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

Meng Li, Yang Xiang, Zhichao Peng

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.

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

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.
Paper Structure (23 sections, 3 theorems, 84 equations, 11 figures, 3 tables, 2 algorithms)

This paper contains 23 sections, 3 theorems, 84 equations, 11 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Full-order model in Eulerian and Lagrangian frames
  4. Linear and nonlinear ROMs in the Eulerian frame
  5. Dynamic mode decomposition
  6. A non-intrusive autoencoder-based ROM
  7. Prediction Failure in the Eulerian Frame
  8. Enhancing compression and prediction via Lagrangian data
  9. Kolmogorov $n$-width in the Lagrangian frame
  10. Strong correlation in the Lagrangian frame
  11. Lagrangian ROMs
  12. Lagrangian data
  13. Lagrangian nonlinear ROM
  14. Offline training based on LagCAE
  15. Online prediction via pDMD
  16. ...and 8 more sections

Key Result

theorem 1

The Kolmogorov $n$-width in the Eulerian frame satisfies where $c>0$ is a constant and $\mathcal{V}_n\subset L^2[0,1]$ is an $n$-dimensional subspace. The Kolmogorov $n$-width in the Lagrangian frame is where $\mathcal{V}_n\subset L^2([0,1];\mathbb{R}^2)$ and $t$ is the varying parameter. Here,

Figures (11)

  • Figure 1: Conventional architecture of the convolutional autoencoder.
  • Figure 2: 1D advection problem $u_t+u_x=0$. (a) Ground-truth solution. The red dashed line at $t=0.8$ separates the training window $t\in[0,0.8]$ from the test window $t\in(0.8,1]$. (b) DMD-based ROM (rank $r=8$). (c) Autoencoder-based ROM (latent dimension $r=8$). For panel (b)-(c), the region on $t\in[0,0.8]$ shows in-window reconstruction results, whereas the solution over $t\in(0.8,1]$ corresponds to out-of-window forecasts.
  • Figure 3: Coherence in $(T_{\text{train}},T]$. The Eulerian-frame coherence decays rapidly in the forecast, indicating weak correlation with the training set (extrapolation), whereas the Lagrangian coherence remains high due to alignment (closer to interpolation).
  • Figure 4: Prediction results of nonlinear ROMs. The red dashed line marks $t=T_{\text{train}}=0.8$, separating reconstruction (left) from prediction (right). The Lagrangian ROM preserves the transported structure during prediction, while the Eulerian ROM exhibits spurious artifacts after $t=0.8$.
  • Figure 5: Latent trajectories during prediction for the 1D advection test. The Lagrangian latent trajectories remain smooth and bounded, while the Eulerian latent trajectories exhibit drift during prediction.
  • ...and 6 more figures

Theorems & Definitions (7)

  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof
  • remark 1