Neural networks provide an efficient, general interpolation method for nonlinear functions of several variables. This paper describes the use of feed-forward neural networks to model global properties of potential energy surfaces from information available at a limited number of configurations. As an initial demonstration of the method, several fits are made to data derived from an empirical potential model of CO adsorbed on Ni(111). The data are error-free and geometries are selected from uniform grids of two and three dimensions. The neural network model predicts the potential to within a few hundredths of a kcal/mole at arbitrary geometries. The accuracy and efficiency of the neural network in practical calculations are demonstrated in quantum transition state theory rate calculations for surface diffusion of CO/Ni(111) using a Monte Carlo/path integral method. The network model is much faster to evaluate than the original potential from which it is derived. As a more complex : test of the method, the interaction potential of H-2 With the Si(100)-2X1 surface is determined as a function of 12 degrees of freedom from energies calculated with the local density functional method at 750 geometries. The training examples are not uniformly spaced and they depend weakly on variables not included in the fit. The neural net model predicts the potential at geometries outside the training set with a mean absolute deviation of 2.1 kcal/mole.