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A self-organizing fuzzy neural network modeling approach using an adaptive quantum particle swarm optimization

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Abstract

To enhance the model’s flexibility, this study proposes a self-organizing fuzzy neural network (SOFNN) modeling methodology based on an adaptive quantum particle swarm optimization algorithm (AQPSO). First, to address the shortcoming of premature convergence of the QPSO algorithm when dealing with complex problems and to acquire the best balance between the exploration and exploitation of the algorithm, a cooperative adaptive adjustment strategy for attractor, coefficient, and boundary is designed. Second, to obtain a suitable number of fuzzy rules and optimal premise parameters, the fitness function is constructed using system accuracy (RMSE) and network complexity (rule number) in the learning process. Simultaneously, an enhanced fuzzy recursive least square (FRLS) algorithm is designed to estimate the output weights of the FNN to identify the nonlinear dynamical system effectively. Furthermore, to ensure that the presented AQPSO-SOFNN can efficiently solve practical engineering problems, Lyapunov stability theory is adopted to prove its convergence in detail. Finally, four testing cases, including identification of Mackey-Glass time series, modeling of concrete compressive strength (CCS), prediction of carbon dioxide, and soft-sensing of effluent total phosphorus (TP), are utilized to verify the usefulness of the proposed AQPSO-SOFNN-based modeling approach. The simulation results of four testing cases demonstrate that the designed AQPSO-SOFNN has high prediction accuracy with a parsimonious network topology. The MATLAB source codes of AQPSO-SOFNN and other comparison algorithms can be downloaded from https://github.com/hyitzhb/AQPSO-SOFNN.git.

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Data availability

The datasets generated during and/or analysed during the current study are available in the Github repository, https://github.com/hyitzhb/AQPSO-SOFNN.git.

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Acknowledgements

This study was supported in part by the Industry University Research Cooperation Project of Jiangsu Province under Grant BY2020247, the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant SJCX21_1509, and the National Training Programs of Innovation and Entrepreneurship for Undergraduates under Grant 202111049233.

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  1. Faculty of Automation, Huaiyin Institute of Technology, Huai’an, 223003, China

    Hongbiao Zhou, Yang Li, Haoyuan Xu, Yan Su & Lianghai Chen

  2. Public Technical Service Platform for Instrumentation Industry of Small and Medium-sized Enterprises in Jiangsu Province, Huai’an, 223003, China

    Hongbiao Zhou

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Appendix A

Appendix A

Deduction process of Eq. (51) using Mathematical induction:

When t = 1,

$$ {\displaystyle \begin{array}{c}{\vartheta}_1\left(\psi, {R}_{\varepsilon}\right)={p}^{(1)}={\varphi}_1\left(\psi \ast +\varepsilon \right)\\ {}\kern6.699996em =1-\left(1-{\varphi}_1\left(\psi \ast +\varepsilon \right)\right)\end{array}} $$
(A1)

That is, the equation holds.

When t = 2,

$$ {\displaystyle \begin{array}{c}{\vartheta}_1\left(\psi, {R}_{\varepsilon}\right)={p}^{(1)}+{p}^{(2)}\\ {}={\varphi}_1\left(\psi \ast +\varepsilon \right)+{\varphi}_2\left(\psi \ast +\varepsilon \right)\left(1-{\varphi}_1\left(\psi \ast +\varepsilon \right)\right)\\ {}={\varphi}_1\left(\psi \ast +\varepsilon \right)+{\varphi}_2\left(\psi \ast +\varepsilon \right)-{\varphi}_1\left(\psi \ast +\varepsilon \right){\varphi}_2\left(\psi \ast +\varepsilon \right)\\ {}=1-1+{\varphi}_1\left(\psi \ast +\varepsilon \right)+{\varphi}_2\left(\psi \ast +\varepsilon \right)-{\varphi}_1\left(\psi \ast +\varepsilon \right){\varphi}_2\left(\psi \ast +\varepsilon \right)\\ {}=1-\left(1-{\varphi}_1\left(\psi \ast +\varepsilon \right)\right)\Big(1-{\varphi}_1\left(\psi \ast +\varepsilon \right)\\ {}=1-\prod \limits_{i=1}^2\left[1-{\varphi}_i\left(\psi \ast +\varepsilon \right)\right]\end{array}} $$
(A2)

The equation also holds.

Assuming t = k, the equation holds, that is

$$ {\displaystyle \begin{array}{c}{\vartheta}_k\left(\psi, {R}_{\varepsilon}\right)={p}^{(1)}+{p}^{(2)}+\cdots +{p}^{(k)}\\ {}\kern4.399998em =1-\prod \limits_{i=1}^k\left[1-{\varphi}_i\left(\psi \ast +\varepsilon \right)\right]\end{array}} $$
(A3)

When t = k + 1,

$$ {\displaystyle \begin{array}{c}{\vartheta}_{k+1}\left(\psi, {R}_{\varepsilon}\right)={\vartheta}_k\left(\psi, {R}_{\varepsilon}\right)+{p}^{\left(k+1\right)}\\ {}=1-\prod \limits_{i=1}^k\left[1-{\varphi}_i\left(\psi \ast +\varepsilon \right)\right]+\\ {}{\varphi}_{k+1}\left(\psi \ast +\varepsilon \right)\prod \limits_{i=1}^k\left[1-{\varphi}_i\left(\psi \ast +\varepsilon \right)\right]\\ {}=1-\left(1-{\varphi}_{k+1}\left(\psi \ast +\varepsilon \right)\right)\prod \limits_{i=1}^k\left[1-{\varphi}_i\left(\psi \ast +\varepsilon \right)\right]\\ {}=1-\prod \limits_{i=1}^{k+1}\left[1-{\varphi}_i\left(\psi \ast +\varepsilon \right)\right]\end{array}} $$
(A4)

Therefore, the equation holds for any t ≥ 1.

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Zhou, H., Li, Y., Xu, H. et al. A self-organizing fuzzy neural network modeling approach using an adaptive quantum particle swarm optimization. Appl Intell 53, 13569–13592 (2023). https://doi.org/10.1007/s10489-022-04133-8

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