Global optimization of atomistic structure enhanced by machine learning

Abstract

Global Optimization with First-principles Energy Expressions (GOFEE) is an efficient method for identifying low energy structures in computationally expensive energy landscapes such as the ones described by density functional theory (DFT), van der Waals-enabled DFT, or even methods beyond DFT. GOFEE relies on a machine learned surrogate model of energies and forces, trained on-thefly, to explore configuration space, eliminating the need for expensive relaxations of all candidate structures using first-principles methods. In this paper we elaborate on the importance of the use of a Gaussian kernel with two length scales in the Gaussian Process Regression (GPR) surrogate model. We further explore the role of the use in GOFEE of the lower confidence bound for relaxation and selection of candidate structures. In addition, we present two improvements to the method: 1) the population generation now relies on a clustering of all low-energy structures evaluated with DFT, with the lowest energy member of each cluster making up the population. 2) the very final relaxations in well-sampled basins of the energy landscape, "the final exploitation steps", are now performed as continued relaxation paths within the first-principles method, to allow for arbitrarily fine relaxations of the best structures, independently of the predictive resolution of the surrogate model. The versatility of the GOFEE method is demonstrated by applying it to identify the lowenergy structures of gas-phase fullerene-type 24-atom carbon clusters and of dome-shaped 18-atom carbon clusters supported on Ir(111).

Global optimization of atomic structures with gradient-enhanced Gaussian process regression

“…In our work, the cutoff radius R R c has values between 4.0 and 8.0 Å, and R α c has values between 3.6 and 4.0 Å, comparable to the radii studied in, e.g., [19]. The fingerprint is very similar to the one used by Bisbo and Hammer [28] but with the additional cut-off function for the radial part.…”

“…Increasing the step length increases the risk of running into problems with an unstable Gaussian process, as observed with Cu 15 . We thus conclude, somewhat at variance with Bisbo and Hammer [28], that adding neighboring data points, as done in L-BEACON-FD, L-BEACON-exact, or GOFEE, is not beneficial in general for the Bayesian approach to global optimization.…”

“…We note that Bisbo and Hammer [28] use a kernel function, which is a sum of two squared-exponential kernels with two different length scales. We tried this, but did not see any systematic improvement by adding an extra length scale.…”

“…for each structure, where we set the parameter κ = 2. This value provides a good balance between exploitation (low energy) and exploration (large uncertainty) [11,28,40]. The structure with the lowest acquisition function is selected for evaluation with the true potential.…”

“…This work follows the pioneering approach for efficient global optimization of atomic structures by Bisbo and Hammer [11,28] with the essential difference that we train our GP regression model with both energies and forces. We also note that the implementation of the approach involves many choices and parameters, where we might differ from Bisbo and Hammer.…”