A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients

TL;DR

This paper introduces a scalable physics-informed neural network framework, scaled TW-PINN, for efficiently computing traveling wave solutions of reaction-diffusion equations with various coefficients, demonstrating superior accuracy and flexibility.

Contribution

The paper presents a novel scaled TW-PINN framework that reduces high-dimensional problems to a one-dimensional form, enabling reuse across different coefficients and dimensions, with proven universal approximation.

Findings

Accurate solutions in 1D and 2D reaction-diffusion equations.

Outperforms existing wave-PINN methods in accuracy and efficiency.

Successfully extended to Fisher's equation with general initial conditions.

Abstract

We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of $n$-dimensional reaction-diffusion equations with various reaction and diffusion coefficients. By applying a scaling transformation with the traveling wave form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation with unit reaction and diffusion coefficients. This reduction leads to the proposed framework, termed scaled TW-PINN, in which a single PINN solver trained on the scaled equation is reused for different coefficient choices and spatial dimensions. We also prove a universal approximation property of the proposed PINN solver for traveling wave solutions. Numerical experiments in one and two dimensions, together with a comparison to the existing wave-PINN method, demonstrate the accuracy, flexibility, and superior performance of scaled TW-PINN. Finally, we explore an extension of the framework to the Fisher's equation with general initial conditions.