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

Robert Szalai

TL;DR

This work identifies reduced order models of forced systems from data using invariant foliations to highlight some fundamental limitations of invariant manifolds and foliations when fitting them to data, that require further mathematics to resolve.

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.

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

TL;DR

This work identifies reduced order models of forced systems from data using invariant foliations to highlight some fundamental limitations of invariant manifolds and foliations when fitting them to data, that require further mathematics to resolve.

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.
Paper Structure (18 sections, 1 theorem, 97 equations, 5 figures)

This paper contains 18 sections, 1 theorem, 97 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Set-up and invariant foliations
  3. Linear dynamics
  4. Invariant foliation
  5. The data and identifying the foliation
  6. Discretisation and shift along the torus
  7. Approximate invariant torus and nearby linear dynamics
  8. Invariant vector bundles about the torus
  9. Finding the invariant foliation
  10. Recovering the invariant manifold
  11. Normal form transformation of the conjugate dynamics
  12. Frequencies and damping ratios
  13. Summary of ROM identification
  14. Examples
  15. Shaw-Pierre oscillator
  16. ...and 3 more sections

Key Result

Theorem 2

Assume an invariant torus eq:TOR-geometry and a linearised system about the torus in the form of eq:ED-linsys-1. Also assume that the linear system has dichotomy spectral intervals $\Sigma_{j}=\left[\alpha_{j},\beta_{j}\right]$, $1\le j\le m$ and that $\max\beta_{j}<1$. Pick one or more spectral int with the condition that $\alpha_{j}\neq0$ for all $j\in\mathcal{I}$ so that linear system eq:ED-lin

Figures (5)

  • Figure 1: Commutative diagrams of a) invariant foliations, b) invariant manifolds, and connection diagrams of c) autoencoders and d) equation-free models. The dashed arrows denote the chain of function composition(s) that involves $\boldsymbol{F}$, the continuous arrows denote the chain of function composition(s) that involves map $\boldsymbol{R}$. The encircled vector space denotes the domain of the defining equation and the boxed vector space is the target of the defining equation. Diagrams (e,f,g,h) represent the defining equations corresponding to diagrams (a,b,c,d). The input (in) is the domain of the defining equation and the output (out) is the target, which must vanish.
  • Figure 2: a) The invariant torus and the invariant manifold at one point along the torus. b) The spectrum of the linear dynamics about the torus. The stars denote the representative eigenvalues as selected by \ref{['eq:BUND-sel']}, the triangle denotes the eigenvalue used for model reduction. c) An incomplete representation of the invariant vector bundle: three out of four coordinates of the first vector that spans the two-dimensional invariant vector bundle.
  • Figure 3: Reduced order models of \ref{['eq:ode-shawpierre']}. a,e) The distribution of data along the leaves of the foliation; b,f) the relative error of the invariance equation over the invariant manifold and over the data; c,g) instantaneous frequency of the vibration predicted by the ROM; d,h) instantaneous damping ratio.
  • Figure 4: a) The invariant torus and the invariant manifold at one point along the torus. b) The spectrum of the linear dynamics about the torus. The stars denote the representative eigenvalues as selected by \ref{['eq:BUND-sel']}, the triangle denotes the eigenvalue use for model reduction. c) An incomplete representation of the invariant vector bundle: three out of four coordinates of the first vector that spans the two-dimensional invariant vector bundle.
  • Figure 5: Reduced order models of \ref{['eq:ode-shawpierre']}. a) The distribution of data along the leaves of the foliation; b) the relative error of the invariance equation over the invariant manifold and over the data; c) instantaneous frequency of the vibration predicted by the ROM; d) instantaneous damping ratio.

Theorems & Definitions (3)

  • Remark 1
  • Theorem 2
  • proof