Machine-learning invariant foliations in forced systems for reduced order modelling

TL;DR

This paper introduces a method to construct reduced order models of forced dynamical systems using invariant foliations, enabling better understanding and prediction of complex forced behaviors.

Contribution

It presents a novel data-driven approach to identify invariant foliations and tori in forced systems, advancing reduced order modeling techniques.

Findings

Successfully identifies invariant foliations from data

Tracks invariant tori and scales invariance equations

Highlights limitations of current invariant manifold fitting methods

Abstract

We identify reduced order models (ROM) of forced systems from data using invariant foliations. The forcing can be external, parametric, periodic or quasi-periodic. The process has four steps: 1. identify an approximate invariant torus and the linear dynamics about the torus; 2. identify a globally defined invariant foliation about the torus; 3. identify a local foliation about an invariant manifold that complements the global foliation 4. extract the invariant manifold as the leaf going through the torus and interpret the result. We combine steps 2 and 3, so that we can track the location of the invariant torus and scale the invariance equations appropriately. We highlight some fundamental limitations of invariant manifolds and foliations when fitting them to data, that require further mathematics to resolve.