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Efficient pseudometrics for data-driven comparisons of nonlinear dynamical systems

Bryan Glaz

TL;DR

The paper addresses the challenge of efficiently quantifying how far nonlinear dynamical systems are from topological conjugacy in a data-driven setting. It develops a Pareto-optimal framework in Koopman eigenfunction space, showing that restricting transformations to the unitary group yields theoretically consistent, polynomial-time solutions for the two critical Pareto points, Cr1 and Cr2. By establishing equivalence between Phi-space and Psi-space formulations and providing explicit, tractable solutions (via Procrustes for r1 and eigenvalue permutation for r2), the authors couple operator and trajectory dissimilarities into a unified metric. The approach is demonstrated on a benchmarking problem and an engineering example comparing biological and artificial actuation, illustrating zero residuals at conjugacy and meaningful, scalable distinctions between dynamical systems, with all computations implemented in MATLAB.

Abstract

Computationally efficient solutions for pseudometrics quantifying deviation from topological conjugacy between dynamical systems are presented. Deviation from conjugacy is quantified in a Pareto optimal sense that accounts for spectral properties of Koopman operators as well as trajectory geometry. Theoretical justification is provided for computing such pseudometrics in Koopman eigenfunction space rather than observable space. Furthermore, it is shown that theoretical consistency with topological conjugacy can be maintained when restricting the search for optimal transformations between systems to the unitary group. Therefore the pseudometrics are based on analytical solutions for unitary transformations in Koopman eigenfunction space. Geometric considerations for the deviation from conjugacy Pareto optimality problem are used to develop scalar pseudometrics that account for all possible optimal solutions given just two Pareto points. The approach is demonstrated on two example problems; the first being a simple benchmarking problem and the second an engineering example comparing the dynamics of morphological computation of biological nonlinear muscle actuators to simplified mad-made (including bioinspired) approaches. The benefits of considering operator and trajectory geometry based dissimilarity measures in a unified and consistent formalism is demonstrated. Overall, the deviation from conjugacy pseudometrics provide practical advantages in terms of efficiency and scalability, while maintaining theoretical consistency.

Efficient pseudometrics for data-driven comparisons of nonlinear dynamical systems

TL;DR

The paper addresses the challenge of efficiently quantifying how far nonlinear dynamical systems are from topological conjugacy in a data-driven setting. It develops a Pareto-optimal framework in Koopman eigenfunction space, showing that restricting transformations to the unitary group yields theoretically consistent, polynomial-time solutions for the two critical Pareto points, Cr1 and Cr2. By establishing equivalence between Phi-space and Psi-space formulations and providing explicit, tractable solutions (via Procrustes for r1 and eigenvalue permutation for r2), the authors couple operator and trajectory dissimilarities into a unified metric. The approach is demonstrated on a benchmarking problem and an engineering example comparing biological and artificial actuation, illustrating zero residuals at conjugacy and meaningful, scalable distinctions between dynamical systems, with all computations implemented in MATLAB.

Abstract

Computationally efficient solutions for pseudometrics quantifying deviation from topological conjugacy between dynamical systems are presented. Deviation from conjugacy is quantified in a Pareto optimal sense that accounts for spectral properties of Koopman operators as well as trajectory geometry. Theoretical justification is provided for computing such pseudometrics in Koopman eigenfunction space rather than observable space. Furthermore, it is shown that theoretical consistency with topological conjugacy can be maintained when restricting the search for optimal transformations between systems to the unitary group. Therefore the pseudometrics are based on analytical solutions for unitary transformations in Koopman eigenfunction space. Geometric considerations for the deviation from conjugacy Pareto optimality problem are used to develop scalar pseudometrics that account for all possible optimal solutions given just two Pareto points. The approach is demonstrated on two example problems; the first being a simple benchmarking problem and the second an engineering example comparing the dynamics of morphological computation of biological nonlinear muscle actuators to simplified mad-made (including bioinspired) approaches. The benefits of considering operator and trajectory geometry based dissimilarity measures in a unified and consistent formalism is demonstrated. Overall, the deviation from conjugacy pseudometrics provide practical advantages in terms of efficiency and scalability, while maintaining theoretical consistency.
Paper Structure (16 sections, 3 theorems, 53 equations, 11 figures, 1 table)

This paper contains 16 sections, 3 theorems, 53 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Data-Driven Koopman Operator Approximation
  3. Observable vector
  4. Identification of the Koopman operator
  5. Topological Conjugacy from the Perspective of Koopman Operators
  6. Deviation from Conjugacy Pseudometrics
  7. Deviation from conjugacy
  8. Simplifications to the Pareto optimality problem
  9. Solutions over the group of unitary transformations
  10. Geometry of Pareto optimality
  11. Analytical solutions in $\Phi$-space
  12. Solutions in $\Psi$-space
  13. Results
  14. Benchmarking example
  15. Engineering example
  16. ...and 1 more sections

Key Result

proposition thmcounterproposition

If two discrete spectra systems $f$ and $g$ are topologically conjugate then is an equivalent condition to Eq. sys_conj and is an equivalent condition to Eq. traj_conj, where the diagonal matrices $\Lambda_{f,g} \in \mathbb{C}^{N_{\Psi} \times N_{\Psi}}$ contain the arbitrarily ordered eigenvalues of $K_{f,g}$, and $W_{f,g} \in GL(N_{\Psi})$ contain the left-egenvectors of $K_{f,g}$ such that t

Figures (11)

  • Figure 1: Geometric bounds on all possible Pareto optimal solutions; the origin corresponds to topological conjugacy
  • Figure 2: Deviations from conjugacy varied over $\alpha$ and $\beta$; (a) $d_{min}$ (b) $d_{avg}$ (c) $d_{max}$
  • Figure 3: Comparisons of $\Psi_f$, $\Psi_g$, and transformed trajectory $T\Psi_f$ for $\alpha,\beta=1$; (a) $T_{C_{r_1}}$, (b)$T_{LSQ}$
  • Figure 4: Conjugacy residuals when optimized independenty and comparison between optimal solutions $C_{r_1}$ and $C_{r_2}$; (a) $r_1(C_{r_1})$, (b) $r_2(C_{r_2})$, (c) $\left\Vert C_{r_1} - C_{r_2} \right\Vert_F/\left\Vert C_{r_2} \right\Vert_F$
  • Figure 5: Comparisons of the $y_1$ component of $\Psi_g$, and transformed trajectory $T\Psi_f$; (a) $\alpha = 1,\beta=1.2$, (b) $\alpha = 1.85,\beta= 0.94$
  • ...and 6 more figures

Theorems & Definitions (15)

  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • proof
  • remark thmcounterremark
  • definition thmcounterdefinition
  • theorem 1
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • ...and 5 more