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.
Highlights
We reconstruct the velocity field of incompressible flows given a finite set of measurements
We introduce the Sparse Fourier divergence-free (SFD-F) approximation based on a discrete L2 projection
Any acceleration, velocity, and pressure fields arising from the incompressible Navier–Stokes equations automatically satisfy the mechanical version of the second law of thermodynamics, whether this property holds for the discretized versions of these equations within a statistical learning framework is unclear
Summary
Machine learning strategies for fluid flows have been extensively developed in recent years. Tempone [4] uses Fourier basis functions coupled with the divergence-free constraint to approximate wind velocity fields. Our method can accurately and adaptively reconstruct divergence-free fields, one would ideally wish to incorporate the Navier–Stokes equations as additional constraints to be satisfied. These equations require the acceleration and pressure measurements or at least one would need to devise an additional model for these quantities. Any acceleration, velocity, and pressure fields arising from the incompressible Navier–Stokes equations automatically satisfy the mechanical version of the second law of thermodynamics, whether this property holds for the discretized versions of these equations within a statistical learning framework is unclear.
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