IEEE Trans Neural Netw Learn Syst. 2022 Nov;33(11):6569-6583. doi: 10.1109/TNNLS.2021.3082560. Epub 2022 Oct 27.
This article presents theoretical results on the multistability of switched neural networks with Gaussian activation functions under state-dependent switching. It is shown herein that the number and location of the equilibrium points of the switched neural networks can be characterized by making use of the geometrical properties of Gaussian functions and local linearization based on the Brouwer fixed-point theorem. Four sets of sufficient conditions are derived to ascertain the existence of 753 equilibrium points, and 432 of them are locally stable, wherein p , p , and p are nonnegative integers satisfying 0 ≤ p+p+p ≤ n and n is the number of neurons. It implies that there exist up to 7 equilibria, and up to 4 of them are locally stable when p=n . It also implies that properly selecting p , p , and p can engender a desirable number of stable equilibria. Two numerical examples are elaborated to substantiate the theoretical results.
本文提出了具有高斯激活函数的切换神经网络在状态相关切换下的多稳定性的理论结果。结果表明，可以利用高斯函数的几何性质和基于布劳威尔不动点定理的局部线性化来确定切换神经网络平衡点的数量和位置。本文推导出了四组充分条件，以确定存在 753 个平衡点，其中 432 个平衡点是局部稳定的，其中 p 、 p 、 p 是非负整数，满足 0 ≤ p + p + p ≤ n ， n 是神经元的数量。这意味着当 p = n 时，存在多达 7 个平衡点，其中多达 4 个是局部稳定的。这也意味着适当选择 p 、 p 、 p 可以产生期望数量的稳定平衡点。通过两个数值例子来说明理论结果。
