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）是科学和工程中建模复杂系统的关键工具。它们将传统的微分和积分概念扩展至非整数阶，能够更精确地表征具有非局部性和记忆依赖特性的过程。这一特性在变量对变化不产生瞬时响应、而是表现出对过去交互强记忆性的系统中尤为重要。基于此思想，并借鉴神经常微分方程（Neural ODEs）的启发，我们提出神经分数阶微分方程（Neural FDE）——一种新颖的深度神经网络架构，可将分数阶微分方程适配至数据动态。本文全面阐述了Neural FDE中采用的数值方法及其架构。数值结果表明，尽管计算成本更高，但Neural FDE在建模具有记忆性或依赖过去状态的系统时可能优于Neural ODE，并能有效应用于学习更复杂的动力系统。