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Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaos

Caroline L. Wormell

Abstract

Extended Dynamic Mode Decomposition (EDMD) is a data-driven tool for forecasting and model reduction of dynamics, which has been extensively taken up in the physical sciences. While the method is conceptually simple, in deterministic chaos it is unclear what its properties are or even what it converges to. In particular, it is not clear how EDMD's least-squares approximation treats the classes of differentiable functions on which chaotic systems act. We develop for the first time a general, rigorous theory of EDMD on the simplest examples of chaotic maps: analytic expanding maps of the circle. To do this, we prove a new, basic approximation result in the theory of orthogonal polynomials on the unit circle (OPUC) and apply methods from transfer operator theory. We show that in the infinite-data limit, the least-squares projection error is exponentially small for trigonometric polynomial observable dictionaries. As a result, we show that forecasts and Koopman spectral data produced using EDMD in this setting converge to the physically meaningful limits, exponentially fast with respect to the size of the dictionary. This demonstrates that with only a relatively small polynomial dictionary, EDMD can be very effective, even when the sampling measure is not uniform. Furthermore, our OPUC result suggests that data-based least-squares projection may be a very effective approximation strategy more generally.

Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaos

Abstract

Extended Dynamic Mode Decomposition (EDMD) is a data-driven tool for forecasting and model reduction of dynamics, which has been extensively taken up in the physical sciences. While the method is conceptually simple, in deterministic chaos it is unclear what its properties are or even what it converges to. In particular, it is not clear how EDMD's least-squares approximation treats the classes of differentiable functions on which chaotic systems act. We develop for the first time a general, rigorous theory of EDMD on the simplest examples of chaotic maps: analytic expanding maps of the circle. To do this, we prove a new, basic approximation result in the theory of orthogonal polynomials on the unit circle (OPUC) and apply methods from transfer operator theory. We show that in the infinite-data limit, the least-squares projection error is exponentially small for trigonometric polynomial observable dictionaries. As a result, we show that forecasts and Koopman spectral data produced using EDMD in this setting converge to the physically meaningful limits, exponentially fast with respect to the size of the dictionary. This demonstrates that with only a relatively small polynomial dictionary, EDMD can be very effective, even when the sampling measure is not uniform. Furthermore, our OPUC result suggests that data-based least-squares projection may be a very effective approximation strategy more generally.
Paper Structure (10 sections, 23 theorems, 134 equations, 3 figures)

This paper contains 10 sections, 23 theorems, 134 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Polynomial approximation result
  3. The dynamics picture
  4. Main result on EDMD
  5. Paper plan
  6. Detailed setup and theorems
  7. Cholesky factorisation
  8. Proof of orthogonal polynomial results
  9. Transfer and Koopman operator results
  10. Proofs of transfer operator bounds

Key Result

Theorem 1.1

Suppose that $\sigma, \tau: \mathbb{Z} \to \mathbb{R}_+$ are Beurling weights with $\tau/\sigma$ decreasing on $\mathbb{N}$. Suppose furthermore that $\mu = h \,\mathrm{d} x$ is a positive measure on $\mathbb{T}$ with $M^{-1} \leq h(x) \leq M$ on $\mathbb{T}$ and $\|\sigma \mathcal{F}[(\log h)']\|_{ where $\mathcal{P}_K$ is the $L^2(\mu)$-orthogonal projection onto trigonometric polynomials of deg

Figures (3)

  • Figure 1: Top left: graph of the circle map $f(x) = 4x - 0.4\sin 6x+0.08\cos 3x \mod 2\pi$. Bottom left: Ruelle-Pollicott resonances (i.e. Perron-Frobenius operator eigenvalues) of $f$ with certain eigenvalues marked in colour. Top right: modulus of EDMD eigenvalues obtained with $\mu$ the physical measure of $f$, with colours corresponding to eigenvalues in the bottom left. Bottom right: exponential convergence with respect to dictionary size of the errors between EDMD eigenvalues and Ruelle-Pollicott responances, with corresponding colours. Ruelle-Pollicott resonances estimated using a Fourier transfer operator discretisation Wormell19.
  • Figure 2: For the circle map given in Figure \ref{['fig:circlemap']}, the absolute value of the left (left plot) and right (right plot) eigenmodes of the Koopman operator approximation $\mathcal{K}_K$ corresponding to the second-largest eigenvalue. The different modes of convergence (respectively in a space of analytic functions, and weakly, as a hyperdistribution) are shown.
  • Figure 3: Convergence of the EDMD operator spectrum for fixed approximation size $K=8$ as the number of data points goes to infinity. In pale lines, the corresponding spectrum.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Corollary 2.2
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['p:CholeskyExists']}
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 35 more