Version 1
: Received: 30 July 2024 / Approved: 31 July 2024 / Online: 31 July 2024 (15:17:27 CEST)
How to cite:
Coelho, C.; Costa, M. F. P.; Ferrás, L. L. Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative. Preprints2024, 2024072572. https://doi.org/10.20944/preprints202407.2572.v1
Coelho, C.; Costa, M. F. P.; Ferrás, L. L. Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative. Preprints 2024, 2024072572. https://doi.org/10.20944/preprints202407.2572.v1
Coelho, C.; Costa, M. F. P.; Ferrás, L. L. Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative. Preprints2024, 2024072572. https://doi.org/10.20944/preprints202407.2572.v1
APA Style
Coelho, C., Costa, M. F. P., & Ferrás, L. L. (2024). Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative. Preprints. https://doi.org/10.20944/preprints202407.2572.v1
Chicago/Turabian Style
Coelho, C., M. Fernanda P. Costa and Luís L. Ferrás. 2024 "Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative" Preprints. https://doi.org/10.20944/preprints202407.2572.v1
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.
Computer Science and Mathematics, Artificial Intelligence and Machine Learning
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
