Efficient pseudometrics for data-driven comparisons of nonlinear dynamical systems

TL;DR

This paper introduces computationally efficient pseudometrics for comparing nonlinear dynamical systems by leveraging Koopman operator theory and geometric considerations, enabling scalable and theoretically consistent analysis.

Contribution

It develops novel pseudometrics based on Koopman eigenfunctions and unitary transformations, with analytical solutions and Pareto optimality, for data-driven comparison of nonlinear systems.

Findings

Pseudometrics are computationally efficient and scalable.

Theoretical justification for using Koopman eigenfunction space.

Demonstrated on benchmark and biological system examples.

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.