  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.
