Unsupervised topological learning for identification of atomic structures

We propose an unsupervised learning methodology with descriptors based on topological data analysis (TDA) concepts to describe the local structural properties of materials at the atomic scale. Based only on atomic positions and without a priori knowledge, our method allows for an autonomous identification of clusters of atomic structures through a Gaussian mixture model. We apply successfully this approach to the analysis of elemental Zr in the crystalline and liquid states as well as homogeneous nucleation events under deep undercooling conditions. This opens the way to deeper and autonomous study of complex phenomena in materials at the atomic scale.

Figure 1

Time-temperature-transformation curve with $1024000$ atoms in a temperature range near the nose, which was obtained over five independent runs for each temperature. Inset: Evolution of the potential energy as a function of time along a nucleation process. See more details in Ref. [21].

Figure 2

(a) Schematic illustration of the first and second neighbor shells of a central particle (in red) with cut-off radii determined by the radial distribution function $g\left(r\right)$ of undercooled Zr at 1250 K. (b) Flowchart illustrating how to define a training set with independent structures. First, all particle positions in the simulation box are randomly ordered and set in a list. Then, the first particle of this list is set as a reference, and all particles which are closer than twice the selected cut-off radius $r_c$ from this reference particle are removed. Changing the reference particle, this process is iteratively repeated until all central particles meet the distance criterion based on $r_c$.

Figure 3

Schematic snapshots of the Vietoris-Rips complexes filtration ${\left({\mathcal{X}}_t\right)}_{t\ge 0}$ applied to a local structure with two neighbor shells. The barcode below encodes the persistence of each topological feature by a line starting at its birth (the radius where its appear) and finishing at its death (the radius where it disappears). The colors correspond to the homological dimension (red, blue, and green for $H_0, H_1$, and $H_2$ respectively) of each topological feature. The insert represents the corresponding PD where each point corresponds to a line and thus its associated topological feature with coordinates (birth, death).

Figure 4

(a) Zr atoms on a bcc lattice and the corresponding PD. (b) Zr atoms on a fcc lattice and the corresponding PD. The homological dimensions $H_0, H_1$, and $H_2$ are depicted in red, blue, and green respectively. As these structures are on periodic lattices, all topological features of the filtration from a specific homological dimension have the same pair (birth, death).

Figure 5

(a) Empirical correlation matrix (absolute value) between several descriptor families: BAA, CNA, TDA, and BOOA. Low values are white, high values are red. We distinguish between the descriptors by black lines, and between levels of homology with a blue line ($H_0$ and $H_1$). (b) PDs of a local atomic structure with one and two neighbor shells. The yellow line corresponds to a threshold computed with the subsampling approach to remove the noise.

Figure 6

ICL criterion on the training set. Its minimum corresponds to the maximum likelihood, which leads to the optimal number of clusters to build the model.

Figure 7

(a) Representation in the real space of the local structures assigned to each cluster $C_i$. (b) Snapshot of the one million atoms in the simulation box after the clustering in the descriptor space. In the inset, the classification obtained from an adaptative CNA method is depicted.

Figure 8

Snapshots at $T=1250$ K of Zr bulk crystal (a) and liquid (b) brought in their inherent local structures. Atoms are colored according to the cluster they belong to (see Fig. 7).

Figure 9

Snapshots of configurations along nucleation at the nose of the TTT of Zr at $T=1250$ K. The local structures in the clusters $C_1$ and $C_2$ identified as carrying the crystalline periodicity are depicted through their central particle in red and pink, respectively.

Figure 10

(a) Representation on a nucleus of the typical translational ordering of the nuclei and their surrounding liquid through the radial density profile of the $C_i$ clusters. The corresponding densities of the bulk crystal and undercooled liquid have been determined at $T=1250$ K and constant ambient pressure. This behavior of the translational ordering is representative of all nuclei in the configurations, even the precritical ones. (b) Typical orientational ordering of the $C_i$ clusters in the configurations through CNA. [555], [544], and [433] stand for fivefold symmetries bonds and, [666] and [444] for bonds relative to a bcc crystal ordering.