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Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator
Hassan Arbabi, Igor Mezić
TL;DR
The paper asks whether DMD can reliably recover spectral properties of the infinite-dimensional Koopman operator from observable data. It applies DMD to Hankel matrices and uses ergodicity to approximate function-space projections, proving convergence of eigenvalues and eigenfunctions and relating SVD to POD. These results extend to trajectories approaching ergodic attractors with physical measures, subject to stated assumptions.
Problem
The paper addresses the challenge of establishing convergence from finite-dimensional DMD spectral approximations to the infinite-dimensional Koopman operator's eigenvalues and eigenfunctions.
Method
The paper applies DMD to Hankel matrices of delay-embedded observables and uses ergodicity and Birkhoff's theorem to approximate function projections from data-vector projections.
Results
For almost every initial condition and infinite observation time, Hankel-DMD eigenvalues converge to Koopman eigenvalues and dynamic modes converge to sampled Koopman eigenfunctions.
Takeaways & Limitations
SVD of ergodically sampled observables converges to POD, yielding least-error truncated representations and supporting representations of dynamics with continuous spectrum.
Takeaways & Limitations
The convergence results assume ergodicity and finite-dimensional Koopman-invariant subspaces, while companion-matrix DMD is unstable for ill-conditioned data bases.
Abstract
from arXiv · showhide
We establish the convergence of a class of numerical algorithms, known as Dynamic Mode Decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the Singular Value Decomposition, which is the central part of most DMD algorithms, converges to the Proper Orthogonal Decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.
1 Introduction
The Koopman operator offers a linear framework for analyzing nonlinear dynamics through observables, while DMD provides data-driven approximations of its spectral properties. This paper studies convergence of DMD methods using ergodicity and Hankel-structured observations.
- The Koopman operator describes observable evolution through a linear transformation, providing a framework for data-driven analysis of high-dimensional nonlinear systems.
- Koopman eigenvalues and eigenfunctions encode dynamical information, including invariant manifolds, stability, and asymptotic phase.
- The paper connects DMD, Koopman mode decomposition, and linear system identification, including equivalence with ERA up to a similarity transformation.
- Dynamic Mode Decomposition is a class of methods for computing Koopman eigenvalues, eigenfunctions, and modes from observable time-series data.
- The paper proves convergence of a class of DMD eigenvalues and eigenfunctions to those of the Koopman operator for ergodic systems.
- The proposed methodology applies DMD to Hankel matrices formed by delay-embedding time-series measurements of observables.
2 Review of Dynamic Mode Decomposition (DMD)
This section reviews companion-matrix, SVD-enhanced, and Exact DMD, which construct finite-dimensional operators from data to extract dynamic eigenvalues and modes. It also identifies numerical-stability and projection issues affecting these methods.
- DMD extracts spatial structures with exponential growth or decay and connects this fluid-flow analysis technique to Koopman modes.
- The three reviewed variants are companion-matrix DMD, SVD-enhanced DMD, and Exact DMD.
- DMD constructs a matrix operator that maps data snapshots forward and obtains dynamic eigenvalues and modes from its spectrum.
- Companion-matrix DMD becomes nonunique for linearly dependent columns and numerically unstable when the data basis is ill-conditioned.
- SVD-enhanced DMD uses orthogonal left singular vectors as a basis, improving projection stability and matching companion-matrix DMD under full-rank, distinct-eigenvalue conditions.
- Exact DMD computes an operator mapping corresponding columns of arbitrary-state data matrices X and Y, extending SVD-enhanced DMD beyond sequential sampling.
- Applying Exact DMD to the sequential matrices used by SVD-enhanced DMD produces the same eigenvalues and modes.
3 Ergodic theory and Hankel-matrix representation of data
The paper interprets Hankel data as sampled Koopman Krylov sequences and proves convergence of Hankel-DMD under ergodicity. Birkhoff averages make data-vector projections approximate observable-space projections.
- The companion-matrix Hankel-DMD proof is intuitive but the method is not well-suited for numerical practice, motivating more suitable alternatives later in the paper.
- The analysis assumes an ergodic, measure-preserving map on a compact invariant set and observables in the associated square-integrable Hilbert space.
- Birkhoff's ergodic theorem makes infinite-time averages of sampled observables converge to spatial averages, allowing data-vector inner products to approximate Hilbert-space inner products.
- Delay-embedded Hankel matrices sample the Krylov sequence [f, Uf, ..., U^n f] of an observable under Koopman evolution.
- A finite-dimensional Koopman-invariant subspace containing the observable enables representation of the restricted Koopman operator by a finite-dimensional matrix.
- For almost every initial condition, as m →∞, companion-matrix Hankel-DMD eigenvalues converge to Koopman eigenvalues and dynamic modes converge to sampled Koopman eigenfunctions.
4 Singular Value Decomposition (SVD) and Proper Orthogonal Decomposition (POD) for ergodic systems
For ergodic systems, applying SVD to time-series data matrices provides convergent approximations to the POD basis and its associated coordinates in the observable space. Applied to Hankel data, this yields a data-driven representation of Koopman evolution, including systems with continuous spectrum.
- Proper Orthogonal Decomposition: POD supplies an orthonormal basis whose truncations minimize the average H-norm error when representing the observables.The singular values quantify the H-norm contribution of the corresponding basis elements.
- Data-driven construction: The computational construction uses a data matrix formed by sampling observables along one ergodic trajectory, equivalent to the transpose of a snapshot matrix.Ergodicity makes the numerical Gramian converge to the observable Gramian, enabling recovery of the POD structure from time-series data.
