Globalizing manifold-based reduced models for equations and data

Introduction

Reduced-order modeling is a widespread technique that seeks to simplify high-dimensional nonlinear systems while retaining their essential dynamical features. Among reduced-order modeling procedures, manifold-based methods have been steadily gaining momentum. This can be largely attributed to the prevalence of data-driven approaches that successfully build on the fundamental manifold hypothesis^{1, 2–3} of machine learning.

Center manifold reduction⁴, geometric singular perturbation theory^5,6 and inertial manifold theory^7,8 all rely on the existence of low-dimensional attracting invariant manifolds in the phase space of a dynamical system. These methods constitute mathematically rigorous examples of nonlinear model reduction and yield truly predictive models. However, for systems encountered in practice that are not close to bifurcations, invariant manifolds can only be realistically constructed when they emanate from a known robust stationary state, such as a hyperbolic fixed point.

In those cases, seeking the invariant manifolds perturbatively and expressing them as local Taylor expansions at the known stationary state is justified. Traditionally, only stable and unstable manifolds, as continuations of the stable and unstable subspaces of the linearization, were approximated in this fashion⁴. The recent theory of spectral submanifolds (SSMs) extends this approach to arbitrary non-resonant spectral subspaces of the linearized system⁹. In particular, SSMs are now known to exist as smooth continuations of stable subspaces (like-mode SSMs) and of subspaces spanned by stable and unstable modes (mixed-mode SSMs)¹⁰. This model reduction approach has been used in a broad range of physical settings to deduce very low-dimensional, mathematically justified polynomial models^9,10.

SSM reduction has been successfully applied to obtain accurate reduced models of nonlinear vibrations observed in high-dimensional finite element models^11,12 and experiments^13,14, multistable fluid flows^15,16, chaotic systems¹⁷, fluid-structure interaction problems^18,19 and control of soft robots²⁰. In an equation-driven setting, SSM reduction starts with the solution of an invariance equation through local Taylor expansions^11,21,22. Data-driven SSM reduction¹³ also uses polynomial expansions to obtain an approximation for SSMs.

SSMs are ideal tools for model reduction because their existence, uniqueness, and smoothness properties are precisely known. Specifically, if the governing equations are analytic, SSMs of attracting fixed points are guaranteed to also possess convergent Taylor series near the...