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Enhancing non-intrusive Reduced Order Models with space-dependent aggregation methods

Anna Ivagnes, Niccolò Tonicello, Paola Cinnella, Gianluigi Rozza

TL;DR

This work introduces a space-dependent aggregation framework that fuses multiple non-intrusive ROMs—each combining a reduction method (POD, AutoEncoder, or PODAE) with an approximation technique (RBF, Gaussian Process Regression, or ANN)—into a single mixed ROM. The key idea is to weigh model predictions spatially via a convex combination, with weights learned from data using Random Forest regression to enable accurate predictions for unseen configurations. The approach is validated on two airfoil test cases (NACA4412 and NACA0012), showing that the mixed ROM consistently improves accuracy over individual ROMs, especially in regions where nonlinear reductions capture sharp features like wakes and shocks. The method provides automatic, region-specific switching between linear and nonlinear reduction strategies and does so with modest offline/online computational costs, offering practical potential for real-time design workflows. The study contributes a robust, data-driven framework for uncertainty quantification and model selection in reduced-order modeling of turbulent flows.

Abstract

In this manuscript, we combine non-intrusive reduced order models (ROMs) with space-dependent aggregation techniques to build a mixed-ROM. The prediction of the mixed formulation is given by a convex linear combination of the predictions of some previously-trained ROMs, where we assign to each model a space-dependent weight. The ROMs taken into account to build the mixed model exploit different reduction techniques, such as Proper Orthogonal Decomposition (POD) and AutoEncoders (AE), and/or different approximation techniques, namely a Radial Basis Function Interpolation (RBF), a Gaussian Process Regression (GPR) or a feed-forward Artificial Neural Network (ANN). The contribution of each model is retained with higher weights in the regions where the model performs best, and, vice versa, with smaller weights where the model has a lower accuracy with respect to the other models. Finally, a regression technique, namely a Random Forest, is exploited to evaluate the weights for unseen conditions. The performance of the aggregated model is evaluated on two different test cases: the 2D flow past a NACA 4412 airfoil, with an angle of attack of 5 degrees, having as parameter the Reynolds number varying between 1e5 and 1e6 and a transonic flow over a NACA 0012 airfoil, considering as parameter the angle of attack. In both cases, the mixed-ROM has provided improved accuracy with respect to each individual ROM technique.

Enhancing non-intrusive Reduced Order Models with space-dependent aggregation methods

TL;DR

This work introduces a space-dependent aggregation framework that fuses multiple non-intrusive ROMs—each combining a reduction method (POD, AutoEncoder, or PODAE) with an approximation technique (RBF, Gaussian Process Regression, or ANN)—into a single mixed ROM. The key idea is to weigh model predictions spatially via a convex combination, with weights learned from data using Random Forest regression to enable accurate predictions for unseen configurations. The approach is validated on two airfoil test cases (NACA4412 and NACA0012), showing that the mixed ROM consistently improves accuracy over individual ROMs, especially in regions where nonlinear reductions capture sharp features like wakes and shocks. The method provides automatic, region-specific switching between linear and nonlinear reduction strategies and does so with modest offline/online computational costs, offering practical potential for real-time design workflows. The study contributes a robust, data-driven framework for uncertainty quantification and model selection in reduced-order modeling of turbulent flows.

Abstract

In this manuscript, we combine non-intrusive reduced order models (ROMs) with space-dependent aggregation techniques to build a mixed-ROM. The prediction of the mixed formulation is given by a convex linear combination of the predictions of some previously-trained ROMs, where we assign to each model a space-dependent weight. The ROMs taken into account to build the mixed model exploit different reduction techniques, such as Proper Orthogonal Decomposition (POD) and AutoEncoders (AE), and/or different approximation techniques, namely a Radial Basis Function Interpolation (RBF), a Gaussian Process Regression (GPR) or a feed-forward Artificial Neural Network (ANN). The contribution of each model is retained with higher weights in the regions where the model performs best, and, vice versa, with smaller weights where the model has a lower accuracy with respect to the other models. Finally, a regression technique, namely a Random Forest, is exploited to evaluate the weights for unseen conditions. The performance of the aggregated model is evaluated on two different test cases: the 2D flow past a NACA 4412 airfoil, with an angle of attack of 5 degrees, having as parameter the Reynolds number varying between 1e5 and 1e6 and a transonic flow over a NACA 0012 airfoil, considering as parameter the angle of attack. In both cases, the mixed-ROM has provided improved accuracy with respect to each individual ROM technique.
Paper Structure (27 sections, 17 equations, 28 figures, 3 tables)

This paper contains 27 sections, 17 equations, 28 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Numerical methods
  3. Non-intrusive Reduced Order Models
  4. Reduction techniques
  5. Reduction through Proper Orthogonal Decomposition
  6. Reduction through AutoEncoder
  7. Reduction through PODAE
  8. Approximation techniques
  9. Radial Basis Function Interpolation
  10. Gaussian Process Regression
  11. Artificial Neural Network
  12. Space-dependent aggregation methods and ROM mixture
  13. Random Forest regression
  14. Numerical results
  15. FOM description and validation
  16. ...and 12 more sections

Figures (28)

  • Figure 1: General structure of an autoencoder.
  • Figure 2: Domain ( \ref{['case1domain']}) and mesh ( \ref{['case1mesh']}) representation for test case 2, with a zoomed representation for the zone around the airfoil ( \ref{['case1mesh-zoom']}).
  • Figure 3: Examples of pressure and velocity magnitude snapshots for the first test case, for $Re\approx 118000.0$ and $Re\approx 518000.0$.
  • Figure 4: Pressure coefficient for the elements of the database considered for the first test case, for both suction and pressure sides of the airfoil.
  • Figure 5: Validation of pressure \ref{['fig1a']} and upper skin friction \ref{['fig1b']} coefficients for the first type of full order simulation. The FOM validation is for NACA 4412 at $\mathrm{Re}=4e5$ and angle of attack $\alpha=5^{\circ}$. The reference simulations are taken from vinuesa2017pressure (DNS) and tabatabaei2022rans (RANS).
  • ...and 23 more figures