Neural Network Method for Solving Time Fractional Diffusion Equations

In this paper, we propose a neural network method to solve time-fractional diffusion equations with Dirichlet boundary conditions by using a combination of machine learning techniques and Method of Lines. We first used the Method of Lines to discretize the equation in the space domain while keeping the time domain continuous, and represent the solution of the diffusion equation using a neural network. Then we used Gauss–Jacobi quadrature to approximate the fractional derivative in the time domain, thereby obtaining the loss function for the neural network. We used TensorFlow to carry out the gradient descent process to train this neural network. We conducted numerical tests in 1D and 2D cases and compared the results with the exact solutions. The numerical tests showed that this method is effective and easy to manipulate for many time-fractional diffusion problems.