Learning primal-dual approach for space-dependent diffusion coefficient identification in fractional diffusion equations

In this paper, we introduce a deep learning neural network approach to precisely identify space-dependent diffusion coefficients within a time-fractional diffusion equation by leveraging a learned primal-dual algorithm. To improve the accuracy of coefficient identification, our approach utilizes supplementary measurement data obtained at the final time. First, a deep neural network is employed to estimate the diffusion coefficient within the direct problem formulation, which is a crucial initial step in effectively characterizing the behavior of the diffusion process. Subsequently, we establish the existence and uniqueness of a weak solution for the corresponding direct problem under a homogeneous Dirichlet boundary condition. By parameterizing the solution using a deep neural network, we reinforce the foundation of our approach. This enables us to reformulate the inverse problem as a non-smooth optimal control problem, and we successfully demonstrate the existence of a solution for this control problem. To this end, we employ three types of primal-dual methods to efficiently address and numerically solve the complex inverse problem. The first primal-dual algorithm used is the classic Chambolle and Pock algorithm. In the second, we replace the dual's proximal operator with a trained convolutional neural network. In the third primal-dual algorithm, both proximal operators are replaced with trained models. We conduct extensive numerical experiments in a two-dimensional domain, and the results highlight the remarkable effectiveness and stability of our proposed method. By demonstrating its applicability and reliability, we underscore the potential of our approach for solving real-world problems related to the identification of diffusion coefficients.