Equivariant Networks for Crystal Structures
Anonymous Author(s)
AfﬁliationAddress email
Abstract
Supervised learning with deep models has tremendous potential for applications inmaterials science.
Recently, graph neural networks have been used in this context,drawing direct inspiration from models for molecules.
However, materials aretypically much more structured than molecules, which is a feature that these modelsdo not leverage.
In this work, we introduce a class of models that are equivariantwith respect to crystalline symmetry groups.
We do this by deﬁning a generalizationof the message passing operations that can be used with more general permutationgroups, or that can alternatively be seen as deﬁning an expressive convolutionoperation on the crystal graph.
Empirically, these models achieve competitiveresults with state-of-the-art on the Materials Project dataset.
Introduction
Deep learning has seen remarkable applications in computational chemistry, both on the side ofmolecular property prediction and molecule generation.
For small molecules, these methods areclose to a level of precision that would make them suitable for practical applications [45].
However,less attention has been put to the problem of designing neural network architectures for the largerassemblies of atoms that constitute materials.
This is a crucial problem, as deep learning couldadvantageously complement computationally demanding ab initio simulation methods like DFT[36, 27].
Random forests and kernel methods have already yielded promising results for concreteapplications in materials science, thanks to a large amount of materials data now available.
Examplesinclude determination of stable structures [70], photovoltaic cells design [67, 33] and discovery ofmagnetic and superconducting materials [39, 57, 16].
In this paper, we address the problem of designing equivariant layers for supervised learning onmaterials.
We focus on crystals, materials characterized by the ordered arrangement of their atoms inlattices.
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 bedescribed 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 theirphysical properties.
Consequently, we suggest that equivariant deep learning provides an appropriateframework to design function approximators for crystals.
Graph Neural Networks (GNNs) have recently been proposed for supervised prediction tasks oncrystals [52, 68, 11].
We hypothesize here that these models may fall short in exploiting crystalsymmetry and that structural information is lost when mapping a crystal structure to a graph.
Inparticular, GNNs are equivariant to the permutation of atoms given as input to the model.
Thiscondition is overly restrictive and amounts to forgetting about the ordered nature of a crystal.
Wepropose to use models equivariant to a product of groups G ⇤ ⇥ S N , where G ⇤ acts at the level of the
Bravais lattice , the underlying periodic grid of the crystal, and S n on the unit cells.
We show that ourproposed equivariant architecture is more expressive than GNNs on this data structure
By using the crystal structure, our approach amounts to deﬁning a group-equivariant convolutionkernel 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 thefollowing: 1) We derive different equivariant models based on the group-theoretical properties ofcrystals.
We show some of the limitations of GNNs for crystal data and propose an alternativedata structure to be used with our architecture.
We perform a rigorous analysis, cleaning, andprocessing of the Materials Project database [31] and share the resulting processed dataset to serveas 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 inductivebiases to be useful in practice [65, 7].
Using models that are invariant or equivariant to the symmetryof the data has proven to be a particularly important inductive bias to promote generalization [8].
Theﬁrst notable application of this idea was the CNN, for which each convolution layer is equivariant totranslation [41].
An alternative parameter-sharing view also appears in early works [54]; while morerecent equivariant networks have used both convolutional [12, 38, 14, 19], and parameter-sharing view[49, 24].
A few notable symmetries considered in recent years are rotation in image and volumetricdata [12, 66, 63], permutation symmetry in sets [71, 48] and graphs [37, 43], as well as Euclidean[58, 9], and rotational symmetry [13, 2, 21, 53, 23].
Deep learning for materials The increasing availability of large materials datasets from highthroughput calculations [18, 31, 35, 46, 29, 10], makes deep learning more and more relevantfor materials science.
Following the successes of GNNs on molecular data, similar models havebeen proposed for materials as alternatives to methods based on feature engineering.
Note that inwhat follows, we refer to GNNs in a general sense that includes message passing neural networks(MPNNs)
[26].Many variants of GNNs exist [34, 60, 69, 43], the underlying idea being for each nodeto aggregate features of neighboring nodes in a permutation invariant, or in the case of [37] in anequivariant way.
The CGCNN [68] and MEGNet models [11] rely on mapping crystal structures tographs and applying GNNs to obtain a prediction.
For the SchNet model, the correspondence withgraphs is less explicit [52], but still present.
