  In a previous paper [Fan Z \textit{et al}. 2021 Phys. Rev. B, \textbf{104},
104309], we developed the neuroevolution potential (NEP), a framework of
training neural network based machine-learning potentials using a natural
evolution strategy and performing molecular dynamics (MD) simulations using the
trained potentials. The atom-environment descriptor in NEP was constructed
based on a set of radial and angular functions. For multi-component systems,
all the radial functions between two atoms are multiplied by some fixed factors
that depend on the types of the two atoms only. In this paper, we introduce an
improved descriptor for multi-component systems, in which different radial
functions are multiplied by different factors that are also optimized during
the training process, and show that it can significantly improve the regression
accuracy without increasing the computational cost in MD simulations.
