Model order reduction via Lie groups

TL;DR

This paper introduces MORLie, a novel model order reduction framework using Lie groups that effectively handles non-equivariant dynamics and outperforms traditional methods in accuracy and efficiency.

Contribution

The paper develops a geometric MOR framework on Lie groups, enabling reduction of complex systems with non-equivariant dynamics and providing new non-intrusive reduction methods.

Findings

MORLie outperforms naive POD in accuracy and dimensionality reduction for deforming body data.

MORLie achieves near state-of-the-art performance in liver motion reconstruction with reduced training time.

The method of freezing is recovered as a special case within the geometric framework.

Abstract

Lie groups and their actions are ubiquitous in the description of physical systems, and we explore implications in the setting of model order reduction (MOR). We present a novel framework of MOR via Lie groups, called MORLie, in which high-dimensional dynamical systems on manifolds are approximated by low-dimensional dynamical systems on Lie groups. In comparison to other Lie group methods we are able to attack non-equivariant dynamics, which are frequent in practical applications, and we provide new non-intrusive MOR methods based on the presented geometric formulation. We also highlight numerically that MORLie has a lower error bound than the Kolmogorov $N$-width, which limits linear-subspace methods. The method is applied to various examples: 1. MOR of a simplified deforming body modeled by noisy point cloud data following a sheering motion, where MORLie outperforms a naive POD approach in terms of accuracy and dimensionality reduction. 2. Reconstructing liver motion during respiration with data from edge detection in MRI scans, where MORLie reaches performance approaching the state of the art, while reducing the training time from hours on a computing cluster to minutes on a mobile workstation. 3. An analytic example showing that the method of freezing is analytically recovered as a special case, showing the generality of the geometric framework.