prediction.
For the SchNet model, the correspondence with graphs is less explicit [56], but still present.
Other approaches combine permutation equivariance with E(3)- equivariance to design models for molecules and materials [5, 55].
Background on crystal symmetry
We ﬁrst start by introducing some principles of crystallography that will be used to derive our main results.
A more comprehensive treatment can also be found in the references [21, 61], whereas basics of group theory are covered in [54] for example.
Lattices A crystal can be described as the periodic and inﬁnite repetition of a pattern in all directions of space.
Crystals are conveniently described using lattices as their underlying structures.
An n -dimensional lattice Λ can be deﬁned as the set of integral combinations of the linearly independent lattice basis vectors a i ∈ R n :
[Equation]
The lattice is entirely speciﬁed by its basis vectors a i .
A lattice is associated with a group of translations T Λ for which the multiplication rule is addition.
This captures the translational symmetry of a crystal.
A lattice Λ also deﬁnes subsets of R n called unit cells .
These subsets have the property of tilling the space when translated by lattice vectors.
Of particular importance is the primitive cell U for the basis a i : [Equation] i
[Equation]
i [Equation] [Equation] In a material, the unit cell comprises a set of atomic positions S
= { ( Z i , x i )
| x i ∈ U } , where the integer Z i is the atomic number and x i the position of the atom.
S can contain an arbitrary number of atoms and must not possess a particular structure.
Together with the lattice Λ , the atomic positions provides a complete description of the crystal structure.
It is often useful to deﬁne the concept of a sublattice .
A sublattice Λ P of Λ is a lattice with basis vectors b i such that Λ P ⊂ Λ .
Correspondingly, the supercell C P is the unit cell associated with the sublattice Λ P , for which C P ⊃ U .
The full lattice is generated by translations of the sublattice by a set of centring vectors { 0 . . .
v s } .
More formally, the sublattice is associated with a normal subgroup T Λ P of lattice translations, which speciﬁes a coset decomposition of the original lattice
T Λ = ( 0 + T Λ P ) ∪ · · · ∪ ( v s + T Λ P ) .
The centring vectors are coset representatives with respect to that decomposition.
It is clear that the centering vectors are also the set of lattice points contained in the supercell C P .
Example 3.1 (Graphene) In graphene, carbon atoms are arranged in a two-dimensional crystal with honeycomb structure.
The underlying lattice Λ has basis vectors a 1 = a(cid:2) √ (cid:3) and a 2 = [Equation] [Equation] [Equation]
[Equation]
, where a is a lattice constant.
The set of atoms in the unit cell is S =
[Equation]
In Fig.1, blue dots represent carbon atoms.
Black and white dots identify points of lattice Λ with unit cell U .
Black points identify a sublattice Λ P deﬁned by basis vectors b 1 and b 2 and with supercell U P .
The centring vectors { 0 , a 1 } correspond to lattice points contained in the supercell.
Space groups Symmetry plays a major role in the description of crystals.
This symmetry is often directly visible through facets in naturally occurring crystals.
Mathematically, it is described by a space group G , the set of isometries that maps a crystal structure to itself.
As isometries, space groups are subgroups of the Euclidean group E ( n ) .
A space group element can be described as a tuple ( W , t ) , where W is the linear part of the transformation and t a translation.
An element maps a vector x ∈ R n to Wx + t .
The multiplication of space group elements is therefore given by ( W 1 , t 1 ) ( W 2 , t 2 ) =
( W 1 W 2 , W 2 t 1 + t 2 ) .
The point group P of a space group is the group obtained from linear part operations in G , which will in general be rotations and reﬂections.
Considering only elements in G for which the linear part is identity ( I , t ) , we obtain the translation subgroup of the space group T .
It is a normal subgroup, which allows deﬁning the factor group G/T isomorphic to the point group P .
The crystallographic restriction theorem guarantees that only certain ﬁnite groups are valid point groups of space groups [18].
In particular, in 2 and 3 dimensions, only n -fold rotations with n ∈ { 2 , 3 , 4 , 6 } are allowed.
Two space groups G and G ′ are said to belong to be of same type if they can be related by a change of coordinate system.
There are 17 space group types in 2 dimensions and 230 in 3 dimensions [61].
For a lattice Λ , we call the Bravais group of the lattice P Λ the set of linear isometries that map Λ to itself.
Bravais groups provide a way to classify lattices according to their symmetry.
