动态系统的数据驱动的线性化
George Haller1, Bálint Kaszás1
1Institute for Mechanical Systems, ETH Zürich, Leonhardstrasse 21, 8092 Zurich, Switzerland.
Nonlinear dynamics
|September 2, 2024
概括
动态模式分解 (DMD) 通过其适用性的新理由得到了改进. 数据驱动线性化 (DDL) 为分析动态系统提供了一种更强大的方法,其性能优于现有的技术.
科学领域:
- 动态系统理论 动态系统理论
- 数据驱动建模数据驱动建模
- 应用数学 应用数学 应用数学
背景情况:
- 动态模式分解 (DMD) 和其变体广泛用于从数据中对动态系统进行线性建模.
- 现有的DMD解释,特别是那些基于库普曼运算符的解释,具有局限性,并基于限制性假设.
- 需要澄清DMD适用的条件,并开发更强大的方法.
研究的目的:
- 为动态模式分解 (DMD) 提供严格的理由,作为主导系统动态的本地领先阶段模型.
- 开发一个新的,更高阶的线性化算法,数据驱动线性化 (DDL),用于分析可观测的动态.
- 与DMD和扩展DMD (EDMD) 相比,为了证明DDL的优越性能.
主要方法:
- 开发了一个理论框架,根据一般可观测的概率为1的条件来证明DMD的合理性.
- 在吸引缓慢光谱子多元体 (SSM) 中构建了主导动态的线性化转换.
- 介绍了数据驱动线性化 (DDL) 算法,这是一个系统的,高阶线性化技术.
主要成果:
- 已确定的条件,在这些条件下,DMD提供了系统动态的有效的本地领先顺序近似值.
- 新的DDL算法系统地线性化了缓慢的SSM内可观察到的动态.
- 在数值和实验数据集上,DDL表现优于DMD和EDMD.
结论:
- 该研究为DMD提供了坚实的理论基础,并介绍了一种更先进的方法,DDL.
- DDL为动态系统分析提供了更准确,更系统的线性化方法.
- 这些发现促进了数据驱动方法的适用性和可靠性,以了解复杂的系统.
相关概念视频
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