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Optimal morphings for model-order reduction for poorly reducible problems with geometric variability

Abbas Kabalan, Fabien Casenave, Felipe Bordeu, Virginie Ehrlacher, Alexandre Ern

TL;DR

This work tackles model-order reduction for PDEs with geometric variability where the solution manifold has a slowly decaying Kolmogorov $N$-width, limiting standard POD-based methods. It introduces optimal morphings that map each sample to a common reference domain, maximizing the energy captured by the first $r$ POD modes of the pulled-back fields; the morphings are found via gradient ascent with an elasticity-based preconditioner and bijectivity-enforcing penalties, including a continuation strategy. After morphing optimization and POD compression, the approach builds a non-intrusive surrogate using Gaussian Process regression to predict both the optimal morphings and the reduced-field coordinates, forming the O-MMGP framework. The method is general to any PDE and geometry type, showing improved data compression, robustness to geometric variability, and efficient many-query predictions across toy problems and CFD-like datasets. Overall, the paper provides a systematic, geometry-aware MOR pipeline that aligns samples to unlock low-dimensional representations and fast, accurate surrogates.

Abstract

We propose a new model-order reduction framework to poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, the solution manifold exhibits a slowly decaying Kolmogorov $N$-width, so that standard projection-based model order reduction techniques based on linear subspace approximations become ineffective. To overcome this difficulty, we introduce an optimal morphing strategy: For each solution sample, we compute a bijective morphing from a reference domain to the sample domain such that, when all the solution fields are pulled back to the reference domain, their variability is reduced. We formulate a global optimization problem on the morphings that maximizes the energy captured by the first $r$ modes of the mapped fields obtained from Proper Orthogonal Decomposition, thus maximizing the reducibility of the dataset. Finally, using a non-intrusive Gaussian Process regression on the reduced coordinates, we build a fast surrogate model that can accurately predict new solutions, highlighting the practical benefits of the proposed approach for many-query applications. The framework is general, independent of the underlying partial differential equation, and applies to scenarios with either parameterized or non-parameterized geometries.

Optimal morphings for model-order reduction for poorly reducible problems with geometric variability

TL;DR

This work tackles model-order reduction for PDEs with geometric variability where the solution manifold has a slowly decaying Kolmogorov -width, limiting standard POD-based methods. It introduces optimal morphings that map each sample to a common reference domain, maximizing the energy captured by the first POD modes of the pulled-back fields; the morphings are found via gradient ascent with an elasticity-based preconditioner and bijectivity-enforcing penalties, including a continuation strategy. After morphing optimization and POD compression, the approach builds a non-intrusive surrogate using Gaussian Process regression to predict both the optimal morphings and the reduced-field coordinates, forming the O-MMGP framework. The method is general to any PDE and geometry type, showing improved data compression, robustness to geometric variability, and efficient many-query predictions across toy problems and CFD-like datasets. Overall, the paper provides a systematic, geometry-aware MOR pipeline that aligns samples to unlock low-dimensional representations and fast, accurate surrogates.

Abstract

We propose a new model-order reduction framework to poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, the solution manifold exhibits a slowly decaying Kolmogorov -width, so that standard projection-based model order reduction techniques based on linear subspace approximations become ineffective. To overcome this difficulty, we introduce an optimal morphing strategy: For each solution sample, we compute a bijective morphing from a reference domain to the sample domain such that, when all the solution fields are pulled back to the reference domain, their variability is reduced. We formulate a global optimization problem on the morphings that maximizes the energy captured by the first modes of the mapped fields obtained from Proper Orthogonal Decomposition, thus maximizing the reducibility of the dataset. Finally, using a non-intrusive Gaussian Process regression on the reduced coordinates, we build a fast surrogate model that can accurately predict new solutions, highlighting the practical benefits of the proposed approach for many-query applications. The framework is general, independent of the underlying partial differential equation, and applies to scenarios with either parameterized or non-parameterized geometries.

Paper Structure

This paper contains 33 sections, 1 theorem, 41 equations, 22 figures, 1 table.

Table of Contents

  1. Abstract
  2. Introduction
  3. Background
  4. Related works
  5. Poorly-reducible problems
  6. Morphing techniques
  7. Contribution
  8. Methodology
  9. Problem setting and notation
  10. Gradient algorithm
  11. Preconditioning
  12. Bijectivity
  13. Continuation for $c_1$
  14. Computational cost and acceleration strategies
  15. Preliminary examples
  16. ...and 18 more sections

Key Result

Lemma 1

Let $\delta_{ij}$ the Kronecker delta. Define $\mathbf{Z}_{ij}[\Phi][\Psi]:= \langle u_i \circ \boldsymbol{\phi}_i , (\Vec{\nabla} u_j \circ \boldsymbol{\phi}_j)\cdot \boldsymbol{\psi}_j\rangle_{\Omega_0}$. Let $\Phi\in \mathbf{M}$, and let $\Psi:= (\boldsymbol{\psi}_i)_{1\leq i \leq n}$ a perturbat

Figures (22)

  • Figure 1: Example of two mappings from the same reference domain onto the same target domain.
  • Figure 2: $u_i$ for three different values of $\beta$.
  • Figure 3: $u_i \circ \boldsymbol{\phi}_i$ for three different values of $\beta$ without using the continuation on $c_1$.
  • Figure 4: $u_i\circ \boldsymbol{\phi}_i$ for three different values of $\beta$, using continuation on $c_1$.
  • Figure 5: $\boldsymbol{\phi}_i$ for three different values of $\beta$ without using continuation on $c_1$.
  • ...and 17 more figures

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Remark 1: Initialization
  • Remark 2: Safeguard check
  • Remark 3: Combining continuation on $c_1$ and $c_2$