Fibonacci Neural Network Approach for Numerical Solutions of Fractional Order Differential Equations

TL;DR

This paper introduces Fibonacci Neural Networks with Fibonacci polynomial activation functions to solve fractional order differential equations, demonstrating improved accuracy over existing methods through empirical testing.

Contribution

The paper presents a novel neural network architecture utilizing Fibonacci polynomials as activation functions for solving differential equations.

Findings

FNN achieves higher accuracy than traditional methods.

The approach effectively solves fractional order differential equations.

Empirical results validate the method's superiority.

Abstract

In this paper, the authors propose the utilization of Fibonacci Neural Networks (FNN) for solving arbitrary order differential equations. The FNN architecture comprises input, middle, and output layers, with various degrees of Fibonacci polynomials serving as activation functions in the middle layer. The trial solution of the differential equation is treated as the output of the FNN, which involves adjustable parameters (weights). These weights are iteratively updated during the training of the Fibonacci neural network using backpropagation. The efficacy of the proposed method is evaluated by solving five differential problems with known exact solutions, allowing for an assessment of its accuracy. Comparative analyses are conducted against previously established techniques, demonstrating superior accuracy and efficacy in solving the addressed problems.