Statistical learning for fluid flows: Sparse Fourier divergence-free approximations
KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
Stochastic Numerics Research Group
Preprint Posting Date2021-07-15
Online Publication Date2021-09-27
Print Publication Date2021-09
Embargo End Date2022-09-27
Permanent link to this recordhttp://hdl.handle.net/10754/672243
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AbstractWe 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.
CitationEspath, L., Kabanov, D., Kiessling, J., & Tempone, R. (2021). Statistical learning for fluid flows: Sparse Fourier divergence-free approximations. Physics of Fluids, 33(9), 097108. doi:10.1063/5.0064862
SponsorsThis 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).
JournalPhysics of Fluids