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

arXiv: https://arxiv.org/abs/2308.04735 · HTML: https://mlsanity.com/paper/2308.04735/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 9 Aug 2023.

## Builds on (most cited first, 12 of 12)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Scaled-YOLOv4: Scaling Cross Stage Partial Network](https://mlsanity.com/paper/2011.08036/) (2020)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh](https://mlsanity.com/paper/1910.08613/) (2019)
- [Physics-informed neural networks to learn cardiac fiber orientation from multiple electroanatomical maps](https://mlsanity.com/paper/2201.12362/) (2022)
- [Learning finite difference methods for reaction-diffusion type equations with FCNN](https://mlsanity.com/paper/2201.01854/) (2022)

## Abstract (arXiv)

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.