Physics-Informed Neural Network solutions for nonlinear wave equations: Applications to Rosenau–Hyman and Sharma–Tasso–Olver equations

Abstract

This study investigates the application of Physics-Informed Neural Networks (PINNs) to nonlinear dispersive wave equations, focusing on the Rosenau-Hyman (RH) and Sharma-Tasso-Olver (STO) models. While PINNs have been extensively validated on benchmark problems such as Burgers’, Korteweg–de Vries (KdV), and Schrödinger equations, their application to nonlinear dispersive systems characterized by nonlinear dispersion, compact support, and higher-order derivative interactions remains comparatively limited. The RH equation supports compactly supported solitary waves (compactons) arising from nonlinear dispersion, introducing derivative irregularities at wave boundaries that pose additional challenges for automatic differentiation and training stability. Similarly, the STO equation incorporates nonlinear advection coupled with third-order derivative interactions, resulting in rich wave dynamics driven by the interplay between steepening and dispersion. The proposed framework embeds the governing PDEs, initial conditions, and boundary conditions into a composite loss function and leverages automatic differentiation to evaluate high-order derivatives in a mesh-free manner. Numerical experiments demonstrate accurate reconstruction of compactons and dispersive solitary waves with strong agreement to analytical solutions across the spatial–temporal domain. The method remains stable under sparse collocation sampling and generalizes well over extended time horizons. Quantitative error analysis confirms high-fidelity approximation of nonlinear wave dynamics. These results extend the applicability of PINNs beyond classical smooth soliton equations to structurally more complex nonlinear dispersive systems and provide a foundation for future extensions to inverse problems, noisy data scenarios, and higher-dimensional models.