Mapping the Chaotic Transitions of the Lorenz Equations With Unsupervised Machine Learning

Abstract

Prior work has explored utilizing machine learning for the Lorenz system in the time domain. In this work, we have focused on applications of machine learning for predicting the onset of chaotic transitions in the Lorenz system. Our methods included the development of a robust numerical solution to the Lorenz equations using a fourth order Runge–Kutta method. We solved the Lorenz equations for a large range of Raleigh ratios from 1 to 1000. We calculated the power spectral density, various descriptive statistics, and a cluster analysis using unsupervised machine learning. To identify behaviors and regions in the data, we utilize unsupervised learning as it is designed to assist in recognizing patterns without being told or trained by prior knowledge. We confirmed the performance of the machine learning system's ability to identify chaotic transitions independent of expert selection of Raleigh ratio ranges. The system correctly identifies the transitional behaviors described in prior mathematical work. The results indicate that the power spectral density is very important for the clustering. We also found that examining machine learning clusters by dimension (x, y, and z) was important to understand many of the facets of the chaotic transitions. The results provide a visual mapping of the regions where chaotic transitions may occur based on variations in the Prandtl number and geometry constant. Unsupervised machine learning may be used as a tool to characterize the transition regions for these geometries, providing new lenses for the heat transfer community.