Partial Differential Equation Solving Model of TgU-Net Based on Golden Spiral Water Flow Optimizer

Recently, theory-guided neural networks have attracted significant attention in solving partial differential equations due to their minimal data requirements and alignment with physical laws. However, selecting the penalty coefficient by incorporating physical laws as penalizing terms in the loss function undoubtedly affects the model's performance. In this paper, we utilize physical laws to guide U-Net neural networks and employ bilevel programming to tune the loss function's hyperparameters, thereby improving the overall model performance. The upper-level variables within the framework are optimized using a water flow optimizer based on the golden spiral eddying function, which enhances population diversity and broadens the search range in the turbulent flow operator. Experimental results confirm the effectiveness of this approach in handling stochastic partial differential equations.