高维非线性动力系统降维理论综述  

Dimension Reduction Theory Review of High-dimensional Nonlinear Dynamical Systems

作　　者：桑瑞涓 龚坚 路宽 靳玉林 张康宇 王衡 Sang Ruijuan;Gong Jian;Lu Kuan;Jin Yulin;Zhang Kangyu;Wang Heng(School of Mechanics and Civil Engineering,Northwestern Polytechnical University,Xi'an 710000,China;Chinese People's Liberation Army 91911 Unit,Sanya 572000,China;School of Mechanical Engineering,Southwest Jiaotong University,Chengdu 611756,China)

机构地区：[1]西北工业大学力学与土木建筑学院,西安710000 [2]中国人民解放军91911部队,三亚572000 [3]西南交通大学机械工程学院,成都611756

出　　处：《动力学与控制学报》2024年第9期1-15,共15页Journal of Dynamics and Control

基　　金：国家自然科学基金资助项目(U2241243,12072263);KGJ国防技术基础国家重点资助项目(JSZL2022213A001);中央高校基本科研业务费(HYGJZN20232)。

摘　　要：工程领域中的结构和机构具有高维、非线性及强耦合等特性,导致其动态行为十分复杂.在相关研究领域中,降维方法对高维复杂非线性的动力学系统研究具有重要意义.它可以降低数据的复杂性,克服动力学系统的维数灾难,提高计算效率;也可以将高维数据的特征进行压缩和重构,提取出其核心特征,更好地揭示数据的内在规律和本质特征;还可以帮助简化模型,降低模型的复杂性,提高模型的稳定性和可解释性.近年来,降维方法体系逐渐发展完善,很多学者利用降维方法实现了高维复杂系统理论研究.基于此,针对非线性高维系统的降维理论进行了综述.重点介绍了基于中心流形理论的降维方法,Lyapunov-Schmidt方法,本征正交分解方法(Proper Orthogonal Decomposition)和非线性Galerkin方法等降维方法的基本思想、应用现状及各自的优缺点.此外,还简要介绍了实际问题中其他降维方法的应用.最后,针对现有降维方法存在的问题,提出了可能的改进方案和未来研究方向的展望.Structures and mechanisms in the engineering field possess characteristics such as high dimensions,nonlinearity,and strong coupling,leading to complex dynamic behaviors.In the related research field,the dimensionality reduction method is of great significance for the study of high-dimensional complex nonlinear dynamical systems.These methods can reduce the complexity of data,overcome the"curse of dimensionality"in dynamical systems,and improve computational efficiency.They can also compress and reconstruct the characteristics of high-dimensional data,extracting core characteristics to better reveal its inherent laws and features.Furthermore,they can simplify models,reduce model complexity,and improve model stability and interpretability.In recent years,the dimension reduction method system has gradually developed and improved,and many scholars have utilized them to achieve theoretical research on high-dimensional complex systems.Based on this,this paper summarizes the dimension reduction theory for nonlinear high-dimensional systems.It focuses on introducing the basic ideas,application status,and advantages and disadvantages of dimension reduction methods such as Central Manifold Theorem dimension reduction method,Lyapunov-Schmidt method,Proper Orthogonal Decomposition method(POD),and nonlinear Galerkin method.Additionally,it briefly introduces the application of other dimension reduction methods in practical problems.Finally,aiming at the problems existing in current dimension reduction methods,it proposes possible improvement plans and prospects for future research directions.

关 键 词：动力学系统 降维方法 中心流形 L-S方法 POD方法 GALERKIN方法 

分 类 号：O322[理学—一般力学与力学基础]