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.

翻译：本文提出了一种创新的物理模型驱动神经网络（PMNN）方法，用于求解时间分数阶微分方程。该方法基于物理模型驱动神经网络建立了时间迭代格式，有效地将深度神经网络（DNNs）与分数阶导数的插值近似相结合。具体而言，在离散化分数阶微分算子后，采用深度神经网络作为桥梁，将插值近似技术与微分方程进行整合。基于这一整合，我们构建了基于神经网络的迭代格式，并通过训练深度神经网络学习该时间迭代格式，从而获得微分方程的近似解。所提出的方法旨在尽可能保留方程内部的物理信息，它在充分利用神经网络强大拟合能力的同时，保持了分数阶微分方程差分格式的高效性。此外，我们通过多个数值实验验证了PMNN方法的有效性与准确性。