Prediction of flow dynamics using point processes

TL;DR

This paper introduces a novel method for predicting the dynamics of continuous-time systems by using crossing times of local cross sections, achieving high accuracy with sufficient data.

Contribution

It proposes a new approach to describe and predict continuous-time system dynamics using crossing times and past observations, independent of cross section placement.

Findings

Accurate prediction of Lorenz system dynamics.

Effective application to wind speed measurements.

Robustness to cross section placement and size.

Abstract

Describing a time series parsimoniously is the first step to study the underlying dynamics. For a time-discrete system, a generating partition provides a compact description such that a time series and a symbolic sequence are one-to-one. But, for a time-continuous system, such a compact description does not have a solid basis. Here, we propose to describe a time-continuous time series using a local cross section and the times when the orbit crosses the local cross section. We show that if such a series of crossing times and some past observations are given, we can predict the system's dynamics with fine accuracy. This reconstructability neither depends strongly on the size nor the placement of the local cross section if we have a sufficiently long database. We demonstrate the proposed method using the Lorenz model as well as the actual measurement of wind speed.