神经网络分岔动力学综述

肖敏; 陆云翔; 虞文武; 郑卫新

doi:10.16383/j.aas.c230789

神经网络分岔动力学综述

doi: 10.16383/j.aas.c230789

1.
南京邮电大学自动化学院、人工智能学院南京 210023 中国
2.
东南大学数学学院南京 211189 中国
3.
西悉尼大学计算机、数据与数学学院悉尼 NSW 2751 澳大利亚

基金项目: 国家自然科学基金 (62073172, 62233004, 62073076), 江苏省自然科学基金 (BK20221329), 江苏省应用数学科学研究中心 (BK20233002), 江苏省研究生科研与实践创新计划 (KYCX23_1060)资助

详细信息

作者简介:
肖敏：南京邮电大学自动化学院、人工智能学院教授. 主要研究方向为非线性控制理论, 复杂网络, 神经网络, 信息网络融合系统, 反常扩散系统. 本文通信作者. E-mail: candymanxm2003@aliyun.com

陆云翔：南京邮电大学自动化学院、人工智能学院博士研究生. 主要研究方向为神经网络动力学, 非线性控制理论, 反应扩散系统. E-mail: miraclemanlyx@163.com

虞文武：东南大学数学学院教授. 2010年获得香港城市大学电子工程系博士学位. 主要研究方向为复杂网络系统协同分析, 控制与优化. E-mail: wwyu@seu.edu.cn

郑卫新：澳大利亚西悉尼大学杰出教授, IEEE Fellow. 主要研究方向为系统辨识, 网络化控制, 多智能体系统, 神经网络, 信号处理. E-mail: w.zheng@westernsydney.edu.au

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出版历程
- 收稿日期: 2023-12-22
- 录用日期: 2024-05-30
- 网络出版日期: 2024-06-30

Overview of Bifurcation Dynamics in Neural Networks

1.
College of Automation and College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
2.
School of Mathematics, Southeast University, Nanjing 211189, China
3.
School of Computer, Data and Mathematical Sciences, Western Sydney University, Sydney NSW 2751, Australia

Funds: Supported by National Natural Science Foundation of China (62073172, 62233004, 62073076), Natural Science Foundation of Jiangsu Province of China (BK20221329), Jiangsu Provincial Scientific Research Center of Applied Mathematics (BK20233002), and Practice Innovation Program of Jiangsu Province (KYCX23_1060)

More Information

Author Bio:
XIAO Min　Professor at the College of Automation and College of Artificial intelligence, Nanjing University of Posts and Telecommunications. His research interest covers nonlinear control theory, complex networks, neural networks, cyber-physical systems, and anomalous diffusion systems. Corresponding author of this paper

LU Yun-Xiang　Ph.D. Candidate at the College of Automation and College of Artificial intelligence, Nanjing University of Posts and Telecommunications. His research interest covers neural networks dynamics, nonlinear control theory, and reaction-diffusion systems

YU Wen-Wu　Professor at the School of Mathematics, Southeast University. He received his Ph.D. degree from the Department of Electrical Engineering, City University of Hong Kong in 2010. His research interest covers collaborative analysis, control and optimization of complex networked systems

ZHENG Wei-Xin　Distinguished professor at Western Sydney University, Australia. IEEE Fellow. His research interest covers system identification, networked control systems, multi-agent systems, neural networks, and signal processing

