Sparse learning of stochastic dynamical equations

Boninsegna, Lorenzo; Nüske, Feliks; Clementi, Cecilia

doi:10.1063/1.5018409

Cited by 203 publications

(218 citation statements)

References 28 publications

Supporting

Mentioning

216

Contrasting

Order By: Relevance

“…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.…”

mentioning

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

Self Cite

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

mentioning

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

Self Cite

187

160

View full text Add to dashboard Cite

show abstract

“…Comparing the results for λ = 0.01, we observe that the solutions are slightly sparser for λ = 0.2 and λ = 0.1 (e.g., coefficients corresponding to the basis φ 1 ≡ 1 in Table 10), while the approximation of the true coefficients is better when λ is smaller (i.e., λ = 0.01, underlined coefficients in Table 10). Finally, we point out that the solution could be further improved if necessary, by using thresholding techniques (i.e., removing unimportant basis functions) [10] or cross-validation techniques (i.e., tuning λ) [8].…”

Section: Example 2: Predator-prey Systemmentioning

confidence: 99%

“…Let us first review related work and summarize the contributions of this paper. The reconstruction of the governing equations from data using sparsity constraints is getting more and more attention, see [53,10,37,14] for methods pertaining to ordinary differential equations (ODEs) and [8] for learning stochastic differential equations (SDEs). For chemical and biological reaction systems, the problem of estimating unknown parameters has been well studied when the systems are modeled both as ODEs [28,3] and as continuous-time Markov chain processes [1,41,9,54], while the reconstruction of the entire chemical reaction networks, i.e., finding parsimonious models, has only been considered when the systems are modeled as ODE systems [51,37,14].…”

Section: Introductionmentioning

confidence: 99%

“…Thirdly, we provide a theoretical justification of the proposed data-driven method. Note that, although different data-driven methods using sparsity [53,10,8] have been developed in the literature for different types of dynamical systems, the theoretical analysis of these methods is largely incomplete (see [49]). We expect the theoretical analysis presented in the current work to shed light on the characteristic properties of other data-driven methods as well.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Learning Chemical Reaction Networks from Trajectory Data

Zhang¹,

Klus²,

Conrad³

et al. 2019

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

We develop a data-driven method to learn chemical reaction networks from trajectory data. Modeling the reaction system as a continuous-time Markov chain and assuming the system is fully observed, our method learns the propensity functions of the system with predetermined basis functions by maximizing the likelihood function of the trajectory data under l 1 sparse regularization. We demonstrate our method with numerical examples using synthetic data and carry out an asymptotic analysis of the proposed learning procedure in the infinite-data limit.

show abstract

“…Sparse learning of dynamical equations has been studied in Refs. [33,34]. Most of the previous work was aimed at recovering correct thermodynamics of the reduced system, that is, the distribution sampled by the effective dynamics should equal the distribution of the projected original process.…”

Section: Introductionmentioning

confidence: 99%

Coarse-graining molecular systems by spectral matching

Nüske

Boninsegna

Clementi

2019

The Journal of Chemical Physics

Self Cite

View full text Add to dashboard Cite

Coarse-graining has become an area of tremendous importance within many different research fields. For molecular simulation, coarse-graining bears the promise of finding simplified models such that long-time simulations of large-scale systems become computationally tractable. While significant progress has been made in tuning thermodynamic properties of reduced models, it remains a key challenge to ensure that relevant kinetic properties are retained by coarse-grained dynamical systems. In this study, we focus on data-driven methods to preserve the rare-event kinetics of the original system, and make use of their close connection to the low-lying spectrum of the system's generator. Building on work by Crommelin and Vanden-Eijnden, SIAM Multiscale Model. Simul. (2011), we present a general framework, called spectral matching, which directly targets the generator's leading eigenvalue equations when learning parameters for coarse-grained models. We discuss different parametric models for effective dynamics and derive the resulting data-based regression problems. We show that spectral matching can be used to learn effective potentials which retain the slow dynamics, but also to correct the dynamics induced by existing techniques, such as force matching.

show abstract

Sparse learning of stochastic dynamical equations

Cited by 203 publications

References 28 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

Learning Chemical Reaction Networks from Trajectory Data

Coarse-graining molecular systems by spectral matching

Contact Info

Product

Resources

About