Building effective models from sparse but precise data: Application to an alloy cluster expansion model

A common approach in computational science is to use a set of highly precise but expensive calculations to parameterize a model that allows less precise but more rapid calculations on larger-scale systems. Least-squares fitting on a model that underfits the data is generally used for this purpose. For arbitrarily precise data free from statistic noise, e.g., ab initio calculations, we argue that it is more appropriate to begin with an ensemble of models that overfit the data. Within a Bayesian framework, a most likely model can be defined that incorporates physical knowledge, provides error estimates for systems not included in the fit, and reproduces the original data exactly. We apply this approach to obtain a cluster expansion model for the ${\text{CaZr}}_{1-x}{\text{Ti}}_x{\text{O}}_3$ solid solution.

Figure 1

(Color online) The ellipsoids schematically represent equiprobability surfaces of a many-dimensional prior probability distribution for $P^0\left(J\right)$. The plane represents the constraints on $J$ given by the results of an ab initio total-energy calculation. The $\left(N-1\right)$-dimensional ellipsoid sliced by the plane gives an equiprobability surface for the posterior distribution of $P^{\left(1\right)}\left(J\right)$; the point marked $J^{\left(1\right)}$, represents the most likely solution for $J$.

Figure 2

(Color online) (a) Representative CZT structure; (b) common 80-atom supercell for CZT energy calculations highlighted in bold.

Figure 3

(Circles) Convergence of the mean and maximum predicted errors on energies of configurations not included in the fit, as a function of the number of configurations included in fit. (Squares) Convergence of actual errors for a selected subset of 37 structures not included in the fit. (Triangles) Predictive ability of CV method.