严格证明Hopfield神经网络的稳定性

研究如下的Hopfield神经网络Cidxi(t)dt=−1Rxi(t)+∑j=1nTijgj(xj(t))+Ii, (i=1,2,⋯ ,n)(1)C_{i} \frac{\mathrm{d} x_{i}(t)}{\mathrm{d} t}=-\frac{1}{R} x_{i}(t)+\sum_{j=1}^{n} T_{i j} g_{j}\left(x_{j}(t)\right)+I_{i},

老实人小李

1865人浏览 · 2020-10-14 18:04:56

老实人小李 · 2020-10-14 18:04:56 发布

研究如下的Hopfield神经网络
$C_{i} \frac{\mathrm{d} x_{i}(t)}{\mathrm{d} t}=-\frac{1}{R} x_{i}(t)+\sum_{j=1}^{n} T_{i j} g_{j}\left(x_{j}(t)\right)+I_{i},\;(i=1,2, \cdots, n) \tag{1}$

其中 $T=[T_{ij}]_{n×n}$ 是网络连接权系数矩阵， $C_i>0,R_i>0$ 及 $I_i$ 为常数， $gi(⋅)g_i(\cdot)$ 为可微的严格单调上升的有界函数，且 $lim⁡l→+∞gi(s)=±1\lim\limits_{l\rightarrow +\infty}g_i(s)=\pm1$

我们称 $x_1^*,x_2^*,...,x_n^*)$ 为网络(1)的平衡态，若 $x_1^*,x_2^*,...,x_n^*)$ 满足：
$-\frac{1}{R} x_{i}^*(t)+\sum_{j=1}^{n} T_{i j} g_{j}\left(x_{j}^*(t)\right)+I_{i}=0,\;(i=1,2, \cdots, n)$

证明

定理2. 若连接权系数矩阵T是对称的，则网络(1)是稳定的.
证明. 我们先证明网络的一切解有界.由式(1)有
$x_{i}(t)=x_{i}(0) \mathrm{e}^{-\frac{1}{C_{i} R_{i}}}+\frac{1}{C_{i}} \int_{0}^{t} \mathrm{e}^{-\frac{1}{C_{i} R_{i}}(t-s)}\left(\sum_{j=1}^{n} T_{i j} g_{j}\left(x_{j}(s)\right)+I_{i}\right) \mathrm{d} s$

于是，对于一切 $t≥0t\ge0$ 有
$\begin{aligned} \left|x_{i}(t)\right| & \leqslant\left|x_{i}(0)\right| \mathrm{e}^{-\frac{1}{C_{i} R_{i}}}+\frac{1}{C_{i}} \int_{0}^{t} \mathrm{e}^{-\frac{1}{C_{i} R_{i}}(t-s)}\left|\sum_{j=1}^{n} T_{i j} g_{j}\left(x_{j}(s)\right)+I_{i}\right| \mathrm{d} s \\ & \leqslant\left|x_{i}(0)\right| \mathrm{e}^{-\frac{1}{C_{i} R_{i}} }+R_{i}\left(\sum_{j=1}^{n}\left|T_{i j}\right|+I_{i}\right) \\ & \leqslant\left|x_{i}(0)\right|+R_{i}\left(\sum_{j=1}^{n}\left|T_{i j}\right|+I_{i}\right),(i=1,2, \cdots, n) \end{aligned}$

构造Hopfield能量函数
$E(x)=-\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} T_{i j} v_{i} v_{j}-\sum_{i=1}^{n} I_{i} v_{i}+\sum_{i=1}^{n} \frac{1}{R_{i}} \int_{0}^{v_{i}} g_{i}^{-1}(\theta) \mathrm{d} \theta$

其中 $v_i=g_i(x_i)$ . 计算函数E沿着网络(1)的轨线的导数
$\begin{aligned} \frac{\mathrm{d} E\left(x(t)\right)}{\mathrm{d} t} &\left.\right|_{(1)}=-\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} T_{i j}\left(\frac{\mathrm{d} v_{i}(t)}{\mathrm{d} t} v_{j}(t)+v_{i}(t) \frac{\mathrm{d} v_{j}(t)}{\mathrm{d} t}\right) \\ &-\sum_{i=1}^{n}\left(I_{i}+\frac{1}{R_{i}} x_{i}(t)\right) \frac{\mathrm{d} v_{i}(t)}{\mathrm{d} t} \\ =&-\sum_{i=1}^{n} \frac{\mathrm{d} v_{i}(t)}{\mathrm{d} t}\left(-\frac{1}{R_{i}} x_{i}(t)+\sum_{j=1}^{n} T_{ij} g_{j}\left(x_{j}(t)\right)+I_{i}\right) \\ =&-\sum_{i=1}^{n} C_{i} \frac{\mathrm{d} g_{i}\left(x_{i}(t)\right)}{\mathrm{d} x_{i}(t)}\left(\frac{\mathrm{d} x_{i}(t)}{\mathrm{d} t}\right)^{2} \\ \leqslant & 0 \end{aligned}$