Multi-soliton solutions and data-driven discovery of higher-order Burgers' hierarchy equations with physics informed neural networks

Abstract

The Burgers hierarchy consists of nonlinear evolutionary partial differential equations (PDEs) with progressively higher-order dispersive and nonlinear terms. Notable members of this hierarchy are the Burgers equation and the Sharma-Tasso-Olver equation, which are widely applied in fields such as plasma physics, fluid mechanics, optics, and biophysics to describe nonlinear waves in inhomogeneous media. Various soliton and multi-soliton solutions to these equations have been identified and the fission and fusion of solitons have been studied using analytical and numerical techniques. Recently, deep learning methods, particularly Physics-Informed Neural Networks (PINNs), have emerged as a new approach for solving PDEs. These methods use deep neural networks to minimize PDE residuals while fitting relevant data. Although PINNs have been applied to equations like Burgers' and Korteweg-de Vries, higher-order members of the Burgers hierarchy remain unexplored in this context. In this study, we employ a PINN algorithm to approximate multi-soliton solutions of linear combinations of equations within the Burgers hierarchy. This semi-supervised approach encodes the PDE and relevant data, determining PDE parameters and resolving the linear combination to discover the PDE that describes the data. Additionally, we employ gradient-enhanced PINNs (gPINNs) and a conservation law, specific to the generic Burgers' hierarchy, to improve training accuracy. The results demonstrate the effectiveness of PINNs in describing multi-soliton solutions within the generic Burgers' hierarchy, their robustness to increased levels of data noise, and their limited yet measurable predictive capabilities. They also verify the potential for training refinement and accuracy improvement using enhanced approaches in certain cases, while enabling the discovery of the PDE model that describes the observed solitary structures.