Data-Driven Operator Theoretic Methods for Global Phase Space Learning

Nandanoori, Sai Pushpak; Sinha, Subhrajit; Yeung, Enoch

doi:10.23919/acc45564.2020.9147220

Cited by 14 publications

(6 citation statements)

References 61 publications

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“…since any function f ∈ G(X ) can be decomposed into disjoint components owing to A, B being disjoint. Reference [245] gives further theoretical backing to this process, as well an incremental and a non-incremental version of the data-driven DMD procedure built in this fashion. For practical purposes, the assumption that A and B are truly dynamically separate is not needed; rather, it is simply sufficient to choose pairs of DMD trajectory samples in a trajectory from two disjoint sets in order to construct such restricted approximations that can be stitched [304].…”

Section: Action-angle Coordinates and Global Linearizationmentioning

confidence: 99%

Modern Koopman Theory for Dynamical Systems

Brunton¹,

Budišić²,

Kaiser³

et al. 2021

Preprint

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The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinitedimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated in terms of measurements, making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.

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Section: Action-angle Coordinates and Global Linearizationmentioning

confidence: 99%

Modern Koopman Theory for Dynamical Systems

Brunton¹,

Budišić²,

Kaiser³

et al. 2021

Preprint

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show abstract

“…Nevertheless, several efficient approaches have been developed to construct a finite-dimensional approximation of the Koopman operator, including dynamic mode decomposition (DMD) [13,14], extended DMD [15], robust DMD [16], and deep DMD (deepDMD) [17][18][19][20]. Koopman operator-based techniques have found many applications in analysis of nonlinear systems, especially in fluid mechanics [13], synthetic biology [20][21][22] and energy systems [20,[23][24][25][26][27][28]. Majority of the data-driven KOT applications in power systems have focused on nonlinear modal analysis (via the use of Koopman modes), e.g.coherency detection [24,25], attack identification [26] or anomaly diagnosis [27], with a few also considering time-series prediction [20,27,29].…”

Section: Introductionmentioning

confidence: 99%

Graph Neural Network and Koopman Models for Learning Networked Dynamics: A Comparative Study on Power Grid Transients Prediction

Nandanoori¹,

Guan²,

Kundu³

et al. 2022

Preprint

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Continuous monitoring of the spatio-temporal dynamic behavior of critical infrastructure networks, such as the power systems, is a challenging but important task. In particular, accurate and timely prediction of the (electro-mechanical) transient dynamic trajectories of the power grid is necessary for early detection of any instability and prevention of catastrophic failures. Existing approaches for prediction of dynamic trajectories either rely on the availability of accurate physical models of the system, use computationally expensive time-domain simulations, or are applicable only at local prediction problems (e.g., a single generator). In this paper, we report the application of two broad classes of data-driven learning models -along with their algorithmic implementation and performance evaluation -in predicting transient trajectories in power networks using only streaming measurements and the network topology as input. One class of models is based on the Koopman operator theory which allows for capturing the nonlinear dynamic behavior via an infinite-dimensional linear operator. The other class of models is based on the graph convolutional neural networks which are adept at capturing the inherent spatio-temporal correlations within the power network. Transient dynamic datasets for training and testing the models are synthesized by simulating a wide variety of load change events in the IEEE 68-bus system, categorized by the load change magnitudes, as well as by the degree of connectivity and the distance to nearest generator nodes. The results confirm that the proposed predictive models can successfully predict the post-disturbance transient evolution of the system with a high level of accuracy.

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“…Literature Review: The Koopman operator [1] provides an alternative representation of the evolution of dynamical systems in terms of observables rather than system trajectories. Despite possible nonlinearies in the system, the eigenfunctions of the Koopman operator evaluated on the system's trajectories have linear temporal evolution, and this leads to efficient numerical methods used in complex system analysis [2], [3], estimation [4], control [5]- [8], and robotics [9], [10], to name a few. Despite these appealing applications, the infinite-dimensional nature of the operator This work was supported by ONR Award N00014-18-1-2828.…”

Section: Introductionmentioning

confidence: 99%

Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition

Haseli¹,

Cortés²

2022

Preprint

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Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the action of the Koopman operator on a linear function space spanned by a dictionary of functions. The accuracy of EDMD model critically depends on the quality of the particular dictionary's span, specifically on how close it is to being invariant under the Koopman operator. Motivated by the observation that the residual error of EDMD, typically used for dictionary learning, does not encode the quality of the function space and is sensitive to the choice of basis, we introduce the novel concept of consistency index. We show that this measure, based on using EDMD forward and backward in time, enjoys a number of desirable qualities that make it suitable for data-driven modeling of dynamical systems: it measures the quality of the function space, it is invariant under the choice of basis, can be computed in closed form from the data, and provides a tight upper-bound for the relative root mean square error of all function predictions on the entire span of the dictionary.

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Data-Driven Operator Theoretic Methods for Global Phase Space Learning

Cited by 14 publications

References 61 publications

Modern Koopman Theory for Dynamical Systems

Modern Koopman Theory for Dynamical Systems

Graph Neural Network and Koopman Models for Learning Networked Dynamics: A Comparative Study on Power Grid Transients Prediction

Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition

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