An adaptive, training-free reduced-order model for convection-dominated problems based on hybrid snapshots

TL;DR

This paper introduces an adaptive, training-free reduced-order modeling approach for convection-dominated problems that combines high-fidelity and reduced models dynamically, achieving significant computational speedups without extensive training.

Contribution

The method adaptively integrates high-fidelity and reduced models without prior training, leveraging local coherence to improve accuracy and efficiency in convection-dominated problems.

Findings

Achieves up to tenfold speedup in simulations.

Effectively combines HDM and ROM evaluations dynamically.

Reduces spurious oscillations with spatial filtering.

Abstract

The vast majority of reduced-order models (ROMs) first obtain a low dimensional representation of the problem from high-dimensional model (HDM) training data which is afterwards used to obtain a system of reduced complexity. Unfortunately, convection-dominated problems generally have a slowly decaying Kolmogorov n-width, which makes obtaining an accurate ROM built solely from training data very challenging. The accuracy of a ROM can be improved through enrichment with HDM solutions; however, due to the large computational expense of HDM evaluations for complex problems, they can only be used parsimoniously to obtain relevant computational savings. In this work, we exploit the local spatial and temporal coherence often exhibited by these problems to derive an accurate, cost-efficient approach that repeatedly combines HDM and ROM evaluations without a separate training phase. Our approach obtains solutions at a given time step by either fully solving the HDM or by combining partial HDM and ROM solves. A dynamic sampling procedure identifies regions that require the HDM solution for global accuracy and the reminder of the flow is reconstructed using the ROM. Moreover, solutions combining both HDM and ROM solves use spatial filtering to eliminate potential spurious oscillations that may develop. We test the proposed method on inviscid compressible flow problems and demonstrate speedups up to an order of magnitude.