Bayesian Active Machine Learning for Cluster Expansion Construction

The Cluster expansion (CE) is a powerful method for representing the energetics of alloys from a fit to first principles energies. However, many common fitting methods are computationally demanding and do not provide the guarantee that the system's ground states are preserved. Thus, we have developed an efficient implementation of a Bayesian algorithm for cluster expansion built upon the method proposed by Cockayne and van de Walle (2010), which ensures all the input structural energies are fitted perfectly while reducing computational cost. This method also enjoys favorable convergence properties as it allows user to incorporate physics-based priors on the magnitude of the interactions. We have also made multiple improvements over this approach. First, we propose a procedure based on optimizing the hyper-parameters of the prior to improve the predictive power of the CE. Secondly, we devise an efficient algorithm to calculate the cross-validation (CV) score in linear time. Third, we use an active machine learning scheme that autonomously searches for new ground state structures to incorporate in the CE training set. Finally, all mechanisms described above have been integrated into the Alloy Theoretic Automated Toolkit (ATAT), thus allowing users to seamlessly adopt the new method. As performance tests, we calculate the phase diagram of the Fe-Ir system and study the short range order (SRO) in an equimolar MoNbTaVW system. We find that we typically need about 1/5 of the structures to reach the same precision of the CE constructed via traditional methods.