Traveling Wave Solutions of Partial Differential Equations via Neural Networks

TL;DR

This paper introduces a neural network-based method to approximate traveling wave solutions and unknown wave speeds in partial differential equations, demonstrating convergence and accuracy across several models.

Contribution

A novel neural network approach that simultaneously approximates traveling wave solutions and unknown wave speeds, with proven convergence under mild conditions.

Findings

Neural network solutions converge to analytic solutions as loss decreases.

Accurate wave speed approximation achieved for Keller-Segel, Allen-Cahn, and Lotka-Volterra models.

Method effectively reduces error in approximating traveling waves in PDEs.

Abstract

This paper focuses on how to approximate traveling wave solutions for various kinds of partial differential equations via artificial neural networks. A traveling wave solution is hard to obtain with traditional numerical methods when the corresponding wave speed is unknown in advance. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter. We proved that under a mild assumption, the neural network solution converges to the analytic solution and the free parameter accurately approximates the wave speed as the corresponding loss tends to zero for the Keller-Segel equation. We also demonstrate in the experiments that reducing loss through training assures an accurate approximation of the traveling wave solution and the wave speed for the Keller-Segel equation, the Allen-Cahn model with relaxation, and the Lotka-Volterra competition model.