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.

本文重点研究如何通过人工神经网络逼近各种偏微分方程的行波解。当相应的波速事先未知时，传统的数值方法很难得到行波解。我们提出了一种通过神经网络和额外的自由参数来近似行波解和未知波速的新方法。我们证明，在一个温和的假设下，神经网络解收敛到解析解，自由参数准确地逼近波速，因为 Keller-Segel 方程的相应损失趋于零。