A Modified Sigma-Pi-Sigma Neural Network with Adaptive Choice of Multinomials

Sigma-Pi-Sigma neural networks (SPSNNs) as a kind of high-order neural networks can provide more powerful mapping capability than the traditional feedforward neural networks (Sigma-Sigma neural networks). In the existing literature, in order to reduce the number of the Pi nodes in the Pi layer, a special multinomial P_s is used in SPSNNs. Each monomial in P_s is linear with respect to each particular variable sigma_i when the other variables are taken as constants. Therefore, the monomials like sigma_i^n or sigma_i^n sigma_j with n>1 are not included. This choice may be somehow intuitive, but is not necessarily the best. We propose in this paper a modified Sigma-Pi-Sigma neural network (MSPSNN) with an adaptive approach to find a better multinomial for a given problem. To elaborate, we start from a complete multinomial with a given order. Then we employ a regularization technique in the learning process for the given problem to reduce the number of monomials used in the multinomial, and end up with a new SPSNN involving the same number of monomials (= the number of nodes in the Pi-layer) as in P_s. Numerical experiments on some benchmark problems show that our MSPSNN behaves better than the traditional SPSNN with P_s.