  We perform a sparse identification of nonlinear dynamics (SINDy) for
low-dimensionalized complex flow phenomena. We first apply the SINDy with two
regression methods, the thresholded least square algorithm (TLSA) and the
adaptive Lasso (Alasso) which show reasonable ability with a wide range of
sparsity constant in our preliminary tests, to a two-dimensional single
cylinder wake at $Re_D=100$, its transient process, and a wake of two-parallel
cylinders, as examples of high-dimensional fluid data. To handle these high
dimensional data with SINDy whose library matrix is suitable for
low-dimensional variable combinations, a convolutional neural network-based
autoencoder (CNN-AE) is utilized. The CNN-AE is employed to map a
high-dimensional dynamics into a low-dimensional latent space. The SINDy then
seeks a governing equation of the mapped low-dimensional latent vector.
Temporal evolution of high-dimensional dynamics can be provided by combining
the predicted latent vector by SINDy with the CNN decoder which can remap the
low-dimensional latent vector to the original dimension. The SINDy can provide
a stable solution as the governing equation of the latent dynamics and the
CNN-SINDy based modeling can reproduce high-dimensional flow fields
successfully, although more terms are required to represent the transient flow
and the two-parallel cylinder wake than the periodic shedding. A nine-equation
turbulent shear flow model is finally considered to examine the applicability
of SINDy to turbulence, although without using CNN-AE. The present results
suggest that the proposed scheme with an appropriate parameter choice enables
us to analyze high-dimensional nonlinear dynamics with interpretable
low-dimensional manifolds.
