In this paper, a new fractional physical information neural network with an adaptive learning rate (Adaptive-FPINN) is proposed for solving time fractional diffusion equations. Firstly, the time fractional derivative in Caputo sense is evaluated by two different time discretization schemes, first-order finite difference scheme (L1) and Piecewise Quadratic Interpolation (PQI) in the sense of the Hadamard finite-part integral. Then the adaptive learning rate residual network is used in the traditional physical information neural network, which greatly reduces the errors along the boundaries by automatically adjusting the weights of different loss terms, so as to obtain a higher precision solution. Besides, a triangular activation function acting as Fourier transform is applied in the first hidden layer of Adaptive-FPINN, which improve the accuracy of the neural network greatly. Finally, one-dimensional, two-dimensional and three-dimensional time-fractional diffusion equations have been solved using Adaptive-FPINN to illustrate the accuracy and efficiency. Besides, the Adaptive-FPINN can balance the gradients of different loss terms in the loss function, and the trigonometric activation function has a clear impact on reducing the training time.

Keywords: adaptive learning rate, time fractional diffusion equation, physical information neural network, activation function