Neural Network Method For Fractional Differential Equations

In recent years,fractional differential equations have been widely used,and most fractional nonlinear differential equations cannot obtain analytical solutions.However,in practical applications,analytical solutions are often needed for theoretical analysis.Therefore,it is important to develop analytical solution methods.With the development of machine learning,neural networks have been proven efficient to approximate arbitrary nonlinear functions and have certain advantages in solving fractional nonlinear equations.This thesis mainly studies neural network methods to solve two types of fractional differential equations and discuss their applications in parameter estimation and prediction.The specific content is:Firstly,this thesis proposes a neural network method for solving time fractional Fokker-Planck equation,constructing a trial solution that satisfies the initial boundary value conditions,and then transforming the original equation solving problem into a minimization problem.The Caputo derivative is approximated byL₁ numerical scheme.The Adam method is adopted for neural network training.Finally,Analytical approximate solutions are obtained and compared with the exact solution to demonstrate the effectiveness of the method.Secondly,a multi-layer neural network method for solving fractional nonlinear differential equations is proposed.Parallel computing is used to search for the optimal structure of neural network.Analytical approximate solution are obtained.Comparison with other commonly methods shows that the method has higher accuracy.Finally,this thesis applies neural network method to deep learning based on fractional Logistic equations.With the experimental data,the gradient descent method is used to determine parameters such as fractional order and growth rate.The proposed neural network method is applied to prediction,and the prediction results are in good agreement with the real values.