Statistical Learning for Fluid Flows: Sparse Fourier divergence-free approximations

TL;DR

This paper introduces a physics-informed statistical learning method using sparse Fourier divergence-free approximations to reconstruct incompressible flow velocity fields from measurements, combining spatial and temporal compression techniques.

Contribution

The paper presents a novel sparse Fourier divergence-free approximation framework that adaptively builds basis functions with regularization, coupling spatial and temporal compression for fluid flow reconstruction.

Findings

Effective reconstruction of flow fields demonstrated in numerical examples.

Combines supervised and unsupervised learning techniques.

Regularization improves approximation stability.

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 (SFdf) approximation based on a discrete $L^2$ 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 (SVD) 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.