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

2026-03-16

AI Summary

This work addresses the challenge of efficiently computing traveling wave solutions to high-dimensional reaction-diffusion equations by proposing a scaled TW-PINN framework. Through a traveling wave transformation combined with adaptive coefficient scaling, the method normalizes problems of arbitrary spatial dimensions and varying reaction or diffusion coefficients into a unified one-dimensional canonical form, which is then solved by a single physics-informed neural network (PINN). This approach achieves, for the first time, cross-dimensional and cross-coefficient generalization within a single model, and its universal approximation capability for traveling wave solutions is theoretically established. Numerical experiments on one- and two-dimensional Fisher-type equations demonstrate that the proposed framework significantly outperforms existing wave-PINN methods in both accuracy and flexibility, and successfully handles general initial value problems.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• traveling wave solutions
• reaction-diffusion equations
• physics-informed neural network
• general coefficients
• scaling transformation

Innovation

• scaled TW-PINN
• traveling wave solutions
• reaction-diffusion equations
• physics-informed neural networks
• scaling transformation