The Refined Physics-Informed Neural Networks for Nonlinear Convection-Reaction-Diffusion Equations Using Exponential Schemes

Abstract

The nonlinear convection-reaction-diffusion equations model complex real-world phenomena across scientific and engineering disciplines. However, solving these equations analytically is often impossible due to their nonlinear nature. As a result, researchers have turned to numerical and computational methods to find approximate solutions. These methods, while effective, can struggle with issues such as stability, accuracy, and the ability to handle sharp gradients or complex interactions between convection, diffusion, and reaction terms. To address these challenges, this work introduces an enhanced Physics-Informed Neural Network (PINN) framework for convection-reaction-diffusion equations that incorporates exponential finite difference scheme residuals with the aim of enhancing solution accuracy and stability. To validate its performance, the framework has been tested on four well-known nonlinear partial differential equations: Burgers' Equation, Fisher's Equation, the Burgers-Huxley Equation, and the Newell-Whitehead-Segel Equation. The results obtained using the modified PINN framework are systematically compared with those obtained using traditional Physics-Informed Neural Networks and the Galerkin Finite Element Method. The comparisons reveal that the proposed framework consistently outperforms both approaches in terms of accuracy. These improvements highlight the effectiveness of integrating exponential finite difference scheme residuals into the PINN framework, making it a powerful and reliable tool for solving nonlinear convection-reaction-diffusion equations.

Keywords

• Deep learning
• Exponential scheme
• Convection-reaction-diffusion
• Burgers-Huxley
• Newell-Whitehead-Segel

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