Other approaches combine permutation equivariancewith E(3)-equivariance to design models for molecules and materials [5, 51].
Background on crystal symmetry
We ﬁrst start by introducing some principles of crystallography that will be used to derive our mainresults.
The exposition presented in the present work is inspired by [56].
A comprehensive treatmentcan also be found in [20], whereas basics of group theory are covered in [50] for example.
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 ⇤ can be deﬁned as the set of integral combinations of the linearly independent lattice basis vectors a i
[Equation]
n :
[Equation]
The lattice is entirely speciﬁed by its basis vectors a i .
A lattice is associated with a group oftranslations T ⇤ for which the multiplication rule is addition.
This captures the translational symmetryof a crystal.
A lattice ⇤ also deﬁnes subsets of R n called unit cells .
These subsets have the propertyof tilling the space when translated by lattice vectors.
Of particular importance is the primitive cell Ufor the basis a i :
[Equation]
In a material, the unit cell comprises a set of atomic positions S
= { ( Z i , x i )
| x
i 2 U } , where the 83 integer Z i is the atomic number and x i the position of the atom.
S can contain an arbitrary numberof atoms and must not possess a particular structure.
Together with the lattice ⇤ , the atomic positionsprovides a complete description of the crystal structure.
It is often useful to deﬁne the concept of a sublattice .
A sublattice ⇤ P of ⇤ is a lattice with basisvectors b i such that ⇤ P ⇢ ⇤ .
Correspondingly, the supercell C P is the unit cell associated withthe sublattice ⇤ P , for which C P  U .
The full lattice is generated by translations of the sublatticeby a set of centring vectors { 0 . . .
v s } .
More formally, the sublattice is associated with a normalsubgroup T ⇤ P of lattice translations, which speciﬁes a coset decomposition of the original lattice
T ⇤ =
( 0 + T ⇤ P ) [ · · · [ ( v s + T ⇤ P ) .
The centring vectors are coset representatives with respect to 92 that decomposition.
It is clear that the centering vectors are also the set of lattice points contained inthe supercell C P .
Example 3.1 (Graphene) In graphene, carbon atoms are arranged in a two-dimensional crystal with honeycomb structure.
The underlying lattice ⇤ has basis vectors a 1 = a⇥ 3 p⇤ and
[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 ⇤ with unit cell U .
Black points identify a sublattice ⇤ 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.
[Figure] Symmetry plays a major in their description of crystals.
This symmetry is often directly visiblethrough facets in naturally occurring crystals.
Mathematically, it is described by a space group G , theset of isometries that maps a crystal structure to itself.
As isometries, space groups are subgroupsof 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 2 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 partoperations in G , which will in general be rotations and reﬂections.
Considering only elements in Gfor which the linear part is identity ( I , t ) , we obtain the translation subgroup of the space group T .
Itis 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 pointgroups of space groups [17].
In particular, in 2 and 3 dimensions, only n -fold rotations withn 2 { 2 , 3 , 4 , 6 } are allowed.
Two space groups G and G 0 are said to belong to be of same type if theycan be related by a change of coordinate system.
There are 17 space group types in 2 dimensions and230 in 3 dimensions [56].
For a lattice ⇤ , we call the Bravais group of the lattice P ⇤ the set of linear isometries that map ⇤ toitself.
Bravais groups provide a way to classify lattices according to their symmetry.
We say that twolattices ⇤ and ⇤ 0 belong to the same Bravais type if their Bravais groups are the same matrix groupswhen written for suitable basis vectors of each lattice.
The 5 Bravais types in 2 dimensions and the14 in 3 dimensions are enumerated in the Appendix.
The full symmetry group G ⇤ of lattice a ⇤ isgiven by the semidirect product of its translation
Consider the space group G a crystal structure with lattice ⇤ and unit cell U .
From the deﬁnition of acrystal structure, it is clear that the translation subgroup of G will be T ⇤ .
However, the point groupof G will, in general, not be P = P ⇤ .
This is because the unit cell may have less symmetry than theunderlying lattice, which will result in P ✓ P ⇤ (see example 3.2).
This is the reason why the numberof 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 ⇤ 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 modelby addressing these limitations.
Typical models for crystalline data [68, 11] will build a graph fromatoms in the unit cell and assign a feature vector h
i 2 R n to each.
