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

Abstract

Cluster expansion (CE) is effective in modeling the stability of metallic alloys, but sometimes cluster expansions fail. Failures are often attributed to atomic relaxation in the DFT-calculated data, but there is no metric for quantifying the degree of relaxation. Additionally, numerical errors can also be responsible for slow CE convergence. We studied over one hundred different Hamiltonians and identified a heuristic, based on a normalized mean-squared displacement of atomic positions in a crystal, to determine if the effects of relaxation in CE data are too severe to build a reliable CE model. Using this heuristic, CE practitioners can determine a priori whether or not an alloy system can be reliably expanded in the cluster basis. We also examined the error distributions of the fitting data. We find no clear relationship between the type of error distribution and CE prediction ability, but there are clear correlations between CE formalism reliability, model complexity, and the number of significant terms in the model. Our results show that the size of the errors is much more important than their distribution.

Accurate representation of formation energies of crystalline alloys with many components

“…Indeed, Fig. 1 shows that when the atomic sizes are significantly different, the accuracy of fitting the energy of the relaxed configurations is significantly less than that of the unrelaxed configurations-this is similar to how cluster expansion behaves [8]. This can be rectified in part by introducing, for instance, the explicit dependence of the energy on the average compositiondependent equilibrium volume of the lattice [7].…”

“…Accurate computational prediction of the mixing enthalpy and configuration entropy would be very instrumental in studying HEAs, as it is hard to experimentally explore different compositions of five or more elements due to combinatorial complexity. The state-of-the-art methodology of computationally assessing the stability of multicomponent crystalline alloys is based on cluster expansion [5,6,7,8], allowing to fit formation energies of binary systems over the entire range of compositions, ternary and quaternary systems [9,10,11,12] over, typically, some subrange of the composition range, and quinary systems at specific points of the composition range [13].…”

“…Data-driven models assume a flexible functional form of the interaction energy with many (hundred or more) free parameters that are fitted from the quantummechanical data. The major drawback of cluster expansion is the "relaxation error" [8] which refers to low ability of cluster expansion to account for changes in mixing enthalpy related to the displacements of atoms to their equilibrium positions due to, for instance, the mismatch in their atomic radii. This is probably the reason why cluster expansion is not very successful in handling systems with a large number of components.…”

Accelerating high-throughput searches for new alloys with active learning of interatomic potentials

“…but is not useful when this is not the case. Its accuracy also converges slowly when atomic size mismatch is not negligible 20 . Additionally, more classical machine-learning algorithms such as decision trees 21 , support vector machines 22 , and other ML algorithms 23,24 have been tried.…”

SAMPLE: Surface structure search enabled by coarse graining and statistical learning

“…that only a small fraction of clusters is relevant for describing the property of interest (e.g. energy) [40]. Conversely, SAMPLE's energy model is not sparse.…”