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A Goal-Oriented Adaptive Sampling Procedure for Projection-Based Reduced-Order Models with Hyperreduction

Calista Biondic, Siva Nadarajah

TL;DR

The work tackles the computational bottleneck of projection-based reduced-order models (PROMs) for CFD by integrating energy-conserving sampling and weighting (ECSW) hyperreduction with a goal-oriented adaptive sampling framework that uses dual-weighted residual indicators. By applying ECSW to both the residual and the Jacobian within a least-squares Petrov-Galerkin (LSPG) PROM and by updating a hyperreduction-aware error indicator, the approach achieves substantial offline and online cost reductions while maintaining controlled accuracy for the lift on NACA0012 airfoils across subsonic and transonic regimes. The main contributions include (i) adapting ECSW training in residual- and Jacobian-based forms for Petrov-Galerkin PROMs, (ii) extending the adaptive sampling procedure with a hyperreduction-aware dual-weighted residual error, and (iii) demonstrating significant reduction in mesh size and computation with acceptable error bounds, validating the method on multiple NACA0012 test cases. The results indicate that hyperreduction, when coupled with GO adaptive sampling, can deliver efficient, error-controlled ROMs suitable for design optimization and real-time predictions, with future work extending to unsteady, 3D problems and more advanced training strategies.

Abstract

Projection-based reduced-order models (PROMs) have demonstrated accuracy, reliability, and robustness in approximating high-dimensional, differential equation-based computational models across many applications. For this reason, it has been proposed as a tool for high-querying parametric design problems like those arising in modern aircraft design. Since aerodynamic simulations can be computationally expensive, PROMs offer the potential for more rapid estimations of high-fidelity solutions. However, the efficiency can still be tied to the dimension of the full-order model (FOM), particularly when projected quantities must be frequently recomputed due to non-linearities or parameter dependence. In the case of Petrov-Galerkin models, the projected residual and Jacobian are re-evaluated at every Newton iteration, thereby limiting the anticipated cost improvements. Hyperreduction is one of the tools available to approximate these quantities and address this issue. This work tests the energy-conserving sampling and weighting (ECSW) method as a potential approach for hyperreduction. It will be incorporated into the work in a previous article {10.1016/j.compfluid.2025.106568} which had developed an adaptive sampling procedure for building a reduced-order model (ROM) with a controlled functional error. The impacts of hyperreduction on computational cost and accuracy will be studied using the NACA0012 airfoil.

A Goal-Oriented Adaptive Sampling Procedure for Projection-Based Reduced-Order Models with Hyperreduction

TL;DR

The work tackles the computational bottleneck of projection-based reduced-order models (PROMs) for CFD by integrating energy-conserving sampling and weighting (ECSW) hyperreduction with a goal-oriented adaptive sampling framework that uses dual-weighted residual indicators. By applying ECSW to both the residual and the Jacobian within a least-squares Petrov-Galerkin (LSPG) PROM and by updating a hyperreduction-aware error indicator, the approach achieves substantial offline and online cost reductions while maintaining controlled accuracy for the lift on NACA0012 airfoils across subsonic and transonic regimes. The main contributions include (i) adapting ECSW training in residual- and Jacobian-based forms for Petrov-Galerkin PROMs, (ii) extending the adaptive sampling procedure with a hyperreduction-aware dual-weighted residual error, and (iii) demonstrating significant reduction in mesh size and computation with acceptable error bounds, validating the method on multiple NACA0012 test cases. The results indicate that hyperreduction, when coupled with GO adaptive sampling, can deliver efficient, error-controlled ROMs suitable for design optimization and real-time predictions, with future work extending to unsteady, 3D problems and more advanced training strategies.

Abstract

Projection-based reduced-order models (PROMs) have demonstrated accuracy, reliability, and robustness in approximating high-dimensional, differential equation-based computational models across many applications. For this reason, it has been proposed as a tool for high-querying parametric design problems like those arising in modern aircraft design. Since aerodynamic simulations can be computationally expensive, PROMs offer the potential for more rapid estimations of high-fidelity solutions. However, the efficiency can still be tied to the dimension of the full-order model (FOM), particularly when projected quantities must be frequently recomputed due to non-linearities or parameter dependence. In the case of Petrov-Galerkin models, the projected residual and Jacobian are re-evaluated at every Newton iteration, thereby limiting the anticipated cost improvements. Hyperreduction is one of the tools available to approximate these quantities and address this issue. This work tests the energy-conserving sampling and weighting (ECSW) method as a potential approach for hyperreduction. It will be incorporated into the work in a previous article {10.1016/j.compfluid.2025.106568} which had developed an adaptive sampling procedure for building a reduced-order model (ROM) with a controlled functional error. The impacts of hyperreduction on computational cost and accuracy will be studied using the NACA0012 airfoil.
Paper Structure (30 sections, 69 equations, 36 figures, 4 tables, 6 algorithms)

This paper contains 30 sections, 69 equations, 36 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Model Reduction
  3. Proper Orthogonal Decomposition
  4. Projection-Based Reduced-Order Model
  5. Least-squares Petrov-Galerkin Projection
  6. Hyperreduction
  7. Energy-conserving Sampling and Weighting Method
  8. Reduced Mesh Set Selection
  9. Residual-based ECSW Training Process
  10. Jacobian-based ECSW Training Process
  11. Non-Negative Least-Squares Problem
  12. Hyperreduced ROM Solution Evaluation with ECSW
  13. Adaptive Sampling Procedure
  14. Overview of the Adaptive Sampling Procedure
  15. Dual-Weighted Residual Errors
  16. ...and 15 more sections

Figures (36)

  • Figure 1: Example Solution of NACA0012 Airfoil Test Case
  • Figure 2: Snapshot and ROM Points with Estimated and True Error for the ROM
  • Figure 3: Snapshot and ROM Points with Estimated and True Error for the HROM w/o Hyper-DWR
  • Figure 4: Snapshot and ROM Points with Estimated and True Error for the HROM w/ Hyper-DWR
  • Figure 5: Maximum and Mean Error Estimate at the ROM Points during each Sampling Iteration for the Three Models
  • ...and 31 more figures