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An innovative efficient and fast neural networks in which hidden neurons are constructed based on Fourier series expansions (FSNN), half-range cosine (FCSNN) and sine expansions (FSSNN) are proposed and tested for linear and nonlinear regulation problems. The results of numerical examples using FSNN are compared with those obtained from traditional linear regression (LP), nonlinear regression (NLP), backward propagation neural networks (BPANN) and radial basis function neural networks (RBFNN). The results obtained from FSNN agree well with those obtained from LP, NLP, BPANN and RBFNN and show global approximation features to the fitting data. Only a few hidden neurons are required to obtain very good and fast convergence of regression as compared with BPANN and RBFNN.

This paper presents classification of linearly separable and non-separable problems using neural networks in which hidden neurons are constructed based on double Fourier series expansions (FSNN). The results of numerical examples including classification problems of logical AND, logical XOR, cows and wolves, as well as 3-category problem such as IRIS classification. All the FSNN results are compared with those obtained from backward propagation neural networks (BPANN) and radial basis function neural networks (RBFNN). Root mean squared errors (RMSE) of the algorithms during the training process are also compared. The classification results obtained from FSNN agree well with those obtained from BPANN and RBFNN. Only a few hidden neurons in FSNNs are required for very good and fast convergence of training as compared with BPANN and RBFNN.