Chen Shyan-Shiou
Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan.
IEEE Trans Neural Netw. 2011 Oct;22(10):1557-65. doi: 10.1109/TNN.2011.2163080. Epub 2011 Aug 12.
In this paper, we have three goals: the first is to delineate the advantages of a variably delayed system, the second is to find a more intuitive Lyapunov function for a delayed neural network, and the third is to design a delayed neural network for a quadratic cost function. For delayed neural networks, most researchers construct a Lyapunov function based on the linear matrix inequality (LMI) approach. However, that approach is not intuitive. We provide a alternative candidate Lyapunov function for a delayed neural network. On the other hand, if we are first given a quadratic cost function, we can construct a delayed neural network by suitably dividing the second-order term into two parts: a self-feedback connection weight and a delayed connection weight. To demonstrate the advantage of a variably delayed neural network, we propose a transiently chaotic neural network with variable delay and show numerically that the model should possess a better searching ability than Chen-Aihara's model, Wang's model, and Zhao's model. We discuss both the chaotic and the convergent phases. During the chaotic phase, we simply present bifurcation diagrams for a single neuron with a constant delay and with a variable delay. We show that the variably delayed model possesses the stochastic property and chaotic wandering. During the convergent phase, we not only provide a novel Lyapunov function for neural networks with a delay (the Lyapunov function is independent of the LMI approach) but also establish a correlation between the Lyapunov function for a delayed neural network and an objective function for the traveling salesman problem.
在本文中，我们有三个目标：第一个是阐述可变延迟系统的优势，第二个是为延迟神经网络找到一个更直观的李雅普诺夫函数，第三个是为二次代价函数设计一个延迟神经网络。对于延迟神经网络，大多数研究者基于线性矩阵不等式（LMI）方法构造李雅普诺夫函数。然而，那种方法并不直观。我们为延迟神经网络提供了一个备选的李雅普诺夫函数。另一方面，如果我们首先给定一个二次代价函数，我们可以通过适当地将二阶项分为两部分来构造一个延迟神经网络：一个自反馈连接权重和一个延迟连接权重。为了证明可变延迟神经网络的优势，我们提出了一个具有可变延迟的瞬态混沌神经网络，并通过数值模拟表明该模型应比陈 - 相原模型、王模型和赵模型具有更好的搜索能力。我们讨论了混沌阶段和收敛阶段。在混沌阶段，我们简单地给出了具有恒定延迟和可变延迟的单个神经元的分岔图。我们表明可变延迟模型具有随机特性和混沌游走。在收敛阶段，我们不仅为具有延迟的神经网络提供了一个新颖的李雅普诺夫函数（该李雅普诺夫函数独立于LMI方法），而且还建立了延迟神经网络的李雅普诺夫函数与旅行商问题的目标函数之间的关联。
