Simultaneous fitting of a potential-energy surface and its corresponding force fields using feedforward neural networks.

An improved neural network (NN) approach is presented for the simultaneous development of accurate potential-energy hypersurfaces and corresponding force fields that can be utilized to conduct ab initio molecular dynamics and Monte Carlo studies on gas-phase chemical reactions. The method is termed as combined function derivative approximation (CFDA). The novelty of the CFDA method lies in the fact that although the NN has only a single output neuron that represents potential energy, the network is trained in such a way that the derivatives of the NN output match the gradient of the potential-energy hypersurface. Accurate force fields can therefore be computed simply by differentiating the network. Both the computed energies and the gradients are then accurately interpolated using the NN. This approach is superior to having the gradients appear in the output layer of the NN because it greatly simplifies the required architecture of the network. The CFDA permits weighting of function fitting relative to gradient fitting. In every test that we have run on six different systems, CFDA training (without a validation set) has produced smaller out-of-sample testing error than early stopping (with a validation set) or Bayesian regularization (without a validation set). This indicates that CFDA training does a better job of preventing overfitting than the standard methods currently in use. The training data can be obtained using an empirical potential surface or any ab initio method. The accuracy and interpolation power of the method have been tested for the reaction dynamics of H+HBr using an analytical potential. The results show that the present NN training technique produces more accurate fits to both the potential-energy surface as well as the corresponding force fields than the previous methods. The fitting and interpolation accuracy is so high (rms error=1.2 cm(-1)) that trajectories computed on the NN potential exhibit point-by-point agreement with corresponding trajectories on the analytic surface.