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.

本文提出了神经分数阶微分方程（Neural FDE），一个新颖的深度神经网络结构，通过调整微分方程来适应数据的动态，从而在具有记忆或依赖于过去状态的系统建模中可能优于神经常微分方程（Neural ODE），并且可以有效地应用于学习更复杂的动力系统。