Structural analysis based on unsupervised learning: Search for a characteristic low-dimensional space by local structures in atomistic simulations

Owing to the advances in computational techniques and the increase in computational power, atomistic simulations of materials can simulate large systems with higher accuracy. Complex phenomena can be observed in such state-of-the-art atomistic simulations. However, it has become increasingly difficult to understand what is actually happening and mechanisms, for example, in molecular dynamics (MD) simulations. We propose an unsupervised machine learning method to analyze the local structure around a target atom. The proposed method, which uses the two-step locality preserving projections (TS-LPP), can find a low-dimensional space wherein the distributions of data points for each atom or groups of atoms can be properly captured. We demonstrate that the method is effective for analyzing the MD simulations of crystalline, liquid, and amorphous states and the melt-quench process from the perspective of local structures. The proposed method is demonstrated on a silicon single-component system, a silicon-germanium binary system, and a copper single-component system.

Figure 1

Schematic of unsupervised learning based structural analysis.

Figure 2

Procedure of the two-step locality preserving projection (TS-LPP) method. The input data matrix $X$ is constructed by the $M$-dimensional feature vector ${\mathbf{x}}_i$ defined by the LAAF descriptor sampled from molecular dynamics simulations. Through a similarity graph with an adjacency matrix $W$, similar data are embedded to the closer positions in low-dimensional space, and a linear mapping matrix $Y$ is obtained by solving the generalized eigenvalue problem. In TS-LPP, first, the embedded data in the middle-dimensional space, which is larger than the target dimension, is calculated. Subsequently, dimension reduction to the final target dimension is performed again. In the low-dimensional space, we extract the relations of each atom in terms of the local structure using the TS-LPP method.

Figure 3

Two dimensional synthetic data where the two classes are widely distributed along the $y$ axis and the distance between the classes in the $x$ axis is gradually decreased as (a) 1.2, (b) 0.2, and (c) 0.0. The original data, PCA, LPP, and TS-LPP results are shown.

Figure 4

Two melt-quench processes to prepare the amorphous states.

Figure 5

(a) Distributions after dimension reduction to $d_{\mathrm{r}}=2$ by the TS-LPP method depending on $R_{\mathrm{a}}$ and $R_{\mathrm{c}}$. Blue, red, and green indicate crystalline, liquid, and amorphous states, respectively. The values of the hyperparameters $\sigma$ and $d_{\mathrm{m}}$ are determined by the Calinski-Harabasz score when the cluster value is fixed at three. When $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$, perfect clustering is obtained, and the result is surrounded by a red dotted line. (b) Cluster number dependence on the distributions and hyperparameters when $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$ by the TS-LPP method. The results are almost the same with as in the three-cluster case.

Figure 6

Distributions after dimension reduction to $d_{\mathrm{r}}=2$ by the PCA and LPP methods when $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$. Blue, red, and green indicate the crystalline, liquid, and amorphous states, respectively. The optimal value of $\sigma$ is denoted in the LPP case when the clustering number is fixed at three.

Figure 7

Ratio of the agreement between the predicted and correct states for new data points by PCA, LPP, and TS-LPP for $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$ by the $k$-nearest-neighbor method. The 2D spaces shown in Figs. 5 and 6 are used. Datapoints of the liquid state at 5000 K and the amorphous states 2a and 2b at 300 K are not included in the training data set.

Figure 8

(a) Distributions of the selected three features of 52nd, 76th, and 100th LAAF descriptors when $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$. Blue, red, and green indicate the crystalline, liquid, and amorphous states, respectively. The data points of each state are distributed on a plane. The orientations of the normal vectors of the planes look similar and the difference of the positions of the planes are small. Distributions in 2D space by (b) TS-LPP, (c) LPP, and (d) PCA are plotted. The optimal values of $\sigma$ and $d_{\mathrm{m}}$ are denoted when the clustering number is fixed at three. For this data set, TS-LPP is better for distinguishing crystalline states from liquid and amorphous states.

Figure 9

(a) Data points for the ten randomly selected atoms in the 2D space by TS-LPP with $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$, which are colored in orange (upper figures), and characteristic snapshots (lower figures) in the melt-quench process. In the upper figures, the blue, red, and green points are regarded as crystalline, liquid, and amorphous states in the target data set, respectively. In the lower figures, to classify each atom, the $k$-nearest-neighbor method in the 2D space by TS-LPP is used. (b) Kinetic energy corresponding to the temperature of the system as a function of time. (c) Percentage of atoms belonging to each state.

Figure 10

Distributions of data points for amorphous states 2a and 2b at 300 K, and the CRN results mapped to the 2D space constructed by (a) TS-LPP method and (b) PCA, when $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$. (c) Enlarged view of the amorphous states of TS-LPP. The region is denoted by the dotted line in (a). (d) Distributions of data points for 1000 atoms in amorphous 1a depending on the coordination number (C.N.). The region is denoted by the dotted line in (a).

Figure 11

Distributions of datapoints for liquid states at 5000 and 9000 K, amorphous states 1a and 1b at 300 K in the low-dimensional space constructed by (a) TS-LPP and (b) PCA, when $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$. Distributions of datapoints for quench states and amorphous states at 900 K in addition to the cases in (a) and (b) by (c) TS-LPP and (d) PCA.

Figure 12

Distributions after dimension reduction to $d_{\mathrm{r}}=2$ by the TS-LPP method for (a) silicon and (b) germanium. Distributions obtained by PCA for (c) silicon and (d) germanium. LAAF descriptors for binary systems with $R_{\mathrm{a}}=4\AA$ and $R_{\mathrm{c}}=5\AA$ are used.

Figure 13

Distributions in 2D space obtained by (a) TS-LPP and (b) PCA for $R_{\mathrm{a}}=3\AA$ and $R_{\mathrm{c}}=3\AA$ for Cu system. The fcc and hcp structures are prepared as a crystalline states.