Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems

By Enoch Yeung, Soumya Kundu, Nathan Oken Hodas. American Control Conference, 2017

479 citations recorded by the source corpus.

The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode decomposition; this requires a combinatorially large basis set to adequately describe many nonlinear systems of interest, e.g. cyber-physical infrastructure systems, biological networks, social systems, and fluid dynamics. Often the dictionaries generated for these problems are manually curated, requiring domain-specific knowledge and painstaking tuning. In this paper we introduce a computational framework for learning Koopman operators of nonlinear dynamical systems using deep learning. We show that this novel method automatically selects efficient deep dictionaries, requiring much lower dimensional dictionaries while outperforming state-of-the-art methods. We benchmark this method on partially observed nonlinear systems, including the glycolytic oscillator and show it is able to predict on test data quantitatively 100 steps into the future, using only a single timepoint as an initial condition, and quantitative oscillatory behavior 400 steps into the future.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a02e0608b79a4b3670324839160073a8f49d77f9

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Citations in the corpus

Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks (2018, background): “Recently, data-driven approximations of Koopman operators, including dynamic mode decomposition [26], diﬀusion maps [26], delay co-ordinates [8, 4, 1], and neural networks [27, 23, 15, 14, 17], have gained signiﬁcant attention.”
Enhancing computational fluid dynamics with machine learning (2021, background): “DMD has also been extended to nonlinear coordinate embed-dings through the use of deep-autoencoder networks [126–130].”
Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, methodology): “Though dynamic mode decomposition [178] is the most common technique for approximating the Koopman operator, many recent approaches have been made to approximate Koopman operator embeddings with deep learning models that outperform existing methods [150, 163, 171, 180, 188, 190, 244, 262, 286].”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, methodology): “After the first submission of this manuscript (in May 2017), several similar approaches to learning data transform for Koopman analysis have been proposed [41, 42, 43, 44, 45].”
Data-driven discovery of coordinates and governing equations (2019, unknown): “Meth-ods such as dynamic mode decomposition (DMD) (4, 5, 9), extended DMD (48), and time-delay DMD (49) build linear models for dynamics, but these methods rely on a proper set of coordinates for linearization.”
Deep Dynamical Modeling and Control of Unsteady Fluid Flows (2018, background): “A number of recent works have studied whether deep learning can be used to learn this set of observables automatically, thereby circumventing the need to hand-specify a dictionary of functions [19], [34]–[36].”
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems (2020, methodology): “Though dynamic mode decomposition [197] is the most common technique for approximating the Koopman operator, many recent approaches have been made to approximate Koopman operator embeddings with deep learning models that outperform existing methods [170, 185, 191, 200, 209, 210, 260, 284, 307].”
Linearly-Recurrent Autoencoder Networks for Learning Dynamics (2017, unknown): “nt neural network, furnishing our approximation of the Koopman operator on the subspace of observables. This approach diers from recent eorts that use neural networks to learn dictionaries for EDMD [31,14] in that we employ a second neural network to reconstruct the full state. This allows us to avo”