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

TL;DR

This paper introduces FCNNs, a neural network architecture that learns finite difference schemes for reaction-diffusion equations, achieving accurate predictions with limited and noisy data, and generalizing to unseen initial conditions.

Contribution

The paper proposes a novel FCNN architecture that learns finite difference methods for reaction-diffusion equations, improving data efficiency and robustness to noise.

Findings

FCNNs learn finite difference schemes with few data.

FCNNs achieve low relative errors on reaction-diffusion evolutions.

FCNNs generalize well to unseen initial conditions and noisy data.

Abstract

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.