Going Deeper with Five-point Stencil Convolutions for Reaction-Diffusion Equations

TL;DR

This paper introduces deep five-point stencil convolutional neural networks (FCNNs) for reaction-diffusion equations, enabling larger time steps than traditional methods while maintaining accuracy, thus improving stability and efficiency.

Contribution

The paper proposes deep FCNNs with large receptive fields that surpass CFL condition limits, offering a novel approach to stable and accurate time evolution prediction in reaction-diffusion equations.

Findings

Deep FCNNs maintain accuracy with larger time steps.

Traditional FDMs blow up beyond CFL threshold.

Deep FCNNs outperform FDMs in stability.

Abstract

Physics-informed neural networks have been widely applied to partial differential equations with great success because the physics-informed loss essentially requires no observations or discretization. However, it is difficult to optimize model parameters, and these parameters must be trained for each distinct initial condition. To overcome these challenges in second-order reaction-diffusion type equations, a possible way is to use five-point stencil convolutional neural networks (FCNNs). FCNNs are trained using two consecutive snapshots, where the time step corresponds to the step size of the given snapshots. Thus, the time evolution of FCNNs depends on the time step, and the time step must satisfy its CFL condition to avoid blow-up solutions. In this work, we propose deep FCNNs that have large receptive fields to predict time evolutions with a time step larger than the threshold of the CFL condition. To evaluate our models, we consider the heat, Fisher's, and Allen-Cahn equations with diverse initial conditions. We demonstrate that deep FCNNs retain certain accuracies, in contrast to FDMs that blow up.