Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition

By Naoya Takeishi, Yoshinobu Kawahara, Takehisa Yairi. Neural Information Processing Systems, 2017

468 citations recorded by the source corpus.

Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral analysis; however, we often need to prepare nonlinear observables manually according to the underlying dynamics, which is not always possible since we may not have any a priori knowledge about them. In this paper, we propose a fully data-driven method for Koopman spectral analysis based on the principle of learning Koopman invariant subspaces from observed data. To this end, we propose minimization of the residual sum of squares of linear least-squares regression to estimate a set of functions that transforms data into a form in which the linear regression fits well. We introduce an implementation with neural networks and evaluate performance empirically using nonlinear dynamical systems and applications.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/49ee811e5c7af5980d3dfe791f6b18e0080ad566

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

Citations in the corpus

Deep Dynamical Modeling and Control of Unsteady Fluid Flows (2018, methodology): “the loss ‖Ỹ − ( Ỹ X̃† ) X̃‖(2)F is minimized), and proposed learning such a mapping with deep neural networks [19].”
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders (2018, background): “[69, 56, 47, 53], which apply autoencoders to learn approximate invariant subspaces of the Koopman operator; and Ref.”
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].”
Variational Approach for Learning Markov Processes from Time Series Data (2017, methodology): “Some recent methods extend EDMDDL for modeling Koopman operators of deterministic systems [58,31,44], and solve the first problem by using the prediction error between the observed xt and that predicted by the low-dimensional model.”
Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems (2017, methodology): “The ideas in this paper are complementary to the recent work presented in [10] and [11], [12], [13].”
Data-driven discovery of coordinates and governing equations (2019, methodology): “Several recent works have focused on the use of deep learning methods to discover the proper coordinates for DMD and extended DMD [31, 32, 79].”
Universal Differential Equations for Scientific Machine Learning (2020, methodology): “[59] Naoya Takeishi, Yoshinobu Kawahara, and Takehisa Yairi.”
Linearly-Recurrent Autoencoder Networks for Learning Dynamics (2017, unknown): “Ours and concurrent approaches utilizing nonlinear decoder neural networks [30, 16] enable learning of very small sets of features that carry rich information about the state and evolve linearly in time.”
Transformer for Partial Differential Equations' Operator Learning (2022, methodology): “…done, the system can evolve in time using Recurrent Neural Networks (RNNs) such as LSTM (Wiewel et al., 2019) or even linear propagators based on the assumptions of learning Koopman operator (Morton et al., 2018; Li et al., 2020a; Lusch et al., 2018; Takeishi et al., 2017; Pan & Duraisamy, 2020).”