Parametrization of analytic interatomic potential functions using neural networks

A generalized method that permits the parameters of an arbitrary empirical potential to be efficiently and accurately fitted to a database is presented. The method permits the values of a subset of the potential parameters to be considered as general functions of the internal coordinates that define the instantaneous configuration of the system. The parameters in this subset are computed by a generalized neural network (NN) with one or more hidden layers and an input vector with at least 3n-6 elements, where n is the number of atoms in the system. The Levenberg-Marquardt algorithm is employed to efficiently affect the optimization of the weights and biases of the NN as well as all other potential parameters being treated as constants rather than as functions of the input coordinates. In order to effect this minimization, the usual Jacobian employed in NN operations is modified to include the Jacobian of the computed errors with respect to the parameters of the potential function. The total Jacobian employed in each epoch of minimization is the concatenation of two Jacobians, one containing derivatives of the errors with respect to the weights and biases of the network, and the other with respect to the constant parameters of the potential function. The method provides three principal advantages. First, it obviates the problem of selecting the form of the functional dependence of the parameters upon the system's coordinates by employing a NN. If this network contains a sufficient number of neurons, it will automatically find something close to the best functional form. This is the case since Hornik et al., Neural Networks 2, 359 (1989) have shown that two-layer NNs with sigmoid transfer functions in the first hidden layer and linear functions in the output layer are universal approximators for analytic functions. Second, the entire fitting procedure is automated so that excellent fits are obtained rapidly with little human effort. Third, the method provides a procedure to avoid local minima in the multidimensional parameter hyperspace. As an illustrative example, the general method has been applied to the specific case of fitting the ab initio energies of Si(5) clusters that are observed in a molecular dynamics (MD) simulation of the machining of a silicon workpiece. The energies of the Si(5) configurations obtained in the MD calculations are computed using the B3LYP procedure with a 6-31G(**) basis set. The final ab initio database, which comprises the density functional theory energies of 10 202 Si(5) clusters, is fitted to an empirical Tersoff potential containing nine adjustable parameters, two of which are allowed to be the functions of the Si(5) configuration. The fitting error averaged over all 10 202 points is 0.0148 eV (1.43 kJ mol(-1)). This result is comparable to the accuracy achieved by more general fitting methods that do not rely on an assumed functional form for the potential surface.