Edge features encoding distancemight also be used.
For convenience, this graph can be represented as a sparse tensor H 2 R N 2 ⇥ n ,where N is the number of atoms in the unit cell, and in which node and edge features are encoded ondiagonal and off-diagonal entries, respectively.
A GNN is a function approximator f : R N 2 ⇥ n !
Ythat produce a prediction, by successive application of layers  : R N 2 ⇥ n ` !
R N 2 ⇥ n ` +1 , where ` isthe layer count.
GNNs are usually stack of layers that are permutation equivariant
[Equation]
3) where P ( g ) is the permutation matrix associated with the group element g .
While P ( g ) permutesnodes, P ( g ) ⌦ P ( g ) permutes the vectorized adjacency matrix.
In other words, our use of Kroneckerproduct is due to the equality vec ( PAP > )
=
( P ⌦ P ) vec ( A ) for A 2 R N ⇥ N .
This captures the fact that these models are designed to be insensitive to node ordering in H .
Thematerial is, in a sense, treated as if it was a molecule.
Much of the crystal structure is forgottenbecause it cannot be captured by ordering the nodes in a speciﬁc way and is only encoded in positionsor distances.
We argue that permutation equivariance is an overly strong requirement and was mainlyused for practical reasons.
When possible, it should be more beneﬁcial to use a model equivariant tothe actual symmetry of the data; in this case, the crystal space groups, which will, in general, be muchsmaller than the symmetric group.
Being equivariant to a smaller group results in less restrictiveparameter sharing and a more expressive model.
Note that this requirement is different from that of E (3) -equivariance characteristic of some architectures [58, 51, 5].
In these models, the E ( n ) group only acts on the position x i 2 R 3 part of eachatom’s feature vector.
By contrast, the symmetry group of a crystal structure maps the crystal to itself,and its action is a permutation.
Therefore, in this work, we choose to concentrate on space groupequivariance.
4.2 Invariance and symmetry breaking
Even if the arrangement of atoms in a crystal is symmetric, this does not have to carry over to alllocal properties of the material.
Spontaneous symmetry breaking is common in materials and crucial to describing phenomena such as magnetism and superconductivity [40, 6].
Building a graph onlyat the unit cell level does not allow to capture local properties that differ across unit cells.
Using asupercell of multiple unit cells does not sufﬁce to solve this problem since equivariant models havethe property that identical inputs will be mapped to identical outputs [55, 72].
Input representation
Following the arguments of Section 4 we consider the set of atoms in a supercell instead of onlyin the unit cell to increase the expressivity of the model.
Since this increases the computationalcomplexity of the method, we choose to keep the supercells small and deﬁne them with sublatticevectors b i = 2 a i .
The supercell is therefore 8 times larger than the unit cell, with centring [-vectorsnP 
-]vectors
[Equation] o .
For each atom, we build a feature vector with a one-hot encoding of theatomic number and a one-hot encoding identiﬁcation of the unit cell it belongs to using the closestcentering vector.
We keep track of the index of each atom within the unit cell, and the index of theatoms’ unit cell h ( a,i ) =
[ onehot ( Z i ) , onehot ( a )] , where a 2 { 1 , . .
, 8 } and i 2 N , where N isthe number of atoms in the unit cell.
The encoding of the unit cell allows to break the symmetrybetween atoms mapped into each other by lattice translations.
5.2 Graph
We construct a graph from atoms in the supercell, encoding relative distances between atoms asedge features.
Inspired by [52], an edge feature vector is built from the distance between atoms [Equation] ⌘ .
The vector of Gaussian centers µµµ and  arehyperparameters.
This can be seen as “soft" binning of interatomic distances.
Using this approach,we use scalar features in contrast to methods that use position vectors.
This has the beneﬁt ofsimplicity while still allowing a complete description of the input structure [64, 4, 61].
Sparsity has proven to be a useful inductive bias in graph representation learning [25], and it is alsobeneﬁcial in reducing computational complexity.
However, atomic bonds are not unambiguouslydeﬁned in crystals [15, 1].
We choose to follow a similar approach to [30]: an edge is drawn betweenatoms 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 .
25 Å.
This approach has the advantage of being physically soundand producing graphs that are relatively sparse.
