Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries.
Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries.
复制标题
使用学习库中的稀疏传感器位置减少动力系统的非线性模型。
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
J. Kutz
中科院分区：
文献类型：
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作者：
S. Sargsyan;S. Brunton;J. Kutz
我们展示了稀疏采样和降维的合成，在一定范围内的分岔参数的非线性动力系统的特征和模型。首先，我们使用经典的适当的正交分解，以暴露占主导地位的低秩相干结构，构建模态库。在这里，图书馆的非线性项也被构造，以利用离散经验插值方法和投影，允许近似的非线性项从一个稀疏的网格点。选定的网格点被证明是有效的传感和测量位置，用于表征非线性动力系统的基本动力学，稳定性和分叉。经验插值点和稀疏表示的使用有利于一个家庭的本地降阶模型为每个物理制度，而不是一个高阶的全球模型，这具有物理解释相干结构之间的能量转移的好处。所提倡的方法还允许在计算速度和存储器需求方面的数量级改进。为了说明该方法，离散插值点和非线性模态库用于稀疏表示，以分类和重建复杂的Ginzburg-Landau方程的动态分岔制度。它还表明，点测量的非线性比线性测量传感器噪声时，是更有效的。
We demonstrate the synthesis of sparse sampling and dimensionality reduction to characterize and model nonlinear dynamical systems over a range of bifurcation parameters. First, we construct modal libraries using the classical proper orthogonal decomposition in order to expose the dominant low-rank coherent structures. Here, libraries of the nonlinear terms are also constructed in order to take advantage of the discrete empirical interpolation method and projection that allows for the approximation of nonlinear terms from a sparse number of grid points. The selected grid points are shown to be effective sensing and measurement locations for characterizing the underlying dynamics, stability, and bifurcations of nonlinear dynamical systems. The use of empirical interpolation points and sparse representation facilitates a family of local reduced-order models for each physical regime, rather than a higher-order global model, which has the benefit of physical interpretability of energy transfer between coherent structures. The method advocated also allows for orders-of-magnitude improvement in computational speed and memory requirements. To illustrate the method, the discrete interpolation points and nonlinear modal libraries are used for sparse representation in order to classify and reconstruct the dynamic bifurcation regimes in the complex Ginzburg-Landau equation. It is also shown that point measurements of the nonlinearity are more effective than linear measurements when sensor noise is present.