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.

本文提出一种创新的物理模型驱动神经网络（PMNN）方法，用于求解分数阶微分方程。该方法通过将深度神经网络与分数导数的插值逼近技术相结合，建立了一个时间迭代框架，并通过训练神经网络学习这个迭代框架，从而获得微分方程的近似解。通过数值实验验证了 PMNN 方法的效率和准确性。