摘要

摘要: 自1982年著名的Hopfield神经网络问世以来, 神经网络的分岔动力学受到了学术界的广泛关注. 本文回顾了四类经典神经网络的数学模型和它们在各个领域的应用. 接着, 综述了近三十年来关于整数阶神经网络、分数阶神经网络、超数域神经网络以及反应扩散神经网络分岔动力学的相关研究成果. 分析了诸多组合因素, 包括节点规模、耦合情形、拓扑结构、系统阶次、复值、四元数、八元数、扩散、时滞、随机性、脉冲、忆阻、激活函数等对神经网络分岔动力学的影响, 并展示了神经网络在多个领域的广泛应用. 最后, 在人工智能、大数据、深度学习等新技术的冲击下, 对神经网络分岔动力学所面临的挑战以及未来的研究方向进行了总结和展望.
- 神经网络 /
- 时滞 /
- 非线性动力学 /
- 稳定性 /
- 分岔 /
- 周期性 /
- 混沌
Abstract: Since the introduction of the renowned Hopfield neural network in 1982, the bifurcation dynamics of neural networks has garnered significant academic attention. This paper provides an overview of the mathematical models of four types of classical neural networks and their applications in various fields. Subsequently, we summarize the research results on the bifurcation dynamics of integer-order neural networks, fractional-order neural networks, supernumerary-domain neural networks, and reaction-diffusion neural networks in the past three decades. We analyze the effects of various combinations of factors, including node size, coupling, topology, system order, complex value, quaternion, octonion, diffusion, time delay, stochasticity, impulse, memristor, and activation function, on the bifurcation dynamics of neural networks, and demonstrate the wide applications of neural networks in various fields. Finally, in light of the impact of new technologies such as artificial intelligence, big data, and deep learning, we summarize the challenges and potential research directions concerning neural network bifurcation dynamics.
- Neural networks /
- time delay /
- nonlinear dynamics /
- stability /
- bifurcation /
- periodicity /
- chaos

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图 1 Hopfield神经网络电路图

Fig. 1 Circuit of Hopfied neural network

下载: 全尺寸图片幻灯片

图 2 细胞神经网络中单个细胞电路图

Fig. 2 Circuit of a single cell in cellular neural network

下载: 全尺寸图片幻灯片

图 3 双向联想记忆神经网络拓扑图

Fig. 3 Topology of bidirectional associative memory neural network

下载: 全尺寸图片幻灯片

图 4 高维耦合下环型拓扑神经网络的发展历程

Fig. 4 Development of ring topology neural networks under high-dimensional coupling

下载: 全尺寸图片幻灯片

图 5 高维耦合下混合型拓扑神经网络的发展历程

Fig. 5 Development of hybrid topology neural networks under high-dimensional coupling

下载: 全尺寸图片幻灯片

表 1 神经网络模型分类

Table 1 Classification for neural network models

神经网络类型		代表性文献	应用领域	特点
IONNs	少节点全连接	[20, 34−35]	嵌入式系统实时系统边缘计算低功耗设备	结构简单, 计算速度较快适用于低功耗、资源受限的实时系统灵活性不足、精度不高、适用范围受限
	少节点非全连接	[41, 45]
	多节点Ring	[70−71, 77]
	多节点Star	[26, 78−79]
	多节点Hybrid	[82−83]
FONNs	少节点耦合	[98−99, 101]	信号处理动态系统建模时间序列预测	具有记忆和遗传特性、适用于非平稳信号处理适用于建模复杂的非线性系统和时间序列数据计算复杂度较高、训练过程比较困难
FONNs	高维耦合	[75, 81, 104]	信号处理动态系统建模时间序列预测	具有记忆和遗传特性、适用于非平稳信号处理适用于建模复杂的非线性系统和时间序列数据计算复杂度较高、训练过程比较困难
SDNNs	CVNNs	[108, 122, 124]	信号处理通信系统量子计算	能够更好地处理复数数据、提高数据表示能力训练复杂度较高、需要特殊的数学处理技巧
	QVNNs	[125, 127]
	OVNNs	[134−135]
RDNNs	少节点耦合	[150, 156−157]	模式生成自组织系统模拟	能够模拟物理世界中各类反应扩散过程计算复杂度较高、训练过程困难
RDNNs	高维耦合	[158−160]	模式生成自组织系统模拟	能够模拟物理世界中各类反应扩散过程计算复杂度较高、训练过程困难

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