Surrogate normal-forms for the numerical bifurcation and stability analysis of navier-stokes flows via machine learning

Abstract

Inspired by the Equation-Free paradigm, we propose an ``embed-learn-lift'' framework for constructing minimal-dimensional surrogate ROMs for the numerical analysis of high-fidelity Navier-Stokes simulations, even in the presence of symmetries that standard machine-learning surrogates often fail to preserve. The framework consists of four main stages. First, manifold learning (here both POD and Diffusion Maps) is used to uncover the intrinsic geometry and dimensionality of the latent space underlying the high-dimensional spatio-temporal Navier-Stokes dynamics across the parameter space. Second, we construct ROMs (here, via Gaussian Process regression (GPR)) of minimal dimension -- by learning the evolution equations directly on the identified latent space. Third, we exploit the toolkit of numerical bifurcation analysis to construct bifurcation diagrams and perform systematic stability analysis directly in the latent coordinates. This enables, for example, the efficient continuation of branches of limit cycles emerging from Andronov-Hopf and Neimark-Sacker bifurcations, together with the computation of limit-cycles periods and stability properties via Floquet multipliers. Such analysis is effectively intractable for the full Navier-Stokes equations. Finally, by solving the pre-image problem in manifold learning, we reconstruct the bifurcating steady and time-periodic states in the original high-dimensional physical space, thus closing the ``lift'' step of the pipeline. We show that DMs-based ROMs allow for a computationally efficient and accurate numerical bifurcation and stability analysis, thus outperforming the widely used POD-ROMs by providing a geometrically consistent parametrization and correctly identifying the intrinsic dimension even in the presence of secondary instabilities, highlighting the need for nonlinear manifold learning methods in CFD.

Figures (19)

• Figure 1: Four-stage data-driven framework for bifurcation and stability analysis of fluid flows in latent spaces: (a) projection of high-dimensional Navier-Stokes solutions to the latent space (manifold learning, here using Diffusion Maps (DMs) and Proper Orthogonal Decomposition (POD)); (b) modelling of the latent dynamics via machine learning surrogates (here, Gaussian Process regression (GPR)); (c) computation of the latent bifurcation diagram and stability analysis; (d) reconstruction of the physical space solutions (pre-image problem, here using the $k$-NN algorithm and POD).
• Figure 2: Schematic representations of the three benchmark two-dimensional configurations: wake flow past a circular cylinder (a); planar sudden-expansion channel flow (b); fluidic pinball (c).
• Figure 3: Instantaneous contour maps of $u$ ((a) and (c)) and $v$ ((b) and (d)) velocity components of the cylinder flow for $Re=20$ ((a)-(b)) and $Re=60$ ((c)-(d)). Panel (e) reports the instantaneous spatial distribution of $u$ on the $y=0$ axis for $Re=20$ (red curve), $Re=40$ (blue curve), and $Re=60$ (green curve). Panel (f) shows the temporal evolution of $u$ at the streamwise location $(\tilde{x},\tilde{y}) = (7,0)$ for the same values of Reynolds number.
• Figure 4: Instantaneous contour maps of $u$ ((a) and (c)) and $v$ ((b) and (d)) velocity components of the sudden-expansion channel for $Re=30$ ((a)-(b)) and $Re=60$ ((c)-(d)). The white curves in panels (a)-(d) denote the flow locations where $u=0$, namely the wake borders. Panels (e)-(f) report the time evolution of the upper ($x^+_w$, black) and lower ($x^-_w$, red) wake region extensions for $Re=30$ and $Re=60$, respectively.
• Figure 5: Instantaneous contour maps of $v$ velocity component of the pinball flow for $Re=95$ (a) and $Re=105$ (b). The temporal evolution of $u$ and $v$ at the streamwise location $(\tilde{x},\tilde{y}) = (7,0)$ is shown for $Re=95$ in panels (c)-(d), and for $Re=105$ in panels (e)-(f), respectively.