Work

Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative

Year: 2024

Type: article

Abstract: Neural Fractional Differential Equations (Neural FDE) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an initial value problem. During the learning process, both the order of the derivative and the parameters of the neural network must be optimised. In this work, we investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network component. Based on our findings, we perform a numerical analysis to examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Results show that the neural network on the right-hand side of the Neural FDE struggles to adjust its parameters to fit the FDE to the data dynamics for any given order of the fractional derivative. Consequently, Neural FDEs do not require a unique α value, instead, they can use a wide range of α values to fit data. This flexibility is beneficial when fitting to given data is required, and the underlying physics is not known.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract8090529

Authors: Cecília Coelho, M. Fernanda P. Costa, Luís L. Ferrás

Institutions: Centro de Matemática, University of Minho, Universidade do Porto, Centro de Estudos de Fenómenos de Transporte

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,069

Funders: Ministério da Ciência, Tecnologia e Ensino Superior, Fundação para a Ciência e a Tecnologia