Sampling strategy in efficient potential energy surface mapping for predicting atomic diffusivity in crystals by machine learning

We propose a machine-learning-based method for efficiently predicting atomic diffusivity in crystals, in which the potential energy surface (PES) of a diffusion carrier is partially evaluated by first-principles calculations. To preferentially evaluate the region of interest governing the atomic diffusivity, a statistical PES model based on a Gaussian process (GP-PES) is constructed and updated iteratively from known information on already-computed potential energies. In the proposed method, all local energy minima (stable and metastable sites) and elementary processes of atomic diffusion (atomic jumps) are explored on the predictive mean of the GP-PES. The uncertainty of jump frequency in each elementary process is then estimated on the basis of the variance of the GP-PES. The acquisition function determining the next grid point to be computed is designed to reflect the impacts of the uncertainties of jump frequencies on the uncertainty of the macroscopic atomic diffusivity. A numerical solution of the master equation is here employed to readily estimate the atomic diffusivity, which enables us to design the acquisition function reflecting the centrality of each elementary process.

Figure 1

A basin surrounded by the four saddle surfaces (yellow lines) on a synthetic 2D-PES.

Figure 2

(a) Schematic diagram of the GP-PES along the trajectory of an elementary process from the initial point $i$ to the final point $j$ through the saddle point $s$. The red line and the pale blue area denote the predictive mean and the confidence interval, respectively. (b) Candidates of points $i^{″}$ and $s^{″}$ on the predictive mean of the GP-PES (two-dimensional PES), which are adjacent to the initial and saddle points, respectively. Note that the grid points on the trajectory of the elementary process are excluded from the $i^{″}$ and $s^{″}$ candidates.

Figure 3

(a) All irreducible grid points in the asymmetric unit of the $c\text{−}\mathrm{BaZr}{\mathrm{O}}_3$ crystal. (b) The global minimum points (pale red spheres) and the trajectories (pale blue spheres) of three types of elementary processes for proton jumps in $c\text{−}\mathrm{BaZr}{\mathrm{O}}_3$ (oscillatory proton transfer, proton rotation, and proton hopping). The yellow surface denotes the PE isosurface (isosurface level: 0.3 eV vs the global minimum point).

Figure 4

The predicted diffusion coefficient of protons at 1000 K in $c\text{−}\mathrm{BaZr}{\mathrm{O}}_3$ with the uncertainty as a function of the number of PE computations. The first ten PE computations corresponds to the initial points sampled at random. The red line is the diffusion coefficient on the predictive mean of the GP-PES. The pale red region denotes the uncertainty of the predicted diffusion coefficient estimated from the 95% confidence interval of the GP-PES.

Figure 5

The sampled grid points (white and black spheres), local minima (pale red spheres) and trajectories of elementary processes for proton jumps (pale blue spheres) on the predictive mean of the GP-PES after 10, 15, 20, 25, 30, and 35 PE computations. The white and black spheres correspond to the initial random sampling and the subsequent preferential sampling, respectively. The yellow surface denotes the PE isosurface on the predictive mean of the GP-PES (isosurface level: 0.3 eV vs the global minimum point).

Figure 6

The comparison between the present method and the previous method [10] in the number of times that the global minimum and three saddle points in $c\text{−}\mathrm{BaZr}{\mathrm{O}}_3$ were sampled in the ten trials.

Figure 7

(a) All irreducible grid points in the asymmetric unit of the $t\text{−}\mathrm{BaTi}{\mathrm{O}}_3$ crystal. (b) The six local energy minima (small spheres bonding to single oxide ions) in $t\text{−}\mathrm{BaTi}{\mathrm{O}}_3$. The relative potential energies at the local energy minima are shown in parentheses. The yellow surface denotes the PE isosurface (isosurface level: 0.5 eV vs the global minimum point).

Figure 8

The predicted diffusion coefficients of protons at 1000 K in $t\text{−}\mathrm{BaTi}{\mathrm{O}}_3$ with the uncertainty as a function of the number of PE computations. The first ten PE computations correspond to the initial points sampled at random. The red and blue lines are the diffusion coefficients in the $ab$ plane and along the $c$ axis on the predictive mean of the GP-PES, respectively. The pale red and blue regions denote the uncertainties of the predicted diffusion coefficients estimated from the 95% confidence interval of the GP-PES.

Figure 9

The comparison between the present method and the previous method [10] in the number of times that the true local minima and saddle points in $t\text{−}\mathrm{BaTi}{\mathrm{O}}_3$ were sampled in the ten trials.