[Paper Review] A Data-Driven Surrogate Modeling Approach for Time-Dependent Incompressible Navier-Stokes Equations with Dynamic Mode Decomposition and Manifold Interpolation

Martin W. Hess, Annalisa Quaini+1 more|arXiv (Cornell University)|Jan 26, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper presents a data-driven surrogate modeling approach for time-dependent incompressible Navier-Stokes equations using a multi-step framework combining proper orthogonal decomposition (POD), dynamic mode decomposition (DMD), and manifold interpolation. The method accurately captures time-periodic and chaotic flows in the Rayleigh-Bénard cavity problem, even recovering frequencies not present in the training data, achieving engineering accuracy (2–15% mean error) in high Grashof number regimes near turbulence.

ABSTRACT

This work introduces a novel approach for data-driven model reduction of time-dependent parametric partial differential equations. Using a multi-step procedure consisting of proper orthogonal decomposition, dynamic mode decomposition and manifold interpolation, the proposed approach allows to accurately recover field solutions from a few large-scale simulations. Numerical experiments for the Rayleigh-Bénard cavity problem show the effectiveness of such multi-step procedure in two parametric regimes, i.e.~medium and high Grashof number. The latter regime is particularly challenging as it nears the onset of turbulent and chaotic behaviour. A major advantage of the proposed method in the context of time-periodic solutions is the ability to recover frequencies that are not present in the sampled data.

Motivation & Objective

• To develop a non-intrusive, data-driven reduced order model (ROM) for time-dependent parametric PDEs with complex dynamics, including bifurcations and periodicity.
• To address the challenge of interpolating solutions across parameter values where the temporal frequency of the solution differs from sampled data, especially in high Grashof number regimes approaching turbulence.
• To enable accurate online prediction of field solutions at new parameter values using only high-fidelity simulation data from a few sampled parameters.
• To overcome limitations of standard ROMs, ANNs, neural ODEs, and sparse identification in capturing frequency shifts in bifurcating systems.
• To validate the method on the Rayleigh-Bénard cavity problem, a benchmark problem with Hopf bifurcations and complex time-periodic behavior.

Proposed method

• Apply proper orthogonal decomposition (POD) to extract spatial modes and reduce the dimensionality of high-fidelity Navier-Stokes solutions.
• Use dynamic mode decomposition (DMD) on POD coefficients to extract temporal dynamics, including frequencies and growth rates.
• Interpolate the DMD operators across parameter values using manifold interpolation to handle non-linear parameter dependence.
• Construct a surrogate model by combining POD modes with interpolated DMD dynamics, enabling online prediction at new parameters.
• Use a multi-stage offline-online decomposition: offline computation of POD modes, DMD operators, and manifold interpolation; online evaluation on low-cost devices.
• Apply a non-flat metric or complex DMD in future extensions to improve stability and accuracy of interpolation.

Experimental results

Research questions

• RQ1 Can a data-driven ROM accurately predict time-periodic solutions in parametric PDEs when the solution frequency at online parameters differs from that in the training data?
• RQ2 How can DMD-based dynamics be effectively interpolated across a parameter domain with bifurcations, especially when the temporal period changes significantly?
• RQ3 To what extent can manifold interpolation of DMD operators preserve the correct frequency and amplitude of time-periodic solutions in high Grashof number regimes?
• RQ4 Can this multi-step approach outperform standard ROMs, ANNs, and sparse identification techniques in capturing complex, non-stationary behavior near chaotic regimes?
• RQ5 Is the method robust enough to achieve engineering accuracy in both medium and high Grashof number regimes, including near-turbulent flows?

Key findings

• The proposed ROM achieves mean relative errors between 2% and 15% in the L² and L∞ norms across both medium and high Grashof number regimes, with errors reduced compared to direct comparison with full-order solutions.
• At Gr = 674.92×10³, the method achieves 3.5%–5% mean and maximum errors in L² and L∞ norms, capturing all major flow features despite phase discrepancies in some rolls.
• At Gr = 683.985×10³, the error increases to 14% mean and 25% maximum in L∞ norm due to phase misalignment in the middle roll, indicating sensitivity to phase accuracy in complex flows.
• The method successfully recovers frequencies not present in the sampled data, enabling accurate simulation at new parameter values even when the temporal dynamics are shifted.
• Higher-order POD modes exhibit complex, non-sinusoidal behavior that is poorly captured by standard DMD, but the method prioritizes accurate representation of dominant modes, which is sufficient for engineering accuracy.
• The approach outperforms direct comparison with full-order solutions at fixed time steps in the high Gr regime, where full-order comparisons yield up to 35% error, while the ROM maintains under 15% mean error.