In this article, we investigate both forward and backward problems for coupled systems of time-fractional diffusion equations, encompassing scenarios of strong coupling. For the forward problem, we establish the well-posedness of the system, leveraging the eigensystem of the corresponding elliptic system as the foundation. When considering the backward problem, specifically the determination of initial values through final time observations, we demonstrate a Lipschitz stability estimate, which is consist with the stability observed in the case of a single equation. To numerically address this backward problem, we refer to the explicit formulation of Tikhonov regularization to devise a multi-channel neural network architecture. This innovative architecture offers a versatile approach, exhibiting its efficacy in multidimensional settings through numerical examples and its robustness in handling initial values that have not been trained.

翻译：本文研究了时间分数阶扩散方程耦合系统的正反问题，涵盖强耦合情形。针对正问题，我们以相应椭圆系统的特征系统为基础，建立了该系统的适定性。在考虑反问题——即通过终值观测确定初始值时——我们证明了一个与单方程情形一致的Lipschitz稳定性估计。为数值求解该反问题，我们依据Tikhonov正则化的显式表述设计了一种多通道神经网络架构。这一创新架构提供了通用方法，通过数值算例展示了其在多维场景中的有效性，并证明了其对未经训练的初始值具有处理鲁棒性。