Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces
Koopman decomposition is a nonlinear 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 a 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. Furthermore, 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-airwake flow. more »
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Publication Date:
NSF-PAR ID:
10284085
Journal Name:
Journal of Fluid Mechanics
Volume:
917
ISSN:
0022-1120
1. Using the newly introduced occupation kernels,'' the present manuscript develops an approach to dynamic mode decomposition (DMD) that treats continuous time dynamics, without discretization, through the Liouville operator. The technical and theoretical differences between Koopman based DMD for discrete time systems and Liouville based DMD for continuous time systems are highlighted, which includes an examination of these operators over several reproducing kernel Hilbert spaces.