Efficient Training of the Machine-Learning Interatomic Potential Based on an Artificial Neural Network for Estimating the Helmholtz Free Energy of Alkali Metals

Although the Helmholtz free energy can be accurately calculated using the thermodynamic integration (TI) method based on first-principles molecular dynamics (FPMD) simulations, the computational cost of this method is very high. By training the data obtained by FPMD simulations based on an artificial neural network (ANN), an interatomic potential with a very low computational cost can be constructed. To learn the FPMD data efficiently, it is important to elucidate the relation between the number of training data and accuracy of the free energy obtained by the TI method. We statistically evaluated the minimum number of training data by constructing multiple ANN potentials to accurately estimate the Helmholtz free energy of rubidium. The root mean square errors (RMSEs) of energy, atomic force, and pressure between the ANN and FPMD converged as the number of training data increased. The radial distribution function of the FPMD can be reproduced, even if the RMSEs do not converge. However, the number of training data for which the RMSEs converge is required to calculate the free energy with sufficient accuracy. This study demonstrated that the quality of an ANN potential can be estimated by statistically investigating the dependence of the RMSEs on the number of training data.