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Time-Delay Observables for Koopman: Theory and Applications

Mason Kamb, Eurika Kaiser, Steven L. Brunton, J. Nathan Kutz

arXiv:1810.01479v2math.NAmath.DS

TL;DR

Nonlinear Koopman analysis requires useful finite-dimensional observables despite the theory’s infinite-dimensional setting. This paper uses time-delay convolutional coordinates, derives universal analytic Koopman representations, and shows that SVD-based coordinates connect these representations to DMD while providing exact finite-dimensional guarantees for certain systems. The method generally gives superior or comparable approximations, with limitations under spectral crowding and unsuitable basis or embedding choices.

  • Problem

    Koopman methods make nonlinear dynamics linear in an infinite-dimensional observable space, but selecting a finite subspace that accurately represents the system remains difficult.

  • Method

    The paper represents dynamics with convolutional coordinates formed by projecting time-delay observables onto basis functions, using trajectory SVDs to obtain finite-dimensional coordinates.

  • Results

    For a fixed basis, the Koopman representation in convolutional coordinates is system-independent; SVD-basis dynamics match DMD estimates, and the first N coordinates exactly encode systems with order-N Koopman mode expansions.

  • Takeaways & Limitations

    Time-delay convolutional coordinates provide a theoretically grounded observable family for Koopman approximation, with SVD coordinates offering a practical finite-dimensional choice.

  • Takeaways & Limitations

    Finite approximations can become unreliable under spectral crowding, while valid representations also depend on suitable basis functions and embedding choices.

Abstract

from arXiv · show

Nonlinear dynamical systems are ubiquitous in science and engineering, yet analysis and prediction of these systems remains a challenge. Koopman operator theory circumvents some of these issues by considering the dynamics in the space of observable functions on the state, in which the dynamics are intrinsically linear and thus amenable to standard techniques from numerical analysis and linear algebra. However, practical issues remain with this approach, as the space of observables is infinite-dimensional and selecting a subspace of functions in which to accurately represent the system is a nontrivial task. In this work we consider time-delay observables to represent nonlinear dynamics in the Koopman operator framework. We prove the surprising result that Koopman operators for different systems admit universal (system-independent) representations in these coordinates, and give analytic expressions for these representations. In addition, we show that for certain systems a restricted class of these observables form an optimal finite-dimensional basis for representing the Koopman operator, and that the analytic representation of the Koopman operator in these coordinates coincides with results computed by the dynamic mode decomposition. We provide numerical examples to complement our results. In addition to being theoretically interesting, these results have implications for a number of linearization algorithms for dynamical systems.

1 Introduction

The paper addresses the difficulty of finding useful Koopman observables for nonlinear systems by using time-delay and convolutional coordinates. It establishes system-independent linear representations, identifies an SVD-based finite basis with several guarantees, and reports generally successful numerical approximations with specific failure cases.

  • Motivation: Koopman theory linearizes nonlinear dynamics in an infinite-dimensional observable space, but selecting eigenfunctions or finite observable subspaces remains difficult.Existing DMD, variable-selection, and EDMD approaches require suitable observables or prior choices.
  • Related work: Time-delay embedding augments state information with history and has been used for attractor reconstruction, system identification, and Koopman eigenfunction computation.Prior work includes ERA, singular-vector projections of delay embeddings, and HAVOK analysis.
  • Contribution: Convolutional coordinates generalize HAVOK by projecting time-delay coordinates onto an infinite orthonormal basis, yielding linear dynamics independent of the underlying system for a fixed basis.This representation is exact in the infinite-dimensional case, whereas finite truncations are generally poor without a suitable basis.
  • Contribution: Analytically computed linear dynamics on an SVD basis match those estimated by a DMD-type algorithm.
  • Contribution: For systems with a Koopman mode expansion of order N, the first N convolutional coordinates exactly encode the dynamics and span the associated Koopman eigenfunctions.The paper states that no other family of observable functions is known to provide a similar guarantee.
  • Contribution: The SVD basis also diagonalizes the signal autocorrelation structure and enables substantially faster DMD computation for these observables.
  • Results: Numerical examples generally produce superior or comparable Koopman approximations, while failures occur for nearly degenerate eigenvalues and small delay windows.

2 Background

The background develops Koopman operators as linear actions on observable functions and reviews DMD, EDMD, and time-delay principal-component methods for approximating them. It emphasizes that observable selection and computational scale govern approximation quality, while SVD principal components provide an optimal low-rank reconstruction basis.

  • Koopman operators: A nonlinear flow induces a Koopman operator that advances observable functions through composition with the flow map.The operators are linear even when the underlying state-space dynamics are nonlinear.
  • Koopman operators: Koopman eigenfunctions provide invariant coordinates in which the operator acts finitely and linearly, and a complete eigenfunction basis can solve observable dynamics exactly.
  • Dynamic Mode Decomposition: DMD estimates a linear propagator from successive snapshot matrices, then computes its eigendecomposition and associated modes.The method can use truncated SVD coordinates to reduce the computational cost of large operators.
  • Dynamic Mode Decomposition: EDMD applies DMD to nonlinear observables and estimates a model of the form g^-1 ◦ K ◦ g, with classic DMD as the identity-observable special case.
  • Dynamic Mode Decomposition: EDMD operators converge to the Koopman projection onto the span of the chosen observables with infinite data, but generic finite selections may omit relevant dynamics and large dictionaries can overfit.
  • Time Delay Embedding: Time-delay principal components are normalized and uncorrelated, and their first r components give the least-squares best rank-r approximation of the trajectory space.They are obtained from the singular vectors of a Hankel matrix and can be truncated using singular-value thresholds.

