Zhang Yinyan, Li Shuai, Weng Jian, Liao Bolin
IEEE Trans Neural Netw Learn Syst. 2024 Jan;35(1):559-569. doi: 10.1109/TNNLS.2022.3175899. Epub 2024 Jan 4.
As a type of recurrent neural networks (RNNs) modeled as dynamic systems, the gradient neural network (GNN) is recognized as an effective method for static matrix inversion with exponential convergence. However, when it comes to time-varying matrix inversion, most of the traditional GNNs can only track the corresponding time-varying solution with a residual error, and the performance becomes worse when there are noises. Currently, zeroing neural networks (ZNNs) take a dominant role in time-varying matrix inversion, but ZNN models are more complex than GNN models, require knowing the explicit formula of the time-derivative of the matrix, and intrinsically cannot avoid the inversion operation in its realization in digital computers. In this article, we propose a unified GNN model for handling both static matrix inversion and time-varying matrix inversion with finite-time convergence and a simpler structure. Our theoretical analysis shows that, under mild conditions, the proposed model bears finite-time convergence for time-varying matrix inversion, regardless of the existence of bounded noises. Simulation comparisons with existing GNN models and ZNN models dedicated to time-varying matrix inversion demonstrate the advantages of the proposed GNN model in terms of convergence speed and robustness to noises.
作为一种被建模为动态系统的递归神经网络（RNN），梯度神经网络（GNN）被认为是一种用于静态矩阵求逆且具有指数收敛性的有效方法。然而，在时变矩阵求逆方面，大多数传统的GNN只能以残余误差跟踪相应的时变解，并且在存在噪声时性能会变差。目前，归零神经网络（ZNN）在时变矩阵求逆中占据主导地位，但ZNN模型比GNN模型更复杂，需要知道矩阵时间导数的显式公式，并且在数字计算机中实现时本质上无法避免求逆运算。在本文中，我们提出了一种统一的GNN模型，用于处理静态矩阵求逆和时变矩阵求逆，具有有限时间收敛性且结构更简单。我们的理论分析表明，在温和条件下，所提出的模型对于时变矩阵求逆具有有限时间收敛性，无论是否存在有界噪声。与现有的专门用于时变矩阵求逆的GNN模型和ZNN模型的仿真比较证明了所提出的GNN模型在收敛速度和抗噪声鲁棒性方面的优势。
