Input vector optimization of feed-forward neural networks for fitting ab initio potential-energy databases

The variation in the fitting accuracy of neural networks (NNs) when used to fit databases comprising potential energies obtained from ab initio electronic structure calculations is investigated as a function of the number and nature of the elements employed in the input vector to the NN. Ab initio databases for H2O2, HONO, Si-5, and H2C=CHBr were employed in the investigations. These systems were chosen so as to include four-, five-, and six-body systems containing first, second, third, and fourth row elements with a wide variety of chemical bonding and whose conformations cover a wide range of structures that occur under high-energy machining conditions and in chemical reactions involving cis-trans isomerizations, six different types of two-center bond ruptures, and two different three-center dissociation reactions. The ab initio databases for these systems were obtained using density functional theory/B3LYP, MP2, and MP4 methods with extended basis sets. A total of 31 input vectors were investigated. In each case, the elements of the input vector were chosen from interatomic distances, inverse powers of the interatomic distance, three-body angles, and dihedral angles. Both redundant and nonredundant input vectors were investigated. The results show that among all the input vectors investigated, the set employed in the Z-matrix specification of the molecular configurations in the electronic structure calculations gave the lowest NN fitting accuracy for both Si-5 and vinyl bromide. The underlying reason for this result appears to be the discontinuity present in the dihedral angle for planar geometries. The use of trigometric functions of the angles as input elements produced significantly improved fitting accuracy as this choice eliminates the discontinuity. The most accurate fitting was obtained when the elements of the input vector were taken to have the form R-ij(-n), where the R-ij are the interatomic distances. When the Levenberg-Marquardt procedure was modified to permit error minimization with respect to n as well as the weights and biases of the NN, the optimum powers were all found to lie in the range of 1.625-2.38 for the four systems studied. No statistically significant increase in fitting accuracy was achieved for vinyl bromide when a different value of n was employed and optimized for each bond type. The rate of change in the fitting error with n is found to be very small when n is near its optimum value. Consequently, good fitting accuracy can be achieved by employing a value of n in the middle of the above range. The use of interparticle distances as elements of the input vector rather than the Z-matrix variables employed in the electronic structure calculations is found to reduce the rms fitting errors by factors of 8.86 and 1.67 for Si-5 and vinyl bromide, respectively. If the interparticle distances are replaced with input elements of the form R-ij(-n) with n optimized, further reductions in the rms error by a factor of 1.31 to 2.83 for the four systems investigated are obtained. A major advantage of using this procedure to increase NN fitting accuracy rather than increasing the number of neurons or the size of the database is that the required increase in computational effort is very small. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3431624]