Adaptive fractional physical information neural network based on PQI scheme for solving time-fractional partial differential equations

In this paper, an accurate fractional physical information neural network with an adaptive learning rate (adaptive-fPINN-PQI) was first proposed for solving fractional partial differential equations. First, piecewise quadratic interpolation (PQI) in the sense of the Hadamard finite-part integral was introduced in the neural network to discretize the time-fractional derivative in the Caputo sense. Second, the adaptive learning rate residual network was constructed to keep the network from being stuck in the locally optimal solution, which automatically adjusts the weights of different loss terms, significantly balancing their gradients. Additionally, different from the traditional physical information neural networks, this neural network employs a new composite activation function based on the principle of Fourier transform instead of a single activation function, which significantly enhances the network's accuracy. Finally, numerous time-fractional diffusion and time-fractional phase-field equations were solved using the proposed adaptive-fPINN-PQI to demonstrate its high precision and efficiency.