Modified feed-forward neural network structures and combined-function-derivative approximations incorporating exchange symmetry for potential energy surface fitting

The classical interchange (permutation) of atoms of similar identity does not have an effect on the overall potential energy. In this study, we present feed-forward neural network structures that provide permutation symmetry to the potential energy surfaces of molecules. The new feed-forward neural network structures are employed to fit the potential energy surfaces for two illustrative molecules, which are H(2)O and ClOOCl. Modifications are made to describe the symmetric interchange (permutation) of atoms of similar identity (or mathematically, the permutation of symmetric input parameters). The combined-function-derivative approximation algorithm (J. Chem. Phys. 2009, 130, 134101) is also implemented to fit the neural-network potential energy surfaces accurately. The combination of our symmetric neural networks and the function-derivative fitting effectively produces PES fits using fewer numbers of training data points. For H(2)O, only 282 configurations are employed as the training set; the testing root-mean-squared and mean-absolute energy errors are respectively reported as 0.0103 eV (0.236 kcal/mol) and 0.0078 eV (0.179 kcal/mol). In the ClOOCl case, 1693 configurations are required to construct the training set; the root-mean-squared and mean-absolute energy errors for the ClOOCl testing set are 0.0409 eV (0.943 kcal/mol) and 0.0269 eV (0.620 kcal/mol), respectively. Overall, we find good agreements between ab initio and NN prediction in term of energy and gradient errors, and conclude that the new feed-forward neural-network models advantageously describe the molecules with excellent accuracy.