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Sigma-Pi-Sigma neural networks (SPSNNs) as a kind of high-order neuralnetworks can provide more powerful mapping capability than the traditionalfeedforward neural networks (Sigma-Sigma neural networks). In the existingliterature, in order to reduce the number of the Pi nodes in the Pi layer, aspecial multinomial P_s is used in SPSNNs. Each monomial in P_s is linear withrespect to each particular variable sigma_i when the other variables are takenas constants. Therefore, the monomials like sigma_i^n or sigma_i^n sigma_j withn>1 are not included. This choice may be somehow intuitive, but is notnecessarily the best. We propose in this paper a modified Sigma-Pi-Sigma neuralnetwork (MSPSNN) with an adaptive approach to find a better multinomial for agiven problem. To elaborate, we start from a complete multinomial with a givenorder. Then we employ a regularization technique in the learning process forthe 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 (= thenumber of nodes in the Pi-layer) as in P_s. Numerical experiments on somebenchmark problems show that our MSPSNN behaves better than the traditionalSPSNN with P_s.

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