Artificial Neural Networks for Solving Partial Differential Equations: A Deep Learning Approach

Abstract

This study explores the use of artificial neural networks (ANNs) to solve nonlinear partial differential equations (PDEs) arising in reaction-diffusion systems. Two variants of the Newell-Whitehead-Segel equation are examined, each involving distinct nonlinearitiesone cubic and the other quadraticalong with different boundary conditions. The NeuroDiffEq framework is employed, which encodes initial and boundary conditions directly into the neural network architecture. Learning is driven by minimizing the residuals of the PDE, eliminating the need for labeled data. Numerical experiments, supported by comparisons with analytical solutions and 3D surface plots, demonstrate the methods accuracy. The results highlight the capability of neural networks to approximate complex PDE solutions effectively, offering a mesh-free alternative to conventional numerical approaches. This study underscores the potential of physics-informed neural networks to address a broad class of nonlinear PDEs with both precision and efficiency.