基于一类新型激活函数的递归神经网络的多稳定性分析  

Multistability analysis of recurrent neural networks with a new type of activation functions

作　　者：闫维昕 刘洋 王震 YAN Weixin;LIU Yang;WANG Zhen(College of Mathematics and Systems Science,Shandong University of Science and Technology,Qingdao 266590,China)

机构地区：[1]山东科技大学数学与系统科学学院,山东青岛266590

出　　处：《山东科技大学学报（自然科学版）》2024年第2期90-101,共12页Journal of Shandong University of Science and Technology(Natural Science)

基　　金：国家自然科学基金项目(62173214);山东省自然科学基金项目(ZR2021MF003)。

摘　　要：本研究基于一类新型连续锯齿型激活函数研究了递归神经网络(Hopfield神经网络)的多稳定性。首先,通过区间剖分法、Brouwer不动点定理证明了基于该激活函数的n维神经网络模型至少具有7n个平衡点,并运用对角占优矩阵理论、局部稳定性判定定理等方法证明了其中4n个平衡点是局部指数稳定的,剩余的平衡点是不稳定的。其次,通过增加锯齿型激活函数的峰值点将激活函数推广到更一般的情况,得到了n维神经网络在含有k个峰值点的连续锯齿型激活函数中至少具有(2k+1)n个平衡点,其中(k+1)n个平衡点为局部稳定的。本研究设计的激活函数相较于现有的一些激活函数会产生更多的稳定平衡点,并且在增加峰值点的过程中不会增加神经网络的计算复杂度。最后,通过两个具体的数值算例验证了本研究结果的有效性。In this paper,the multistability of Hopfield neural network was studied based on a new type of activation function,the continuous sawtooth activation function.Firstly,the n-neural network model based on this activation function was proved to have at least 7 n equilibrium points through interval partition method and Brouwer’s fixed point theorem.By using the diagonally dominant matrix theory and local stability judgment theorem,4 n equilibrium points were proved to be locally exponentially stable while the remaining equilibrium points to be unstable.Secondly,the activation function was generalized to a more general situation by adding the peak points of the sawtooth activation function.The n-neural network was found to have at least(2k+1)n equilibrium points in the continuous sawtooth activation function with k peak points,of which(k+1)n equilibrium points are local stable.Compared with the existing activation functions,the activation function designed in this paper can generate more stable equilibrium points and will not increase the computational complexity of the neural network in the process of increasing the number of peak points.Finally,two numerical examples were presented to demonstrate the validity of the results.

关 键 词：HOPFIELD神经网络 多稳定性 平衡点 连续锯齿型激活函数 

分 类 号：N941.3[自然科学总论—系统科学] N941.7