# A Short Introduction to the Koopman Representation of Dynamical Systems

arXiv: https://arxiv.org/abs/2205.08048 · HTML: https://mlsanity.com/paper/2205.08048/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2022.

## Built on by (most cited first, 1 of 1)

- [Bilinear World Models: Learning Representations with Structured Dynamics for Efficient Control](https://mlsanity.com/paper/2609.36305/) (2026)

## Abstract (arXiv)

The Koopman representation is an infinite dimensional linear representation of linear or nonlinear dynamical systems. It represents the dynamics of output maps (aka observables), which are functions on the state space whose evaluation is interpreted as an output. Conceptually simple derivations and commentary on the Koopman representation are given. We emphasize an important duality between initial conditions and output maps of the original system, and those of the Koopman representation. This duality is an important consideration when this representation is used in data-driven applications such as the Dynamic Mode Decomposition (DMD) and its variants. The adjoint relation between the Koopman representation and the transfer operator of mass transport is also shown.