Efficient Global Structure Optimization with a Machine-Learned Surrogate Model

We propose a scheme for global optimization with first-principles energy expressions of atomistic structure. While unfolding its search, the method actively learns a surrogate model of the potential energy landscape on which it performs a number of local relaxations (exploitation) and further structural searches (exploration). Assuming Gaussian processes, deploying two separate kernel widths to better capture rough features of the energy landscape while retaining a good resolution of local minima, an acquisition function is used to decide on which of the resulting structures is the more promising and should be treated at the first-principles level. The method is demonstrated to outperform by 2 orders of magnitude a well established first-principles based evolutionary algorithm in finding surface reconstructions. Finally, global optimization with first-principles energy expressions is utilized to identify initial stages of the edge oxidation and oxygen intercalation of graphene sheets on the Ir(111) surface.

Figure 1

Example of a surrogate guided structure search in a two dimensional search space. The two artificial dimensions are constructed by perturbing naphthalene as depicted in the top left. The FP energy landscape is shown in the top right. In the bottom, the surrogate landscape resulting from $N=3$, 4, and 5 training structures is shown. Note that training structures 4 and 5 result from local relaxation in the surrogate landscape.

Figure 2

Key elements of GOFEE. (i) Creation and FP evaluation of random initial structures. (ii) Addition of new FP data to training database and training of surrogate model. (iii) Generation of multiple new candidate structures by applying mutation operations to a diversified population of the best structures, currently found in the search. (iv) Local relaxation of the population and all new candidates, $N_c$ in total, using the surrogate model. (v) Selection of the most promising relaxed candidate using the acquisition function. (vi) Single-point FP evaluation of the chosen structure and, to benefit from the rich information content of the FP forces, another on the same structure perturbed slightly along the forces. This is the bottleneck in the search, $t_{\mathrm{FP}}>t_{\mathrm{ML}}$. Finally, the search is carried out by repeating steps (ii)–(vi). Steps (iii)–(iv) are easily parallelized by handling each new candidate on separate computer cores.

Figure 3

The ${\mathrm{SnO}}_2(110)-(4\times 1)$ test system. (a) Crosses: The optimized kernel widths after 200 visited structures of 20 GOFEE restarts. Contour: the log marginal likelihood [39], for fixed ${\lambda }_1$ and ${\lambda }_2$ and optimal ${\theta }_0$, averaged over the same 20 datasets. (b) GM of the system. (c) Success curves for finding the GM when using GOFEE with fixed (gray) and optimized (blue) hyperparameters compared to an EA. (d) 2D principal component analysis visualization of structures visited in three different GOFEE restarts with $\kappa =1$, 2, 4, respectively. The two principal components were found from a dataset of $\approx 4000$ structures from 10 GOFEE restarts, on which they explained 77% of the variance.

Figure 4

(a),(b) The most stable structures from the search and their energies shown for the edges with 0–5 carbon atoms less than the perfect edge. (c),(d) Structures and energies for oxygen added to the perfect edge. Structures ($E$, $G$) are made by hand, motivated by the trend from structures ($B-D$). Finally, lowest energy $CI-EB$ curves are given for (e) inserting a surface adsorbed oxygen atom to the perfect edge ($A$), (f) inserting the third oxygen atom, and (g) intercalation of an oxygen atom below the edge of structure ($E$). In the search, the second layer of iridium is kept fixed, while the upper layer is allowed to relax a maximum of half a covalent distance away from the bulk positions.