  Data-driven modeling for nonlinear fluid flows using sparse convolution-based
mapping into a feature space where the dynamics are Markov linear is explored
in this article. The underlying principle of low-order models for fluid systems
is identifying convolutions to a feature space where the system evolution (a)
is simpler and efficient to model and (b) the predictions can be reconstructed
accurately through deconvolution. Such methods are useful when real-time models
from sensor data are needed for online decision making. The Markov linear
approximation is popular as it allows us to leverage the vast linear systems
machinery. Examples include the Koopman operator approximation techniques and
evolutionary kernel methods in machine learning. The success of these models in
approximating nonlinear dynamical systems is tied to the effectiveness of the
convolution map in accomplishing both (a) and (b) mentioned above. To assess
this, we perform in-depth study of two classes of sparse convolution operators:
(i) a pure data-driven POD-convolution that uses left singular vectors of the
data snapshots - a staple of Koopman approximation methods and (ii) a sparse
Gaussian Process (sGP) convolution that combines sparse sampling with a
Gaussian kernel embedding an implicit feature map to an inner product
reproducing kernel Hilbert space.
