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.
On the Assessment of Domain Decomposition Approaches in Extended Physics-Informed Neural Networks for the Shock-Dominated Burgers Equation
FLUX - Thermal systems Lab, Department of Aeronautical Engineering, School of Engineering of São João da Boa Vista, Universidade Estadual Paulista（圣若昂达博阿维斯塔工程学院航空工程系FLUX热系统实验室，圣保罗州立大学）
Bayesian Reasoning for Physics Informed Neural Networks
Institute for Theoretical Physics, University of Wroc aw（沃拉夫大学理论物理研究所）；Institute of Low Temperature and Structure Research（低温与结构研究所）；Polish Academy of Sciences（波兰科学院）
From Mesh to Neural Nets: A Multi-Method Evaluation of Physics-Informed Neural Networks and Galerkin Finite Element Method for Solving Nonlinear Convection-Reaction-Diffusion Equations
School of Interdisciplinary Science, Beijing Institute of Technology（北京理工大学交叉科学学院）；School of Mechatronical Engineering, Beijing Institute of Technology（北京理工大学机电工程学院）；Beijing Institute of Technology (Zhuhai)（北京理工大学（珠海））
