Solving Coupled Time Fractional Partial Differential Equations Based On Physical Information Neural Network

In recent decades,machine learning algorithms have made great and stable progress in the fields of fluid mechanics,big data fitting analysis,and digital image processing.Its thinking method can be traced back to 1936,the mathematical expression formed by the linear and nonlinear structure of the neural network in the machine learning algorithm has a basis function slice structure similar to the finite element method,and the relevant researchers regard the single-hidden layer neural network as a slice structure similar to the finite element method,and then use the deep learning model of this structure to solve fractional partial differential equations.In this paper,a time fractional coupled partial differential model solver is proposed based on fractional physical information neural network.The solver adopts a second-order backward difference format discrete time fractional operator in the time direction,and does not process the spatial direction,which reduces the storage requirements of the computer by discretizing the fractional operator and calling the automatic differential algorithm on classical calculus,thereby improving the computational efficiency.This paper considers the twodimensional coupled time fractional KGZ model and the more complex generalized secondorder magnetofluid time fractional coupled model,and the influence of factors including the number of discrete points(series),the weight of the penalty term,the size of the learning rate,the network structure(the number of neurons and the number of hidden layers),the index of the fractional and other factors on the convergence accuracy of the method are considered,and the experiment shows that the error accuracy can reach e-2,and the structure of the paper is:In Chapter 1,the research background and basic knowledge of fractional calculus and neural networks are introduced.In Chapter 2,the time fractional coupled partial differential model solver is proposed to solve the two-dimensional coupled fractional KGZ equation with the help of fractional physical information neural network algorithm,and the influence of different parameters and network structures on the convergence accuracy of the solver is discussed.Chapter 3 borrows the solver proposed in Chapter 2 to consider a more complex generalized second-order magnetofluid fractional coupled model,and the convergence accuracy under different parameters is also discussed.The results show that the fractional physical information neural network is effective in solving the two-dimensional coupled time fractional KGZ equation and the generalized second-order magnetic fluid time fractional decoupled model.