Learning the Grain Boundary Manifold: Tools for Visualizing and Fitting Grain Boundary Properties

Abstract

Representation of materials data is an important challenge in property prediction. Current representations of grain boundary structure-property relationships are limited to high symmetry paths through the 5-D space of experimentally measurable crystallographic parameters. We develop a method to visualize and fit grain boundary properties along arbitrary paths in 5-space that leverages the recently developed octonion metric to interpolate between grain boundaries along shortest paths. First, we consider the purely geometric problem of grain boundary dimensionality reduction: how can we optimally visualize the high dimensional connectivity of grain boundary space in 3-D? We show that a reduced representation of grain boundary space based on the octonion metric and standard dimensionality reduction techniques recovers and extends common knowledge about grain boundary geometry. Second, we show that an octonion-based kernel regression method with one fitting parameter can predict grain boundary energy in the canonical Olmsted dataset to within 1% RMSE and mobility to within 10% RMSE. Open source tutorials for grain boundary dimensionality reduction and regression are made available online.