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

TL;DR

The paper tackles solving the Newell-Whitehead-Segel (NWS) and Allen-Cahn equations with physics-informed neural networks (PINN) and benchmarks their accuracy and efficiency against spline-based collocation methods. A PINN loss combines initial, boundary, and residual terms, with the residual defined as $\mathcal{R} = u_t - m u_{xx} - n u - o u^p - \eta(x,t,u,u_x)$ and total loss $\mathcal{L}(\Theta) = \alpha \mathcal{L}_{init}(\Theta) + \beta \mathcal{L}_{bound}(\Theta) + \gamma \mathcal{L}_{res}(\Theta)$, optimized by Adam or L-BFGS, using GELU activations. Results show PINN achieves higher pointwise accuracy than multiple spline variants across both problems, while computation times for 10{,}000 points scale roughly as $O(n)$. This demonstrates PINN’s potential as an accurate and efficient PDE solver in physics and engineering, suitable for data-informed PDE solution workflows.

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.

Figures (18)

• Figure 1: Physics Informed Neural Network for NWS Equation
• Figure 2: Activation functions
• Figure 3: 3D surfaces of exact solution (left) and PINN solution(right) for Problem 1 where $\lambda = 0.1$
• Figure 4: Heat maps plots representing the surface solutions for Problem 1 using both the exact method (left) and the PINN method (right)
• Figure 5: Gradient lines plots representing the flow of gradient vectors at solution surface. where gradient lines of exact solution is on left side and the PINN method is on right side