Neural network Approximations for Reaction-Diffusion Equations -- Homogeneous Neumann Boundary Conditions and Long-time Integrations

TL;DR

This paper advances neural network methods for solving reaction-diffusion equations, focusing on implementing Neumann boundary conditions and enabling stable long-time integrations through domain splitting techniques.

Contribution

It introduces four neural network approaches for Neumann boundary conditions and a domain splitting method for long-time integration, improving accuracy and stability.

Findings

Domain splitting is crucial for long-time stability.

Different boundary conditions enhance numerical accuracy.

Neural network methods effectively approximate reaction-diffusion solutions.

Abstract

Reaction-Diffusion systems arise in diverse areas of science and engineering. Due to the peculiar characteristics of such equations, analytic solutions are usually not available and numerical methods are the main tools for approximating the solutions. In the last decade, artificial neural networks have become an active area of development for solving partial differential equations. However, several challenges remain unresolved with these methods when applied to reaction-diffusion equations. In this work, we focus on two main problems. The implementation of homogeneous Neumann boundary conditions and long-time integrations. For the homogeneous Neumann boundary conditions, we explore four different neural network methods based on the PINN approach. For the long time integration in Reaction-Diffusion systems, we propose a domain splitting method in time and provide detailed comparisons between different implementations of no-flux boundary conditions. We show that the domain splitting method is crucial in the neural network approach, for long time integration in Reaction-Diffusion systems. We demonstrate numerically that domain splitting is essential for avoiding local minima, and the use of different boundary conditions further enhances the splitting technique by improving numerical approximations. To validate the proposed methods, we provide numerical examples for the Diffusion, the Bistable and the Barkley equations and provide a detailed discussion and comparisons of the proposed methods.