In order to accurately capture non-local properties and long-term memory effects, this study combines the tempered fractional-order operator with delayed neural networks to investigate its stability, leveraging the introduced decay term of the tempered fractional-order operator. Firstly, the discrete-time tempered fractional-order neural networks model (DTFNNs) is presented. Furthermore, in an effort to better understand the dynamic behavior of complex systems, solutions to discrete-time tempered fractional non-homogeneous equations are obtained. The stability conditions for systems are subsequently established, contributing novel insights to the field. To validate the robustness of these conditions, numerical experiments are conducted, underscoring the practical relevance of the proposed model.

为了准确捕获非局部属性和长期记忆效应，本研究将调和分数阶算子与延迟神经网络相结合，利用调和分数阶算子引入的衰减项来研究其稳定性。首先，提出了离散时间调和分数阶神经网络模型（DTFNN）。此外，为了更好地理解复杂系统的动态行为，获得了离散时间缓和分数非齐次方程的解。随后建立了系统的稳定性条件，为该领域提供了新颖的见解。为了验证这些条件的稳健性，进行了数值实验，强调了所提出模型的实际相关性。