# Extraction and Prediction of Coherent Patterns in Incompressible Flows through Space-Time Koopman Analysis

## TL;DR

This paper introduces a data-driven method using Koopman operator theory and diffusion maps to detect, analyze, and predict coherent patterns in complex incompressible fluid flows, with applications to vortex and Lorenz systems.

## Contribution

It develops a novel space-time Koopman analysis framework that captures global flow patterns and enables model-free predictions of flow evolution.

## Key findings

- Successfully identified coherent structures in vortex flows.
- Predicted flow evolution in Lorenz 96 systems.
- Demonstrated effectiveness of the method on complex fluid flows.

## Abstract

We develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time-dependent fluid flows driven by ergodic dynamical systems. Our approach is based on representations of the generators of the Koopman and Perron-Frobenius groups of operators governing the evolution of observables and probability measures on Lagrangian tracers, respectively, in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm. These operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place, and as a result their eigenfunctions correspond to global space-time coherent patterns under a skew-product dynamical system. Moreover, using this data-driven representation of the generators in conjunction with Leja interpolation for matrix exponentiation, we construct model-free prediction schemes for the evolution of observables and probability densities defined on the tracers. We present applications to periodic Gaussian vortex flows and aperiodic flows generated by Lorenz 96 systems.