deepFDEnet: A Novel Neural Network Architecture for Solving Fractional Differential Equations

TL;DR

This paper introduces deepFDEnet, a neural network architecture that effectively solves various fractional differential equations using Gaussian integration and L1 discretization, demonstrating high accuracy across multiple equation types.

Contribution

The paper presents a novel neural network architecture specifically designed for solving fractional differential equations with high precision, incorporating Gaussian integration and L1 discretization techniques.

Findings

Successfully solves fractional ODEs, integrodifferential, and PDEs

Achieves high accuracy in diverse fractional equations

Demonstrates versatility and effectiveness of the proposed method

Abstract

The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the proposed design. In each equation, a deep neural network is used to approximate the unknown function. Three forms of fractional differential equations have been examined to highlight the method's versatility: a fractional ordinary differential equation, a fractional order integrodifferential equation, and a fractional order partial differential equation. The results show that the proposed architecture solves different forms of fractional differential equations with excellent precision.