Sparse representation for a potential energy surface

We propose a simple scheme to estimate the potential energy surface (PES) for which the accuracy can be easily controlled and improved. It is based on model selection within the framework of linear regression using the least absolute shrinkage and selection operator (LASSO) technique. Basis functions are selected from a systematic large set of candidate functions. The sparsity of the PES significantly reduces the computational cost of evaluating the energy and force in molecular dynamics simulations without losing accuracy. The usefulness of the scheme for describing the elemental metals Na and Mg is clearly demonstrated.

Figure 1

Relationship between atomic positions and total energy in the linear model for the PES of structure $i$. It is assumed that the energy of atom $j$, $E^{(i,j)}$, and the set of basis functions for atom $j$, $\left\{b_{n,a}^{(i,j)}\right\}$, have a linear relationship. The set of basis functions is here calculated from the set of pair distances $\left\{R_{jk}^{\left(i\right)}\right\}$ smaller than the cutoff radius $R_c$, which are obtained from the positions of all atoms.

Figure 2

RMSEs for the test data of the PESs constructed by linear ridge regression using various types of basis functions with $a_{\max }=3$, $R_c=7.0$, and $\lambda =0.001$ (a) for Na and (b) for Mg. RMSEs optimized by the LASSO are also shown (c) for Na and (d) for Mg. For the Bessel and Neumann basis sets, the number of basis functions is controlled by the maximum order of the basis functions. For the Gaussian basis set, the number of basis functions is controlled by the maximum values of the internal parameters of the Gaussian.

Figure 3

(a) Energies predicted by the LASSO PES and DFT for Mg, measured from the energy of the ideal hcp structure. (b) Differences between energies predicted by the LASSO PES and DFT.

Figure 4

(a) Phonon dispersion and (b) specific heat at a constant volume for hcp Mg obtained from the LASSO PES (solid lines) compared with those obtained by the DFT calculation (dashed lines). To evaluate the dynamical matrix, each atomic position is displaced by 0.01 Å.