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On the lifting and reconstruction of nonlinear systems with multiple invariant sets

Shaowu Pan, Karthik Duraisamy

TL;DR

This work addresses the challenge of applying Koopman operator theory to nonlinear systems with multiple disjoint invariant sets by showing that discontinuous observables can realize a weak but globally valid linear reconstruction across basins. It develops a symmetry-aware framework that reduces lift dimension and enables data-efficient learning, supported by a general decomposition theorem and concrete demonstrations on the unforced Duffing oscillator and the Lorenz attractor. The key contributions include a mechanistic explanation for weak linear reconstruction with discontinuous observables, a symmetry-based extension that yields compact Koopman representations, and empirical evidence that symmetry-constrained EDMD improves generalization and data efficiency. The results have practical implications for scalable Koopman learning in multi-attractor dynamics across engineering and physical systems.

Abstract

The Koopman operator provides a linear perspective on non-linear dynamics by focusing on the evolution of observables in an invariant subspace. Observables of interest are typically linearly reconstructed from the Koopman eigenfunctions. Despite the broad use of Koopman operators over the past few years, there exist some misconceptions about the applicability of Koopman operators to dynamical systems with more than one disjoint invariant sets (e.g., basins of attractions from isolated fixed points). In this work, we first provide a simple explanation for the mechanism of linear reconstruction-based Koopman operators of nonlinear systems with multiple disjoint invariant sets. Next, we discuss the use of discrete symmetry among such invariant sets to construct Koopman eigenfunctions in a data efficient manner. Finally, several numerical examples are provided to illustrate the benefits of exploiting symmetry for learning the Koopman operator.

On the lifting and reconstruction of nonlinear systems with multiple invariant sets

TL;DR

This work addresses the challenge of applying Koopman operator theory to nonlinear systems with multiple disjoint invariant sets by showing that discontinuous observables can realize a weak but globally valid linear reconstruction across basins. It develops a symmetry-aware framework that reduces lift dimension and enables data-efficient learning, supported by a general decomposition theorem and concrete demonstrations on the unforced Duffing oscillator and the Lorenz attractor. The key contributions include a mechanistic explanation for weak linear reconstruction with discontinuous observables, a symmetry-based extension that yields compact Koopman representations, and empirical evidence that symmetry-constrained EDMD improves generalization and data efficiency. The results have practical implications for scalable Koopman learning in multi-attractor dynamics across engineering and physical systems.

Abstract

The Koopman operator provides a linear perspective on non-linear dynamics by focusing on the evolution of observables in an invariant subspace. Observables of interest are typically linearly reconstructed from the Koopman eigenfunctions. Despite the broad use of Koopman operators over the past few years, there exist some misconceptions about the applicability of Koopman operators to dynamical systems with more than one disjoint invariant sets (e.g., basins of attractions from isolated fixed points). In this work, we first provide a simple explanation for the mechanism of linear reconstruction-based Koopman operators of nonlinear systems with multiple disjoint invariant sets. Next, we discuss the use of discrete symmetry among such invariant sets to construct Koopman eigenfunctions in a data efficient manner. Finally, several numerical examples are provided to illustrate the benefits of exploiting symmetry for learning the Koopman operator.
Paper Structure (10 sections, 1 theorem, 20 equations, 5 figures)

This paper contains 10 sections, 1 theorem, 20 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background on Koopman operators
  3. Linear reconstruction for nonlinear systems with multiple disjoint invariant sets
  4. Role of symmetry
  5. Results
  6. Impact of Symmetry on Linear Reconstruction of the Unforced Duffing Oscillator
  7. Exploiting symmetry improves generalization performance
  8. Example: symmetry-constrained EDMD for unforced Duffing oscillator
  9. Example: EDMD trained on symmetry-augmented data for chaotic Lorenz attractor
  10. Conclusions

Key Result

Theorem 4.5

\newlabelthm:10 Given a $\Gamma$-equivariant discrete-time dynamical system $x_{n+1}=F(x_n)$ defined on a manifold $\mathcal{M}$, $n \in \mathbb{N}$ with $J\in \mathbb{N}$ disjoint invariant sets $\{\mathcal{M}_j\}_{j=1}^{J}$ of which the union has full measure in $\mathcal{M}$, if for each set $\ where $\chi_{\mathcal{M}_1}$ is the indicator function for $\mathcal{M}_1$ and $K$ is the finite-dim

Figures (5)

  • Figure 1: Mechanism of lifting with $\Phi(x)$ into higher dimension for Duffing oscillator, which has two fixed asymptotically stable points (while the third one is unstable at the origin). Note that the above observables $\phi_1,\phi_2,\phi_3$ in the figure is only for the ease of illustration since it is just one of the acceptable observables rather than the one we obtained in the actual training. Reconstruction is performed via $\Psi(x)$, which can be linear or nonlinear.
  • Figure 1: Training data for EDMD on the unforced Duffing oscillator. 49 trajectories for training while blue trajectories end up in the positive equilibrium and red ones in the negative equilibrium.
  • Figure 2: Comparison of generalization performance between vanilla EDMD and symmetry-constrained EDMD over a variety of observables. The comparison is based on averaged mean-squared-error over 100 unseen trajectories with initial condition uniformly distributed within the domain and performed across different choice of observables $\Phi(x)$. Hyperparameter of $\Phi(x)$ from left to right are: number of centers, number of pairs of frequencies, maximum order of polynomials.
  • Figure 3: Distributions of three different training data of chaotic Lorenz system projected on $x-y$, $y-z$ and $x-z$ planes. raw data refers to the original trajectory collected. symmetry-augmented data refers to double the size of the original data by rotating the data $pi$ around $z$ axis. symmetry-augmented data halved refers to double the size of halved original data by rotating the data $\pi$ around $z$ axis. The blue dots correspond to the training data in each case. The light green dots are the unseen testing data, which is generated by further integrating the Lorenz system continuing from the last time step of raw data.
  • Figure 4: Generalization performance of vanilla EDMD on three different datasets from the chaotic Lorenz attractor. Since this is a chaotic system, long-term point-wise mean-squared error is less meaningful. Thus we compute the average mean-squared error conditioned in the horizon of future state prediction. Although all models work well in the short time, both two EDMD models trained on symmetry-augmented data performs better in the longer horizon.

Theorems & Definitions (10)

  • Definition 4.1: $\Gamma$-equivariant
  • Remark 4.2
  • Example 1: Unforced Duffing Oscillator
  • Remark 4.3
  • Remark 4.4
  • Theorem 4.5
  • Proof 1
  • Remark 4.6
  • Remark 4.7
  • Remark 4.8