Deep learning data-driven multi-soliton dynamics and parameters discovery for the fifth-order Kaup–Kuperschmidt equation

Abstract

In this paper, we extend the multi-layer physics-informed neural networks (PINNs) to successfully learn the data-driven multi-soliton solutions and discover the coefficients of fifth-order Kaup–Kuperschmidt (KK) equation with the aid of multi-soliton data. In the forward problems, the corresponding accuracies for one-, two- and three-soliton solutions are O(10−6), O(10−4) and O(10−3), respectively. Moreover, some basic elements affecting the performance of PINNs are analyzed in detail, such as activation functions, collocation points, optimizers, neural network structures, and so on. For the inverse problems, the coefficients of the equation are discovered by the data of one-, two- and three-soliton solutions, respectively. Meanwhile, the robustness of the PINNs algorithm is explored under different noises. The accuracy of the identified coefficients can reach O(10−3) when 1% initial noise is added to the training data. And the prediction accuracy can still reach O(10−2) even if 3% initial noise is added. These numerical experiments not only show the effectiveness of PINNs to solve and discover the KK equation with some known information, but also embody the applicability to reveal the multi-soliton dynamic behaviors in higher-order nonlinear wave equations.