Viera-Martin E, Gómez-Aguilar J F, Solís-Pérez J E, Hernández-Pérez J A, Escobar-Jiménez R F
Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos Mexico.
CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos Mexico.
Eur Phys J Spec Top. 2022;231(10):2059-2095. doi: 10.1140/epjs/s11734-022-00455-3. Epub 2022 Feb 12.
In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.
在这项工作中，开展了一项关于使用分数阶微积分（FC）理论的人工神经网络（ANNs）的文献分析，以总结人工神经网络的主要特征和应用。人工神经网络是一种在多个科学和工程领域中使用的数学建模工具。分数阶微积分主要应用于人工神经网络，有三个不同目标，例如系统稳定、系统同步和参数训练，采用优化算法。分数阶微积分和一些控制策略已被成功用于实现人工神经网络的同步和稳定。为说明这一事实，本手稿总结了各研究工作中系统的架构、控制策略和使用的分数阶导数，还展示了所达成的目标。关于使用优化算法进行参数训练的问题，本手稿还介绍了系统类型、涉及的分数阶导数以及用于训练人工神经网络参数的优化算法。在文献中发现的大多数涉及人工神经网络和分数阶微积分的工作中，作者们专注于通过同步和稳定来控制系统。此外，本手稿详细回顾了人工神经网络与分数阶微积分在医学、密码学、图像处理、机器人等多个领域的最新应用。还纳入了诸如混沌分析、函数逼近、传热过程、周期性和耗散性等应用方面的工作。在论文快结尾时，向研究界推荐了一些关于涉及分数阶微积分的人工神经网络的未来研究主题。通过文献综述，我们得出结论，卡普托导数是用于解决人工神经网络问题时最常使用的导数，因为其初始值与整数阶微分方程具有相同形式。
