Parametric Analysis and Validation of Machine Learning in Chaotic Transitions of the Lorenz System

Abstract

Abstract We are continuing an examination of unsupervised machine learning methods in natural convection systems. We are leveraging the Lorenz system to consider turbulent transitions and how machine learning models assist understanding these transitions. The Lorenz equations were selected to test the machine learning algorithms due to the relative simplicity of the equations, and the well-documented chaotic behavior. We developed a robust numerical solution to the Lorenz equations using a fourth order Runge-Kutta method with a time step of 0.001 seconds. We solved the Lorenz equations for a large range of Raleigh ratios from 1–1000. We also used a finer time resolution and larger Rayleigh range than prior work. 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. In prior work we found that the automated cluster analysis aligns well with well known key transition regions of the convection system. In this paper we further explore the inputs to the machine learning system and new methods of analysis. We further validated our methods with a windowing approach to the Rayleigh ratio range. By running this methodology over multiple Raleigh ratio ranges, the system is able to locate transitional regions. 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 correct 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. This simplified model serves as a test case for more complex natural convection flows. This unsupervised learning approach can be utilized on other systems where numerical analysis is computationally difficult. The results provide a visual mapping of the regions where chaotic transitions may occur based on variations in the Prandtl number and geometry constant. We plan to apply these methods to more complex natural convection systems and compare to experimental data. 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.