We experiment with another approach to build graphsand detail the resulting graph properties in Appendix A.1.
To preserve translational invariance for atoms at the boundary of the supercell, edges are initiallyalso drawn to atoms outside the supercell.
Then, if an edge points outside the supercell, its head ismapped to the corresponding representative node inside the supercell.
This is analogous to circularpadding in image processing.
Equivariant crystal networks
Consider a crystal structure with space group G C 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 ⇡ Sg and ⇡ Ug are the permutations associated with group element g on the supercells and withinthe unit cells respectively.
If a dataset contains only samples that share the same crystal structure, then a model equivariant withrespect to G C can readily be used.
However, the case of a dataset with multiple different crystalstructures that would be used in a typical supervised learning setting is more challenging for tworeasons.
First, the samples may have different space groups, which would require different modelsthat would each be trained with a fraction of the data.
Second, the group action may be different even [Figure] for the same space group.
This is because the unit cells may have different structures and numbers ofatoms.
A solution to address the second issue is to consider equivariance to a direct product of groups
G ⇤ ⇥ S N , where the symmetry group of the lattice G ⇤ acts across unit cells and S N within unit cells:
[Equation]
In this way, differences in unit cell structures are dealt with by the symmetric group, where parametersharing can handle variable-sized inputs [71].
We still have to accommodate 14 different groupactions G ⇤ corresponding to the different Bravais lattices.
To avoid using a different model for eachlattice, we propose two groups to deal with all the lattices.
The ﬁrst option is to consider the leastsymmetric Bravais lattice of primitive triclinic type and use its symmetry group P ¯1 =
T ⇤ o C 2 .
Thisgroup is a subgroup of all the other lattice symmetry groups.
The second option we consider is tosimply use the symmetric group S P , which is an overgroup of all the lattice symmetry groups.
Thisleaves us with a hierarchy of groups :
[Equation]
In our experiments, we use the two groups in this hierarchy, which offer different levels of expressivity.
Having deﬁned the group action, we can now build the Equivariant Crystal Network (ECN).
We willseek to use the message passing framework, which has demonstrated good performance on moleculardata,
[Equation]
and generalize it to obtain equivariance to other groups than the symmetric group.
The idea is to deﬁne parameter-sharing patterns for functions  e and  h , such that there can bemultiple ones while still retaining equivariance.
Following [49], we ﬁrst deﬁne the parametersharing pattern of the set of input nodes N , with respect to group G as the colored bipartite graph⌦ ⌘ ( N , ↵ ,  ) , with the edge-color function ↵ : N ⇥ N ! { 1 , . . .
, C e } and node-color function 218  :
N ! { 1 , . . .
, C h } .
We also consider the action of the group G on edges ⌦ as g · ( i, j ) .
=( ⇡
g ( i ) , ⇡ g ( j ))
8 g 2 G .
We deﬁne the orbit G · ( i, j ) of edge ( i, j ) as the set of edges in which itcan be moved to by the group action : G · ( i, j ) .
= { g · ( i, j ) | g 2 G } .
A similar deﬁnition applies tothe orbit of a node, G · i .
= { ⇡ g ( i ) | g 2 G } .
We then make the following claim:
Claim 6.1 The layer deﬁned by the C e functions  ↵ ( i,j ) e and the C n functions   ( i ) h
[Equation]
[Equation]
is G-equivariant if the parameter-sharing pattern ⌦ respects the equivariance condition:
[Equation]
[Equation]
The proof of this claim follows in Appendix A.2.
The computational process for producing the patternis to ﬁnd the orbit of G-action on the edges (nodes) and assign the same color to all edges (nodes)in the same orbit [49].
The computational cost of this orbit-ﬁnding process grows linearly with thenumber of edges (nodes) [28].
This layer generalizes both MPNNs and equivariant multilayer perceptrons like CNNs.
The
MPNN is recovered with G = S n and a standard CNN with circular convolution with G
= T ⇤ , ↵ ( i,j ) e
[Equation]
[Equation]
We now consider a product group G ⇥ H , acting according to Eq.
From Claim 1 of [62], weknow that the equivariant linear map for this group will be the Kronecker product of equivariant mapsfor individual groups; see also [44].
The reformulation for parameter-sharing patterns is the following.
