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能量函数在神经网络的研究中有着非常重要的作用．人们普遍认为：只要能量函数沿着网络的解是下降的，能量函数的导数为零的点是网络的平衡态，能量函数有下界，则网络是稳定的且网络的平衡态为能量函数的极小点．文中取反例说明上述条件不能保证网络的稳定性，并取例说明即使网络稳定也不能保证网络的平衡态为能量函数的极小点．证明了在网络具有上述条件的能量函数的情况下网络稳定的充分必要条件是网络的解有界．讨论了网络的平衡态与能量函数的极小点的关系．进一步完善了能量函数的方法．作为应用，严格证明了Ｈｏｐｆｉｅｌｄ神经网络的收敛性，并讨论了一个能用于计算实对称矩阵最大特征值对应的全部特征向量的神经网络．
Energy function plays a very important role in the research of neural network. It is generally accepted that as long as the energy function decreases along the network, the point at which the derivative of the energy function is zero is the equilibrium state of the network, the energy function has a lower bound, the network is stable, and the equilibrium state of the network is the energy function smaller. In this paper, an example is given to illustrate that the above conditions can not guarantee the stability of the network, and an example is given to show that even if the network is stable, the equilibrium state of the network can not be guaranteed to be the minimum point of the energy function. It is proved that the necessary and sufficient condition of the network stability under the condition that the network has the energy function of the above conditions is the solution boundedness of the network. The relationship between the equilibrium state of the network and the minimum point of the energy function is discussed. A further refinement of the energy function approach. As an application, the convergence of Hopfield neural networks is strictly proved, and a neural network that can be used to compute all the eigenvectors corresponding to the largest eigenvalues of real symmetric matrices is discussed.