We say that two lattices Λ and Λ ′ belong to the same Bravais type if their Bravais groups are the same matrix groups when written for suitable basis vectors of each lattice.
The Bravais types in 2 dimensions and the 14 in 3 dimensions are enumerated in the Appendix.
The full symmetry group G Λ of lattice a Λ is given by the semidirect product of its translation group with its
Bravais group : G Λ = T Λ ⋊ P Λ .
Consider the space group G a crystal structure with lattice Λ and unit cell U .
From the deﬁnition of a crystal structure, it is clear that the translation subgroup of G will be T Λ .
However, the point group of G will, in general, not be P = P Λ .
This is because the unit cell may have less symmetry than the underlying lattice, which will result in P ⊆ P Λ (see example 3.2).
This is the reason why the number of space groups is much larger than the number of Bravais lattices.
Example 3.2 (Wallpaper pattern)
The wallpaper in Figure 2 is described by a square Bravais lattice Λ and unit cell U with 4-fold rotational symmetry.
The square lattice has a symmetry group P 4 m , with D 4 Bravais group.
However, the unit cell reduces the symmetry of the overall pattern since it does not have reﬂection axes.
The symmetry group of the wallpaper is, therefore, P 4 , with point group C 4 .
Limitations of GNNs
In this section, we examine some of the limitations of GNNs on crystalline data, motivating our model by addressing these limitations.
Typical models for crystalline data [74, 12] will build a graph from atoms in the unit cell and assign a feature vector h i ∈ R n to each.
Edge features encoding distance might also be used.
For convenience, this graph can be represented as a sparse tensor H ∈ R N 2 × n , where N is the number of atoms considered, and in which node and edge features are encoded on diagonal and off-diagonal entries, respectively.
A GNN is a function approximator f : R N 2 × n → Y that produce a prediction, by successive application of layers ϕ : R N 2 × n ℓ →
R N 2 × n ℓ +1 , where ℓ is the layer index.
Expressivity GNNs are usually stack of layers that are permutation equivariant
[Equation]
where P ( g ) is the permutation matrix associated with the group element g .
While P ( g ) permutes nodes, P ( g ) ⊗ P ( g ) permutes the vectorized adjacency matrix.
Our use of Kronecker product is due to the equality vec ( PAP ⊤ ) =
( P ⊗ P ) vec ( A ) for A ∈ R N × N .
This captures the fact that these models are designed to be insensitive to node ordering in H .
The material is, in a sense, treated as if it was a molecule.
Much of the crystal structure is forgotten because it cannot be captured by ordering the nodes in a speciﬁc way and is only encoded in positions or distances.
We argue that permutation equivariance is an overly strong requirement and was mainly used for practical reasons.
When possible, it should be more beneﬁcial to use a model equivariant to the actual symmetry of the data; in this case, the crystal space groups, which will, in general, be much smaller than the symmetric group.
Being equivariant to a smaller group results in less restrictive parameter sharing and a more expressive model.
Note that this requirement is different from that of E (3) -equivariance characteristic of some architectures [63, 55, 5].
In these models, the E ( n ) group only acts on the position x i ∈ R 3 part of each atom’s feature vector.
By contrast, the symmetry group of a crystal structure maps the crystal to itself, and its action is a permutation.
It therefore offers an alternative to building more powerful architectures for these systems that does not suppose that coordinate information is available.
In particular, this approach is more suitable for abstract condensed matter systems, like spin and freefermion lattice models, in which coordinates are not relevant.
Moreover, space groups are subgroups of the Euclidean group; equivariance to space groups is thus less restrictive.
Therefore, in this work, we choose to concentrate on space group equivariance.
Note that equivariance (and not only invariance) is important, as expressive invariant functions can be built by composing equivariant layers with an output pooling layer.
Equivariant functions can also be used to predict local properties like magnetization and charge distribution for example.
Invariance and symmetry breaking Even if the arrangement of atoms in a crystal is symmetric, this does not have to carry over to all local properties of the material.
Spontaneous symmetry breaking is common in materials and crucial to describing phenomena such as magnetism and superconductivity [43, 6].
Building a graph only at the unit cell level does not allow to capture local properties that differ across unit cells.
Using a supercell of multiple unit cells does not sufﬁce to solve this problem since permutation equivariant models have the property that equal input elements will be mapped to equal outputs elements [60, 78].
Input representation
Supercell Following the arguments of Section 4 we consider the set of atoms in a supercell instead of only in the unit cell, and add explicity symmetry breaking to increase the representational power of the model.
