This paper wants to increase our understanding and computational know-how for time--varying matrix problems and Zhang Neural Networks (ZNNs). These neural networks were invented for time or single parameter--varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. Zhang Neural Network methods have become a backbone for solving discretized sensor driven time--varying matrix problems in real-time, in theory and in on--chip applications for robots, in control theory and other engineering applications in China. They have become the method of choice for many time--varying matrix problems that benefit from or require efficient, accurate and predictive real--time computations. A typical discretized Zhang Neural Network algorithm needs seven distinct steps in its initial set-up. The construction of discretized Zhang Neural Network algorithms starts from a model equation with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi--step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes of discretized Zhang Neural Network algorithms for time varying matrix problems typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized Zhang Neural network based algorithms highly competitive with ordinary differential equation initial value analytic continuation methods for function given data that are designed to work adaptively. .

翻译：本文旨在加深对时变矩阵问题及张神经网络（ZNNs）的理解与计算实践。这类神经网络于2001年左右在中国为时变或单参数可变矩阵问题而发明，其几乎所有进展均源自且多数仍源于诞生地。张神经网络方法已成为实时求解离散化传感器驱动时变矩阵问题的核心手段，在理论上及中国机器人芯片应用、控制理论及其他工程应用中均有体现。对于众多需要或受益于高效、精准及预测性实时计算的时变矩阵问题，该方法已成为首选方案。典型的离散化张神经网络算法在初始设置中需经历七个明确步骤。离散化ZNN算法的构建始于一个模型方程及其关联误差方程，并规定误差函数呈指数衰减。随后，将误差函数微分方程与收敛的前瞻有限差分公式结合，生成一种独特的多步求解器，可依据当前及历史状态与解数据可靠地预测系统未来状态。处理时变矩阵问题的离散化ZNN算法Matlab代码通常包含每时间步求解一个线性方程组并对已有数据进行一次递归操作。这使得基于离散化ZNN的算法在面对函数给定数据且需自适应运行的常微分方程初值解析延拓方法时，具有高度竞争力。