Knowledge-transfer-based cost-effective search for interface structures: A case study on fcc-Al [110] tilt grain boundary

Determining the atomic configuration of an interface is one of the most important issues in materials science research. Although theoretical simulations are effective tools, an exhaustive search is computationally prohibitive due to the high degrees of freedom of the interface structure. In the interface structure search, multiple energy surfaces created by a variety of orientation angles need to be explored, and the necessary computational costs for different angles vary substantially owing to significant variations in the supercell sizes. In this paper, we introduce two machine-learning concepts, called transfer learning and cost-sensitive search, to the interface-structure search. As a case study, we demonstrate the effectiveness of our method, called cost-sensitive multitask Bayesian optimization, using the fcc-Al [110] tilt grain boundary. Four microscopic parameters, the three-dimensional rigid body translation, and the number of atomic columns, are optimized by transferring knowledge of energy surfaces among different orientation angles. We show that transferring knowledge of different energy surfaces can accelerate the structure search, and that considering the cost variations further improves the total efficiency.

Figure 1

(a) Atomic configuration of the GB supercell of the fcc-Al [110] $\mathrm{\Sigma }9$ symmetric tilt GB. The red and blue balls denote Al atoms in the (110) and (220) atomic layers. Based on this GB model, four microscopic parameters are optimized. One of two grains is rigidly shifted with $\mathrm{\Delta }X,\mathrm{\Delta }Y$, and $\mathrm{\Delta }Z$, and an atomic pair within the cutoff distance $d_{\mathrm{cut}}$ is replaced with a single atom located at the center of the original pair. (b) Calculated stable GB energies as a function of the rotation angle for fcc-Al [110] tilt GB. (c) Energy surfaces created by RBTs for the angles ${141}^{\circ }$ (top), ${134}^{\circ }$ (bottom left), and ${145}^{\circ }$ (bottom right). For illustrative purpose, we here only show two-dimensional RBTs on $X$ and $Y$. The structural units are also shown along with the surfaces in which the units are denoted as A and B. The red and blue balls denote Al atoms in (110) and (220) atomic layers. The markers on the surfaces indicate their minimums. The numbers of atoms in the supercells, which determine computational cost, are written in red.

Figure 2

Schematic illustration of our proposed method. Our method transfers knowledge of observed GB models in different tasks (illustrated as red arrows). Machine learning constructs a probabilistic approximation of the energy surfaces based on information shared across the tasks, which results in better approximation accuracy compared to that realized by solving all tasks separately. We also consider the cost discrepancy for given tasks by evaluating cost effectiveness of each candidate GB model, which accelerates the search by reducing the number of model samplings for the high cost energy surfaces.

Figure 3

A schematic illustration of MGP. The left three-dimensional plot (a) shows the MGP mean function surface ${\mu }^{\left(\mathrm{MT}\right)}$ in the joint space of the GB model descriptor $\mathit{x}$ and the task-specific descriptor $\mathit{z}$ (the black points are the sampled GB models). The center plots (b) show the surfaces for the three different tasks neighboring each other in the task axis. The blue lines are mean functions, the blue shaded are standard deviations, and dashed red lines are underlying true functions. Since information on the sampled GB models is shared, all the mean functions of MGB provide better approximations for the true functions compared with the separated estimation of each task illustrated in (c).

Figure 4

An illustrative example of the proposed method for the synthetic four tasks $t=1,...,4$ with cost $C_1=10,C_2=100,C_3=20$, and $C_4=80$. In iteration 0, the initial points are randomly set. Our method first investigates the low cost surfaces $t=1$ and 3 as indicated by the acquisition function (green). In iteration 5, with the increase in the low cost surface points, uncertainty of the Gaussian process model is reduced even for the high cost surfaces $t=2$ and 4 in which no additional points are sampled yet. Then, the acquisition function values for the high cost surfaces become relatively large because the possible energy improvement in the low cost surfaces is not significant compared to that in iteration 0. In iteration 10, the small energy points in the high cost surfaces are identified with a small number of model samplings.

Figure 5

The GB energy gaps from the minimums to the identified structure by each method. The vertical axis is the mean for all angles. The shaded region represents the standard deviation for five runs.

Figure 6

The GB energy gaps for the $O\left(M^3\right)$ cost setting.

Figure 7

Stable GB energy as a function of the rotation angle. The solid line represents the true stable energy obtained by computing all GB models exhaustively. The dotted line corresponds to the average energy at the initial step of Bayesian optimization. The dashed line represents the average energy obtain by CMB with the cost value of $100000$, which is about 0.3% $\left(\approx 100000/33458160\right)$ of the cost of the exhaustive search.