PMNN:Physical Model-driven Neural Network for solving time-fractional
  differential equations
Zhiying Ma Jie Hou Wenhao Zhu Yaxin Peng Ying Li
In this paper, an innovative Physical Model-driven Neural Network (PMNN)
method is proposed to solve time-fractional differential equations. It
establishes a temporal iteration scheme based on physical model-driven neural
networks which effectively combines deep neural networks (DNNs) with
interpolation approximation of fractional derivatives. Specifically, once the
fractional differential operator is discretized, DNNs are employed as a bridge
to integrate interpolation approximation techniques with differential
equations. On the basis of this integration, we construct a neural-based
iteration scheme. Subsequently, by training DNNs to learn this temporal
iteration scheme, approximate solutions to the differential equations can be
obtained. The proposed method aims to preserve the intrinsic physical
information within the equations as far as possible. It fully utilizes the
powerful fitting capability of neural networks while maintaining the efficiency
of the difference schemes for fractional differential equations. Moreover, we
validate the efficiency and accuracy of PMNN through several numerical
experiments.