Bayesian optimization of chemical composition: a comprehensive framework and its application to $R$Fe$_{12}$-type magnet compounds

We propose a framework for optimization of the chemical composition of multinary compounds with the aid of machine learning. The scheme is based on first-principles calculation using the Korringa-Kohn-Rostoker method and the coherent potential approximation (KKR-CPA). We introduce a method for integrating datasets to reduce systematic errors in a dataset, where the data are corrected using a smaller and more accurate dataset. We apply this method to values of the formation energy calculated by KKR-CPA for nonstoichiometric systems to improve them using a small dataset for stoichiometric systems obtained by the projector-augmented-wave (PAW) method. We apply our framework to optimization of $R$Fe$_{12}$-type magnet compounds (R$_{1-\alpha}$Z$_{\alpha}$)(Fe$_{1-\beta}$Co$_{\beta}$)$_{12-\gamma}$Ti$_{\gamma}$, and benchmark the efficiency in determination of the optimal choice of elements (R and Z) and ratio ($\alpha$, $\beta$ and $\gamma$) with respect to magnetization, Curie temperature and formation energy. We find that the optimization efficiency depends on descriptors significantly. The variable $\beta$, $\gamma$ and the number of electrons from the R and Z elements per cell are important in improving the efficiency. When the descriptor is appropriately chosen, the Bayesian optimization becomes much more efficient than random sampling.