Since this increases the computational complexity of the method, we choose to keep the supercells small and deﬁne them with sublattice vectors b i = 2 a i .
The supercell is therefore 8 times larger than the unit cell, with centring [-vectorsnP 
 i-]vectors nPi m i a i | m i ∈ { 0 , 1 } o .
For each atom, we build a feature vector with a one-hot encoding of the atomic number and a one-hot encoding identiﬁcation of the unit cell it belongs to using the corresponding centering vector.
We keep track of the index of each atom within the unit cell, and the index of the atoms’ unit cell h ( a,i ) =
[ onehot ( Z i ) , onehot ( a )] , where a ∈ { 1 , . .
, 8 } and i ∈ C , where C is the number of atoms in the unit cell.
The encoding of the unit cell allows to break the symmetry between atoms mapped into each other by lattice translations.
Graph We construct a graph from atoms in the supercell, encoding relative distances between atoms as edge features.
Inspired by [56], an edge feature vector is built from the distance between atoms d ij = ∥ x i − x j ∥ as e ij = exp [Equation] (cid:17) .
The vector of Gaussian centers µµµ and γ are hyperparameters.
We use this approach to facilitate comparaison to previous works [56, 74], although Bessel encondings [38] could also be considered.
It can be seen as “soft" binning of interatomic distances.
Using this approach, we use only distance features in contrast to methods that use position vectors.
This has the beneﬁt of simplicity while still allowing a complete description of the input structure [69, 4, 66].
Sparsity has proven to be a useful inductive bias in graph representation learning [26], and it is also beneﬁcial in reducing computational complexity.
However, atomic bonds are not unambiguously deﬁned in crystals [16, 1].
We choose to follow a similar approach to [32]: an edge is drawn between atoms if they share a Voronoi face and if the distance between atoms is smaller than the sum of atomic Cordero radii plus a cutoff ∆ = 0 .
5 Å.
This approach has the advantage of being physically sound and producing graphs that are relatively sparse.
We provide more details on the graph-building strategy and compute metrics on the resulting graphs in Appendix A.1.
To preserve translational invariance for atoms at the boundary of the supercell, edges are initially also drawn to atoms outside the supercell.
Then, if an edge points outside the supercell, its head is mapped to the corresponding representative node inside the supercell.
This is analogous to circular padding in image processing. [Figure]
Equivariant crystal networks
Product groups Consider a crystal structure with space group G and in which each atom has a feature vector h i .
Then the space group acts as a permutation of the input atoms
[Equation]
where π Sg and π Ug are the permutations associated with group element g on the supercells and within the unit cells respectively.
If a dataset contains only samples that share the same crystal structure, then a model equivariant with respect to G can readily be used.
However, the case of a dataset with multiple different crystal structures that would be used in a typical supervised learning setting is more challenging for two reasons.
First, samples may have different space groups, which would require different models, each be trained with a fraction of the data.
Second, the group action may be different even for the same space group.
This is because the unit cells may have different structures and numbers of atoms.
A solution to address these issues is to consider equivariance to a direct product of groups G Λ × S C , where the symmetry group of the lattice G Λ acts across unit cells and S C , the symmetric group, acts within unit cells:
[Equation]
In this way, differences in unit cell structures are dealt with by the symmetric group, where parameter-sharing can handle variable-sized inputs [77].
We still have to accommodate 14 different group actions G Λ corresponding to the different Bravais lattices.
To avoid using a different model for each lattice, we propose two groups to deal with all the lattices.
The ﬁrst option is to consider the least symmetric Bravais lattice of primitive triclinic type and use its symmetry group P ¯1 =
T Λ ⋊ C 2 .
This group is a subgroup of all the other lattice symmetry groups.
The second option we consider is to simply use the symmetric group S Λ that is the symmetric group across unit cells, which is an overgroup of all the lattice symmetry groups.
This leaves us with a hierarchy of groups, with S N , the symmetric group over all atoms of a supercell, being the largest :
[Equation]
In our experiments, we use the two groups in this hierarchy for different levels of expressivity.
Equivariant message passing Having deﬁned the group action, we can now build the Equivariant
Crystal Network (ECN).
We will seek to use the message passing framework, which has demonstrated good performance on molecular data, and generalize it to obtain equivariance to other groups than the symmetric group.
