Associative memories based on delayed fractional-order neural networks and application to explaining-lesson skills assessment of normal students: from the perspective of multiple $ \mathit O(t^{-\alpha}) $ stability

Jiang-Wei Ke; Jin-E Zhang; Jiang-Wei Ke; Jin-E Zhang

doi:10.3934/math.2024847

AIMS Mathematics

2024, Volume 9, Issue 7: 17430-17452. doi: 10.3934/math.2024847

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Research article

Associative memories based on delayed fractional-order neural networks and application to explaining-lesson skills assessment of normal students: from the perspective of multiple $ \mathit O(t^{-\alpha}) $ stability

Jiang-Wei Ke ,
Jin-E Zhang ^,

School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

Received: 17 March 2024 Revised: 24 April 2024 Accepted: 15 May 2024 Published: 21 May 2024
MSC : 34A08

This paper discusses associative memories based on time-varying delayed fractional-order neural networks (DFNNs) with a type of piecewise nonlinear activation function from the perspective of multiple $ \mathit O(t^{-\alpha}) $ stability. Some sufficient conditions are gained to assure the existence of $ 5^n $ equilibria for $ n $-neuron DFNNs with the proposed piecewise nonlinear activation functions. Additionally, the criteria ensure the existence of at least $ 3^n $ equilibria that are locally multiple $ \mathit O(t^{-\alpha}) $ stable. Furthermore, we apply these results to a more generic situation, revealing that DFNNs can attain $ (2k+1)^n $ equilibria, and among them, $ (k+1)^n $ equilibria are locally $ \mathit O(t^{-\alpha}) $ stable. Here, the parameter $ k $ is highly dependent on the sinusoidal function frequency in the expanded activation functions. Such DFNNs are well-suited to synthesize high-capacity associative memories; the design process is given via singular value decomposition. Ultimately, four illustrative examples, including applying neurodynamic associative memory to the explaining-lesson skills assessment of normal students, are supplied to validate the efficacy of the results.
Citation: Jiang-Wei Ke, Jin-E Zhang. Associative memories based on delayed fractional-order neural networks and application to explaining-lesson skills assessment of normal students: from the perspective of multiple $ \mathit O(t^{-\alpha}) $ stability[J]. AIMS Mathematics, 2024, 9(7): 17430-17452. doi: 10.3934/math.2024847

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Abstract

This paper discusses associative memories based on time-varying delayed fractional-order neural networks (DFNNs) with a type of piecewise nonlinear activation function from the perspective of multiple $ \mathit O(t^{-\alpha}) $ stability. Some sufficient conditions are gained to assure the existence of $ 5^n $ equilibria for $ n $-neuron DFNNs with the proposed piecewise nonlinear activation functions. Additionally, the criteria ensure the existence of at least $ 3^n $ equilibria that are locally multiple $ \mathit O(t^{-\alpha}) $ stable. Furthermore, we apply these results to a more generic situation, revealing that DFNNs can attain $ (2k+1)^n $ equilibria, and among them, $ (k+1)^n $ equilibria are locally $ \mathit O(t^{-\alpha}) $ stable. Here, the parameter $ k $ is highly dependent on the sinusoidal function frequency in the expanded activation functions. Such DFNNs are well-suited to synthesize high-capacity associative memories; the design process is given via singular value decomposition. Ultimately, four illustrative examples, including applying neurodynamic associative memory to the explaining-lesson skills assessment of normal students, are supplied to validate the efficacy of the results.

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