Efficient Prediction of Grain Boundary Energies from Atomistic Simulations via Sequential Design

Abstract

With the goal of improving data based materials design, it is shown that by a sequential design of experiment scheme the process of generating and learning from the data can be combined to discover the relevant sections of the parameter space. The application is the energy of grain boundaries as a function of their geometric degrees of freedom, calculated from a simple model, or via atomistic simulations. The challenge is to predict the deep cusps of the energy, which are located at irregular intervals of the geometric parameters. Existing sampling approaches either use large sets of datapoints or a priori knowledge of the cusps' positions. By contrast, the authors' technique can find unknown cusps automatically with a minimal amount of datapoints. Key point is a Kriging interpolator with Matérn kernel to estimate the energy function. Using the jackknife variance, the next point in the sequential design is a compromise between sampling the region of largest fluctuations and avoiding a clustering of datapoints. In this way, the cusps of the energy can be found within only a few iterations, and refined as desired. This approach will be advantageous for any application with strong, localized fluctuations in the values of the unknown function.

Towards active learning: A stopping criterion for the sequential sampling of grain boundary degrees of freedom

“…Finally, an atomistic simulation is conducted at this new point, and an updated Kriging model is fitted to the augmented data set (previous design + new point). We refer to [1] for more details.…”

“…Roughly speaking it monitors two quantities: the development of the energy profile and the number of cusps. The results part of this paper in Section 3 starts with a validation of the stopping criterion by post-processing the data of the 1D STGB subspaces from [1]. Here it is demonstrated that the criterion makes the sampling more efficient, which means that we can achieve the same accuracy with fewer atomistic simulations (Section 3.1).…”

“…Recently, Kroll et al [1] have proposed a method along these lines. It combines a statistical sampling of the parameter space via a sequential design of experiment approach with a Kriging interpolator to estimate the energy function.…”

“…In addition to the new stopping criterion for sequential sampling of grain boundary energies, this work extends the methodology of [1] for the 1D subspace of symmetrical tilt grain boundaries (energy as a function of misorientation) to the two-dimensional subspace of energies as function of GB plane inclination. This creates the additional challenges of how to choose an initial design for the fundamental zone (FZ) of the 2D subspace, and how to properly interpolate the energy on a suitable grid.…”

“…on the fact that this energy is a function of the five geometric degrees of freedom (DOF) of the grain boundaries. To access this parameter space in an efficient way and discover energy cusps in unexplored regions, a method was recently established, which combines atomistic simulations with statistical methods [1]. This sequential sampling technique is now extended in the spirit of an active learning algorithm by adding a criterion to decide when the sampling is advanced enough to stop.…”