Neural Network Models of Potential Energy Surfaces:  Prototypical Examples

Neural networks can be used generate potential energy hypersurfaces by fitting to a data set of energies at discrete geometries, as might be obtained from ab initio calculations. Prior work has shown that this method can generate accurate fits in complex systems of several dimensions. The present paper explores fundamental properties of neural network potential representations in some simple prototypes of one, two, and three dimensions. Optimal fits to the data are achieved by adjusting the network parameters using an extended Kalman filtering algorithm, which is described in detail. The examples provide insight into the relationships between the form of the function being fit, the amount of data needed for an adequate fit, and the optimal network configuration and number of neurons needed. The quality of the network interpolation is substantially improved if gradients as well as the energy are available for fitting. The fitting algorithm is effective in providing an accurate interpolation of the underlying potential function even when random noise is added to the data used in the fit.