On-the-fly machine learning force field generation: Application to melting points

An efficient and robust on-the-fly machine learning force field method is developed and integrated into an electronic-structure code. This method realizes automatic generation of machine learning force fields on the basis of Bayesian inference during molecular dynamics simulations, where the first-principles calculations are only executed, when new configurations out of already sampled datasets appear. The developed method is applied to the calculation of melting points of Al, Si, Ge, Sn and MgO. The applications indicate that more than 99% of the first-principles calculations are bypassed during the force field generation. This allows the machine to quickly construct first-principles datasets over wide phase spaces. Furthermore, with the help of the generated machine learning force fields, simulations are accelerated by a factor of thousand compared with first-principles calculations. Accuracies of the melting points calculated by the force fields are examined by thermodynamic perturbation theory, and the examination indicates that the machine learning force fields can quantitatively reproduce the first-principles melting points.

Figure 1

Flowchart of our on-the-fly machine learning force field generation scheme.

Figure 2

(a) Radial and (b) angular descriptors.

Figure 3

Errors during a finite temperature simulation of rocksalt MgO containing 64 atoms at 300 K. (a) Actual difference between the FP forces and the forces calculated by the MLFF. Solid line shows the root mean square difference averaged over all atoms, and the dashed line shows a quarter of maximum difference on a single atom. (b) The maximum Bayesian error [the second term of Eq. (30)] of the force on a single atom. (c) The maximum spilling factor on a single atom. Open circles and black small dots indicate steps 3 and 4, respectively. Dark gray triangles indicate the Bayesian errors stored as ${\sigma }_{\text{max},I}$. These are used to determine the threshold for the Bayesian error ${\epsilon }_{\mathrm{BE}}$ as explained in Sec. 2e. Gray dashed lines indicate the threshold for the Bayesian error.

Figure 4

Total (black), kinetic (light gray), and potential (dark gray) energies of solid MgO containing 216 atoms at 300 K during the first 1 ps MD simulation with the microcanonical ensemble using the generated force field. The energies were shifted to provide zero at the beginning.

Figure 5

Flowchart of the decision step whether to perform FP simulations or not. The symbols $\vec{\sigma }$ and $\vec{s}$ denote the vectors containing the Bayesian errors in the forces and spilling factors, respectively, for all atoms. $\left|\right|\vec{x}{\left|\right|}_{\infty }$ denotes the infinity norm (also called supremum norm) of the vector $\vec{x},{\epsilon }_{\mathrm{BE}}$ denotes the criterion for the Bayesian error, and $\mathrm{Var}\left(x\right)$ and $\mathrm{E}\left(x\right)$ refer to the variance and the average of the data $x$. At the beginning of every training simulation, the criterion ${\epsilon }_{\mathrm{BE}}$ is set to zero.

Figure 6

Flowchart of the sparsification and data selection step. The symbol $K$ denotes the matrix comprised of the elements $K(i,j)$ defined as Eq. (14). The leverage scoring ${\omega }_i$ is calculated by Eq. (E5) in Appendix pp5.

Figure 7

Mean absolute errors in the energy per atom $\left(\mathrm{meV}{\mathrm{atom}}^{-1}\right)$ (a), force $\left(\mathrm{eV}{\AA }^{-1}\right)$ and stress tensor (GPa) predicted by the force fields on 200 configurations of solids and liquids at the melting points.

Figure 8

Melting point $T_m$ calculated by the interface pinning method using the machine learning force fields and the thermodynamic perturbation corrections vs the energy difference $\mathrm{\Delta }E$ between the $\alpha$- and $\beta$-tin structures obtained from DFT calculations at 0 K. $\mathrm{\Delta }E$ is defined to be negative when the $\alpha$-tin structure is more stable than the $\beta$-tin structure.

Figure 9

(a) Real (root mean square error) (gray line) and Bayesian error (black line) on forces (a) and (b) histogram of training data on forces during the training mainly on force data.