Controlling Rayleigh-B´enard Convection through Regression Neural Network*

Rayleigh-Bénard thermal turbulent convection depends primarily on two dimensionless numbers: the Rayleigh (Ra) number related to the vigor of convection, and the Prandtl (Pr) number, the ratio between viscous and thermal diffusivities. Other parameters are boundary and initial conditions [1]. Here we simulate the Rayleigh-Bénard convection by using the Lattice Boltzmann Method (LBM) [2] to create training and testing images for a Deep Neural Network (DNN) [3] using regression to simultaneously estimate Ra and Pr from a single snapshot of temperature and velocities of the convective field. Training and testing ranges for Pr and Ra are [1 − 128] and [10⁵ − 10⁹] respectively. We verify the ability of the network to simultaneously predict Ra and Pr both for values never seen by the network within the training range, as well as beyond the training range (Pr ∈ [0.35 − 362] and Ra ∈ [10⁴ − 10¹⁰]). Results show that the distribution of the predicted values for Ra and Pr is centered around the correct value for Ra and Pr within the training range and that the estimation precision decays outside it. Ra predictions are more precise than Pr across the entire training range, while Pr predictions are systematically lower at high Pr, which can be explained based on the characteristics of the Thermal Navier-Stokes equations. Our analysis suggests a practical tool for monitoring and controlling industrial flow and for studying geophysical flows, including heat transport in the earth’s interiors, ocean, and atmosphere.