We present adapted Zhang Neural Networks (AZNN) in which the parameter settings for the exponential decay constant $\eta$ and the length of the start-up phase of basic ZNN are adapted to the problem at hand. Specifically we study experiments with AZNN for time-varying square matrix factorizations as a product of time-varying symmetric matrices and for the time-varying matrix square roots problem. Differing from generally used small $\eta$ values and minimal start-up length phases in ZNN, we adapt the basic ZNN method to work with large or even gigantic $\eta$ settings and arbitrary length start-ups using Euler's low accuracy finite difference formula. These adaptations improve the speed of AZNN's convergence and lower its solution error bounds for our chosen problems significantly to near machine constant or even lower levels. Parameter-varying AZNN also allows us to find full rank symmetrizers of static matrices reliably, for example for the Kahan and Frank matrices and for matrices with highly ill-conditioned eigenvalues and complicated Jordan structures of dimensions from $n = 2$ on up. This helps in cases where full rank static matrix symmetrizers have never been successfully computed before.

翻译：我们提出自适应张量神经网络（AZNN），该方法将基本ZNN中指数衰减常数$\eta$和启动阶段长度的参数设置适配于具体问题。我们重点研究了AZNN在时变方阵分解为时变对称矩阵乘积问题以及时变矩阵平方根问题中的实验。与ZNN中通常采用的小型$\eta$值和最小启动时长不同，我们通过欧拉低精度有限差分公式，将基本ZNN方法适配为允许使用大型甚至巨型$\eta$设置及任意长度启动阶段。这些改进显著提升了AZNN的收敛速度，并将所选问题的解误差界降低至接近机器常数甚至更优水平。参数自适应AZNN还能可靠地计算静态矩阵的满秩对称化器，例如Kahan矩阵、Frank矩阵，以及具有严重病态特征值和复杂若尔当结构（维度从$n=2$起）的矩阵。这有助于解决以往从未成功计算过的静态矩阵满秩对称化器问题。