The update equations for message passing framework are
[Equation]
The idea is to deﬁne parameter-sharing patterns for functions ϕ e and ϕ h , such that there can be multiple versions while still retaining equivariance.
Following [53], we ﬁrst deﬁne the parametersharing pattern of the set of input nodes N , with respect to group G as the colored bipartite graph Ω ≡ ( N , α, β ) , with the edge-color function α : N × N → { 1 , . . .
, C e } and node-color function β : N → { 1 , . . .
, C h } .
We also consider the action of the group G on edges Ω as g · ( i, j ) .
=
( π g ( i ) , π g ( j ))
∀ g ∈ G .
We deﬁne the orbit G · ( i, j ) of edge ( i, j ) as the set of edges in which it can be moved to by the group action : G · ( i, j ) .
= { g · ( i, j ) | g ∈ G } .
A similar deﬁnition applies to the orbit of a node, G · i .
= { π g ( i ) | g ∈ G } .
We then make the following claim:
Claim 6.1 The layer deﬁned by the C e functions ϕ α ( i,j ) e and the C n functions ϕ β ( i ) h
[Equation]
is G-equivariant if the parameter-sharing pattern Ω respects the equivariance condition:
[Equation]
The proof of this claim follows in Appendix A.2.
In words, the group action on the graph creates node and edge orbits, and we use a different copy of ϕ e and ϕ h for each edge and node orbit, respectively. The computational process for producing the pattern is to ﬁnd the orbit of G-action on the edges (nodes) [53], and the computational cost of this orbit-ﬁnding process grows linearly with the number of edges (nodes) [30].
This layer generalizes both MPNNs and equivariant multilayer perceptrons, such as CNNs.
The
MPNN is recovered with G = S n and a standard CNN with circular convolution with G = T Λ , ϕ α ( i,j ) e (cid:0) h ti , h tj , e ij (cid:1)
[Equation]
i
[Equation]
[Equation]
ti and ϕ
[Equation]
We now consider a product group G × H , acting according to Eq.
From Claim 1 of [67], the equivariant linear map for this group is the Kronecker product of equivariant maps for individual groups; see also [47].
The reformulation for parameter-sharing patterns is the following.
If parameters-sharing patterns Ω 1 and Ω 2 satisfy the equivariance condition for G and H respectively, then the parameter-sharing pattern Ω =
( N × M , α, β ) satisﬁes the equivariant condition if
[Equation]
and
[Equation] [Equation]
This simply means that a new color is deﬁned in the product pattern for each possible combination of colors in the original patterns.
Example 6.1 demonstrates this idea with a simple example. We choose functions ϕ α ( i,j ) e and ϕ β ( i ) h to be MLPs with one hidden layer.
In addition, we add a weighting factor to the edge aggregation 9, as this as been shown to be beneﬁcial by [74, 55]: m i =
X j ∈ N i e ij m ij , where e ij = ϕ a ( m ij ) , (15) and ϕ a is simply a linear layer.
This change does not affect the equivariance of the model.
Example 6.
In our running example, the parameter-sharing pattern for the message passing is produced by the Kronecker product of the pattern for the P 2 group (see Appendix A.3 for a similar example with the P 6 m group) and the pattern for the symmetric group S 2 as shown in the ﬁgure below (Top left).
However, we only need to keep the colors for which there is a corresponding edge (Top right).
Message passing is used on the resulting edge/node colored graph where similar colored nodes and edges use the same functions in message passing.
Model implementation
Network architecture We aimed to keep the architecture of our ECN model as simple as possible.
The model receives crystal structure graphs as input, with one-hot encoded feature vectors for each atom.
We use 128-dimensional embeddings and keep the same dimension for hidden layers throughout the network.
The ECN consists of 6 layers of the equivariant message passing operation 6.1.
This is followed by a mean-pooling operation over node embeddings of each graph and a twolayer MLP that outputs the ﬁnal prediction.
Mean pooling is preferred to other options because we only predict intensive physical quantities.
These properties are "per-atom" and do not depend on the choice of supercell.
Therefore, selecting a pooling operation that respects this invariance makes sense.
If we were to predict extensive quantities, sum pooling would be the preferred option.
Input features Many choices to encode atomic features are possible and have been suggested in the literature.
We perform experiments over these variants.
For each atom, we encode its type in a feature vector.
We consider two strategies, using only information available from the periodic table.
The ﬁrst one is to use only the atomic number.
The second one is to use the one-hot encoded group and period numbers for each atom.