- Convergence of SVD to POD: SVD of ergodic samplings converges to the POD decomposition of the underlying observable ensemble.As observation length grows, singular values, right singular vectors, and sampled left singular vectors converge to their POD counterparts.
- Representation of dynamics: Applying SVD to a Hankel matrix approximates an orthonormal basis for the Krylov sequence [f, Uf, …, U^nf] and represents Koopman evolution through principal coordinates.This lifts coordinates from the state space to the space of observables.
- Numerical example: For the Lorenz attractor, a Hankel matrix with m = 10000 and n = 500 produces approximate basis functions and principal coordinates from the observable f(z) = z1.The computed basis functions and associated singular values show little change for m ≥10000.
- Numerical example: Although Lorenz principal-coordinate evolution is linear, its growing Krylov sequence prevents any finite-dimensional linear system from describing that evolution.For this mixing attractor, only the constant Koopman eigenfunction belongs to the discrete spectrum, while the Krylov sequence is generally n+1-dimensional.
5 Convergence of Exact DMD and extension to multiple observables
Exact DMD converges to Koopman spectral quantities under ergodic sampling when observables span a finite-dimensional invariant subspace. Hankel delay embeddings extend the method to single and multiple observables, with sufficient delays guaranteeing convergence.
- Exact DMD formulation: Exact DMD represents the Koopman operator on an invariant subspace through the unique matrix A satisfying AX = Y.The uniqueness follows when X spans the relevant space and Y is linearly consistent with X.
- Convergence results: For ergodic systems, dynamic eigenvalues converge to Koopman eigenvalues as the number of observations increases.The convergence holds for almost every initial condition under the stated invariant-subspace assumptions.
- Convergence results: The scaled projected modes √mχj converge to samples of the corresponding Koopman eigenfunctions along the trajectory.The result follows from convergence of the candidate functions and the DMD eigenvectors.
- Ergodic attractors: The same convergence result applies in the basin of an ergodic attractor when its invariant measure is physical, the restricted observables span an invariant subspace, and observables are continuous.The conclusion holds for ν-almost every initial condition in the basin.
- Hankel extensions: Delay-embedding a single observable makes F = [f, Uf, ..., U^(n−1)f], reducing Exact DMD to SVD-enhanced DMD on shifted Hankel matrices.Multiple observables can similarly be combined through blocks of Hankel matrices.
- Hankel extensions: For multiple observables, choosing delay lengths l, q > k + 1 guarantees Exact DMD convergence when k is the invariant-subspace dimension containing the observables.Scaling may be needed when observables have substantially different norms, preventing smaller POD components from being discarded by SVD truncation.
6 Numerical application of Hankel-DMD method
Hankel-DMD applies DMD to delay-embedded observable data to approximate Koopman spectral properties on ergodic attractors. Numerical examples cover periodic and quasi-periodic cavity flows, Van der Pol asymptotic phase, and multiple observables.
- Method: Hankel-DMD applies DMD to Hankel matrices formed from delayed time-series measurements on observables.The method uses composite Hankel matrices and a truncated SVD before extracting approximate Koopman eigenvalues and eigenfunctions.
- Method: The number of computed modes depends on signal length and the dimension of the observable’s subspace.
- Convergence: Ergodic-average convergence is generally O(1/m) for periodic and quasi-periodic attractors, c/√m for strongly mixing systems, and rate-unestablished for general ergodic systems.The paper reports that a few hundred samples can accurately determine periodic frequencies, while a few thousand suffice for a 2-torus attractor.
- Periodic cavity flow: At Re = 13000, Hankel-DMD computes periodic cavity-flow eigenfunctions with mean squared error of 10^-5 or smaller for the six highest-vector-energy eigenfunctions.The periodic-flow computation uses 200 kinetic-energy samples with a 0.1-second sampling interval.
- Quasi-periodic cavity flow: At Re = 16000, 6500 kinetic-energy samples support quasi-periodic cavity-flow computations whose frequencies and eigenfunctions agree well with prior results.The trajectory is treated as lying on a parameterized 2-torus, and computed trajectory values are extended across the torus by interpolation.
- Van der Pol oscillator: For the Van der Pol oscillator, Hankel-DMD agrees with a Koopman eigenfunction obtained by Fourier averaging with known frequency.The associated eigenfunction is used to represent asymptotic phase along trajectories.
- Multiple observables: Using two observables preserves comparable frequency and eigenvalue accuracy while capturing new eigenfunctions.The additional observable is the stream function alongside kinetic energy.
7 Summary and future work
The paper connects ergodic-theoretic projection limits with convergent DMD and SVD/POD constructions. It also presents observable-space representations for chaotic dynamics and notes future investigation of dissipative Koopman eigenvalues.
- Summary: For ergodic-attractor systems, DMD algorithms converge through approximation of function projections by vector projections using Birkhoff’s ergodic theorem.
- Summary: SVD of Hankel-embedded time series is connected to POD of observables and can construct an orthonormal basis from ergodic trajectory data.
- Summary: The paper introduces an observable-space representation of chaotic dynamics on mixing attractors for analysis and control purposes.
- Summary: Hankel-DMD computes Koopman spectra from a small number of observables and trajectories in high-dimensional systems such as fluid flows.
- Future work: The method shows promise for computing dissipative Koopman eigenvalues inside the unit circle, which the authors leave for future articles.
Time series data and MATLAB codes
The numerical-example time-series data and MATLAB codes are available from the authors’ stated online resource.
- The time-series data and MATLAB codes for the numerical examples are available at the listed UCSB resource.