# Learning finite difference methods for reaction-diffusion type equations with FCNN

arXiv: https://arxiv.org/abs/2201.01854 · HTML: https://mlsanity.com/paper/2201.01854/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 32 references that are in the map.
- First posted on arXiv 4 Jan 2022.

## Builds on (most cited first, 3 of 3)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)

## Built on by (most cited first, 3 of 3)

- [Characteristics-Informed Neural Networks for Forward and Inverse Hyperbolic Problems](https://mlsanity.com/paper/2212.14012/) (2022)
- [Convolutional Autoencoders, Clustering and POD for Low-dimensional Parametrization of Navier-Stokes Equations](https://mlsanity.com/paper/2302.01278/) (2023)
- [Going Deeper with Five-point Stencil Convolutions for Reaction-Diffusion Equations](https://mlsanity.com/paper/2308.04735/) (2023)

## Abstract (arXiv)

In recent years, Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations alongside numerical methods because PINNs can be trained without observations and deal with continuous-time problems directly. In contrast, optimizing the parameters of such models is difficult, and individual training sessions must be performed to predict the evolutions of each different initial condition. To alleviate the first problem, observed data can be injected directly into the loss function part. To solve the second problem, a network architecture can be built as a framework to learn a finite difference method. In view of the two motivations, we propose Five-point stencil CNNs (FCNNs) containing a five-point stencil kernel and a trainable approximation function for reaction-diffusion type equations including the heat, Fisher's, Allen-Cahn, and other reaction-diffusion equations with trigonometric function terms. We show that FCNNs can learn finite difference schemes using few data and achieve the low relative errors of diverse reaction-diffusion evolutions with unseen initial conditions. Furthermore, we demonstrate that FCNNs can still be trained well even with using noisy data.