Optimization of PINN Loss Function and Solution of Partial Differential Equation Based on Dynamic Weighting With Subconstraints and Multiregion Adaptive Sampling

Traditional numerical methods have limitations such as grid dependence and poor adaptability to high-dimensional problems when solving partial differential equations. Physics-Informed Neural Network (PINN) also faces defects such as constraint fitting imbalance, insufficient sampling of key areas, and gradient anomalies in loss function design. In this regard, this article proposes an improved model based on multidimensional optimization of the loss function. This model innovatively designs a subconstraint dynamic weighting strategy, achieving balanced optimization of the governing equations, initial conditions, and boundary conditions through initial loss magnitude calibration and exponential decay factors. A multiregion adaptive sampling mechanism is constructed, dividing the solution domain based on the physical importance of each region and configuring sampling densities differently to ensure higher sampling resources for critical boundaries and complex regions. The results showed that the improved MSE loss function used in the PINN reduced the loss value to 0.0021 after 5000 iterations. The RMSE of its key section was only 0.0026, which was significantly better than the MAE and Huber loss. In the Cook Panel problem, the MDE was as low as 2.8% and the average displacement error was 1.9% under the balanced weight configuration. The MDE of the improved PINN in 3D scenes was 40.5% lower than that of the standard PINN, while maintaining acceptable computing efficiency. The improved PINN can effectively solve the inherent defects of traditional models and provide a new way for efficient and high-precision solution of partial differential equations in complex engineering scenarios.