The crystal structure of an inorganic material is a regular and periodic arrangement of atoms in 3D space. This periodic arrangement is usually described by a minimum repeatable pattern called a unit cell. The atoms in a unit cell have no explicit order, and their quantity varies from one to hundreds in number.

Because ML models generally accept fixed-dimensional and consistently ordered tensors for processing, determining crystal structures via the models is challenging.

To address this problem, we represent a crystal structure as a continuous vector field tied to 3D space, rather than as a discrete set of atoms. The proposed approach, named neural structure fields (NeSF), uses two types of vector fields: position field and species field. These fields implicitly represent the positions and species of the atoms in the unit cell, respectively.

To illustrate the concept of the NeSF, assume that we are given target material information as a fixed-dimensional vector, $\bm{z}$, and consider the problem of recovering a crystal structure from $\bm{z}$. Input $\bm{z}$ may specify, for example, information on the crystal structure of a material or some desired criteria for materials to be produced.

The NeSF uses a neural network, $f$, as a vector field on 3D Cartesian coordinates $\bm{p}$, conditioned on the target material information, $\bm{z}$:

$$\bm{s} = f(\bm{p}, \bm{z})$$

In the position field, network $f$ is trained to output a 3D vector pointing from query point $\bm{p}$ to its nearest atom position $\bm{a}$ in the crystal structure of interest. Thus, we expect output $\bm{s}$ to be $\bm{a} - \bm{p}$. If the position field is ideally trained, we can retrieve position $\bm{a}$ of the nearest atom at any query point $\bm{p}$ as $\bm{p} + f(\bm{p}, \bm{z})$.

Analogously, the species field is trained to output a categorical probability distribution that indicates the species of the nearest atom.

As an application of the NeSF, we present an autoencoder of crystal structures. In this autoencoder, crystal structures are embedded into vectors $\bm{z}$ (called latent vectors) via an encoder, and then the NeSF acts as a decoder to recover the input crystal structures from $\bm{z}$.

Autoencoders are typically used to learn latent vector representations of data via self-supervised learning, in which the input data can supervise the learning via a reconstruction loss.

Benchmark

The NeSF-based autoencoder outperforms an existing voxel-based method, ICSG3D (Court et al., 2020), in reconstruction accuracies.

Interpolation of Materials

Interestingly, the learned latent space with the NeSF enables us to produce interpolated crystal structures between a pair of materials.