A machine learning approach for solving inverse Stefan problem

In this paper, we propose a numerical scheme by using Least Squares Support Vector Regression (LS-SVR) for the simulation of the inverse Stefan problem, which has ill-posedness issues. The purpose of this paper is to express the temperature distribution in a homogeneous environment with a phase change. In the proposed machine learning approach, we apply the unconditionally stable Crank-Nicolson method to decrease the computational cost and reduce one of the dimensions. Therefore, we solve an ODE equation at each time step. The training points of the network are chosen as the Chebyshev roots, which have a normal distribution, and our constructed roots, which we describe more precisely later. In the proposed method, the regularization parameter of the SVM aims to overcome the instability issues, leading to convergent approximation. For the given method, both the primal and dual forms are investigated. The dual form of the problem is written in matrix form. Finally, some numerical examples are provided to illustrate the effectiveness and accuracy of the proposed method.