在潜在空间中对混乱系统的稳定性分析
Elise Özalp1, Luca Magri1,2,3
1Department of Aeronautics, Imperial College London, South Kensington Campus, London, SW7 2BX UK.
Nonlinear dynamics
|April 14, 2025
概括
本研究介绍了一种数据驱动的潜空间方法,使用卷积自编码器回声状态网络 (CAE-ESN) 来解决混乱的部分微分方程. 该方法准确地推断系统动态,并从观测数据中预测稳定性质.
科学领域:
- 计算物理 计算物理
- 应用数学 应用数学 应用数学
- 机器学习 机器学习
背景情况:
- 部分微分方程 (PDEs) 建模复杂的系统,经常呈现混乱的解决方案.
- 数据驱动的方法提供了一种新的方法来解决PDEs,通过推断压缩的潜空间中的动态.
- 传统的方法对混乱系统来说可能是计算密集的.
研究的目的:
- 为了证明隐性空间方法可以解决混乱的PDEs并预测系统稳定性.
- 将卷积自编码器回声状态网络 (CAE-ESN) 应用于混乱系统.
- 验证CAE-ESN推断利亚普诺夫指数和共变的利亚普诺夫向量 (CLV) 的能力.
主要方法:
- 使用卷积自编码器回声状态网络 (CAE-ESN) 进行数据压缩和时间动态推断.
- 将CAE-ESN应用于各种混乱政权中的混乱Kuramoto-Sivashinsky方程.
- 扩展CAE-ESN以分析流动力学,并将结果与没有雅可比式方法进行比较.
主要成果:
- 该CAE-ESN成功地确定了观测数据的低维潜空间表示.
- 准确推断不同吸引子的低维多元体内的利亚普诺夫指数和CLV.
- 该模型有效地保留了混乱系统吸引力的几何结构.
结论:
- 基于CAE-ESN的潜在空间方法可以作为混乱系统的有效减少顺序模型.
- 该方法准确地预测系统动态,并从数据中推断出关键的稳定性.
- CAE-ESN为分析复杂,混乱的物理系统提供了一个强大的工具.
相关概念视频
01:28Stability
67
The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
67
01:24Pole and System Stability
219
The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's...
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's...
219
01:23Linear time-invariant Systems
187
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...
187
01:27State Space Representation
151
The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
Consider an RLC circuit, a...
151
01:21Linear Approximation in Time Domain
56
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,...
56
01:23Stability of Equilibrium Configuration
414
Understanding the stability of equilibrium configurations is a fundamental part of mechanical engineering. In any system, there are three distinct types of equilibrium: stable, neutral, and unstable.
A stable equilibrium occurs when a system tends to return to its original position when given a small displacement, and the potential energy is at its minimum. An example of a stable equilibrium is when a cantilever beam is fixed at one end and a weight is attached to the other end. If the weight...
A stable equilibrium occurs when a system tends to return to its original position when given a small displacement, and the potential energy is at its minimum. An example of a stable equilibrium is when a cantilever beam is fixed at one end and a weight is attached to the other end. If the weight...
414