Solving the Degasperis–Procesi Equation Boundary Value Problem Using Physics-Informed Neural Networks

Abstract

The Degasperis– Procesi (DP) equation, a prominent nonlinear partial differential equation with important applications in fluid dynamics and wave propagation, poses significant challenges when seeking accurate numerical solutions, especially in the context of boundary value problems. This article presents a novel approach for solving the boundary value problem associated with the DP equation utilizing Physics-Informed Neural Networks (PINNs). Traditional numerical methods often struggle to capture the complex wave dynamics and steep gradients inherent in the DP equation. In contrast, PINNs leverage the power of artificial neural networks and physical principles to learn and approximate the underlying solution without relying on a predefined grid. This enables the accurate representation of the solution in regions where traditional methods falter. Our methodology involves training a neural network to minimize the residual of the DP equation along with enforcing boundary conditions, leading to a data-driven solution that seamlessly incorporates both the governing equation and boundary constraints. We demonstrate the effectiveness of this approach on a range of challenging DP equation boundary value problems. Through comprehensive numerical experiments, we showcase the ability of PINNs to accurately predict the solution of the DP equation while maintaining stability and computational efficiency. Furthermore, we highlight the flexibility of the approach in handling various boundary conditions and complex geometries. This article contributes to the growing body of research in the application of machine learning techniques to solve challenging partial differential equations, particularly for problems where traditional numerical methods encounter limitations.