基于神经网络的病态线性方程组求解

提出了一种基于神经网络的病态线性方程组求解方法。将病态线性方程组的一般系数矩阵转化为对称正定矩阵，然后将此方程组的求解转化为一个无约束优化问题。以此优化问题的目标函数作为神经网络的能量函数，利用最速下降原理构造神经网络的动力学方程，并证明该神经网络系统的稳定性。从而把原病态线性方程组的求解问题转化为一个等价的神经网络优化问题。最后通过两个算例的数值仿真求解以及与其他求解方法的比较，验证了该方法的可行性与有效性。

Neural network method in solving ill-conditioned linear system of equations

This paper presents a novel method based on neural network to solve ill-conditioned linear system of equations. First the general coefficient matrix is changed into a symmetric positive one. Then the solving of these equations is changed into a nonconstraint optimization problem. The target function of the optimization problem is served as energy function of neural network and the dynamic equation of neural network is constructed with the most rapid falling principle. And the proposed neural network is proved to be stable. The solving problem for the original group of ill-conditioned linear equations is transformed into an equivalent one which is to solve the neural network optimization. The method is proved to be reliable and effective by the theoretical analysis and computer simulation results of two examples.