Enhancing Future Prediction of Linear and Nonlinear Reduced-Order Models for Transport-Dominated Problems Using Lagrangian Data

TL;DR

This paper introduces Lagrangian-based reduced-order models that significantly improve the accuracy and stability of future predictions in transport-dominated problems by leveraging Lagrangian data and dynamics.

Contribution

The paper develops two novel Lagrangian ROMs that outperform Eulerian-based models in predicting transport-dominated systems, addressing the Kolmogorov barrier.

Findings

Lagrangian representation reduces Kolmogorov n-width faster.

Lagrangian ROMs outperform Eulerian ROMs in accuracy.

Lagrangian models show enhanced stability in predictions.

Abstract

Designing effective reduced-order models (ROMs) for parametrized transport-dominated problems remains challenging because of the well-known Kolmogorov barrier. Autoencoder-based nonlinear ROMs have been developed to improve the compression ability for such systems. However, despite their stronger compression ability, autoencoder-based ROMs constructed in the Eulerian frame may fail to accurately predict future solutions, due to the poor coherence between historical and future solutions in the Eulerian frame. In contrast, we show that representing transport-dominated dynamics in the Lagrangian frame can lead to a significantly faster decay of the Kolmogorov n-width and improve coherence between historical and future solutions. Building on these insights, we develop two non-intrusive ROMs leveraging Lagrangian data: a Lagrangian autoencoder-based ROM and a Lagrangian parametric dynamic mode decomposition. Numerical experiments demonstrate that these Lagrangian ROMs achieve more accurate and stable future predictions than their Eulerian counterparts.