The study of wave-like propagation of information in nonlinear and dispersive media is a complex phenomenon. In this paper, we provide a new approach to studying this phenomenon, paying special attention to the nonlinear solitary wave problem of the Korteweg-De Vries (KdV) equation. Our proposed algorithm is based on the traveling wave transformation of the KdV equation, which reduces the dimensionality of the system, enabling us to obtain a highly accurate solution with fewer data. The proposed algorithm uses a Lie-group-based neural network trained via the Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization method. Our experimental results demonstrate that the proposed Lie-group-based neural network algorithm can simulate the behavior of the KdV equation with high accuracy while using fewer data. The effectiveness of our method is proved by examples.
Keywords: BFGS; KdV equation; Lie groups; deep learning; partial differential equations.