Data-driven model reduction of agent-based systems using the Koopman generator

Niemann, Jan-Hendrik; Klus, Stefan; Schütte, Christof

doi:10.1371/journal.pone.0250970

Cited by 11 publications

(6 citation statements)

References 49 publications

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“…Differentiating nodes by their community is frequently used in the literature, e.g. [27,41,22]. The well-known stochastic block model, which we discuss in more detail in section 4.4, generates random graphs that exhibit this clustered structure.…”

Section: Classes and Collective Variablesmentioning

confidence: 99%

Large population limits of Markov processes on random networks

Marvin¹,

Heitzig²,

Koltai³

et al. 2022

Preprint

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We consider time-continuous Markovian discrete-state dynamics on random networks of interacting agents and study the large population limit. The dynamics are projected onto low-dimensional collective variables given by the shares of each discrete state in the system, or in certain subsystems, and general conditions for the convergence of the collective variable dynamics to a mean-field ordinary differential equation are proved. We discuss the convergence to this mean-field limit for a continuous-time noisy version of the so-called "voter model" on Erdős-Rényi random graphs, on the stochastic block model, as well as on random regular graphs. Moreover, a heterogeneous population of agents is studied. For each of these types of interaction networks, we specify the convergence conditions in dependency on the corresponding model parameters.

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Section: Classes and Collective Variablesmentioning

confidence: 99%

Large population limits of Markov processes on random networks

Marvin¹,

Heitzig²,

Koltai³

et al. 2022

Preprint

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show abstract

“…Moreover, the techniques developed here are used in the computational part of the recently developed framework [12]. The infinitesimal generator approach has also been successfully used for learning stochastic models from aggregated trajectory data [13,14], as well as for inverse modelling of Markov jump processes [15]. The stochastic framework is not considered in this paper, but its results apply to systems defined by stochastic differential equations as described in [8].…”

Section: Contributions and Organisation Of The Papermentioning

confidence: 99%

“…The column pivoting has the rank revealing property and the triangular factor is diagonally dominant in a very strong sense, see e.g., [23,30] and (13).…”

Section: Pivoted Qr Factorization Based Preconditionermentioning

confidence: 99%

Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

Drmač

Mezić

Mohr³

2021

Mathematics

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Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems.

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“…eigenfunctions contain information about the locations of the metastable states. Methods for detecting metastability also have important applications in climate science or agent-based modeling, see, for instance, [50,51,52,53,54,55].…”

Section: Introductionmentioning

confidence: 99%

Koopman-Based Spectral Clustering of Directed and Time-Evolving Graphs

Klus

Conrad

2022

J Nonlinear Sci

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While spectral clustering algorithms for undirected graphs are well established and have been successfully applied to unsupervised machine learning problems ranging from image segmentation and genome sequencing to signal processing and social network analysis, clustering directed graphs remains notoriously difficult. Two of the main challenges are that the eigenvalues and eigenvectors of graph Laplacians associated with directed graphs are in general complex-valued and that there is no universally accepted definition of clusters in directed graphs. We first exploit relationships between the graph Laplacian and transfer operators and in particular between clusters in undirected graphs and metastable sets in stochastic dynamical systems and then use a generalization of the notion of metastability to derive clustering algorithms for directed and time-evolving graphs. The resulting clusters can be interpreted as coherent sets, which play an important role in the analysis of transport and mixing processes in fluid flows. Graphic Abstract

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Data-driven model reduction of agent-based systems using the Koopman generator

Cited by 11 publications

References 49 publications

Large population limits of Markov processes on random networks

Large population limits of Markov processes on random networks

Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

Koopman-Based Spectral Clustering of Directed and Time-Evolving Graphs

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