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Symbolic Regression of Data-Driven Reduced Order Model Closures for Under-Resolved, Convection-Dominated Flows

Simone Manti, Ping-Hsuan Tsai, Alessandro Lucantonio, Traian Iliescu

TL;DR

The paper addresses under-resolved, convection-dominated flow simulations by enhancing reduced order models with data-driven closures. It introduces symbolic regression–based ROM closures (SR-ROM) within a data-driven variational multiscale ROM framework to yield interpretable, parsimonious, accurate, generalizable, and robust models. Comparative experiments on flow past a cylinder and lid-driven cavity demonstrate that SR-ROM achieves superior predictive accuracy and robustness with far fewer parameters than linear, quadratic, or neural-network closures. This approach offers a practical path to reliable, efficient ROM closures for turbulent-like flows and paves the way for parametric and cross-Re applications. The work highlights the potential of SR to produce compact, physically interpretable closure forms that generalize well beyond training data.

Abstract

Data-driven closures correct the standard reduced order models (ROMs) to increase their accuracy in under-resolved, convection-dominated flows. There are two types of data-driven ROM closures in current use: (i) structural, with simple ansatzes (e.g., linear or quadratic); and (ii) machine learning-based, with neural network ansatzes. We propose a novel symbolic regression (SR) data-driven ROM closure strategy, which combines the advantages of current approaches and eliminates their drawbacks. As a result, the new data-driven SR closures yield ROMs that are interpretable, parsimonious, accurate, generalizable, and robust. To compare the data-driven SR-ROM closures with the structural and machine learning-based ROM closures, we consider the data-driven variational multiscale ROM framework and two under-resolved, convection-dominated test problems: the flow past a cylinder and the lid-driven cavity flow at Reynolds numbers Re = 10000, 15000, and 20000. This numerical investigation shows that the new data-driven SR-ROM closures yield more accurate and robust ROMs than the structural and machine learning ROM closures.

Symbolic Regression of Data-Driven Reduced Order Model Closures for Under-Resolved, Convection-Dominated Flows

TL;DR

The paper addresses under-resolved, convection-dominated flow simulations by enhancing reduced order models with data-driven closures. It introduces symbolic regression–based ROM closures (SR-ROM) within a data-driven variational multiscale ROM framework to yield interpretable, parsimonious, accurate, generalizable, and robust models. Comparative experiments on flow past a cylinder and lid-driven cavity demonstrate that SR-ROM achieves superior predictive accuracy and robustness with far fewer parameters than linear, quadratic, or neural-network closures. This approach offers a practical path to reliable, efficient ROM closures for turbulent-like flows and paves the way for parametric and cross-Re applications. The work highlights the potential of SR to produce compact, physically interpretable closure forms that generalize well beyond training data.

Abstract

Data-driven closures correct the standard reduced order models (ROMs) to increase their accuracy in under-resolved, convection-dominated flows. There are two types of data-driven ROM closures in current use: (i) structural, with simple ansatzes (e.g., linear or quadratic); and (ii) machine learning-based, with neural network ansatzes. We propose a novel symbolic regression (SR) data-driven ROM closure strategy, which combines the advantages of current approaches and eliminates their drawbacks. As a result, the new data-driven SR closures yield ROMs that are interpretable, parsimonious, accurate, generalizable, and robust. To compare the data-driven SR-ROM closures with the structural and machine learning-based ROM closures, we consider the data-driven variational multiscale ROM framework and two under-resolved, convection-dominated test problems: the flow past a cylinder and the lid-driven cavity flow at Reynolds numbers Re = 10000, 15000, and 20000. This numerical investigation shows that the new data-driven SR-ROM closures yield more accurate and robust ROMs than the structural and machine learning ROM closures.

Paper Structure

This paper contains 14 sections, 29 equations, 12 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Full Order Model (FOM)
  3. Reduced Order Models (ROMs)
  4. Galerkin ROM (G-ROM)
  5. Machine Learning Variational Multiscale ROMs (ML-VMS-ROMs)
  6. VMS-ROM with Linear Regression Closure Models (LR-ROM)
  7. Data-Driven Variational Multiscale ROMs (d2-VMS-ROM)
  8. VMS-ROM with Symbolic Regression Closure Models (SR-ROM)
  9. VMS-ROM with Neural Network Closure Models (NN-ROM)
  10. Numerical Results
  11. 2D Flow Past a Cylinder
  12. 2D Lid-Driven Cavity
  13. Conclusions and Future Work
  14. CRediT Authorship Contribution Statement

Figures (12)

  • Figure 3.1: Schematic of the ML-VMS-ROMs considered in this paper.
  • Figure 3.2: Schematic of the symbolic regression approach used to model ROM closures (SR-ROM).
  • Figure 4.1: 2D flow past a cylinder. Velocity magnitude snapshots for $\rm Re=400$ (left) and $\rm Re=500$ (right).
  • Figure 4.2: 2D flow past a cylinder at $\rm Re=400$ and $\rm Re=500$. Comparison of the kinetic energy of the FOM and the ROMs for the reduced space dimension $r=2,3,4,5$.
  • Figure 4.3: 2D flow past a cylinder. $\text{rMSE}_{\text{test}}$ as a function of the number of parameters of the model. For the SR-ROM and the NN-ROM, the mean values out of 5 independent runs are reported. Each point is marked using the following legend: point for $r=2$; triangle for $r=3$; square for $r=4$; pentagon for $r=5$.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5