The cluster expansion is a popular surrogate model for alloy modeling to avoid costly quantum mechanical simulations. As its practical implementations require approximations, its use trades efficiency for accuracy. Furthermore, the coefficients of the model need to be determined from some known data set (training set). These two sources of error, if not quantified, decrease the confidence we can put in the results obtained from the surrogate model. This paper presents a framework for the determination of the cluster expansion coefficients using a Bayesian approach, which allows for the quantification of uncertainties in the predictions. In particular, a relevance vector machine is used to automatically select the most relevant terms of the model while retaining an analytical expression for the predictive distribution. This methodology is applied to two binary alloys, SiGe and MgLi, including the temperature dependence in their effective cluster interactions. The resulting cluster expansions are used to calculate the uncertainty in several thermodynamic quantities: ground state line, including the uncertainty in which structures are thermodynamically stable at 0 K, phase diagrams and phase transitions. The uncertainty in the ground state line is found to be of the order of meV/atom, showing that the cluster expansion is reliable to ab initio level accuracy even with limited data. We found that the uncertainty in the predicted phase transition temperature increases when including the temperature dependence of the effective cluster interactions. Also, the use of the bond stiffness versus bond length approximation to calculate temperature dependent properties from a reduced set of alloy configurations showed similar uncertainty to the approach where all training configurations are considered but at a much reduced computational cost.
Aldegunde, M., Zabaras, N., & Kristensen, J. (2016). Quantifying uncertainties in first-principles alloy thermodynamics using cluster expansions. Journal of Computational Physics, 323, 17–44. https://doi.org/10.1016/j.jcp.2016.07.016