Traveling Wave Solutions of Partial Differential Equations via Neural Networks

2021/01/21 by Sung Woong Cho, Cho, Sung Woong, Hyung Ju Hwang +3
Physics and Astronomy · Mathematics · Computer Science · #Model Reduction and Neural Networks #Numerical methods for differential equations #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2101.08520

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.

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