Data augmentation for the POD formulation of the parametric laminar incompressible Navier-Stokes equations

TL;DR

The paper tackles the computational burden of constructing accurate reduced-order models for parametric steady incompressible Navier–Stokes equations by introducing physics-informed data augmentation. It develops pairing-based snapshot augmentation and two augmentation strategies: a solenoidal geometric average to preserve mass conservation, and an Oseen-based enhancement to enforce both mass and momentum conservation. Results across 2D jet, 2D lid-driven cavity, and 3D microswimmer tests show that Oseen-enhanced augmentation substantially improves POD–RB accuracy with modest additional cost, outperforming standard POD–RB, especially for intermediate parameter values. The approach offers a practical path to accurate, efficient ROMs for multi-query parametric flows and provides a basis for extending to transient and turbulent regimes.

Abstract

A posteriori reduced-order models (ROM), e.g. based on proper orthogonal decomposition (POD), are essential to affordably tackle realistic parametric problems. They rely on a trustful training set, that is a family of full-order solutions (snapshots) representative of all possible outcomes of the parametric problem. Having such a rich collection of snapshots is not, in many cases, computationally viable. A strategy for data augmentation, designed for parametric laminar incompressible flows, is proposed to enrich poorly populated training sets. The goal is to include in the new, artificial snapshots emerging features, not present in the original basis, that do enhance the quality of the reduced basis (RB) constructed using POD dimensionality reduction. The methodologies devised are based on exploiting basic physical principles, such as mass and momentum conservation, to construct physically-relevant, artificial snapshots at a fraction of the cost of additional full-order solutions. Interestingly, the numerical results show that the ideas exploiting only mass conservation (i.e., incompressibility) are not producing significant added value with respect to the standard linear combinations of snapshots. Conversely, accounting for the linearized momentum balance via the Oseen equation does improve the quality of the resulting approximation and therefore is an effective data augmentation strategy in the framework of viscous incompressible laminar flows. Numerical experiments of parametric flow problems, in two and three dimensions, at low and moderate values of the Reynolds number are presented to showcase the superior performance of the data-enriched POD-RB with respect to the standard ROM in terms of both accuracy and efficiency.

Figures (13)

• Figure 1: Pairing strategy of uniformly sampled snapshots in the parametric space. (a) One parameter, three snapshots. (b) Two parameters, five snapshots. Each point corresponds to one snapshot in the initial dataset. Lines denote the pairing of snapshots. Each snapshot is paired with its nearest neighbor in each direction and the number of lines denote the number of combinations to be employed to generate artificial snapshots.
• Figure 2: Geometry of the domain and boundary partition.
• Figure 3: Velocity streamlines in the wake of the cylinder (area of interest $[7,13]\times [6,10]$) for representative values of the parameters ${\text{Re}} \in [5,30]$ and $\gamma\in[0,4]$ (jet velocity). The color scales indicate the velocity module.
• Figure 4: Relative errors of the POD-RB approximation, measured in the Euclidean norm, of velocity and pressure (top), drag and lift (bottom) for different strategies of data augmentation, with parametrized ${\text{Re}} \in [5,30]$ and fixed $\gamma = 4$.
• Figure 5: Relative errors of the POD-RB approximation, measured in the Euclidean norm, of velocity and pressure (top), drag and lift (bottom) for different strategies of data augmentation, with parametrized $\gamma \in [0,4]$ and fixed ${\text{Re}} = 30$ .

Theorems & Definitions (2)

• Remark 1
• Remark 2