3 Convolutional Coordinates

The paper defines convolutional coordinates by projecting local trajectory segments onto an orthonormal basis, then shows that their Koopman dynamics have an analytic representation determined by the basis rather than the underlying system. The representation is exact under stated convergence and domain assumptions, while discrete-time extensions require stronger analyticity conditions.

  • Coordinate construction: Convolutional coordinates are projections of local measurement trajectories onto orthonormal basis functions over a time window.They are formed by convolving basis functions with the observable trajectory and can be interpreted as coordinates on the time-delay embedded state.
  • Representation conditions: The basis expansion reconstructs the delayed observable in an L2 sense generally, and pointwise when observables and bases provide stronger regularity such as continuity with Chebyshev polynomials.The theorem assumes uniform convergence of the expansion over the selected domain.
  • System-independent dynamics: For appropriately chosen bases and observables, the Koopman generator acts on convolutional coordinates through a simple analytic representation independent of the system dynamics.The representation depends on the selected basis functions rather than the underlying dynamical system or measurement functions.
  • Representation conditions: The validity of the representation depends on selecting a suitable domain and basis, with Fourier expansions requiring restrictive periodicity while Chebyshev bases offer broader convergence flexibility.For suitable bases, the window length can be arbitrary in theory, although finite sampling can affect reconstruction quality in practice.
  • Representation conditions: Convolutional coordinates form a Koopman-invariant subspace when the system has an invariant measure and the basis expansions converge appropriately.Without trajectories remaining in the domain, existence of the generator representation alone does not guarantee Koopman invariance.
  • Discrete-time extension: Discrete-time dynamics admit an analogous representation only under analyticity and analytic-continuation assumptions on both trajectories and basis functions.The paper notes that these assumptions are substantially stronger than those required for the continuous-time theorem and may limit usefulness.

4 SVD Convolutional Coordinates

SVD convolutional coordinates provide a data-driven finite representation of delay-embedded dynamics, with analytic Koopman structure that connects directly to DMD and EDMD. Under finite-mode conditions, leading coordinates can be optimal and recover Koopman-invariant dynamics, while truncation quality depends on regularity and window length.

  • Motivation: Finite sampling and coordinate limits make basis selection essential because arbitrary finite bases can distort the estimated spectrum and produce unstable eigenvalues.The finite-dimensional spectrum depends strictly on the chosen basis, so a problem-specific basis is needed.
  • Analytic and Data-Driven Representations: The analytic Koopman representation in convolutional coordinates matches least-squares estimates and the maps obtained by EDMD and DMD on those coordinates.The discrete-time linear map minimizing the one-step prediction error coincides with DMD, while the closest linear approximation is the exact rank-r truncation of the analytic Koopman operator.
  • Spectral Recovery: If an observable is a finite linear combination of r Koopman eigenfunctions, the first r convolutional coordinates exactly span a Koopman-invariant subspace and recover the associated eigenfunctions.For general observables, finite exact recovery is not guaranteed, but coefficients may decay so finite truncations can still approximate the dynamics.
  • Optimality: The first r SVD coordinates form the closest r-dimensional subspace to the full delay-coordinate space and provide the best least-squares approximation to finite-trajectory dynamics.This optimality result underlies the use of SVD coordinates for truncated Koopman representations.
  • Spectral Dynamics: In the long-data limit, the first n SVD convolutional coordinates yield an antisymmetric map under the stated singular-value assumptions, with spectrum matching the underlying map and approaching the imaginary axis.This suggests effectiveness for systems whose Koopman operators have purely imaginary spectra.
  • Approximation and Limiting Bases: For bounded-variation or analytic delay kernels, singular values and approximation errors decay with rank, while SVD bases approach Legendre or Fourier bases in short- and long-window limits.The long- and short-window limiting bases do not generally diagonalize the correlation matrix, because off-diagonal correlations can remain O(1).
  • Computing SVD Convolutional Coordinates: The SVD of a Hankel matrix yields convolutional coordinates from delay-embedded trajectory data, and a partial SVD can efficiently retain the dynamically informative directions.The Hankel matrix has dimension DN × M, and computing rank r ≪ min(N, M) is proposed as an efficient alternative to a full or thin SVD.

5 Applications to Dynamical Systems

Applications show that SVD-based convolutional coordinates recover Koopman spectral structure and can outperform or match alternative observables across nonlinear systems, while requiring sufficiently large delay windows and careful handling of spectral crowding.

