Representation of compounds for machine-learning prediction of physical properties

The representations of a compound, called “descriptors” or “features”, play an essential role in constructing a machine-learning model of its physical properties. In this study, we adopt a procedure for generating a set of descriptors from simple elemental and structural representations. First, it is applied to a large data set composed of the cohesive energy for about 18 000 compounds computed by density functional theory calculation. As a result, we obtain a kernel ridge prediction model with a prediction error of 0.041 eV/atom, which is close to the “chemical accuracy” of 1 kcal/mol (0.043 eV/atom). A prediction model with an error of 0.071 eV/atom of the cohesive energy is obtained for the normalized prototype structures, which can be used for the practical purpose of searching for as-yet-unknown structures. The procedure is also applied to two smaller data sets, i.e., a data set of the lattice thermal conductivity for 110 compounds computed by density functional theory calculation and a data set of the experimental melting temperature for 248 compounds. We examine the effect of the descriptor sets on the efficiency of Bayesian optimization in addition to the accuracy of the kernel ridge regression models. They exhibit good predictive performances.

Figure 1

Schematic of how to generate compound descriptors. First, each atom in a compound is characterized by $N_x$ representations. The collection of atoms in the compounds is written as a representation matrix $\mathit{X}$. Then the representation matrix is regarded as the data distribution in an $N_x$-dimensional space. To transform the distribution into descriptors, representative quantities are introduced to characterize the data distribution such as its mean, standard deviation, skewness, kurtosis, and covariance.

Figure 2

Comparison of cohesive energy calculated by DFT calculation and that calculated by the KRR prediction model. Only one test data set is shown. Descriptor sets are composed of (a) the means of elemental representations, (b) the means of elemental and PRDF representations, (c) the means, SDs, and covariances of elemental and PRDF representations, and (d) the means, SDs, and covariances of elemental and 20 trigonometric GRDF representations. The mean of the PRDF corresponds to the RDF. Structure representations are computed from the optimized structure for each compound.

Figure 3

Dependence of the prediction error on the number of training data. The standard deviation for 20 trials is also shown. A set of 20 trigonometric GRDFs and 20 BOPs is used as the structural representations. Open circles and filled circles show the prediction error of the KRR model without and with consideration of the covariances of the elemental and structural representations, respectively. The numbers of descriptors are 122 and 1952 in the former and latter cases, respectively.

Figure 4

Behavior of Bayesian optimization for the LTC data in finding PbClBr, CuCl, and LiI. The best LTC is shown, along with the number of samples (iterations) during the Bayesian optimization and random search.

Figure 5

Behavior of Bayesian optimization for the melting temperature data in finding AlN, SiC, and MgO.