Parametrization of interatomic potential functions using a genetic algorithm accelerated with a neural network

A genetic algorithm (GA) can be used to fit highly nonlinear functional forms, such as empirical interatomic potentials from a large ensemble of data. The performance of a GA for fitting such functional forms is enhanced through an approach that is based on the use of a neural network (NN) to accelerate the computation of the fitness function for the GA. Application of the new approach for fitting an ensemble of potentials computed from ab initio calculations to a specified functional form (here, the Tersoff potential functional form is used as an example) has shown that the computational efficiency achieved through the use of a NN can reduce computational time by over two orders of magnitude. The potentials estimated from functions thus fitted were within 0.1% of the actual potential values. Specifically, the mean squared error (MSE) on molecular potentials was $<{10}^{-5}{\mathrm{eV}}^2$ for fitting Tersoff potentials, and $<0.0025{\mathrm{eV}}^2$ for fitting ab initio potential energies of isolated, 5-atom silicon clusters. Furthermore, since the potential was fitted to a physically meaningful Tersoff functional form, the resulting potential function appears to have the ability to extrapolate over a reasonable range of the parameter space, and may have a better accuracy in estimating the forces compared to that obtained from neural networks, which are often highly inaccurate when extrapolated. Hence, the method can be useful for rendering various molecular dynamics (MD) simulations more tractable. It is also apparent, based on the present investigation, that a Tersoff potential, albeit with different (GA parametrized) coefficients, is adequate for representing the ab initio potentials of 5-atom Si clusters.

Figure 1

Internal coordinates: (a) bond distance, (b) bond angle, (c) dihedral angle, and (d) configuration of 5 silicon atoms.

Figure 2

(Color online) Variation of the objective function O[P] with (a) $\beta$ and $d$ (b) $n$ and $d$ and (c) $c$ and $d$, respectively, showing large plateau regions.

Figure 3

(Color online) Flowchart of the approach used for GA optimization.

Figure 4

(Color online) Comparison of the NN output with the objective function O[P] for the testing set: (a) first stage and (b) fourth stage, respectively.

Figure 5

(Color online) (a) Variation of the mutation rate during each stage with number of iterations. (b) Variation of the crossover rate during each stage with the number of iterations.

Figure 6

(Color online). Variation of MSE with the number of iterations at various stages for parametrization of Tersoff potential.

Figure 7

Comparison of the potential computed using Tersoff $V\left(x\right)$ and GA optimized parameters $V\left[P\right]\left(x\right)$. Energies are given in eV.

Figure 8

(Color online) (a) Snapshot of MD simulation of nanometric cutting of silicon showing atomic configuration of Si workpiece and the diamond tool used. (b) Histogram showing the distribution of various Si clusters in the primary and secondary deformation zones ahead of the cutting tool in nanometric cutting as shown in (a).

Figure 9

(a)–(i) Histograms of the number of configurations as a function of internal coordinates, (a)–(d) bond distance $\left(r\right)$, (e)–(g) bond angles $\left(\theta \right)$, and (h)–(i) dihedral angle $\left(\varphi \right)$.

Figure 10

(Color online). Comparison of the NN output with the objective function $\mathrm{O}\left[P\right]$ for the testing set at (a) first and (b) fourth stages, respectively.

Figure 11

(Color online) Variation of objective function $\mathrm{O}\left[P\right]$ (here, MSE) at various stages of GA fitting of ab initio potentials to Tersoff functional form.

Figure 12

(Color online) Comparison of the ab initio potential computed using Gaussian 03 $V\left(x\right)$ and GA-optimized parameters for Tersoff potential $V\left[P\right]\left(x\right)$.

Figure 13

(Color online) Variation of NN- and GA-Tersoff-estimates of ab initio potential with bond distances compared against the variation of actual ab initio potential.