关键词: data-driven, PINNs, residual-based adaptive refinement, fifth-order KdV equation

Abstract: In the field of nonlinear partial differential equations (PDEs), the fifth-order Korteweg-De Vries (KdV) equation serves as a fundamental model with significant physical implications, extending the classical KdV framework through the incorporation of high-order spatial derivatives to capture strong dispersion effects. However, the inherent nonlinearity and complexity of this PDE present substantial challenges for obtaining accurate numerical solutions. To address these issues, this paper proposes a residual-based adaptive refinement physics-informed neural networks (RAR-PINNs) method. This approach synergizes the nonlinear approximation capability of PINNs with a residual-driven adaptive sampling strategy. By dynamically redistributing training points according to the magnitude of the PDE residuals, RAR-PINNs effectively concentrate computational resources on ``critical regions'', such as soliton peaks and high-gradient zones, where errors are predominant. Furthermore, we construct a composite physics-informed loss function that incorporates initial and boundary conditions, PDE residuals, and, in an enhanced variant, energy conservation laws, to further improve solution fidelity. Numerical experiments on two variants of the fifth-order KdV equation demonstrate that RAR-PINNs significantly outperform conventional PINNs in terms of both accuracy (reducing relative errors by one to two orders of magnitude) and computational efficiency. The conservation-law-enhanced version of the model yields even higher precision, underscoring the efficacy and robustness of the proposed method. This study establishes a powerful deep learning framework for tackling complex PDEs with sharp or singular solution structures.

Key words: data-driven, PINNs, residual-based adaptive refinement, fifth-order KdV equation