PINNs with hybrid residual-driven adaptive sampling and weighted loss for the (2+1)-dimensional gpKP equation

To address the challenges of predicting complex solutions and parameter inversion for high-dimensional nonlinear partial differential equations (PDEs), this study proposes an adaptive sampling and loss weighting physics-informed neural network (ASW-PINN) method, overcoming the limitations of traditional PINNs in capturing regions with steep variations. The ASW-PINN method employs a hybrid residual-driven adaptive sampling strategy (combining PDE residuals and their gradients) to dynamically enhance sampling density in high-error regions, while integrating a weighted loss function to balance optimization priorities across physical constraints. We conduct a series of experimental comparisons between the traditional PINNs and the ASW-PINN method, using the (2+1)-dimensional generalized potential Kadomtsev-Petviashvili equation as an example. The experimental results clearly demonstrate the effectiveness of the ASW-PINN method in improving accuracy, with the relative $L_2$ error of the final results reduced in predicting all four solutions. For parameter inversion tasks, the ASW-PINN method accurately identifies equation parameters under noise contamination, highlighting its robustness. Furthermore, key factors affecting the performance of neural networks are discussed in detail, including the number of extra data points added and the architecture of the neural network.