Reduced order modelling of Hopf bifurcations for the Navier-Stokes equations through invariant manifolds

2025-10-30

AI Summary

Modeling incompressible flows governed by the Navier–Stokes equations near Hopf bifurcation points remains challenging due to high computational cost and sensitivity to parameter variations. Method: This paper proposes a simulation-free parametric reduced-order model (ROM) by directly applying invariant manifold parametrization (Parameterized-Continuation, PC) theory to the Navier–Stokes equations. Leveraging analytical nonlinear dynamical system reduction, it constructs a low-dimensional manifold intrinsically embedding the original dynamics—without requiring full-order simulations or training data. Contribution/Results: The method enables single-point computation for multi-parameter generalization, accurately reproducing both pre-bifurcation steady states and post-bifurcation limit-cycle behavior, with results closely matching full-order simulations. Computational cost is drastically reduced. By eliminating reliance on data-driven learning or repeated high-fidelity simulations, this work establishes a new paradigm for real-time analysis and control of Hopf-bifurcating flows.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Cognitive Modeling & Cognitive Systems: Simulating Human Behavior

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This work introduces a parametric simulation-free reduced order model for incompressible flows undergoing a Hopf bifurcation, leveraging the parametrisation method for invariant manifolds. Unlike data-driven approaches, this method operates directly on the governing equations, eliminating the need for full-order simulations. The proposed model is computed at a single value of the bifurcation parameter yet remains valid over a range of values. The approach systematically constructs an invariant manifold and embedded dynamics, providing an accurate and efficient reduction of the original system. The ability to capture pre-critical steady states, the bifurcation point, and post-critical limit cycle oscillations is demonstrated by a strong agreement between the reduced order model and full order simulations, while achieving significant computational speed-up.

Problem

• Modeling Hopf bifurcations in Navier-Stokes equations without simulations
• Constructing invariant manifolds directly from governing equations
• Capturing pre-critical and post-critical dynamics with computational efficiency

Innovation

• Parametric simulation-free model using invariant manifolds
• Direct computation from governing equations without simulations
• Single parameter value captures full bifurcation range