Neural Fractional Differential Equations

TL;DR

This paper introduces Neural FDEs, a neural network architecture based on fractional differential equations, capable of modeling complex systems with memory effects more effectively than Neural ODEs, despite higher computational costs.

Contribution

It presents the design and implementation of Neural FDEs, extending Neural ODEs to fractional calculus for better modeling of memory-dependent systems.

Findings

Neural FDEs outperform Neural ODEs in systems with memory effects.

Neural FDEs effectively learn complex dynamical systems.

Neural FDEs are more computationally demanding.

Abstract

Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise representation of processes characterised by non-local and memory-dependent behaviours. This property is useful in systems where variables do not respond to changes instantaneously, but instead exhibit a strong memory of past interactions. Having this in mind, and drawing inspiration from Neural Ordinary Differential Equations (Neural ODEs), we propose the Neural FDE, a novel deep neural network architecture that adjusts a FDE to the dynamics of data. This work provides a comprehensive overview of the numerical method employed in Neural FDEs and the Neural FDE architecture. The numerical outcomes suggest that, despite being more computationally demanding, the Neural FDE may outperform the Neural ODE in modelling systems with memory or dependencies on past states, and it can effectively be applied to learn more intricate dynamical systems.