A three-stage framework combining neural networks and Monte Carlo tree search for approximating analytical solutions to the Thomas–Fermi equation

Abstract

This study presents an innovative framework that integrates physics-informed neural networks with Monte Carlo tree search to develop an approximate analytical solution for the Thomas–Fermi equation. The framework operates in three stages. Initially, a physics-informed neural network is used to generate a numerical approximation of the Thomas–Fermi equation. Subsequently, the Monte Carlo tree search algorithm identifies an analytical expression that closely approximates the numerical solution from the first stage, resulting in an initial analytical solution. In the final stage, the physics-informed neural network is employed once more to optimize the coefficients of the expression found by Monte Carlo tree search, further refining the accuracy of the solution. Experimental results validate the effectiveness of this approach, demonstrating its capability to solve the challenging and nonlinear Thomas–Fermi equation, for which an exact analytical solution is not available.