DeepAI AI Chat
Log In Sign Up

Linearly-Recurrent Autoencoder Networks for Learning Dynamics

by   Samuel E. Otto, et al.
Princeton University
Princeton Consultants, Inc.

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 over-fitting 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 over-specified EDMD systems in feature space. Nonlinear reconstruction using partially linear multi-kernel 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.


page 24

page 25

page 27

page 28

page 32


Koopman Operator Theory for Nonlinear Dynamic Modeling using Dynamic Mode Decomposition

The Koopman operator is a linear operator that describes the evolution o...

A dynamical systems based framework for dimension reduction

We propose a novel framework for learning a low-dimensional representati...

Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability

The Koopman operator has emerged as a powerful tool for the analysis of ...

Computing the Invariant Distribution of Randomly Perturbed Dynamical Systems Using Deep Learning

The invariant distribution, which is characterized by the stationary Fok...

Deep Learning of Conjugate Mappings

Despite many of the most common chaotic dynamical systems being continuo...

Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces

Koopman decomposition is a non-linear generalization of eigen decomposit...

Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators

Although projection-based reduced-order models (ROMs) for parameterized ...