  We reconstruct the velocity field of incompressible flows given a finite set
of measurements. For the spatial approximation, we introduce the Sparse Fourier
divergence-free (SFdf) approximation based on a discrete $L^2$ projection.
Within this physics-informed type of statistical learning framework, we
adaptively build a sparse set of Fourier basis functions with corresponding
coefficients by solving a sequence of minimization problems where the set of
basis functions is augmented greedily at each optimization problem. We
regularize our minimization problems with the seminorm of the fractional
Sobolev space in a Tikhonov fashion. In the Fourier setting, the
incompressibility (divergence-free) constraint becomes a finite set of linear
algebraic equations. We couple our spatial approximation with the truncated
Singular Value Decomposition (SVD) of the flow measurements for temporal
compression. Our computational framework thus combines supervised and
unsupervised learning techniques. We assess the capabilities of our method in
various numerical examples arising in fluid mechanics.
