Neural Fractional Differential Equations
This work addresses modeling challenges in science and engineering for systems with non-local and memory-dependent behaviors, representing an incremental advancement over Neural ODEs.
The authors tackled the problem of modeling complex systems with memory-dependent behaviors by proposing Neural Fractional Differential Equations (Neural FDEs), a novel deep neural network architecture that adapts fractional differential equations to data dynamics, and found that it may outperform Neural ODEs in such systems despite higher computational demands.
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.