  In this work, we further develop the Physics-informed Spectral Learning
(PiSL) by Espath et al. \cite{Esp21} based on a discrete $L^2$ projection to
solve the discrete Hodge--Helmholtz decomposition from sparse data. Within this
physics-informed 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. Moreover, our PiSL computational
framework enjoys spectral (exponential) convergence. We regularize the
minimization problems with the seminorm of the fractional Sobolev space in a
Tikhonov fashion. In the Fourier setting, the divergence- and curl-free
constraints become a finite set of linear algebraic equations. The proposed
computational framework combines supervised and unsupervised learning
techniques in that we use data concomitantly with the projection onto
divergence- and curl-free spaces. We assess the capabilities of our method in
various numerical examples including the `Storm of the Century' with satellite
data from 1993.
