Methods for a Partial Differential Equation Discovery: Application to Physical and Engineering Problems

Abstract

The paper presents two methods for discovering differential equations from available data. The first method uses a genetic algorithm with evolutionary optimization, while the second method employs the best subset selection procedure and the Bayesian information criterion. Both methods have been improved to work with noisy and highly sparse data. Diverse techniques for numerical differentiation are proposed, including neural network data approximation and an algorithm for selecting differentiation steps. The proposed approaches are applied to solve physical and engineering problems. As a physical application, the problem of pulsed heating of a viscous liquid by a submerged wire of circular cross section is considered. As an engineering application, the problem of the motion of the arc root along the hollow cylindrical electrode of the alternating current plasma torch is taken. The efficiency of applying approaches for discovering heat transfer models in the form of a partial differential equation and the possibility of the methods to indicate the change in the regimes of the ongoing process are shown. The employment of the model generation approach in the form of a differential equation based on experimental data on the motion of the arc root in the plasma torch made it possible to solve the complex hybrid problem of determining the spatio-temporal distributions of the plasma-forming gas parameters.