Understanding the geometry of transport: diffusion maps for Lagrangian trajectory data unravel coherent sets

TL;DR

This paper introduces a diffusion map-based method to identify long-lived coherent sets in dynamical systems from sparse trajectory data, revealing low-dimensional structures and aiding automated transport analysis.

Contribution

It extends diffusion maps to trajectory space, enabling extraction of coherent sets with minimal prior knowledge and demonstrating convergence to theoretical transfer operator frameworks.

Findings

Method successfully identifies coherent sets in simulated data

Converges to theoretical transfer operator in the infinite data limit

Effective on real-world Lagrangian trajectory data

Abstract

Dynamical systems often exhibit the emergence of long-lived coherent sets, which are regions in state space that keep their geometric integrity to a high extent and thus play an important role in transport. In this article, we provide a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. Our method can be seen as an extension of diffusion maps to trajectory space, and it allows us to construct "dynamical coordinates" which reveal the intrinsic low-dimensional organization of the data. The only a priori knowledge about the dynamics that we require is a locally valid notion of distance, which renders our method highly suitable for automated data analysis. We show convergence of our method to the analytic transfer operator framework of coherence in the infinite data limit, and illustrate its potential on several two- and three-dimensional examples as well as real world data.