UP-dROM : Uncertainty-Aware and Parametrised dynamic Reduced-Order Model, application to unsteady flows

TL;DR

UP-dROM introduces a probabilistic, parametrised reduced-order model designed for unsteady flows by fusing nonlinear dimensionality reduction with a Transformer-based temporal model. The VAE provides a latent, uncertainty-aware representation, while cross-attention couples external parameters into the dynamics, enabling robust generalisation across regimes. Uncertainty quantification is integrated into the forecasting pipeline, yielding space-, time-, and parameter-space confidence measures that correlate with model performance and guide adaptive retraining. The approach demonstrates accurate interpolation and controlled extrapolation for flow around a cylinder and offers a practical, computationally efficient path for uncertainty-aware, parameter-sensitive ROMs in fluid dynamics.

Abstract

Reduced order models (ROMs) play a critical role in fluid mechanics by providing low-cost predictions, making them an attractive tool for engineering applications. However, for ROMs to be widely applicable, they must not only generalise well across different regimes, but also provide a measure of confidence in their predictions. While recent data-driven approaches have begun to address nonlinear reduction techniques to improve predictions in transient environments, challenges remain in terms of robustness and parametrisation. In this work, we present a nonlinear reduction strategy specifically designed for transient flows that incorporates parametrisation and uncertainty quantification. Our reduction strategy features a variational auto-encoder (VAE) that uses variational inference for confidence measurement. We use a latent space transformer that incorporates recent advances in attention mechanisms to predict dynamical systems. Attention's versatility in learning sequences and capturing their dependence on external parameters enhances generalisation across a wide range of dynamics. Prediction, coupled with confidence, enables more informed decision making and addresses the need for more robust models. In addition, this confidence is used to cost-effectively sample the parameter space, improving model performance a priori across the entire parameter space without requiring evaluation data for the entire domain.

Figures (15)

• Figure 1: UP-dROM architecture with $q$ the look-back window
• Figure 2: Predicted kinetic energy signal using UP-dROM compared with the ground truth.
• Figure 3: Schematic of a bifurcation diagram with respect to the varying Reynolds number. The two inserted plots are the kinetic energy signals of two simulations performed at $Re=50$ and $Re = 90$, before and after the critical Reynolds number, respectively.
• Figure 4: UP-dROM parametrised by external excitation variable $\xi$.
• Figure 5: Inferred and true kinetic energy signals in the in-distribution and out-of-distribution training data, superimposed on a sketch of the bifurcation plot with varying Reynolds number.