  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.
