Abstract
This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and overfitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of overspecified EDMD systems in feature space. Nonlinear reconstruction using partially linear multikernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto-Sivashinsky equation.
Original language | English (US) |
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Pages (from-to) | 558-593 |
Number of pages | 36 |
Journal | SIAM Journal on Applied Dynamical Systems |
Volume | 18 |
Issue number | 1 |
DOIs | |
State | Published - 2019 |
All Science Journal Classification (ASJC) codes
- Analysis
- Modeling and Simulation
Keywords
- Data-driven analysis
- High-dimensional systems
- Koopman operator
- Neural networks
- Nonlinear systems
- Reduced-order modeling