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Non intrusive method for parametric model order reduction using a bi-calibrated interpolation on the Grassmann manifold

M. Oulghelou, C. Allery

TL;DR

This work tackles the challenge of non-intrusive interpolation for nonlinear parametrized problems by leveraging subspace interpolation on the Grassmann manifold. Building on the ITSGM framework, it introduces Bi-CITSGM, which interpolates spatial and temporal POD bases, interpolates singular values with cubic splines, and calibrates the resulting bases with two orthogonal matrices to preserve POD structure, all without requiring access to governing equations. The method achieves real-time predictions for untrained parameters, demonstrated on a flow past a cylinder with Reynolds numbers in the range [90,450], showing velocity errors around 1% and pressure errors under 6%, outperforming intrusive ITSGM/Galerkin in speed. The approach is broadly applicable to numerical and experimental parametrized data and offers a practical path for fast, accurate non-intrusive reduced order models in engineering applications.

Abstract

Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and leads generally to incorrect results. However, instead of using a straight forward interpolation on solutions, reduced order models can be interpolated. More particularly, Amsallem and Farhat proposed an efficient POD reduced order model interpolation technique based on differential geometry tools. This approach, named in this paper ITSGM (Interpolation On a Tangent Space of the Grassmann Manifold), allows through the passage to the tangent space of the Grassmann manifold, to approximate accurately the reduced order basis associated to a new untrained parameter. This basis is used afterwards to build the interpolated ROM describing the temporal dynamics by performing the Galerkin projection on the high fidelity model. Such Galerkin ROMs require to access to the underlying high fidelity model, leading by that to intrusive ROMs. In this paper, and contrary to the ITSGM/Galerkin approach, we propose a non-intrusive reduced order modeling method which is independent of the governing equations. This method is named through this paper Bi-CITSGM (Bi-Calibrated ITSGM). It consists first to interpolate the spatial and temporal POD sampling bases considered as representatives of points on Grassmann manifolds, by the ITSGM method. Then, the resulting bases modes are reclassified by introducing two orthogonal matrices. These calibration matrices are determined as analytical solutions of two optimization problems. Results on the flow problem past a circular cylinder where the parameter of interpolation is the Reynolds number, show that for new untrained Reynolds number values, the developed approach produces satisfyingly accurate solutions in a real computational time.

Non intrusive method for parametric model order reduction using a bi-calibrated interpolation on the Grassmann manifold

TL;DR

This work tackles the challenge of non-intrusive interpolation for nonlinear parametrized problems by leveraging subspace interpolation on the Grassmann manifold. Building on the ITSGM framework, it introduces Bi-CITSGM, which interpolates spatial and temporal POD bases, interpolates singular values with cubic splines, and calibrates the resulting bases with two orthogonal matrices to preserve POD structure, all without requiring access to governing equations. The method achieves real-time predictions for untrained parameters, demonstrated on a flow past a cylinder with Reynolds numbers in the range [90,450], showing velocity errors around 1% and pressure errors under 6%, outperforming intrusive ITSGM/Galerkin in speed. The approach is broadly applicable to numerical and experimental parametrized data and offers a practical path for fast, accurate non-intrusive reduced order models in engineering applications.

Abstract

Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and leads generally to incorrect results. However, instead of using a straight forward interpolation on solutions, reduced order models can be interpolated. More particularly, Amsallem and Farhat proposed an efficient POD reduced order model interpolation technique based on differential geometry tools. This approach, named in this paper ITSGM (Interpolation On a Tangent Space of the Grassmann Manifold), allows through the passage to the tangent space of the Grassmann manifold, to approximate accurately the reduced order basis associated to a new untrained parameter. This basis is used afterwards to build the interpolated ROM describing the temporal dynamics by performing the Galerkin projection on the high fidelity model. Such Galerkin ROMs require to access to the underlying high fidelity model, leading by that to intrusive ROMs. In this paper, and contrary to the ITSGM/Galerkin approach, we propose a non-intrusive reduced order modeling method which is independent of the governing equations. This method is named through this paper Bi-CITSGM (Bi-Calibrated ITSGM). It consists first to interpolate the spatial and temporal POD sampling bases considered as representatives of points on Grassmann manifolds, by the ITSGM method. Then, the resulting bases modes are reclassified by introducing two orthogonal matrices. These calibration matrices are determined as analytical solutions of two optimization problems. Results on the flow problem past a circular cylinder where the parameter of interpolation is the Reynolds number, show that for new untrained Reynolds number values, the developed approach produces satisfyingly accurate solutions in a real computational time.

Paper Structure

This paper contains 14 sections, 42 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Some preliminary results of the Grassmann manifold
  3. Subspaces interpolation using ITSGM method
  4. Bi-CITSGM method
  5. Problem statement
  6. Description of the method
  7. Computational complexity
  8. Numerical example : flow past a cylinder
  9. Problem description
  10. Construction of the POD sampling bases
  11. Application of the Bi-CITSGM to predict the flow for untrained Reynolds number values
  12. Conclusions
  13. Review of POD
  14. POD reduced order model of the flow past a cylinder

Figures (7)

  • Figure 1: Two-dimensional domain and boundary conditions for the problem of flow past a circular cylinder.
  • Figure 2: POD eigenvalues and corresponding ratio with respect to trained Reynolds numbers. $q$ is referred to as the number of POD modes.
  • Figure 3: POD (left) and calibrated ITSGM (right) velocity modes associated to the untrained Reynolds number $Re = 195$.
  • Figure 4: POD (left) and calibrated ITSGM (right) pressure modes associated to the untrained Reynolds number $Re = 195$.
  • Figure 5: Lift and drag coefficients $C_D$ and $C_L$ obtained by Bi-CITSGM and ITSGM/Galerkin with those obtained by the high fidelity model at the untrained Reynolds number $Re = 195$.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Remark 1