Forward and backward problems for coupled subdiffusion systems

TL;DR

This paper studies forward and backward problems for coupled time-fractional diffusion systems, establishing well-posedness, stability, and proposing a neural network-based numerical method for initial value reconstruction.

Contribution

It introduces a novel neural network approach with Tikhonov regularization for solving the backward problem in coupled subdiffusion systems, demonstrating effectiveness in multidimensional cases.

Findings

Well-posedness of the forward coupled system established.

Lipschitz stability for the backward problem demonstrated.

Neural network method effectively reconstructs initial values in numerical tests.

Abstract

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 consistent 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.