  • 5.1 Linear Systems: Linear-system experiments illustrate that SVD delay embeddings preserve the spectrum of the underlying dynamics in convolutional coordinates.The section uses finite-dimensional linear systems to demonstrate the method and issues introduced by discrete signals.
  • 5.1 Linear Systems: When frequencies are close relative to the delay window, nearly degenerate eigenvalues make the SVD-based model ill-conditioned and error-sensitive.The recommended robust alternative is computing DMD from full convolutional-coordinate trajectories rather than directly from basis vectors.
  • 5.2 The Van der Pol Oscillator: For van der Pol dynamics, the Koopman embedding captures nonlinear frequency shifts and harmonics beyond the weakly nonlinear regime covered by asymptotic expansions.The asymptotic frequency shift is ω = 1 to ω = (1 + 7ϵ2/16), with a third harmonic generated at O(ϵ).
  • 5.2 The Van der Pol Oscillator: With a sufficiently large delay window, SVD-coordinate eigenvalues match dominant van der Pol spectral peaks across weakly to strongly nonlinear regimes.For small delay windows, the estimated spectrum becomes distorted.
  • 5.2 The Van der Pol Oscillator: Off-attractor van der Pol trajectories are reconstructed exactly with 28 SVD convolutional coordinates, whereas polynomial observables of comparable size perform uniformly poorly.The accurate reconstruction requires a window nearly as long as the transient decay time to the attractor.
  • 5.3 Nonlinear Schrödinger Equation: For the nonlinear Schrödinger equation, HAVOK attains the lowest RMS error at truncation rank r = 14 and performs comparably to eDMD with u|u|2.HAVOK test error increases beyond r = 14, while eDMD test error remains steady.
  • 5.4 Understanding Intermittent Forcing in the Lorenz System: In the Lorenz system, analytically derived SVD-coordinate dynamics match DMD-type models and exhibit nearly antisymmetric, mainly first-off-diagonal structure.The coefficient structure explains the forcing behavior associated with models of the leading coordinates.

6 Summary and Discussion

The paper concludes that convolutional coordinates provide system-independent linear Koopman representations, with SVD-based coordinates offering efficient and parsimonious finite-dimensional approximations. Their approximation quality beyond the demonstrated settings remains incompletely characterized.

  • 6 Summary and Discussion: Convolutional coordinates naturally linearize dynamics, and their Koopman representation depends on the chosen basis rather than the underlying system.The representation is derived analytically from the basis functions and their derivatives.
  • 6 Summary and Discussion: SVD convolutional coordinates align analytically derived dynamics with DMD approximations and provide an optimally parsimonious basis for finite discrete-spectrum systems.The SVD basis is given by autocorrelation eigenvectors, equivalently Hankel singular vectors.
  • 6 Summary and Discussion: The quality and error bounds of these approximations for broader classes of systems remain conjectural and require further analysis.The authors identify this characterization as a direction for future research.

Appendix A Delay Embedded Dynamics in Fourier and Legendre Bases

The appendix derives coefficient structures for Fourier and Legendre bases used to represent delay-embedded dynamics. Orthogonality imposes sparsity conditions on the Legendre-basis operator coefficients.

  • Fourier Basis: The Fourier basis {eπint/τ} provides an explicit starting point for deriving convolutional-coordinate coefficients.These basis-specific coefficients can be substituted into the general expression for Ajk.
  • Legendre Basis: Legendre polynomials are orthogonal over the interval and alternate between even and odd parity.The appendix uses a rescaled Legendre basis that is orthonormal on [−τ, τ].
  • Legendre Basis: The derivatives of the rescaled Legendre functions determine the coefficient formulas governing their convolutional dynamics.The appendix introduces the derivatives and corresponding coefficients before simplifying them using orthogonality.
  • Legendre Basis: For j ≥ k, Legendre orthogonality forces Ajk = 0; for j < k, the coefficients reduce to a simpler expression.The coefficient structure therefore has a triangular sparsity constraint.
  • Legendre Basis: The parity condition contributes an additional zero when (j + k) is even.

Appendix B Fast Computation of Singular Vector Observables from the Autocorrelation Function

The appendix replaces a full SVD of a large Hankel matrix with autocovariance-based computations for the singular-vector observables. A truncated Taylor expansion reduces the required inner products from N^2 to nmax, with analogous extensions for multivariate signals.

  • Computational bottleneck: The naive computation forms the SVD of an N × m Hankel matrix, costing O(N^2m + m^2N + min(N^3, m^3)).Here, N is the delay-embedding length and m is the number of snapshots.
  • Autocovariance formulation: The singular-vector basis functions U can be obtained by diagonalizing the autocovariance matrix instead of computing the full SVD.This targets only the basis functions needed for the observables.
  • Reduced computation: Truncating the Taylor series at nmax reduces the computation from N^2 inner products of length-m vectors to nmax such inner products.This provides a significant cost saving over the naive SVD.
  • Autocovariance formulation: Translation invariance makes the autocovariance depend on relative delay, enabling a Taylor expansion in p − q.The autocovariance satisfies A(p, q) = A(p + t, q + t).
  • Multivariate extension: For a state of dimension k, the approach requires k^2nmax inner products rather than the naive k^2N^2.The same reduction extends to multivariate signals through a tensor-valued autocorrelation and componentwise Taylor expansions.
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