Crystal structure prediction accelerated by Bayesian optimization

We propose a crystal structure prediction method based on Bayesian optimization. Our method is classified as a selection-type algorithm which is different from evolution-type algorithms such as an evolutionary algorithm and particle swarm optimization. Crystal structure prediction with Bayesian optimization can efficiently select the most stable structure from a large number of candidate structures with a lower number of searching trials using a machine learning technique. Crystal structure prediction using Bayesian optimization combined with random search is applied to known systems such as NaCl and ${\mathrm{Y}}_2{\mathrm{Co}}_{17}$ to discuss the efficiency of Bayesian optimization. These results demonstrate that Bayesian optimization can significantly reduce the number of searching trials required to find the global minimum structure by 30–40% in comparison with pure random search, which leads to much less computational cost.

Figure 1

Flow chart of crystal structure prediction by BO in cryspy.

Figure 2

Schematic procedure of crystal structure prediction by BO. Red circles are calculated (training) data, and red broken lines are the true function. Blue solid lines show the predicted energy functions with Gaussian processes. The shaded area represents the predicted values plus and minus two times the standard deviation, corresponding to the 95% confidence region. Black vertical dashed lines indicate the descriptors of remaining candidate structures. The remaining candidates with lower predicted values and large variance are preferentially selected.

Figure 3

Result of crystal structure prediction using a random search for NaCl. The unit cell includes eight NaCl formula units. Total-energy differences from the most stable structure are plotted as a function of number of trials within 20 eV/cell.

Figure 4

Frequency distribution of number of trials required to find the rocksalt structure with BO for NaCl. Two hundred BO simulations were carried out. The result of random selection obtained from the expected values in frequency theory is also shown as open circles.

Figure 5

Correlation between energy and structure distance for NaCl. The structure distance is based on Euclidean distance from the most stable structure. The energy is measured from that of the most stable structure. Triangles and circles represent the data of optimized structures and initial structures of the 23 good candidates, respectively. Only the data within 30 eV/cell are plotted.

Figure 6

${\mathrm{Th}}_2{\mathrm{Zn}}_{17}$ structure of ${\mathrm{Y}}_2{\mathrm{Co}}_{17}$. (a) A view from the [111] direction and (b) an overall view of the rhombohedral unit cell are rendered by vesta [41]. The unit cell contains 19 atoms. Large green and small blue atoms represent Y and Co, respectively.

Figure 7

Result of crystal structure prediction using random search for ${\mathrm{Y}}_2{\mathrm{Co}}_{17}$. The unit cell includes one formula unit (19 atoms). Total-energy differences from the most stable structure are plotted as a function of number of trials within 20 eV/cell.

Figure 8

Frequency distribution of number of trials required to find the ${\mathrm{Th}}_2{\mathrm{Zn}}_{17}$ structure with BO for ${\mathrm{Y}}_2{\mathrm{Co}}_{17}$. Two hundred BO simulations were carried out. The result of random selection obtained from the expected values in frequency theory is also shown as open circles.

Figure 9

Correlation between energy and structure distance for ${\mathrm{Y}}_2{\mathrm{Co}}_{17}$. The structure distances are based on Euclidean distance from the most stable structure. The energy is measured from that of the most stable structure. Triangles and circles represent the data of optimized structures and initial structures of four good candidates, respectively. Only the data within 60 eV/cell are plotted.