Methods of Partial Differential Equation Discovery: Application to Experimental Data on Heat Transfer Problem

Abstract

The paper presents two effective methods for discovering process models in the form of partial differential equations based on an evolutionary algorithm and an algorithm for the best subset selection. The methods are designed to work with sparse and noisy data and implement various numerical differentiation techniques, including piecewise local approximation using multidimensional polynomial functions, neural network approximation, and an additional algorithm for selecting differentiation steps. To verify the algorithms, the experiment is carried out on pulsed heating of a viscous liquid (glycerol) by a submerged horizontal cylindrical heat source. Temperature measurements are taken only at six points, which makes the data very sparse. The noise level ranges from 0.2 to 1% of the observed maximum temperature. The algorithms can successfully restore the structure of the heat transfer equation in cylindrical coordinates and determine the thermal diffusivity coefficient with an error of 2.5–20%, depending on the algorithm type and heating mode. Additional synthetic setups are employed to analyze the dependence of accuracy on the noise level. Results also demonstrate the algorithms’ ability to identify underlying processes such as convective motion.