Bayesian Ensemble Approach to Error Estimation of Interatomic Potentials

Using a Bayesian approach a general method is developed to assess error bars on predictions made by models fitted to data. The error bars are estimated from fluctuations in ensembles of models sampling the model-parameter space with a probability density set by the minimum cost. The method is applied to the development of interatomic potentials for molybdenum using various potential forms and databases based on atomic forces. The calculated error bars on elastic constants, gamma-surface energies, structural energies, and dislocation properties are shown to provide realistic estimates of the actual errors for the potentials.

Figure 1

Model selection using test and training sets. The cost is plotted as function of the total number of potential parameters. MEMT potentials with up to 40 parameter are fitted while the FS and the MEAM have a fixed number of parameters. The minimum test error is obtained with 21 parameters with one parameter in the density function, $9$ parameters in the pair potential, $6$ parameters in the angular $p$ term, and $5$ parameters in the angular $d$ term. The relatively poor performance of the MEAM potential is also seen in tantalum [19]; its screening method is not useful on the bulk systems we have used.

Figure 2

Ensemble distributions of the structural energy difference between fcc and bcc. Distributions are calculated using the MEMT potential for different ensemble temperatures. In the inset means ($\mu$) and standard deviations ($\sigma$) are plotted as function of the temperature.

Figure 3

Cumulative error distributions of force components and the observables in Table . The cumulative distribution defined in Eq. (2) is calculated for each of the three potentials using the force components as the set of observables. The ensembles are generated at a temperature of $T_0$. Furthermore, the distribution obtained with all the observables in Table  is also shown. The good agreement with the Gaussian error function is an indication that the calculated standard deviations provide a good estimate of the actual errors.