[Paper Review] Registration-based model reduction in complex two-dimensional geometries

Tommaso Taddei, Lei Zhang|arXiv (Cornell University)|Jan 25, 2021

Model Reduction and Neural Networks32 references4 citations

TL;DR

This paper presents a registration-based model order reduction framework for complex two-dimensional parametric PDEs in non-trivial geometries, such as annular domains and general 2D shapes. By introducing a geometry-aware mapping via spectral element approximation and polar transformation, the method transforms solution manifolds into forms amenable to linear compression, significantly improving reduced-order model accuracy for problems with sharp gradients or complex boundaries.

ABSTRACT

We present a general -- i.e., independent of the underlying equation -- registration procedure for parameterized model order reduction. Given the spatial domain $Ω\subset \mathbb{R}^2$ and the manifold $\mathcal{M}= \{ u_μ : μ\in \mathcal{P} \}$ associated with the parameter domain $\mathcal{P} \subset \mathbb{R}^P$ and the parametric field $μ\mapsto u_μ \in L^2(Ω)$, our approach takes as input a set of snapshots $\{ u^k \}_{k=1}^{n_{ m train}} \subset \mathcal{M}$ and returns a parameter-dependent bijective mapping $Φ: Ω imes \mathcal{P} o \mathbb{R}^2$: the mapping is designed to make the mapped manifold $\{ u_μ \circ Φ_μ: \, μ\in \mathcal{P} \}$ more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition $\{ Ω_{q} \}_{q=1} ^{N_{ m dd}}$ of the domain $Ω$ such that $Ω_1,\ldots,Ω_{N_{ m dd}}$ are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.

Motivation & Objective

• Address the limitation of existing registration-based model order reduction methods, which are restricted to domains diffeomorphic to the unit square.
• Extend the registration framework to handle annular domains through a polar coordinate transformation and spectral approximation.
• Develop a partitioned spectral element approach to generalize registration to arbitrary two-dimensional geometries with smooth boundaries.
• Enable effective linear compression of solution manifolds with slowly decaying Kolmogorov N-widths in complex domains.
• Demonstrate the method’s effectiveness on challenging parametric PDEs, including heat transfer, potential flow, and transonic flow with shock formation.

Proposed method

• Introduce a parametric bijective mapping $\Phi: \Omega \times \mathcal{P} \to \mathbb{R}^2$ that reparameterizes the domain to align solution manifolds for better linear approximation.
• For annular domains, apply a polar transformation $\Psi: \widehat{\Omega}_{\text{pol}} \to \Omega$ to map the annulus to a rectangular reference domain $ (0,1) \times (-1/2, 1/2) $.
• Construct mappings as $ \Phi = \Psi \circ \Phi_{\text{pol}} \circ \Lambda $, where $ \Phi_{\text{pol}} = \text{id} + W_M \mathbf{a} $, with $ W_M $ a linear operator on the reference domain.
• Use a spectral element approximation based on a decomposition $ \{\Omega_q\}_{q=1}^{N_{\text{dd}}} $, each $ \Omega_q $ diffeomorphic to the unit square, to handle general 2D geometries.
• Apply proper orthogonal decomposition (POD) to the mapped solutions $ u_\mu \circ \Phi_\mu $ to construct low-dimensional approximation spaces.
• Leverage RBF regression to predict the mapping coefficients $ \mathbf{a}^\mu $ and solution coefficients $ \boldsymbol{\alpha}^\mu $ for new parameter values.

Experimental results

Research questions

• RQ1 Can registration-based model reduction be extended to annular domains, which are not diffeomorphic to the unit square?
• RQ2 How can spectral element methods be used to generalize registration to arbitrary two-dimensional geometries with smooth boundaries?
• RQ3 To what extent does the proposed registration framework improve the approximation quality of reduced-order models for parametric PDEs with sharp gradients or shocks?
• RQ4 Can the method maintain accuracy and efficiency when applied to high-fidelity, nonlinear PDEs such as transonic Euler flows?
• RQ5 What is the impact of geometry transformation and domain partitioning on the convergence and stability of the reduced-order model?

Key findings

• The proposed method successfully extends registration-based model order reduction to annular domains via a polar transformation and spectral approximation on the reference domain.
• For the transonic flow past a non-symmetric airfoil, the registered ROM achieved significantly lower relative $ L^2 $ prediction errors compared to the unregistered counterpart, especially near shock regions.
• The average relative $ L^2 $ projection error for the registered method dropped below $ 10^{-3} $ for $ N = 20 $ modes in the transonic flow example.
• The relative prediction error for the registered ROM remained below $ 10^{-3} $ across all test parameters, demonstrating robust out-of-sample performance.
• The method enables effective linear compression of solution manifolds in complex geometries where standard POD fails due to slowly decaying Kolmogorov N-widths.
• Numerical results confirm that the mapping $ \Phi_\mu $ effectively aligns solution features (e.g., shocks, boundary layers), making them amenable to linear approximation.