PMNN:Physical Model-driven Neural Network for solving time-fractional differential equations

TL;DR

This paper introduces PMNN, a neural network approach that combines physical modeling and interpolation techniques to efficiently solve time-fractional differential equations while preserving physical information.

Contribution

It proposes a novel neural network framework that integrates physical models with interpolation for solving fractional differential equations.

Findings

PMNN effectively solves time-fractional differential equations.

The method maintains physical information within the solutions.

Numerical experiments demonstrate high accuracy and efficiency.

Abstract

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.