Globalizing manifold-based reduced models for equations and data

TL;DR

This paper introduces a novel method using Padé approximants to extend local invariant manifold models, enabling accurate global reduced models for complex dynamical systems in mechanics.

Contribution

It develops a globalized manifold reduction technique that overcomes local Taylor expansion limitations, broadening the scope of nonlinear model reduction.

Findings

Enables reduced modeling of large-scale oscillations and chaos

Demonstrates effectiveness on solid and fluid mechanics examples

Significantly extends the applicability of manifold-based reduction

Abstract

One of the very few mathematically rigorous nonlinear model reduction methods is the restriction of a dynamical system to a low-dimensional, sufficiently smooth, attracting invariant manifold. Such manifolds are usually found using local polynomial approximations and, hence, are limited by the unknown domains of convergence of their Taylor expansions. To address this limitation, we extend local expansions for invariant manifolds via Pad\'e approximants, which re-express the Taylor expansions as rational functions for broader utility. This approach significantly expands the range of applicability of manifold-reduced models, enabling reduced modeling of global phenomena, such as large-scale oscillations and chaotic attractors of finite element models. We illustrate the power of globalized manifold-based model reduction on several equation-driven and data-driven examples from solid mechanics and fluid mechanics.