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

arXiv: https://arxiv.org/abs/2106.15678 · HTML: https://mlsanity.com/paper/2106.15678/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jun 2021.

## Built on by (most cited first, 6 of 6)

- [On the lifting and reconstruction of nonlinear systems with multiple invariant sets](https://mlsanity.com/paper/2304.11860/) (2023)
- [Online Real-time Learning of Dynamical Systems from Noisy Streaming Data: A Koopman Operator Approach](https://mlsanity.com/paper/2212.05259/) (2022)
- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [Data-driven Modeling of Granular Chains with Modern Koopman Theory](https://mlsanity.com/paper/2411.15142/) (2024)
- [Cluster-Weighted EDMD](https://mlsanity.com/paper/2607.12243/) (2026)
- [fSRD: Fuzzy Spectral Region Decomposition -- Automated Multi Operator Koopman Representations via an Adaptive Spectral Learning Architecture](https://mlsanity.com/paper/2607.17990/) (2026)

## Abstract (arXiv)

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.