Machine learning method for the supplement, correction, and prediction of the nonlinear dynamics in pattern formation

Abstract

The pattern formation and spatial–temporal chaos are interesting issues in nonlinear dynamics. A novel model based on machine learning methods is designed to learn and imitate the pattern evolution in Bénard–Marangoni convection (BM convection). There is a supercritical process, which is an inevitable and unique experimental phenomenon, on the way to chaos in BM convection. A single layer of fluid uniformly heated at the bottom is used as the experimental system. During the experiment, the temperature difference between top and bottom of the liquid layer is increased first to make the system enter the supercritical convection state and then decreased after a while; surface temperature distribution of the liquid layer is measured in real time with an infrared thermal imager, which visualized the formation and re-organization of cellular convection during the supercritical state. The temperature data are used as the material that meets the conditions of machine learning and then the machine learning method in charge of predicting the picture of temperature distribution that it never has seen before in two steps. The experimental data are used to train an auto-encoder model based on convolutional neural networks and an RNN–CNN joint model, in which the former is used for extracting low-dimensional features of the temperature field, and the latter is used for predicting evolution results of the low-dimensional features and recovering them back to the temperature field. The models have finally achieved the objectives of supplementing the missing experimental data and correcting actual experimental data by comparing the actual experimental results with the prediction results of the machine learning approach and theoretical analysis results. On the other hand, active exploration has been undertaken in predicting physical experimental results that have never happened before.