Verifiability of the Data-Driven Variational Multiscale Reduced Order Model

TL;DR

This paper establishes the verifiability of data-driven variational multiscale reduced order models (ROMs), ensuring small closure errors lead to small overall ROM errors, and demonstrates this property through numerical simulations of fluid flow problems.

Contribution

It extends the concept of verifiability to ROM closures, proves that data-driven variational multiscale ROMs are verifiable, and validates this through numerical experiments.

Findings

Verifiability holds for the data-driven variational multiscale ROM.

Small closure model errors imply small overall ROM errors.

Numerical simulations confirm theoretical verifiability results.

Abstract

In this paper, we focus on the mathematical foundations of reduced order model (ROM) closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. Finally, we investigate the verifiability of the data-driven variational multiscale ROM in the numerical simulation of the one-dimensional Burgers equation and a two-dimensional flow past a circular cylinder at Reynolds numbers $Re=100$ and $Re=1000$.