# Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable

## TL;DR

This paper presents a machine-learning approach using reservoir computing to model a macroscopic fluid variable from scalar time-series data without prior physical knowledge, successfully capturing chaotic dynamics.

## Contribution

It introduces a data-driven reservoir computing model that effectively reconstructs fluid dynamics from scalar observations, emphasizing the importance of delay-coordinate parameters.

## Key findings

- The model accurately approximates the actual time-series over various intervals.
- It captures key characteristics of the chaotic invariant set.
- Proper delay-coordinate selection enhances model complexity and accuracy.

## Abstract

We construct a data-driven dynamical system model for a macroscopic variable the Reynolds number of a high-dimensionally chaotic fluid flow by training its scalar time-series data. We use a machine-learning approach, the reservoir computing for the construction of the model, and do not use the knowledge of a physical process of fluid dynamics in its procedure. It is confirmed that an inferred time-series obtained from the model approximates the actual one in each of various time-intervals, and that some characteristics of the chaotic invariant set mimic the actual ones. We investigate the appropriate choice of the delay-coordinate, especially the delay-time and the dimension, which enables us to construct a model having a relatively high-dimensional attractor easily.