The second strategy could beneﬁt from promoting better generalization since embeddings of atoms belonging to the same group or period will show a certain similarity.
Around 24% of the structures retained in the Materials Project dataset have been computed using the Hubbard- U extension of DFT.
Since this information can signiﬁcantly inﬂuence the resulting properties [64] it is added to the atomic feature vector as a binary feature.
Other details We implemented our model using Pytorch [51].
We use a sparse implementation of the equivariant message-passing based on the Pytorch Scatter package [23].
We use the AdamW optimizer [45] with weight decay regularization.
The full hyperparameter setup is provided in Appendix A.5.
Experiments
Materials Project We perform experiments using the Materials Project dataset [33] 1 .
This standard dataset of materials informatics comprises more than 120K materials with a complete speciﬁcation of their crystal structure and some important physical properties obtained with high-throughput
DFT calculations.
The Materials Project dataset has not been initially built to serve as a machine learning benchmark; we perform some preprocessing to make it suitable.
First, since multiple DFT calculations were sometimes performed on close initial conﬁgurations, some samples only show marginal structure differences and resulting properties.
This is exempliﬁed by the compound Li
Mn 2 Co 5 O 16 , which appears 322 times in the database.
Such duplicates can result in training-test leakage.
To prevent this, we consider two structures redundant if they have the same unit cell chemical formula and the same space group.
Amongst a set of duplicate structures, the one with the lowest formation energy is chosen.
Second, we ﬁlter out one-dimensional and two-dimensional materials from the dataset to only keep three-dimensional materials.
Finally, we remove materials for which the unit cell contains more than 50 atoms.
These are often associated with molecular and inorganic crystals with very different properties than the other materials.
Training, validation, and test splits are 80%, 10%, and 10% of the dataset.
We provide statistics on the processed dataset in Appendix A.4.
Following previous work, we predict a few relevant energetic properties: the formation energy E , the Fermi energy E F , and the band gap E g for the subset of insulating materials.
We also predict the binary insulator or conductor character material.
Finally, we also predict the magnetic moment per atom M .
This can be seen as a graph regression or classiﬁcation task.
For regression, training is performed using the mean-squared error (MSE) loss function, but we report the mean-absolute error (MAE).
For classiﬁcation, we use the cross-entropy loss function. [Table] [Figure] [Table]
[Figure]
[Table] [Figure] [Table] We compare the results obtained by our models to baselines [56, 74, 12].
Note that because these papers used different training and test splits and preprocessing schemes (even between themselves) [Table] our results cannot be directly compared.
To alleviate that [Table] we trained our own versions of two of the baselines using our splits and a similar training procedure.
We obtain slightly better or comparable results to the baselines on all targets when evaluated on the same splits.
The S Λ version offers better performance than the P ¯1 model overall, showing that it is more beneﬁcial to lean on the side of having slightly more symmetry than necessary at the cost of some expressivity.
We provide additional results for the model variants in Appendix A.6.
The beneﬁts of the increased expressivity on this task is in the not crucial, which we think can be explained in part by the relatively small size of the Materials Project dataset.
In a larger data regime, we expect that the beneﬁt of increased expressivity will outweight the cost in generalization capability.
Perov-
Finally, we perform experiments using the Perov-5 dataset [9] as provided by [73].
In this dataset, all the materials share the same Perovskite crystal structure.
The task considered is the regression of the heat of formation computed through DFT.
Results are shown on Table 2.The improvement on this dataset is signiﬁcantly more important for the proposed model compared to the baselines than on the Materials Project dataset.
We hypothesize that the fact that all the structures are shared in this dataset allows the model to specialize more efﬁciently leading to better generalization.
[Table]
Conclusion
We have shown how to leverage crystal symmetry to build more expressive and physically motivated neural networks for materials data.
This allows us to obtain a close equivalent of group equivariant convolution on this data structure.
These models show excellent accuracy in supervised property prediction, which supports the idea that symmetry is a useful inductive bias.
Such models could be used for other tasks on materials such as dynamics prediction, if the dynamics approximately preserves the crystal structure.
We also think that these models have signiﬁcant potential on more abstract condensed matter systems such as spin models and free-fermion models.
We have also deﬁned equivariant message passing, a generalization of the MPNN framework that can potentially be used on any data structures for which a group can capture the symmetry in sparse interactions between the basic elements.
One limitation of this approach and area of future improvement is on the computational efﬁciency of the equivariant message passing, which does not beneﬁt from optimized algorithms available for convolutions.
