Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, PR China.
Neural Netw. 2010 Mar;23(2):189-200. doi: 10.1016/j.neunet.2009.11.010. Epub 2009 Dec 2.
In this paper, we investigate the neural networks with a class of nondecreasing piecewise linear activation functions with 2r corner points. It is proposed that the n-neuron dynamical systems can have and only have (2r+1)(n) equilibria under some conditions, of which (r+1)(n) are locally exponentially stable and others are unstable. Furthermore, the attraction basins of these stationary equilibria are estimated. In the case of n=2, the precise attraction basin of each stable equilibrium point can be figured out, and their boundaries are composed of the stable manifolds of unstable equilibrium points. Simulations are also provided to illustrate the effectiveness of our results.
在本文中，我们研究了一类具有 2r 个角点的非递减分段线性激活函数的神经网络。提出在一定条件下，n 神经元动力系统可以且仅可以有(2r+1)(n)个平衡点，其中(r+1)(n)个平衡点是局部指数稳定的，而其他的是不稳定的。此外，还对这些稳定平衡点的吸引域进行了估计。在 n=2 的情况下，可以计算出每个稳定平衡点的精确吸引域，其边界由不稳定平衡点的稳定流形组成。还提供了模拟结果来说明我们结果的有效性。
