Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Koopman operator learning using invertible neural networks

Yuhuang Meng, Jianguo Huang, Yue Qiu

TL;DR

FlowDMD tackles the challenge of learning Koopman embeddings without manually selecting observables by employing a coupling flow invertible neural network (CF-INN) to jointly learn the observable map and its inverse. The method yields a finite-dimensional Koopman representation with a simplified loss that relies on DMD consistency and state reconstruction via the inverse flow, while reducing parameter count due to shared parameters. Across fixed-point, Burgers', and Allen-Cahn benchmarks, FlowDMD outperforms Exact DMD, EDMD, and LIR-DMD in reconstruction accuracy and generalization. The results demonstrate improved interpretability, robustness, and data efficiency for data-driven Koopman embeddings in nonlinear dynamical systems.

Abstract

In Koopman operator theory, a finite-dimensional nonlinear system is transformed into an infinite but linear system using a set of observable functions. However, manually selecting observable functions that span the invariant subspace of the Koopman operator based on prior knowledge is inefficient and challenging, particularly when little or no information is available about the underlying systems. Furthermore, current methodologies tend to disregard the importance of the invertibility of observable functions, which leads to inaccurate results. To address these challenges, we propose the so-called FlowDMD, aka Flow-based Dynamic Mode Decomposition, that utilizes the Coupling Flow Invertible Neural Network (CF-INN) framework. FlowDMD leverages the intrinsically invertible characteristics of the CF-INN to learn the invariant subspaces of the Koopman operator and accurately reconstruct state variables. Numerical experiments demonstrate the superior performance of our algorithm compared to state-of-the-art methodologies.

Koopman operator learning using invertible neural networks

TL;DR

FlowDMD tackles the challenge of learning Koopman embeddings without manually selecting observables by employing a coupling flow invertible neural network (CF-INN) to jointly learn the observable map and its inverse. The method yields a finite-dimensional Koopman representation with a simplified loss that relies on DMD consistency and state reconstruction via the inverse flow, while reducing parameter count due to shared parameters. Across fixed-point, Burgers', and Allen-Cahn benchmarks, FlowDMD outperforms Exact DMD, EDMD, and LIR-DMD in reconstruction accuracy and generalization. The results demonstrate improved interpretability, robustness, and data efficiency for data-driven Koopman embeddings in nonlinear dynamical systems.

Abstract

In Koopman operator theory, a finite-dimensional nonlinear system is transformed into an infinite but linear system using a set of observable functions. However, manually selecting observable functions that span the invariant subspace of the Koopman operator based on prior knowledge is inefficient and challenging, particularly when little or no information is available about the underlying systems. Furthermore, current methodologies tend to disregard the importance of the invertibility of observable functions, which leads to inaccurate results. To address these challenges, we propose the so-called FlowDMD, aka Flow-based Dynamic Mode Decomposition, that utilizes the Coupling Flow Invertible Neural Network (CF-INN) framework. FlowDMD leverages the intrinsically invertible characteristics of the CF-INN to learn the invariant subspaces of the Koopman operator and accurately reconstruct state variables. Numerical experiments demonstrate the superior performance of our algorithm compared to state-of-the-art methodologies.
Paper Structure (20 sections, 27 equations, 14 figures, 3 tables, 1 algorithm)

This paper contains 20 sections, 27 equations, 14 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman operator theory
  4. Dynamic mode decomposition
  5. State reconstruction
  6. Learning Koopman embedding by invertible neural networks
  7. Drawback of AE in the Koopman embedding learning
  8. Structure of CF-INN
  9. Loss function of FlowDMD for Koopman embedding
  10. Numerical experiments
  11. Fixed-point attractor
  12. Burgers' equation
  13. Allen-Cahn equation
  14. Sensitivity study
  15. Sensitivity with respect to the neural network initialization
  16. ...and 5 more sections

Figures (14)

  • Figure 1: Koopman operator and inverse of observable functions
  • Figure 2: Generalization capability test of AE. (a) the training data distribution. (b) the $sin(x)$ test function. (c) S-shaped scatters test. (d) random scatters from 2-d standard normal distribution.
  • Figure 3: The forward and backward directions of ACF and flipped ACF, as well as the structure of an ACF block. Here, the "Id" operation represents the identity mapping.
  • Figure 4: The general framework of FlowDMD.
  • Figure 5: Comparison of four methods for Example \ref{['sec:fixed_point']}. The total relative $L_2$ error of the Exact DMD, EDMD, LIR-DMD, and FlowDMD are 0.2448, 0.08, 0.0111 and 0.0018, respectively.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Definition 1: Koopman operator kutz2016dynamic
  • Definition 2
  • Definition 3: Coupling flow papamakarios2021normalizing
  • Definition 4: Affine coupling flow 9089305
  • Definition 5: Residual coupling flow gomez2017reversible