Koopman Operator Spectrum for Random Dynamical Systems

Črnjarić-Žic, Nelida; Mačešić, Senka; Mezić, Igor

doi:10.1007/s00332-019-09582-z

Cited by 62 publications

(59 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where a = σ σ and ∇ 2 x denotes the Hessian. Properties of the generator associated with non-deterministic dynamical systems are studied in [29]. The function u(t, x) = K t f (x) satisfies the second-order partial differential equation ∂u ∂t = Lu, which is called the Kolmogorov backward equation [30].…”

Section: Non-deterministic Dynamical Systemsmentioning

confidence: 99%

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

Klus

Nüske

Peitz

et al. 2020

Physica D: Nonlinear Phenomena

187

160

View full text Add to dashboard Cite

We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems. arXiv:1909.10638v1 [math.DS] 23 Sep 2019Most of the aforementioned techniques turn out to be strongly related, with the unifying concept being Koopman operator theory [16,17,18]. In what follows, we will focus mainly on the generator of the Koopman operator and its properties and applications. SINDy [5] constitutes a milestone for data-driven discovery of dynamical systems. Because of the close relationship between the vector field of a deterministic dynamical system and its Koopman generator, SINDy is a special case of the framework we will introduce in this study. In [19,20], the authors presented an extension of SINDy to determine eigenfunctions of the Koopman generator. The discovered eigenfunctions are then used for control, resulting in the so-called KRONIC framework. Another extension of SINDy was derived in [21], allowing for the identification of parameters of a stochastic system using Kramers-Moyal formulae.A different avenue towards system identification was taken in [22,23]. Here, the Koopman operator is first approximated with the aid of EDMD, and then its generator is determined using the matrix logarithm. Subsequently, the right-hand side of the differential equation is extracted from the matrix representation of the generator. The relationship between the Koopman operator and its generator was also exploited in [24] for parameter estimation of stochastic differential equations.A method for computing eigenfunctions of the Koopman generator was proposed in [25], where the diffusion maps algorithm is used to set up a Galerkin-projected eigenvalue problem with orthogonal basis elements. Two efficient methods for computing the generator of the adjoint Perron-Frobenius operator based on Ulam's method and spectral collocation were presented in [26]. Provided that a model of the system dynamics is available, the computation of trajectories can be replaced by evaluations of the right-hand side of the system, which is often orders of magnitude faster.The purpose of this study is to present a general framework to compute a matrix approximation of the Koop...

show abstract

Section: Non-deterministic Dynamical Systemsmentioning

confidence: 99%

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

Klus

Nüske

Peitz

et al. 2020

Physica D: Nonlinear Phenomena

187

160

View full text Add to dashboard Cite

show abstract

“…By exploiting the duality between Koopman and P-F operators the work in [19] provides novel naturally structured DMD algorithm for data-driven approximation of both Koopman and P-F operator that preserves positivity and Markov properties of these operators. Recent work has focused on the data-driven approximation of Koopman operator for random dynamical systems (RDS) [20], [21]. In [20] the authors have provided characterization of the spectrum and eigenfunctions of the Koopman operator for discrete and continuous time RDS, while in [21], the authors have provided an algorithm to compute the Koopman operator for systems with both process and observation noise.…”

Section: Introductionmentioning

confidence: 99%

“…Recent work has focused on the data-driven approximation of Koopman operator for random dynamical systems (RDS) [20], [21]. In [20] the authors have provided characterization of the spectrum and eigenfunctions of the Koopman operator for discrete and continuous time RDS, while in [21], the authors have provided an algorithm to compute the Koopman operator for systems with both process and observation noise. The results in [21] claim that DMD algorithm will approximate Koopman operator for RDS in the asymptotic limit of large data set.…”

Section: Introductionmentioning

confidence: 99%

On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems

2019

View full text Add to dashboard Cite

In the paper, we consider the problem of robust approximation of transfer Koopman and Perron-Frobenius (P-F) operators from noisy time series data. In most applications, the time-series data obtained from simulation or experiment is corrupted with either measurement or process noise or both. The existing results show the applicability of algorithms developed for the finite dimensional approximation of deterministic system to a random uncertain case. However, these results hold true only in asymptotic and under the assumption of infinite data set. In practice the data set is finite, and hence it is important to develop algorithms that explicitly account for the presence of uncertainty in data-set. We propose a robust optimization-based framework for the robust approximation of the transfer operators, where the uncertainty in data-set is treated as deterministic norm bounded uncertainty. The robust optimization leads to a min-max type optimization problem for the approximation of transfer operators. This robust optimization problem is shown to be equivalent to regularized least square problem. This equivalence between robust optimization problem and regularized least square problem allows us to comment on various interesting properties of the obtained solution using robust optimization. In particular, the robust optimization formulation captures inherent tradeoffs between the quality of approximation and complexity of approximation. These tradeoffs are necessary to balance for the proposed application of transfer operators, for the design of optimal predictor. Simulation results demonstrate that our proposed robust approximation algorithm performs better than the Extended Dynamic Mode Decomposition (EDMD) and DMD algorithms for a system with process and measurement noise.

show abstract

“…Towards this goal various data-driven methods are proposed for the finite dimensional approximation of these operators [5], [21]- [24], with Dynamic Mode Decomposition (DMD) and extended DMD being the ones which are used extensively. Recent works have also addressed the problem of computing these operators for systems with process and observation noise and for Random Dynamical Systems (RDS) [25]- [28]. In [25] the authors have provided a characterization of the spectrum and eigenfunctions of the Koopman operator for discrete and continuous time RDS, while in [26], the authors have provided an algorithm to compute the Koopman operator for systems with both process and observation noise.…”

Section: Introductionmentioning

confidence: 99%

“…Recent works have also addressed the problem of computing these operators for systems with process and observation noise and for Random Dynamical Systems (RDS) [25]- [28]. In [25] the authors have provided a characterization of the spectrum and eigenfunctions of the Koopman operator for discrete and continuous time RDS, while in [26], the authors have provided an algorithm to compute the Koopman operator for systems with both process and observation noise. In [27], [28] the authors used robust optimization-based techniques to compute the approximate Koopman operator for data sets of finite length and have shown that normal DMD or EDMD and subspace DMD [26] lead to an unsatisfactory approximation of Koopman operator for data sets of finite length.…”

Section: Introductionmentioning

confidence: 99%

On Computation of Koopman Operator from Sparse Data

Sinha

Vaidya

Yeung

2019

2019 American Control Conference (ACC)

View full text Add to dashboard Cite

In this paper we propose a novel approach to compute the Koopman operator from sparse time series data. In recent years there has been considerable interests in operator theoretic methods for data-driven analysis of dynamical systems. Existing techniques for the approximation of the Koopman operator require sufficiently large data sets, but in many applications, the data set may not be large enough to approximate the operators to acceptable limits. In this paper, using ideas from robust optimization, we propose an algorithm to compute the Koopman operator from sparse data. We enrich the sparse data set with artificial data points, generated by adding bounded artificial noise and and formulate the noisy robust learning problem as a robust optimization problem and show that the optimal solution is the Koopman operator with smallest error. We illustrate the efficiency of our proposed approach in three different dynamical systems, namely, a linear system, a nonlinear system and a dynamical system governed by a partial differential equation.

show abstract

Koopman Operator Spectrum for Random Dynamical Systems

Cited by 62 publications

References 44 publications

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems

On Computation of Koopman Operator from Sparse Data

Contact Info

Product

Resources

About