Efficient Real-Time Adaptation of ROMs for Unsteady Flows Using Data Assimilation

TL;DR

The dominant source of error in out-of-sample forecasts stems from distortions of the latent manifold rather than changes in the latent dynamics, allowing for a lightweight, computationally efficient, real-time adaptation procedure with very sparse fine-tuning data.

Abstract

We propose an efficient retraining strategy for a parameterized Reduced Order Model (ROM) that attains accuracy comparable to full retraining while requiring only a fraction of the computational time and relying solely on sparse observations of the full system. The architecture employs an encode-process-decode structure: a Variational Autoencoder (VAE) to perform dimensionality reduction, and a transformer network to evolve the latent states and model the dynamics. The ROM is parameterized by an external control variable, the Reynolds number in the Navier-Stokes setting, with the transformer exploiting attention mechanisms to capture both temporal dependencies and parameter effects. The probabilistic VAE enables stochastic sampling of trajectory ensembles, providing predictive means and uncertainty quantification through the first two moments. After initial training on a limited set of dynamical regimes, the model is adapted to out-of-sample parameter regions using only sparse data. Its probabilistic formulation naturally supports ensemble generation, which we employ within an ensemble Kalman filtering framework to assimilate data and reconstruct full-state trajectories from minimal observations. We further show that, for the dynamical system considered, the dominant source of error in out-of-sample forecasts stems from distortions of the latent manifold rather than changes in the latent dynamics. Consequently, retraining can be limited to the autoencoder, allowing for a lightweight, computationally efficient, real-time adaptation procedure with very sparse fine-tuning data.

Figures (17)

• Figure 1: Hopf bifurcations with latent manifolds at $Re=50$ and $Re = 90$. $U$ velocity field at unstable fixed point and at limit cycle.
• Figure 2: Our Reduced-Order Model encodes and reduces with $\mathcal{E}$ both velocity fields $U$ and $V$ along with the parametrisation $\xi$. The transformer rolls-out the dynamics auto-regressively in the latent space with self-attention and cross-attention with $\xi$. Finally the decoder $\mathcal{D}$ projects the latent states and $\xi$ back to the physical space.
• Figure 3: ROM uncertainty evaluation through ensemble generation $\psi$. The ensemble mean (prediction) is denoted $\bar{\psi}$ and the ensemble variance (uncertainty) by $\tilde{\psi}$.
• Figure 4: Kinetic energy prediction vs ground truth (validation set).
• Figure 5: Comparison between energy distance and uncertainty quantification for different Reynolds numbers.