Solving And Identifying Subdiffusion Problem Via Deep Learning

In recent years,deep learning methods have been widely used among various fields,including computer vision,natural language processing and database system.Simultaneously some works about solving and identifying differential equations via deep learning have some important progress.This thesis will solve and identify subdiffusion problem via deep learning.First,we propose two methods for solving subdiffusion problem: data-driven method and physical learning method.Data-driven method rewrites numerical discretization of the subdiffusion into the Sylester equation and then stack all the columns of the solution of this equation to generate a large column vector matrix and replace this form with the simple form of AX = B,using the simple form to gain the known labels,then uses the neural network of deep learning to approximate these labels.The training process of this method is stable,and it can not only deal with linear problems,but also obtain better numerical results.Physical learning method only inputs data of the spatial item and outputs all data,which makes the input and output of the neural network simpler,and makes all output data into the optimization problem with special discrete numerical discretization.through iterating and updating,finally all data becomes the optimal solution.These two methods can effectively overcome error accumulation and be compatible with other discretizations,as well as also be extended to solve other differential equations.Second,we provide a format for identifying subdiffusion problem and use two special networks for training,then it can be found that this format can effectively respond to different noises and robustly identify the source function,which provides a research basis for effectively identifying fractional models with special structures.