Qiu Binbin, Zhang Yunong, Yang Zhi
IEEE Trans Neural Netw Learn Syst. 2018 Nov;29(11):5767-5776. doi: 10.1109/TNNLS.2018.2805810. Epub 2018 Mar 12.
In this brief, a new one-step-ahead numerical differentiation rule called six-instant -cube finite difference (6I CFD) formula is proposed for the first-order derivative approximation with higher precision than existing finite difference formulas (i.e., Euler and Taylor types). Subsequently, by exploiting the proposed 6I CFD formula to discretize the continuous-time Zhang neural network model, two new-type discrete-time ZNN (DTZNN) models, namely, new-type DTZNNK and DTZNNU models, are designed and generalized to compute the least-squares solution of dynamic linear equation system with time-varying rank-deficient coefficient in real time, which is quite different from the existing ZNN-related studies on solving continuous-time and discrete-time (dynamic or static) linear equation systems in the context of full-rank coefficients. Specifically, the corresponding dynamic normal equation system, of which the solution exactly corresponds to the least-squares solution of dynamic linear equation system, is elegantly introduced to solve such a rank-deficient least-squares problem efficiently and accurately. Theoretical analyses show that the maximal steady-state residual errors of the two new-type DTZNN models have an pattern, where denotes the sampling gap. Comparative numerical experimental results further substantiate the superior computational performance of the new-type DTZNN models to solve the rank-deficient least-squares problem of dynamic linear equation systems.
在本简报中，提出了一种新的一步超前数值微分规则，称为六瞬态立方有限差分（6I CFD）公式，用于一阶导数逼近，其精度高于现有有限差分公式（即欧拉型和泰勒型）。随后，通过利用所提出的6I CFD公式对连续时间张神经网络模型进行离散化，设计并推广了两种新型离散时间ZNN（DTZNN）模型，即新型DTZNNK和DTZNNU模型，以实时计算具有时变秩亏系数的动态线性方程组的最小二乘解，这与现有的关于在满秩系数情况下求解连续时间和离散时间（动态或静态）线性方程组的ZNN相关研究有很大不同。具体而言，巧妙地引入了相应的动态正规方程组，其解恰好对应于动态线性方程组的最小二乘解，以高效准确地解决此类秩亏最小二乘问题。理论分析表明，两种新型DTZNN模型的最大稳态残余误差具有一种模式，其中表示采样间隔。比较数值实验结果进一步证实了新型DTZNN模型在解决动态线性方程组秩亏最小二乘问题方面的优越计算性能。
