Data-Driven Operator Theoretic Methods for Phase Space Learning and Analysis

TL;DR

This paper develops data-driven operator theoretic methods to analyze and learn the global phase space of dynamical systems, leveraging spectral properties of the Koopman operator, symmetry, and conjugacy to identify invariant subspaces and construct the global Koopman operator.

Contribution

It introduces new conditions and methods for discovering invariant subspaces, stitching local spectral data into a global operator, and extending these results to symmetric and conjugate systems.

Findings

Successfully applied to nonlinear systems including a bistable toggle switch.

Demonstrates phase space stitching from local invariant subspaces.

Provides a strategy for designing experiments to approximate the global Koopman operator.

Abstract

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system. We defined conditions for discovering new invariant subspaces in the state space of a dynamical system starting from an invariant subspace based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subspaces in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subspaces of the dynamical system and time-series data from any one of the invariant subspaces. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subspaces can be combined to approximate the global Koopman operator.