Sparse Identification for bifurcating phenomena in Computational Fluid Dynamics

TL;DR

This work tackles the challenge of capturing bifurcations in time-dependent CFD by developing a non-intrusive, data-driven reduced-order model. It integrates SINDy for sparse latent dynamics with autoencoder-based nonlinear reduction and a nested POD basis to efficiently handle parameterized, high-dimensional Navier–Stokes data. The approach is demonstrated on two canonical bifurcations in a sudden-expansion channel: a symmetry-breaking pitchfork and a Hopf bifurcation, achieving accurate bifurcation diagrams, unseen-parameter predictions, and substantial speed-ups over full-order simulations. The results offer a pathway to real-time analysis and deeper dynamical insight, with potential extensions to more complex or compressible flows and to advanced identification techniques.

Abstract

This work investigates model reduction techniques for nonlinear parameterized and time-dependent PDEs, specifically focusing on bifurcating phenomena in Computational Fluid Dynamics (CFD). We develop interpretable and non-intrusive Reduced Order Models (ROMs) capable of capturing dynamics associated with bifurcations by identifying a minimal set of coordinates. Our methodology combines the Sparse Identification of Nonlinear Dynamics (SINDy) method with a deep learning framework based on Autoencoder (AE) architectures. To enhance dimensionality reduction, we integrate a nested Proper Orthogonal Decomposition (POD) with the SINDy-AE architecture, enabling a sparse discovery of system dynamics while maintaining efficiency of the reduced model. We demonstrate our approach via two challenging test cases defined on sudden-expansion channel geometries: a symmetry-breaking bifurcation and a Hopf bifurcation. Starting from a comprehensive analysis of their high-fidelity behavior, i.e. symmetry-breaking phenomena and the rise of unsteady periodic solutions, we validate the accuracy and computational efficiency of our ROMs. The results show successful reconstruction of the bifurcations, accurate prediction of system evolution for unseen parameter values, and significant speed-up compared to full-order methods.

Figures (25)

• Figure 1: Problem geometry showing inlet boundary $\Gamma_\text{in}$ (green), outlet boundary $\Gamma_\text{out}$ (red), and wall boundary $\Gamma_0$ (black).
• Figure 2: Magnitude of the velocity field for two coexisting solutions at $\mu=0.5$ in the bifurcating regime.
• Figure 3: Evolution of the velocity field's magnitude for $\mu = 1.3$, and comparison with solution of the steady problem.
• Figure 4: Evolution of the velocity field's magnitude for $\mu = 0.5$, and comparison with the solution obtained considering the steady problem.
• Figure 5: Evolution over time of the relative distance between two successive iterations for two values of $\mu$, before and after the bifurcation point.

Theorems & Definitions (5)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 4.1: SINDy-AE with nested POD
• Remark 5.1: Error analysis