发现信息Koopman不变子空间的稀疏提升算法
Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces
摘要：
Koopman分解是特征分解的一种非线性推广，在时空动力学分析中得到越来越多的应用。众所周知的技术，如动态模式分解（DMD）及其线性变体提供近似的Koopman算子，并已被广泛应用于许多流体动力学问题。尽管被赋予了一个更丰富的字典的非线性观测，DMD的非线性变体，如扩展/内核动态模式分解（EDMD/KDMD）很少应用于大规模的问题，主要是由于难以辨别的Koopman不变子空间从成千上万的Koopman特征模。为了解决这个问题，我们提出了一个框架的基础上多任务特征学习，以提取最丰富的Koopman不变子空间，通过删除冗余和虚假的Koopman三元组。特别是，我们开发了一个修剪程序，惩罚偏离线性演化。这些算法可以被看作是稀疏促进EDMD/KDMD的扩展。此外，我们将KDMD扩展到连续时间设置，并从非凸优化的角度显示本算法，稀疏促进DMD和经验标准之间的关系。从简单的动力系统到不同雷诺数下的二维圆柱尾流和三维湍流舰船空气尾流的算例都证明了该算法的有效性。后两个问题的设计，使得非常强的非线性瞬态存在，因此需要一个准确的近似Koopman运营商。基本的物理机制进行了分析，重点是表征瞬态动力学。结果与现有的理论阐述和数值逼近。
英文摘要：
Koopman decomposition is a non-linear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on multi-task feature learning to extract the most informative Koopman invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity promoting extensions of EDMD/KDMD. Further, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD, and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship air-wake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analyzed, with an emphasis on characterizing transient dynamics. The results are compared to existing theoretical expositions and numerical approximations.
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