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

TL;DR

It is shown that the domain splitting method is crucial in the neural network approach, for long time integration in Reaction-Diffusion systems, and the use of different boundary conditions further enhances the splitting technique by improving numerical approximations.

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.

Figures (11)

• Figure 1: A neural network of $L$ layers with an input of $x$ and $t$.
• Figure 2: Physical and Computational domains for the Mirror technique. The mirrors are located at $x=100$ and $x=400$.
• Figure 4: Performance of the four methods to solve the diffusion equation with no-flux boundary conditions. The first column, shows the approximation of the solution obtained with NN, for a particular epoch $k$. The second column, shows the error at each point of the mesh when compared with the exact solution. This is to show where the main source of errors are located. The last column shows the evolution of the loss function respect to the epoch.
• Figure 5: Obtained error for the different methods to solve the diffusion equation with no-flux boundary conditions. (A) the $L_{\infty}$ error versus the epoch, (B) A zoom for the first 1000 epochs. The error is obtained over a regular mesh of 400 points in the $x$ direction and 250 points in the $t$ direction.
• Figure 6: Numerical solution of the bistable equation obtained with the Chebyshev pseudospectral method given in oj22 with a fixed grid.