[Paper Review] Koopman Theory and Linear Approximation Spaces

Andrew J. Kurdila, Parag Bobade|arXiv (Cornell University)|Nov 27, 2018

Model Reduction and Neural Networks55 references7 citations

TL;DR

This paper establishes convergence rates for Koopman and Perron-Frobenius operator approximations in data-driven dynamical systems using linear approximation spaces, wavelets, and eigenfunctions. It decomposes error into approximation and sample components, deriving semi-optimal convergence rates of order $(\log m / m)^{2r/(2r+1)}$ under i.i.d. sampling, with bounds dependent on function smoothness and approximation space structure.

ABSTRACT

Koopman theory studies dynamical systems in terms of operator theoretic properties of the Perron-Frobenius operator $\mathcal{P}$ and Koopman operator $\mathcal{U}$ respectively. In this paper, we derive the rates of convergence of approximations of $\mathcal{P}$ or $\mathcal{U}$ that are generated by finite dimensional bases like wavelets, multiwavelets, and eigenfunctions, as well as approaches that use samples of the input and output of the system in conjunction with these bases. We introduce a general class of priors that describe the information available for constructing such approximations and facilitate the error estimates in many applications of interest. These priors are defined in terms of the action of $\mathcal{P}$ or $\mathcal{U}$ on certain linear approximation spaces. The rates of convergence for the estimates of these operators are investigated under a variety of situations that are motivated from associated assumptions in practical applications. When the estimates of these operators are generated by samples, it is shown that the error in approximation of the Perron-Frobenius or Koopman operators can be decomposed into two parts, the approximation error and the sampling error. This result emphasizes that sample-based estimates of Perron-Frobenius and Koopman operators are subject to the well-known trade-off between the bias and variance that contribute to the error, a balance that also features in nonlinear regression and statistical learning theory.

Motivation & Objective

• To establish rigorous convergence rates for approximations of the Koopman and Perron-Frobenius operators in nonlinear dynamical systems.
• To characterize the trade-off between approximation error and sample error in data-driven operator estimation.
• To develop a general framework for priors based on spectral and linear approximation spaces to enable error estimation across diverse applications.
• To analyze the performance of wavelet, multiwavelet, and eigenfunction-based bases in approximating these operators.
• To derive bounds for EDMD-type algorithms using sampled system trajectories, accounting for both bias and variance in estimation.

Proposed method

• Uses a general class of priors defined via the action of the Koopman or Perron-Frobenius operator on linear approximation spaces $A^{r,q}(X)$, with $X$ a Banach space of functions on $\Omega$.
• Applies Hilbert space frameworks $U = L^2_\mu(\Omega)$ and reproducing kernel Hilbert spaces (RKHS) to define the functional setting.
• Derives error decomposition into approximation error (dependent on function smoothness and basis choice) and sample error (dependent on sampling distribution).
• Employs wavelets and multiwavelets that reproduce piecewise polynomials to link results to finite element and spline-based Galerkin methods.
• Analyzes EDMD-type algorithms using empirical regression and exact data-driven methods, with error bounded via $L^2$-norms and sup-norms.
• Uses concentration inequalities to bound the probability of poor sampling outcomes, showing exponential decay in bad set measure with sample size.

Experimental results

Research questions

• RQ1 What are the convergence rates of Koopman and Perron-Frobenius operator approximations when using finite-dimensional bases such as wavelets or eigenfunctions?
• RQ2 How do approximation error and sample error trade off in data-driven estimation of these operators?
• RQ3 What is the best attainable convergence rate for EDMD-type algorithms under i.i.d. sampling, and how does it depend on function smoothness?
• RQ4 How can the approximation space structure (e.g., wavelets, eigenfunctions) be leveraged to derive tight error bounds?
• RQ5 Can the error in sample-based operator estimation be decomposed into interpretable components, and how do they scale with sample size and basis dimension?

Key findings

• The error in EDMD approximations of the Koopman operator is bounded by $\mathbb{E}_{\nu^m}\left(\|\mathcal{U}f - \mathcal{U}^{edmd}_{j,z}f\|_U\right) \lesssim \left(\frac{\log m}{m}\right)^{2r/(2r+1)}$ for i.i.d. samples, which is semi-optimal up to the $\log m$ factor.
• The approximation error is bounded by $O(2^{-rj/2})$ when the function $f$ lies in the spectral approximation space $A^{r,2}(U)$, reflecting dependence on smoothness $r$ and resolution $j$.
• The sample error term is bounded by $\epsilon$ with high probability, where the measure of the bad set of samples decays exponentially with sample size $m$.
• For functions in $\text{Lip}^*(r, L^\infty(\Omega))$, the projection error $\|(I - \Pi_j)f\|_{L^\infty(\Omega)} \approx 2^{-rj}$, linking smoothness to approximation accuracy.
• When the input data are generated by a strongly mixing Markov chain, the effective sample size $e(m)$ replaces $m$ in the rate, yielding $\mathbb{E}_{\mathbb{P}^m_{\{z\}}}\left(\|\mathcal{U}f - \mathcal{U}^{edmd}_{j,z}f\|_U\right) \lesssim \left(\frac{\log e(m)}{e(m)}\right)^{2r/(2r+1)}$.
• The analysis confirms that wavelet and multiwavelet bases with polynomial reproduction properties achieve approximation rates comparable to those of finite element and spline methods in Galerkin schemes.