On the role of nonlinear correlations in reduced-order modelling

This work investigates nonlinear dimensionality reduction as a means of improving the accuracy and stability of reduced-order models of advection-dominated flows. Nonlinear correlations between temporal proper orthogonal decomposition (POD) coefficients can be exploited to identify latent low-dimensional structure, approximating the attractor with a minimal set of driving modes and a manifold equation for the remaining modes. By viewing these nonlinear correlations as an invariant manifold reduction, this least-order representation can be used to stabilize POD-Galerkin models or as a state space for data-driven model identification. In the latter case, we use sparse polynomial regression to learn a compact, interpretable dynamical system model from the time series of the active modal coefficients. We demonstrate this perspective on a quasiperiodic shear-driven cavity flow and show that the dynamics evolves on a torus generated by two independent Stuart-Landau oscillators. The specific approach to nonlinear correlations analysis used in this work is applicable to periodic and quasiperiodic flows, and cannot be applied to chaotic or turbulent flows. However, the results illustrate the limitations of linear modal representations of advection-dominated flows and motivate the use of nonlinear dimensionality reduction more broadly for exploiting underlying structure in reduced-order models.

Automated summary

This work explores the use of nonlinear dimensionality reduction to enhance the accuracy and stability of reduced-order models for advection-dominated flows. By leveraging nonlinear correlations between temporal Proper Orthogonal Decomposition (POD) coefficients, latent low-dimensional structures can be identified and approximated using a minimal set of driving modes and a manifold equation for the remaining modes. This approach can stabilize POD-Galerkin models and serve as a state space for data-driven model identification.