Koopman analysis of Burgers equation
Koopman analysis of Burgers equation
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DOI：
10.1103/physrevfluids.3.071901
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发表时间：
2017-12
影响因子：
2.7
通讯作者：
Jacob Page;R. Kerswell
中科院分区：
文献类型：
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作者：
Jacob Page;R. Kerswell
动态模式分解（DMD）的出现，作为一种实用的方法来尝试一个Koopman模式分解的非线性PDE识别不变集和缓慢衰减的瞬态结构埋在PDE动力学呈现令人兴奋的前景。然而，DMD与Koopman分析之间有许多微妙之处，目前还不清楚Koopman分析对复杂系统（如Navier-Stokes方程）的现实程度。以此为动机，我们在这里提出了一个完整的Koopman分解的速度场的Burgers方程导出明确的表达式的Koopman模式和本征函数-这是第一次这样做的非线性偏微分方程。分解强调了这样一个事实，即不同的可观测量可能需要不同的Koopman特征函数子集来表达它们，并给出了一个很好的例子：（i）Koopman模式是线性相关的，因此在不知道Koopman特征函数的情况下，不能后验地拟合流的快照;（ii）Koopman特征值是高度退化的，这意味着计算的Koopman模式变得依赖于初始条件。作为说明的方式，我们讨论了不同初始条件下的Koopman展开的形式，并评估DMD在运行模拟中提取衰减的非线性相干结构的能力。
The emergence of Dynamic Mode Decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear PDE presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in the PDE dynamics. However, there are many subtleties in connecting DMD to Koopman analysis and it remains unclear how realistic Koopman analysis is for complex systems such as the Navier-Stokes equations. With this as motivation, we present here a full Koopman decomposition for the velocity field in Burgers equation by deriving explicit expressions for the Koopman modes and eigenfunctions - the first time this has been done for a nonlinear PDE. The decomposition highlights the fact that different observables can require different subsets of Koopman eigenfunctions to express them and presents a nice example where: (i) the Koopman modes are linearly dependent and so cannot be fit a posteriori to snapshots of the flow without knowledge of the Koopman eigenfunctions; and (ii) the Koopman eigenvalues are highly degenerate which means that computed Koopman modes become initial-condition dependent. As way of illustration, we discuss the form of the Koopman expansion with various initial conditions and assess the capability of DMD to extract the decaying nonlinear coherent structures in run-down simulations.