学习非线性投影用于使用受约束自动编码器进行动态系统的减少顺序建模
Samuel E Otto1, Gregory R Macchio2, Clarence W Rowley2
1AI Institute in Dynamic Systems, University of Washington, Seattle, Washington 98195, USA.
Chaos (Woodbury, N.Y.)
|November 27, 2023
概括
本研究引入了使用受约束自编码器的新型非线性投影方法,以准确地模拟复杂系统中的短暂动态. 这些技术改善了用于实时控制和预测应用的减少顺序建模.
科学领域:
- 动态系统和控制理论.
- 机器学习用于科学建模
- 流体动力学 流体动力学
背景情况:
- 减少顺序建模 (ROM) 通过使用低维分组来近似非线性动态系统.
- 当前的ROM在过渡后的模式中表现出色,但由于快速的动态和异常的灵敏度,它们在过渡动态中扎.
- 对于实时控制和预测,精确的过渡动态建模至关重要.
研究的目的:
- 为非线性预测开发一个新的框架,准确地捕捉过渡动态.
- 解决现有ROM在建模复杂系统行为方面的局限性.
- 为了实现更好的实时控制和预测能力.
主要方法:
- 引入了受约束的自编码器神经网络,用于学习分组和投影纤维.
- 采用可逆激活函数和双直角重量矩阵来实现编码器解码器一致性.
- 开发了动态感知成本函数,以学习快速动态和非正常性的斜投射纤维.
- 为了分析案例研究,使用了三态流散模型.
主要成果:
- 证明了拟议的非线性投影框架能够捕捉短暂动态的能力.
- 展示了动态意识成本函数在处理非正常效应方面的有效性.
- 验证了流体动力学案例研究的方法,使用分析计算的慢变 manifold.
- 为计算效率高的ROM构造提出的技术,包括在Grassmann变频器上促进稀疏性.
结论:
- 开发的非线性投影框架有效地模拟了复杂系统中的短暂动态.
- 这种方法提高了用于实时控制和预测的减少顺序建模的适用性.
- 未来的工作可以将这些方法扩展到更广泛的科学应用的高维系统.
相关概念视频
01:21Linear Approximation in Time Domain
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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,...
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01:26Linear Approximation in Frequency Domain
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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....
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01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
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01:23Linear time-invariant Systems
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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...
262
01:27State Space Representation
210
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...
210
01:26Feedback control systems
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Feedback control systems are categorized in various ways based on their design, analysis, and signal types.
Linear feedback systems are theoretical models that simplify analysis and design. These systems operate under the principle that their output is directly proportional to their input within certain ranges. For instance, an amplifier in a control system behaves linearly as long as the input signal remains within a specific range. However, most physical systems exhibit inherent nonlinearity...
Linear feedback systems are theoretical models that simplify analysis and design. These systems operate under the principle that their output is directly proportional to their input within certain ranges. For instance, an amplifier in a control system behaves linearly as long as the input signal remains within a specific range. However, most physical systems exhibit inherent nonlinearity...
316