Solving a Class of Thomas-Fermi Equations: A New Solution Concept Based on Physics-Informed Machine Learning

This paper presents a novel physics-informed machine learning approach, designed to approximate solutions to a specific category of Thomas-Fermi differential equations. To tackle the inherent intricacies of solving Thomas-Fermi equations, we employ the quasi-linearization technique, which transforms the original non -linear problem into a series of linear differential equations. Our approach utilizes collocation least -squares support vector regression, leveraging fractional Chebyshev functions for finite interval simulations and fractional rational Chebyshev functions for semi-infinite intervals, enabling precise solutions for linear differential equations across varied spatial domains. These selections facilitate efficient simulations across both finite and semi-infinite intervals. Key contributions of our approach encompass its versatility, demonstrated through successful approximation of solutions for diverse Thomas-Fermi problem types, including those featuring non -local integral terms, Bohr radius boundary conditions, and isolated neutral atom boundary conditions defined on semi-infinite domains. Furthermore, our method exhibits computational efficiency, surpassing classical collocation methods by solving a sequence of positive definite linear equations or quadratic programming problems. Notably, our approach showcases precision, as evidenced by experiments, including the attainment of the initial slope of the renowned Thomas-Fermi equation with an impressive 39 -digit precision.