Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator

Abstract

Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD)51 and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables which spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky partical differential equation as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy, we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.

Document Details

Document Type
Pub Defense Publication
Publication Date
Oct 01, 2017
Source ID
10.1063/1.4993854

Entities

People

  • Erik M. Bollt
  • Felix Dietrich
  • Qianxiao Li
  • Yannís G. Kevrekidis

Organizations

  • Army Research Office
  • Clarkson University
  • Defense Advanced Research Projects Agency
  • National Science Foundation
  • Office of Naval Research
  • Princeton University
  • Technical University of Munich

Tags

Fields of Study

  • Computer science

Readers

  • Adaptive Control and Estimation with Uncertainty in Dynamic Systems.
  • Computer Engineering
  • Distributed Systems and Data Platform Development

Technology Areas

  • AI & ML
  • AI & ML - Machine Learning Algorithms
  • AI & ML - Neural Networks