对于偶联的非线性系统的转换不变库普曼运算子的近似值
Thomas Hochrainer1, Gurudas Kar1
1Institute of Strength of Materials, Graz University of Technology, Kopernikusgasse 24/I, 8010 Graz, Austria.
Chaos (Woodbury, N.Y.)
|August 14, 2024
概括
我们开发了翻译不变扩展动态模式分解 (tieDMD),以改进物理系统的数据驱动建模. tieDMD通过尊重转换对称性来提高非线性动态的重建精度.
科学领域:
- 物理 物理学 物理
- 应用数学 应用数学 应用数学
- 数据科学数据科学数据科学
背景情况:
- 许多物理系统具有翻译不变性,在不同的空间位置上,物理定律保持不变.
- 使用数据驱动方法准确建模非线性动态系统需要结合这些物理对称性.
- 现有的动态模式分解 (DMD) 技术可能无法充分保持转换不变性.
研究的目的:
- 为了引入一种新的方法,翻译不变扩展动态模式分解 (tieDMD),用于分析周期域上的合非线性系统.
- 为了确保库普曼运算符的数据驱动近似值遵守转换对称性.
- 为了评估tieDMD与现有的DMD变体的性能.
主要方法:
- 在里埃空间中利用库普曼运算符的块对角结构来实现翻译不变性.
- 将tieDMD的变体应用于来自相扩散,Burgers,Korteweg-de Vries和FitzHugh-Nagumo方程的数据集.
- 将tieDMD的数据重建能力与传统的线性和非线性DMD方法进行比较.
主要成果:
- tieDMD在各种非线性偏微分方程中展示了卓越的数据重建能力.
- 该方法有效地利用里埃空间,在库普曼运算子近似中强制执行转换对称.
- 在所有测试的数据集中都观察到重建准确度的持续改善.
结论:
- tieDMD提供了一种基于物理的方法来建模具有转化不变性的非线性动态系统.
- 该方法在数据驱动模型的准确性和可靠性方面取得了重大进展.
- tieDMD是分析表现出对称性的复杂物理系统的宝贵工具.
相关概念视频
01:26Linear Approximation in Frequency Domain
88
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
88
01:21Linear Approximation in Time Domain
75
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
75
01:23Linear time-invariant Systems
233
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
233
01:26Transmission-Line Differential Equations
251
Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
251
01:16Vector Transformation in Rotating Coordinate Systems
1.5K
Consider a vector rotating about an axis with an angular velocity, such that its tip sweeps a circular path.
1.5K
01:02¹H NMR: Interpreting Distorted and Overlapping Signals
1.0K
Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are...
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are...
1.0K