Quantifying uncertainties in first-principles alloy thermodynamics using cluster expansions

Abstract

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.

Managing uncertainty in data-derived densities to accelerate density functional theory

“…Evaluating an error or measure of confidence in a data-driven prediction like n(x) is a well studied problem [15,16]. Applications of uncertainty quantification have recently begun appearing in Materials Science, with some even in DFT, such as the linear model exchange correlation functional of Aldegunde et al [17,18,19,20,21]. In this work, we show that useful applications of a predictive uncertainty in n(x) can be realised for just one of many possible approaches.…”

Bayesian uncertainty quantification and information fusion in CALPHAD-based thermodynamic modeling

“…For over a decade, at the elec-tronic/atomic scale, probabilistic parameterization in first principles calculations based on density function theory (DFT) [8,9] remains as some of the most significant works on UQ applied to materials problems. Aldegunde et al [10] have recently used a machine learning-informed Bayesian approach to quantify uncertainties associated to the prediction, via cluster expansions, of the thermodynamic properties of alloys. In that work, the resulting uncertainties are associated to the model parameters as well as the structure of the models themselves, which results from poorly converged cluster expansions due to lack of training data.…”

Robustness of the cluster expansion: Assessing the roles of relaxation and numerical error

“…In certain cases, one can map first-principles results on to a faster Hamiltonian, the cluster expansion (CE) [1][2][3]. Over the past 30 years, CE has been used in combination with firstprinciples calculations to predict the stability of metal alloys [1,2,[5][6][7][8][9][10][11][12][14][15][16], to study the stability of oxides [17][18][19][20][21], and to model interaction and ordering phenomena at metal surfaces [22][23][24][25][26]. Numerical error and relaxation effects decrease the predictive power of CE models.…”