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

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.

💡 Research Summary

This paper addresses a fundamental limitation of reduced‑order models (ROMs) for transport‑dominated partial differential equations (PDEs), namely the difficulty of accurately predicting future states when the solution exhibits strong advection. In the traditional Eulerian framework, snapshots are collected on a fixed spatial grid. Because advective features translate across the grid, snapshots at different times become nearly orthogonal, leading to a slow decay of the Kolmogorov n‑width (dₙ(M_E) ≥ c·n⁻¹/²). Consequently, even sophisticated nonlinear ROMs such as convolutional autoencoders (CAEs) require relatively high latent dimensions to achieve modest accuracy, and they still fail to extrapolate beyond the training time window: the learned manifold does not contain the translated structures that appear later.

The authors propose to work in the Lagrangian frame, where the computational grid moves along characteristic lines. By pairing the moving coordinate χ(x̂,t) with the solution u(χ,t) and treating (χ,u) as a combined state, the solution manifold becomes essentially two‑dimensional regardless of the original spatial dimension. The paper proves (Theorem 1) that the Kolmogorov n‑width in the Lagrangian setting is zero for n ≥ 2, i.e., the manifold can be captured exactly with only two linear modes. This theoretical result explains why a very small number of latent variables suffices to represent the entire solution set.

Building on this insight, two non‑intrusive ROMs are developed:

1. Lagrangian Autoencoder (LAE) – In the offline stage, a convolutional autoencoder is trained on Lagrangian snapshots to learn an encoder Gₑ and decoder G_d. The encoder maps each Lagrangian state to a low‑dimensional latent vector h, while the decoder reconstructs the full Eulerian field. In the online stage, a parametric Dynamic Mode Decomposition (p‑DMD) is applied to the latent dynamics h(t;μ), providing a linear evolution model that can be evaluated for new parameters μ. The decoder then lifts the predicted latent state back to the physical space.

2. Lagrangian Parametric DMD (LP‑DMD) – This method directly constructs a Koopman‑type linear operator A(μ) on the Lagrangian state (χ,u) using snapshot pairs, and interpolates A across the parameter space. No neural network is required; future Lagrangian states are obtained by repeated application of the interpolated operator, followed by mapping back to Eulerian coordinates.

A series of numerical experiments validates the approach. Test cases include:

• 1‑D linear advection‑diffusion and nonlinear Burgers equations with varying diffusion coefficients;
• 2‑D circular advection on a rotating flow;
• Parametric Burgers equation with a wide range of viscosity parameters.

Across all cases, the Lagrangian ROMs (both LAE and LP‑DMD) achieve orders‑of‑magnitude lower relative errors than their Eulerian counterparts for the same latent dimension (e.g., r = 8–16). More importantly, they remain stable over long prediction horizons, whereas Eulerian DMD and Eulerian CAE solutions quickly diverge or produce blank regions where the advected front leaves the training support. Correlation analysis shows that the Pearson coefficient between past and future Lagrangian states stays above 0.9, while it drops below 0.3 in the Eulerian frame, confirming that Lagrangian data turn future prediction into an interpolation problem rather than an out‑of‑distribution extrapolation.

The paper’s contributions are threefold:

1. Theoretical insight – Demonstrating that the Kolmogorov n‑width decays dramatically in the Lagrangian frame, effectively bypassing the Kolmogorov barrier for transport‑dominated problems.
2. Algorithmic development – Introducing a Lagrangian autoencoder combined with parametric DMD, and a purely linear Lagrangian parametric DMD, both of which are non‑intrusive and require only snapshot data.
3. Empirical validation – Providing extensive numerical evidence that the proposed methods outperform state‑of‑the‑art Eulerian ROMs in accuracy, stability, and computational efficiency.

The authors conclude that leveraging Lagrangian information is a powerful strategy for ROM construction in advective systems and suggest future work on extending the framework to multi‑parameter, high‑dimensional, and unstructured‑grid problems, as well as incorporating physics‑informed constraints to further improve robustness.