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Computer Science > Machine Learning

arXiv:2403.02737 (cs)
[Submitted on 5 Mar 2024 (v1), last revised 25 Jul 2024 (this version, v2)]

Title:Neural Fractional Differential Equations

Authors:C. Coelho, M. Fernanda P. Costa, L.L. Ferrás
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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.
Subjects: Machine Learning (cs.LG); Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
MSC classes: G.1, G.1.10, G.4, I.5.1
Cite as: arXiv:2403.02737 [cs.LG]
  (or arXiv:2403.02737v2 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.2403.02737

Submission history

From: Cecília Coelho [view email]
[v1] Tue, 5 Mar 2024 07:45:29 UTC (629 KB)
[v2] Thu, 25 Jul 2024 09:18:24 UTC (1,290 KB)
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