Acknowledgments and Disclosure of Funding
We thank Mehran Shakerinava, Christopher Morris, Joey Bose, Simon Verret and the anonymous reviewers for their valuable comments.
This project is in part supported by the CIFAR AI chairs program and NSERC Discovery.
S.-O. K.’s research is also supported by IVADO and the DeepMind
Scholarship.
Computational resources were provided by Mila and Compute Canada.
sections
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A Appendix [Table] A.1 Graph-building strategies
The graphs were built using the IsayevNN class from the pymatgen [50] package. It implements the commonly used Voronoi tessalation to deﬁne neighbors. Two atoms are considered bonded if they share a face in the Voronoi tessalation of the supercell and their distance is less than the sum of the atomic Cordero radii (a measure of the atomic radius) plus a cutoff ∆ = 0 . 5 Å.
This value of the cutoff was increase compared to [32] to reduce the number of disconnected graphs.
We provide statistics for the graphs obtained by the method described in Section 5.
A hard cutoff on atomic distances of 6 Å is also imposed on atomic distances.
A.2 Proof of Claim 6.
We prove that the function deﬁned by
[Equation]
with
[Equation]
is G -equivariant by proving the equivariance of every step and using the fact that function composition preserves equivariance.
First, we need to show equivariance of the message function on Equation (16)
[Equation]
Using Equation (19), we have
[Equation]
[Equation]
The right-hand side is equal to m π g ( i ) ,π g ( j ) by deﬁnition.
The message aggregation step at Equation (17) is permutation equivariant
[Equation]
Finally, we need to show that the node function on Equation (18) is also equivariant
[Equation]
Using Equation (20), we ﬁnd
[Equation]
The right-hand side is equal to h t +1 π g
( i ) by deﬁnition.
A.3 Parameter sharing patterns
The parameter sharing patterns are computed using a group-theoretical orbit ﬁnding algorithm that has linear complexity in the number of size of the generating sets of a group and in the number of edges in the coloured bipartite graphs [29, 58].
We show the parameter sharing patterns for the different Bravais lattice groups (Figure 7 and Figure 8).
For the 2-dimensional groups, we use a 2 × 2 supercell and for the 3-dimensional groups a 2 × 2 × 2 supercell.
Note that the numbering of the unit cells within the supercell is chosen by convention and can vary for different lattices.
Notice that the parameter-sharing patterns for different groups can be the same.
This is because, for different groups, the group action can induce the same orbits on the bipartite graph.
In particular, for some groups (Figure 7c), the pattern collapses to that of the symmetric group.
This can be undesirable since it reduces expressivity.
This can be alleviated by using larger supercells or eliminated by considering the group acting on itself instead of on supercells, as done in [13].
[Figure]
[Figure]
4 Materials Project dataset
We hereafter report the number of samples in the Materials Project dataset at different levels of the preprocessing scheme :
(a) Full dataset (b) No duplicates (c) No duplicates and unit cell size constraint (d) No duplicates, unit cell size constraint and 3D materials (e) No duplicates, unit cell size constraint, 3D materials and valid graphs
[Figure]
[Figure]
[Figure]
[Figure]
(e) (f)
[Table]
[Table]
[Table]
We also report the mean and standard deviation of each target for the processed dataset
[Table]
A.5 Hyperparameters
We use the same training setup for all the models.
The learning rate is initialized at 1 × 10 − 3 with a scheduler halving it after each 25 epoch plateau on the validation loss.
Training is perfomed for 1000 epochs or until the learning rate reaches 1 × 10 − 6 .
We performed sweeps over learning rates for all the models to verify that this is indeed a setup in which all models train well.
For the ECN models, we optimized the number of layers, the embedding dimension, the weight decay parameter and dropout (without noticing signiﬁcant improvement).
We use 6 layers of message passing operations.
The ϕ α ( i,j ) e and ϕ β ( i ) h functions are 1-hidden layer MLPs.
For the ϕ α ( i,j ) e functions, the weights for the hidden layer and output layers are shared across all α ( i, j ) .
This was found to perform slightly better.
The S WISH activation function is used.
The feature dimension of node embeddings is 100 across all the network.
For edge embeddings, the dimension is set at 20.
For the CGCNN model, we used the same embedding sizes (they were not speciﬁed in the original paper) and 2 convolution layers.
For MEGNet, we used the same architecture setup as in the original paper.
A.6 Supplementary results
[Table]