Going Deeper with Five-point Stencil Convolutions for Reaction-Diffusion
  Equations
Yongho Kim Yongho Choi
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.