arXiv subjects

Ali Haider Shah

Publications and source records attributed to Ali Haider Shah.

2 recordsLinked to original sources

arXiv2025

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

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.

math.AP↗

arXiv2025

Enhancing PINN Performance Through Lie Symmetry Group

This paper presents intersection of Physics informed neural networks (PINNs) and Lie symmetry group to enhance the accuracy and efficiency of solving partial differential equation (PDEs). Various methods have been developed to solve these equations. A Lie group is an efficient method that can lead to exact solutions for the PDEs that possessing Lie Symmetry. Leveraging the concept of infinitesimal generators from Lie symmetry group in a novel manner within PINN leads to significant improvements in solution of PDEs. In this study three distinct cases are discussed, each showing progressive improvements achieved through Lie symmetry modifications and adaptive techniques. State-of-the-art numerical methods are adopted for comparing the progressive PINN models. Numerical experiments demonstrate the key role of Lie symmetry in enhancing PINNs performance, emphasizing the importance of integrating abstract mathematical concepts into deep learning for addressing complex scientific problems adequately.

math.AP↗