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Hybrid ROM-PINN Framework for Closure Modeling in Convection-Dominated Systems

Ferhat Kaya, Birgul Koc, Atakan Aygun, Onur Ata, Ali Karakus

Abstract

Reduced-order models (ROMs) have become an essential tool for reducing the computational cost of fluid flow simulations. While standard ROMs can efficiently approximate laminar flows, their accuracy often suffers in convection-dominated regimes due to the truncation of dynamically important modes. To account for the influence of unresolved scales, ROM closure models are commonly introduced. Classical closure strategies are typically based on phenomenological arguments or analogies with large eddy simulation (LES), often formulated within a variational multiscale (VMS) framework, in which the resolved and unresolved scales are explicitly separated and their interactions are systematically modeled. More recently, advances in data-driven modeling and machine learning have opened new opportunities to construct ROM closures that are both more accurate and more consistent with the underlying physics. In this work, we develop a new ROM closure that combines machine learning with physics-based modeling principles. The closure term is derived within a VMS framework, where the reduced solution space is decomposed into resolved and unresolved components. This VMS-derived closure term is then modeled using PhysicsInformed Neural Networks (PINNs) and incorporated into a newly constructed C-PINN-ROM. The resulting closure leverages high-fidelity data while enforcing physical constraints imposed by the reduced-order equations, thereby ensuring consistency with the underlying dynamics and enhanced robustness in convection-dominated regimes. Through this PINN-based framework, we demonstrate how physics-informed machine learning can substantially improve the accuracy and robustness of ROMs, effectively bridging classical multiscale closure modeling with state-of-the-art data-driven methodologies.

Hybrid ROM-PINN Framework for Closure Modeling in Convection-Dominated Systems

Abstract

Reduced-order models (ROMs) have become an essential tool for reducing the computational cost of fluid flow simulations. While standard ROMs can efficiently approximate laminar flows, their accuracy often suffers in convection-dominated regimes due to the truncation of dynamically important modes. To account for the influence of unresolved scales, ROM closure models are commonly introduced. Classical closure strategies are typically based on phenomenological arguments or analogies with large eddy simulation (LES), often formulated within a variational multiscale (VMS) framework, in which the resolved and unresolved scales are explicitly separated and their interactions are systematically modeled. More recently, advances in data-driven modeling and machine learning have opened new opportunities to construct ROM closures that are both more accurate and more consistent with the underlying physics. In this work, we develop a new ROM closure that combines machine learning with physics-based modeling principles. The closure term is derived within a VMS framework, where the reduced solution space is decomposed into resolved and unresolved components. This VMS-derived closure term is then modeled using PhysicsInformed Neural Networks (PINNs) and incorporated into a newly constructed C-PINN-ROM. The resulting closure leverages high-fidelity data while enforcing physical constraints imposed by the reduced-order equations, thereby ensuring consistency with the underlying dynamics and enhanced robustness in convection-dominated regimes. Through this PINN-based framework, we demonstrate how physics-informed machine learning can substantially improve the accuracy and robustness of ROMs, effectively bridging classical multiscale closure modeling with state-of-the-art data-driven methodologies.
Paper Structure (16 sections, 38 equations, 14 figures, 5 tables, 1 algorithm)

This paper contains 16 sections, 38 equations, 14 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Reduced-Order Modeling
  3. Governing Equations
  4. Galerkin Reduced-Order Model (G-ROM)
  5. Limitations of Galerkin ROMs
  6. Variational Multiscale Reduced-Order Modeling
  7. VMS-ROM Closure Modeling
  8. Physics-Informed Neural Networks for ROM Closure
  9. Physics Informed Neural Networks
  10. PINN for Galerkin ROM Closure
  11. Numerical Results
  12. 2-Dimensional Viscous Burgers Equation
  13. Testing in Extrapolation and Interpolation Regimes
  14. 2-Dimensional Flow Past a Cylinder
  15. Conclusions and Outlook
  16. ...and 1 more sections

Figures (14)

  • Figure 1: A schematic diagram of the proposed PINN based closure model.
  • Figure 2: Burgers equation; relative information content of the complementary POD subspaces $\boldsymbol{X}_r$ and $\boldsymbol{X}_r^\perp$.
  • Figure 3: Burgers equation; average relative $L^2$ velocity errors for different Reynolds numbers and ROM dimension.
  • Figure 4: Burgers equation; POD modes of parametric ROM subspace.
  • Figure 5: Burgers equation; time-dependent coefficients in the interpolation testing regime with $Re = 1500$ and $Re=5500$.
  • ...and 9 more figures