Abstract
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.
Highlights
In this work we show that nonlinear correlations can be exploited to identify and enforce this phase coherence in reduced-order models, as shown schematically in figure 1
The dynamic mode decomposition (DMD)–Petrov–Galerkin model can be truncated differently from the proper orthogonal decomposition (POD)–Galerkin model by selecting a subset of the DMD eigenvectors, so that a = Vα with a ∈ Rr, α ∈ Cs and V ∈ Cr×s, and with V −1 replaced by the pseudoinverse V + in (5.6)
Since this transformation is a rotation in the space of modal coefficients, the dynamics and qualitative behaviour of the DMD–Galerkin models do not change compared with figure 8
Summary
Multiscale structure are characterized by emergent large-scale coherence (Haken 1983; Cross & Hohenberg 1993), generating low-dimensional structure often conceptualized as an attracting or slow manifold. After projecting data from a direct numerical solution of a quasiperiodic shear-driven cavity flow onto a basis of DMD modes, the recently proposed randomized dependence coefficient (Lopez-Paz, Hennig & Schölkopf 2013) allows us to clearly distinguish the active degrees of freedom from correlated higher harmonics and nonlinear cross-talk In this minimal representation, the dynamics occurs on a 2-torus, while the rest of the modes, which arise as triadic interactions of the active variables in the frequency domain, can be expressed as polynomial functions of the dynamically active variables. The restriction to this manifold stabilizes a standard POD-Galerkin model, avoiding both decoherence and energy imbalance This representation is a natural basis for data-driven system identification methods; we apply the sparse identification of nonlinear dynamics (SINDy) algorithm (Brunton et al 2016) and show that the flow can be accurately described by two independent Stuart-Landau equations.
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