From Mesh to Neural Nets: A Multi-Method Evaluation of Physics-Informed Neural Networks and Galerkin Finite Element Method for Solving Nonlinear Convection-Reaction-Diffusion Equations

TL;DR

The paper presents elsarticle.cls, a modern LaTeX class designed for formatting submissions to Elsevier journals. It builds on the standard article.cls and integrates common packages (e.g., natbib, geometry, graphicx) to minimize conflicts with existing packages while preserving kernel behavior. Key contributions include a clear distinction from the older elsart.cls, support for preprint and final-model formats, natbib-based citation handling, and comprehensive frontmatter and theorem-environment tooling. The document also provides practical installation guidance and distribution paths via Elsevier resources and CTAN to facilitate widespread adoption within the TeX ecosystem.

Abstract

Non-linear convection-reaction-diffusion (CRD) partial differential equations (PDEs) are crucial for modeling complex phenomena in fields such as biology, ecology, population dynamics, physics, and engineering. Numerical approximation of these non-linear systems is essential due to the challenges of obtaining exact solutions. Traditionally, the Galerkin finite element method (GFEM) has been the standard computational tool for solving these PDEs. With the advancements in machine learning, Physics-Informed Neural Network (PINN) has emerged as a promising alternative for approximating non-linear PDEs. In this study, we compare the performance of PINN and GFEM by solving four distinct one-dimensional CRD problems with varying initial and boundary conditions and evaluate the performance of PINN over GFEM. This evaluation metrics includes error estimates, and visual representations of the solutions, supported by statistical methods such as the root mean squared error (RMSE), the standard deviation of error, the the Wilcoxon Signed-Rank Test and the coefficient of variation (CV) test. Our findings reveal that while both methods achieve solutions close to the analytical results, PINN demonstrate superior accuracy and efficiency. PINN achieved significantly lower RMSE values and smaller standard deviations for Burgers' equation, Fisher's equation, and Newell-Whitehead-Segel equation, indicating higher accuracy and greater consistency. While GFEM shows slightly better accuracy for the Burgers-Huxley equation, its performance was less consistent over time. In contrast, PINN exhibit more reliable and robust performance, highlighting their potential as a cutting-edge approach for solving non-linear PDEs.