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

TL;DR

This paper introduces an optimal morphing-based model-order reduction method for complex parametric PDE problems with geometric variability, improving reducibility where traditional methods fail.

Contribution

It proposes a novel morphing strategy and a global optimization framework to enhance model reduction for poorly reducible problems with geometric variability.

Findings

Effective reduction of solution variability through morphing

Accurate surrogate models via Gaussian Process regression

Applicable to parameterized and non-parameterized geometries

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.