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We propose a data assimilation method for evaluating the finite-temperature magnetization of a permanent magnet over a high-dimensional composition space. Based on a general framework for constructing a predictor from two data sets including missing values, a practical scheme for magnetic materials is formulated in which a small number of experimental data in limited composition space are integrated with a larger number of first-principles calculation data. We apply the scheme to ${({\mathrm{Nd}}_{1\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremath{-}\ensuremath{\gamma}}{\mathrm{Pr}}_{\ensuremath{\alpha}}{\mathrm{La}}_{\ensuremath{\beta}}{\mathrm{Ce}}_{\ensuremath{\gamma}})}_{2}{({\mathrm{Fe}}_{1\ensuremath{-}\ensuremath{\delta}\ensuremath{-}\ensuremath{\zeta}}{\mathrm{Co}}_{\ensuremath{\delta}}{\mathrm{Ni}}_{\ensuremath{\zeta}})}_{14}\mathrm{B}$. The magnetization in the whole $(\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\delta},\ensuremath{\zeta})$ space at arbitrary temperature is obtained. It is shown that the Co doping does not enhance the magnetization at low temperatures, whereas the magnetization increases with increasing $\ensuremath{\delta}$ above 320 K.

We perform a crystal structure search using Bayesian optimization and evaluate its efficiency with a varying parameter value in the descriptor. Applying the crystal structure search to crystalline silicon shows that the efficiency of the search depends heavily on the parameter value. We find that the efficiency is linked to the distribution of the descriptor. Therefore, we introduce an information measure of the distribution to estimate an appropriate parameter value for performing the crystal structure search efficiently. The measure can also be used to predetermine an appropriate parameter value. The validity of the measure is confirmed with its applications to silicon oxide and yttrium-cobalt alloy.

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.

The magnetic properties and structural stability of ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{14}X$ ($X=\mathrm{B}$, C, N, O, F) are theoretically studied by first-principles calculations focusing on the role of $X$. We find that B reduces the magnetic moment (per formula unit) and magnetization (per volume) in ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{14}\mathrm{B}$. The crystal-field parameter ${A}_{2}^{0}\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}$ of Nd is not enhanced either, suggesting that B has minor roles in the uniaxial magnetocrystalline anisotropy of Nd. These findings are in contrast to the long-held belief that B works positively for the magnetic properties of ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{14}\mathrm{B}$. As $X$ changes from B to C, N, O, and F, both the magnetic properties and stability vary significantly. The formation energies of ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{14}X$ and $\ensuremath{\alpha}$-Fe relative to that of ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{17}X$ are negative for $X=\mathrm{B}$ and C, whereas they are positive when $X$ = N, O, and F. This indicates that B plays an important role in stabilizing the ${\mathrm{Nd}}_{2}{\mathrm{Fe}}_{14}\mathrm{B}$ phase.

We propose a crystal structure prediction method based on Bayesian optimization. Our method is classified as a selection-type algorithm which is different from evolution-type algorithms such as an evolutionary algorithm and particle swarm optimization. Crystal structure prediction with Bayesian optimization can efficiently select the most stable structure from a large number of candidate structures with a lower number of searching trials using a machine learning technique. Crystal structure prediction using Bayesian optimization combined with random search is applied to known systems such as NaCl and ${\mathrm{Y}}_{2}{\mathrm{Co}}_{17}$ to discuss the efficiency of Bayesian optimization. These results demonstrate that Bayesian optimization can significantly reduce the number of searching trials required to find the global minimum structure by 30--40% in comparison with pure random search, which leads to much less computational cost.