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

Physics of Fluids, vol. 33

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.

Authors 4

1. Luis Espath corresponding Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University 1 , Gebäude-1953 1.OG, Pontdriesch 14-16, 52062 Aachen, Germany

   Department of Mathematics, RWTH Aachen University, Gebäude-1953 1.OG, Pontdriesch 14-16, 52062 Aachen, Germany

2. Dmitry I. Kabanov Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University 1 , Gebäude-1953 1.OG, Pontdriesch 14-16, 52062 Aachen, Germany

   Department of Mathematics, RWTH Aachen University, Gebäude-1953 1.OG, Pontdriesch 14-16, 52062 Aachen, Germany

3. Jonas Kiessling

   KTH Royal Institute of Technology

   Affiliation as printed

   2H-Ai AB, Box 5216, 102 45 Stockholm, Sweden

   Institutionen för Matematik, Kungl. Tekniska Högskolan 3 , 100 44 Stockholm, Sweden

   H-Ai AB, Box 5216, 102 45 Stockholm, Sweden

4. Raul F. Tempone Aachen Department of Mathematics

   King Abdullah University of Science and Technology · RWTH Aachen University

   Affiliation as printed

   Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University 4 , Germany

   Department of Mathematics, RWTH Aachen University 1 , Gebäude-1953 1.OG, Pontdriesch 14-16, 52062 Aachen, Germany

   King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE) 5 , Thuwal 23955-6900, Saudi Arabia

   Department of Mathematics, RWTH Aachen University, Gebäude-1953 1.OG, Pontdriesch 14-16, 52062 Aachen, Germany

   King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), Thuwal 23955-6900, Saudi Arabia