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 H2O 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 H2O, 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.