Continuous and Discrete GZN (Gradient-Zhang Neuronet) Handling UTVNES (Underdetermined Temporally-Variant [TV] Nonlinear Equation [NE] Systems)

Addressing the underdetermined temporally-variant nonlinear equation systems (e.g., the complicated fluid equation systems encountered by aircraft in unknown environments) represents a significant theme within science and engineering. This paper introduces gradient-Zhang neuronet (GZN), a novel one that marries the rapid convergence of gradient neuronet (GN) with the high precision of Zhang neuronet (ZN), and links it with Zhang time discretization (ZTD) formulas to achieve the discretization of the GZN algorithms. We thus reduce one pseudo-inverse multiplication per update, compared with high-precision ZN. The results showcase the effectiveness of five GZN algorithms, i.e., the discrete Euler GZN (Euler-GZN) algorithm, the discrete four-instant GZN (FI-GZN) algorithm, the discrete eight-instant GZN (EI-GZN) algorithm, the discrete ten-instant GZN (TI-GZN) algorithm, and the discrete eleven-instant GZN (ElI-GZN) algorithm in solving the underdetermined temporally-variant nonlinear equation systems with different parameters.