Going Deeper with Five-point Stencil Convolutions for Reaction-Diffusion Equations
This work addresses computational efficiency and stability issues in simulating reaction-diffusion equations for researchers in computational physics and machine learning, but it is incremental as it builds on existing FCNN methods.
The authors tackled the challenge of optimizing physics-informed neural networks for reaction-diffusion equations by proposing deep five-point stencil convolutional neural networks (FCNNs) that can predict time evolutions with time steps larger than the CFL condition threshold, demonstrating that these models retain certain accuracies while traditional finite difference methods blow up.
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.