If parameters-sharing patterns ⌦ 1 and ⌦ 2 satisfy the equivariance condition for G and H respectively,then the parameter-sharing pattern ⌦ =
( N ⇥ M , ↵ ,  ) 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 ofcolors in the original patterns.
For our model, we choose functions  ↵ ( i,j ) e and   ( i ) h to be MLPs with one hidden layer.
In addition,we add a weighting factor to the edge aggregation 11, as this as been shown to be beneﬁcial by[68, 51]:
[Equation]
e ij =  a
[Equation]
m ij
[Equation]
17) and
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 implementation7.1 Network architecture
We aimed to keep the architecture of our ECN model as simple as possible.
The model receivescrystal structure graphs as input, with one-hot encoded feature vectors for each atom.
We use 128dimensional 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 bya mean-pooling operation over node embeddings of each graph and a two-layer MLP that outputsthe ﬁnal prediction.
Mean pooling is preferred to other options because we only predict intensivephysical 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 topredict extensive quantities, sum pooling should be the preferred option.
Many choices to encode atomic features are possible and have been suggested in the literature.
Weperform experiments over these variants.
For each atom, we encode its type in a feature vector.
Weconsider two strategies, using only information available from the periodic table.
The ﬁrst one isto use only the atomic number.
The second one is to use the one-hot encoded group and periodnumbers for each atom.
The second strategy could beneﬁt from promoting better generalizationsince 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 usingthe Hubbard-
U extension of DFT.
Since this information can signiﬁcantly inﬂuence the resultingproperties [59] it is added to the atomic feature vector as a binary feature.
We implemented our model using Pytorch [47].
We use a sparse implementation of the equivariantmessage-passing based on the Pytorch Scatter package [22].
We use the AdamW optimizer [42] withweight decay regularization.
The full hyperparameter setup is provided in Appendix A.5.
Experiments
[Table] We perform experiments usingthe Materials Project dataset [31] 272 1 .
This standard dataset of materials informatics comprises morethan 120K materials with a complete speciﬁcation of their crystal structure and some importantphysical 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 weresometimes performed on close initial conﬁgurations, some samples only show marginal structuredifferences and resulting properties.
This is exempliﬁed by the compound Li
Mn 2 Co 5 O 16 , whichappears 322 times in the database.
Such duplicates can result in training-test leakage.
To preventthis, we consider two structures redundant if they have the same unit cell chemical formula and thesame space group.
Amongst a set of duplicate structures, the one with the lowest formation energy ischosen.
Second, we ﬁlter out one-dimensional and two-dimensional materials from the dataset toonly keep three-dimensional materials.
Finally, we remove materials for which the unit cell containsmore than 50 atoms.
These are often associated with molecular and inorganic crystals with verydifferent properties than the other materials.
Training, validation, and test splits are 80%, 10%, and10% 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 predictthe binary insulator or conductor character material.
Finally, we also predict the magnetic momentper atom M .
This can be seen as a graph regression or classiﬁcation task.
For regression, training isperformed 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.
We compare the results obtained by our models to baselines [52, 68, 11].
Note that because theauthors of these papers used different training and test splits and preprocessing schemes, our resultscannot be directly compared.
To alleviate that, we trained our own versions of two of the baselinesusing our splits and a similar training procedure.
We obtain better or comparable results to thebaselines on all targets when evaluated on the same splits.
The S P version offers slightly betterperformance than the P ¯1 model overall, showing that it is more beneﬁcial to lean on the side ofhaving slightly more symmetry than necessary at the cost of some expressivity.
We provide additionalresults for the model variants in Appendix A.6.
[-Perov-
-]
Conclusion
We have shown how to leverage crystal symmetry to build more expressive and physically motivatedneural networks for materials data.
This allows us to obtain a close equivalent of group equivariantconvolution on this data structure.
These models show excellent accuracy in supervised propertyprediction, which supports the idea that symmetry is a useful inductive bias.
Such models could beused for other tasks on materials such as dynamics prediction or even on more abstract condensedmatter systems such as spin models and free-fermion models.
We have also deﬁned equivariant message passing, a generalization of the MPNN framework thatcan potentially be used on any data structures for which a group can capture the symmetry insparse interactions between the basic elements.
One limitation of this approach and area of futureimprovement is on the computational efﬁciency of the equivariant message passing, which does notbeneﬁt from optimized algorithms available for convolutions.
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