Solving Newell-Whitehead-Segel and Allen-Cahn Equations Employing Physics-Informed Neural Networks: A Comparative Analysis with Spline Methods

Abstract: This study focuses on the solution of partial differential equations (PDEs) by using physics-informed neural networks (PINNs). The Newell-Whitehead-Segel (NWS) equation and the Allen-Cahn equation belong to fundamental PDEs used mostly in various scientific disciplines. Different methods, including analytical and numerical approaches, have been proposed for solving these equations alongside the recently introduced PINN method. This study provides a detailed and comprehensive comparison between the developed PINN method and the state-of-the-art spline numerical solution for the NWS and Allen-Cahn equation. Furthermore, the computational time of the trained PINN models is evaluated to determine their computational efficiency. The findings show that PINN is significantly better than spline methods in solving both problems.