Koopman analysis of Burgers equation

Abstract

The emergence of Dynamic Mode Decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear PDE presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in the PDE dynamics. However, there are many subtleties in connecting DMD to Koopman analysis and it remains unclear how realistic Koopman analysis is for complex systems such as the Navier-Stokes equations. With this as motivation, we present here a full Koopman decomposition for the velocity field in Burgers equation by deriving explicit expressions for the Koopman modes and eigenfunctions -the first time this has been done for a nonlinear PDE. The decomposition highlights the fact that different observables can require different subsets of Koopman eigenfunctions to express them and presents a nice example where: (i) the Koopman modes are linearly dependent and so cannot be fit a posteriori to snapshots of the flow without knowledge of the Koopman eigenfunctions; and (ii) the Koopman eigenvalues are highly degenerate which means that computed Koopman modes become initial-condition dependent. As way of illustration, we discuss the form of the Koopman expansion with various initial conditions and assess the capability of DMD to extract the decaying nonlinear coherent structures in run-down simulations.In complex systems such as turbulent fluid flows, it has long been hoped that the state of the system could be decomposed into a series of (spatially) simpler states with known time behaviour to aid understanding and offer opportunities for prediction and control. While this seems to go against the nonlinearity inherent in complex systems, the Koopman operator [1,2], which is an infinite dimensional linear operator that shifts observables (functionals) of the state forward in time, offers some reasons for optimism. Because of its linearity, the Koopman operator possesses eigenfunctions with exponential time behaviour (given by the associated eigenvalue) which appear to capture some essence of the underlying PDE. In particular, neutral eigenfunctions can identify basins of attraction of invariant sets in the underlying dynamical system, while slowly decaying Koopman eigenfunctions identify least damped coherent structures in transient systems.However, until recently, it has been unclear how to analyse the Koopman operator except in the simplest settings. The discovery of Dynamic Mode Decomposition (DMD) [3] has changed this situation by presenting a practical way to extract dynamic modes which have a fixed spatial structure and an exponential dependence on time from numerical and experimental time series [e.g. 4-6]. These modes are (right) eigenvectors of a best-fit linear operator which maps between equispaced-in-time vectors of observables (measurements) from the dataset and can be connected under certain circumstances to (vector) Koopman modes which weight the (scalar) Koopman eigenfunctions in their expansion of the observable vector [7,8]. Beyond the considerable interest in its co...

Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

“…The dynamics (7) are conjugated to the linear diffusion dynamics v = ∂ 2 v/∂t 2 through the socalled Cole-Hopf transformation [23]. As explained in [8,24], this implies that the Koopman spectrum, associated with Burgers dynamics, coincides with the Koopman spectrum associated with linear diffusion dynamics. Thus, it follows that Koopman eigenfunctions are related to the eigenfunctions of the diffusion operator (see Section 3.1) and, in particular, they are of the form ζ(v) = sin(kπx/2), v(x) α (in the new variable v).…”

Deep learning models for global coordinate transformations that linearise PDEs

“…Importantly, many of the advantageous transformations highlighted above attempt to construct Koopman embeddings for the dynamics using neural networks [24,27,29,36,42,43,47]. This is in addition to enriching the observables of DMD [19,23,34,35,37,45,46]. Thus, neural networks have emerged as a highly effective mathematical architecture for approximating complex data [2,13].…”

Koopman mode expansions between simple invariant solutions

“…Koopman eigenfunctions have been obtained analytically in some simple nonlinear ordinary differential equations (e.g. Bagheri 2013;Brunton et al 2016b;Rowley & Dawson 2017) and recently for Burgers' equation which can be linearized by the Cole-Hopf transformation (Page & Kerswell 2018). However, it is unlikely that closed-form expressions for Koopman eigenfunctions of the Navier-Stokes equations can be written down.…”

“…'Kernel' based methods, see Kutz et al 2016a). Furthermore, even if DMD can accurately extract Koopman eigenfunctions, there is no guarantee that these then form a basis for the state variable itself (e.g Brunton et al 2016b;Page & Kerswell 2018). Alongside DMD, other related methods have been proposed to extract Koopman modes from turbulent flows that may circumvent some of these issues.…”