Materials generation is an essential task in material science that aims to discover new materials. While most of the existing models have shown interesting results in simulation, they struggle to produce new original and stable materials. This paper discusses the salient properties required for material generation and studies the dificulties related to material pattern repetition, which impacts the stability of the generated structures.
y Crystalline materials are involved everywhere in our reg modern society. From metal alloy to semi-conductor, sev- En eral technological objects contain crystalline materials.
Discovering new materials remains a dificult task in
material science. While existing algorithms can search for Materials space
new structures in the materials space [
Discovering new materials is a challenging problem. Several methods were recently introduced for
chemBut contrary to most generation problems, theoretical ical simulation either by enhancing current algorithms
chemistry provides a powerful set of tools to analyse with machine learning techniques (e.g. [
Indeed, graph neural networks (GNN) have already
been used for the representation and generation of
organic molecules [
Crystalline systems As molecules, crystalline systems can be defined as coloured point clouds. However, as crystals are periodic structures, additional information about how the point could is repeated in the space is required to represent it. The periodicity of the material can be then represented as a network where a group of points is repeated by discrete translation, which is equivalent to tiling space with a parallelepiped containing a cloud of atoms as illustrated in Figure 2.
As a result, a periodic system can be describe as atomic positions ∈ [0, 1[3 with an associated feature space representing the chemical information of each atom ∈ ℱ and a lattice ∈ GL3(R) representing the periodicity of the material. The infinite point cloud generated by this representation can be defined as ︀{( · ( + ), )︀ | ∈ Z3, 1 ≤ ≤ }︀ ⊆ R3 × (1)
cloud.
Material graph There are multiple ways to define
the graph of a material. Chemical bonds can be used
to build the set of edges, but generally, edges are built
from the atoms under a given threshold distance of from
the nearest neighbourhood [
Material deformation To tackle the crystalline system generation problem, we should define action on the geometry of the material. This action can be seen as an action on the lattice of the material resulting in the updated lattice ′ and action on the atomic positions resulting in the updated atomic position ′.
{︃ ′ = ′ = [ + ] .
(2)
GL3(R) on the lattice and the actions ∈ R3 on the atomic position. The atomic positions are brought back into the lattice of the crystal by truncation.
a material, we should satisfy the equivariance property.
Indeed, translations and rotations have no impact on the material’s properties. Consequently, a machine learning model acting on the material should not be dependent on the orientation and position of the structure but only be dependent on its geometry as shown in Figure 3.
′ ℎ ℎ ℎ ·
ℎ · ′
Actions on the lattice When acts on ℎ the lattice of the crystal, we would restrain to the group of actions deforming the structure. However, any matrix in R3× 3 can be decomposed as a sum of a symmetric and an antisymmetric matrix. As we define the action of in GL(3), the anti-symmetric part of the transformation can be seen as a rotation. But as we discussed earlier, applying rotations on a crystal doesn’t act on a material as the rotated material is equivalent to the original material.
This is illustrated in Figure 4. Consequently, it is better Figure 5: Actions defined on the edges of the graph can be to restrain to GL(3) ∖ SO(3). In other words, to restrain decomposed into action on the atomic positions and action to the subset of matrices of GL(3) that are symmetric. on the lattice of the crystalline material.
Acting on the crystalline structure To act on both the lattice and the atomic position, we can define actions on the edge of the graph. These actions can be then decomposed into a global action on the lattice of the crystal and local action on the atomic position. In order to define these actions, the contributions of all edges are aggregated to compute the action on the lattice. To act on the atomic positions, only the actions of the edges connected to the node are taken into account.
Crystalline materials are dificult to process because of the complexity and the variety of their chemistry, but also because of their repetitive structure. However, materials can be represented as graphs containing both chemical and geometrical information about the structures. Consequently, geometric machine learning techniques such as graph neural networks can be used in a wide variety of tasks including supervised and unsupervised learning. But in order to enhance the generalization capability of machine learning models, some properties such as the symmetry of the actions on the lattice or the equivariance can be beneficial.