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.

翻译：分数阶微分方程（FDEs）是科学与工程中建模复杂系统的重要工具。它将微分与积分的传统概念推广至非整数阶，从而能够更精确地表征具有非局部性和记忆依赖行为的过程。这一特性在系统变量不立即响应变化、而是表现出对过去相互作用的强烈记忆的场合尤为有用。基于此，并受神经常微分方程（Neural ODEs）的启发，我们提出了神经FDE，这是一种新颖的深度神经网络架构，可将FDE适配于数据的动力学特性。本文全面概述了神经FDE中采用的数值方法及其架构。数值结果表明，尽管计算需求更高，神经FDE在建模具有记忆或依赖过去状态的系统时可能优于神经ODE，并能有效应用于学习更复杂的动力系统。