Model reduction on manifolds: A differential geometric framework

TL;DR

This paper introduces a differential geometric framework for model reduction on smooth manifolds, unifying and generalizing existing techniques by emphasizing geometric structures and nonlinear projections.

Contribution

It provides a novel geometric approach to model reduction that captures and extends various existing methods, including structure-preserving and nonlinear projection techniques.

Findings

Framework unifies multiple MOR techniques

Allows for structure preservation in Hamiltonian and Lagrangian systems

Includes data-driven methods for nonlinear projections

Abstract

Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our framework.