Summary It is significant to fit a Gaussian function with the observation data for artificial intelligence or other engineering fields. Considering the influence of noises, this article proposes a nonlinear optimization method for fitting the Gaussian activation functions. By means of the gradient search and the Newton search, a direct gradient‐based iterative algorithm and a direct Newton iterative algorithm are presented for identifying the Gaussian functions. Considering the computational cost, the authors develop a multi‐innovation stochastic gradient algorithm for the noisy Gaussian functions. After introducing a forgetting factor, the parameter estimation accuracy can be further improved. The simulation results indicate that the proposed nonlinear optimization method and gradient‐based algorithms can fit the noisy Gaussian functions very well.

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Constructing an appropriate membership function is significant in fuzzy logic control. Based on the multi‐model control theory, this article constructs a novel kernel function which can implement the fuzzification and defuzzification processes and reflect the dynamic quality of the nonlinear systems accurately. Then we focus on the identification problems of the nonlinear systems based on the kernel functions. Applying the hierarchical identification principle, we present the hierarchical stochastic gradient algorithm for the nonlinear systems. Meanwhile, the one‐dimensional search methods are proposed to solve the problem of determining the optimal step sizes. In order to improve the parameter estimation accuracy, we propose the hierarchical multi‐innovation forgetting factor stochastic gradient algorithm by introducing the forgetting factor and using the multi‐innovation identification theory. The simulation example is provided to test the proposed algorithms from the aspects of parameter estimation accuracy and prediction performance.