A class of improved fractional physics informed neural networks

Physics informed neural networks (PINNs) are a type of deep learning framework which can use the knowledge of physical laws to facilitate the learning of various forward and inverse problems. PINNs have displayed significant efficiency in numerical computation, but there are still some limitations such as their inability in tackling differential equations with oscillations or singular perturbation. To alleviate this difficulty, a new deep learning framework called improved fractional physics informed neural networks (IFPINNs) is proposed in this paper. Compared to PINNs, the calculation in the weight propagation between layers is replaced with nonlinear functions rather than linear ones. This new architecture allows for a better usage of physical information of differential equations, thus leading to an improved fitting quality for the cases with high oscillations or singular perturbation. Under certain conditions, IFPINNs can be degenerated into PINNs. Moreover, this paper investigates the performance of IFPINNs in dealing with fractional differential equations with time-fractional orders ranging from 1 to 2. The numerical results demonstrate the proposed architecture can achieve a higher accuracy for various integral and fractional differential equations.