Discovery of Subdiffusion Problem with Noisy Data via Deep Learning

Data-driven discovery of partial differential equations (PDEs) from observed data in machine learning has been developed by embedding the discovery problem. Recently, the discovery of traditional ODEs dynamics using linear multistep methods in deep learning have been discussed in [Racheal and Du, SIAM J. Numer. Anal. 59 (2021) 429-455; Du et al. arXiv:2103.1148 ]. We extend this framework to the data-driven discovery of the time-fractional PDEs, which can effectively characterize the ubiquitous power-law phenomena. In this paper, identifying source function of subdiffusion with noisy data using L_1 approximation in deep neural network is presented. In particular, two types of networks for improving the generalization of the subdiffusion problem are designed with noisy data. The numerical experiments are given to illustrate the availability with high noise levels using deep learning. To the best of our knowledge, this is the first topic on the discovery of subdiffusion in deep learning with noisy data.