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

The Burgers hierarchy consists of nonlinear evolutionary PDEs with increasingly higher-order dispersive and nonlinear terms, resulting in a rich variety of soliton solutions. Notable members include the Burgers equation and the Sharma-Tasso-Olver equation, which are applied in plasma physics, fluid mechanics, optics, and biophysics to describe nonlinear waves in inhomogeneous media. Various soliton and multi-soliton solutions have been identified, and the fission and fusion of solitons have been explored through analytical and numerical techniques. Recently, deep learning methods, particularly Physics Informed Neural Networks (PINNs), have emerged as powerful tools for solving PDEs. These methods use deep neural networks to minimize PDE residuals and satisfy boundary and initial conditions 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 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 best 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. They also verify the potential for training refinement and accuracy improvement using enhanced approaches in certain cases, while enabling the inference of the PDE model that best describes the observed solitary structures.