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 approximation based on a discrete L2 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 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.
Bibliographical noteKAUST Repository Item: Exported on 2021-10-08
Acknowledged KAUST grant number(s): OSR, URF/1/2281-01-01, URF/1/2584-01-01
Acknowledgements: This work was partially supported by the KAUST Office of Sponsored Research (OSR) under Award Nos. URF/1/2281-01-01 and URF/1/2584-01-01 in the KAUST Competitive Research Grants Program Round 8, the Alexander von Humboldt Foundation, and Coordination for the Improvement of Higher Education Personnel (CAPES).
ASJC Scopus subject areas
- Condensed Matter Physics