Statistical learning for fluid flows: Sparse Fourier divergence-free approximations

Espath, Luis; Kabanov, Dmitry; Kiessling, Jonas; Tempone, Raúl

Abstract

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.

Citation

Espath, L., Kabanov, D., Kiessling, J., & Tempone, R. (2021). Statistical learning for fluid flows: Sparse Fourier divergence-free approximations. Physics of Fluids, 33(9), Article 097108. https://doi.org/10.1063/5.0064862

• Journal Article Type: Article
• Acceptance Date: Sep 4, 2021
• Online Publication Date: Sep 27, 2021
• Publication Date: 2021-09
• Deposit Date: Dec 6, 2022
• Publicly Available Date: Dec 8, 2022
• Journal: Physics of Fluids
• Print ISSN: 1070-6631
• Electronic ISSN: 1089-7666
• Publisher: American Institute of Physics
• Peer Reviewed: Peer Reviewed
• Volume: 33
• Issue: 9
• Article Number: 097108
• DOI: https://doi.org/10.1063/5.0064862
• Keywords: Condensed Matter Physics; Fluid Flow and Transfer Processes; Mechanics of Materials; Computational Mechanics; Mechanical Engineering
• Public URL: https://nottingham-repository.worktribe.com/output/7711318
• Publisher URL: https://aip.scitation.org/doi/10.1063/5.0064862
• Additional Information: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Luis Espath, Dmitry Kabanov, Jonas Kiessling, and Raúl Tempone , "Statistical learning for fluid flows: Sparse Fourier divergence-free approximations", Physics of Fluids 33, 097108 (2021) https://doi.org/10.1063/5.0064862 and may be found at https://aip.scitation.org/doi/10.1063/5.0064862