Adaptive Regularization Strategy for Solving Ill-conditioned Matrices via Zeroing Neural Network

Abstract

In the fields of scientific computing and engineering control, zero-form neural networks (ZNNs) are widely used for solving time-varying matrices due to their excellent continuous tracking capabilities. However, when the input matrix exhibits highly ill-conditioned characteristics, the basic ZNN and its variants combined with traditional static regularization face an insurmountable contradiction between “solution accuracy” and “numerical robustness.” This paper systematically reviews the theoretical bottlenecks of ZNNs in solving ill-conditioned matrices and delves into the evolution and core mechanisms of adaptive regularization strategies. By comparing dynamic strategies based on open-loop time-varying decay mechanisms and closed-loop state error feedback mechanisms, the study points out that the adaptive regulation law driven by real-time residuals can provide strong damping to truncate error divergence in extremely singular transients and automatically fall back when the system returns to a benign state to eliminate steady-state truncation errors caused by excessive smoothing, thus exhibiting excellent global disturbance rejection and tracking performance under complex time-varying conditions. Meanwhile, this paper objectively evaluates the limitations of existing adaptive strategies in terms of hyperparameter empirical tuning, increased single-step computational overhead, and theoretical proof of convergence at predetermined time, and looks forward to the future development trend of constructing rigorous analytical design criteria and exploring lightweight evolutionary architectures.