# Three-Dimensional Hyperbolic Mesh Generation Method Based on the Neural Network
## 4 Discussion  
In this paper, the mesh generation scheme based on the output range-constrained PINNs was applied to the three-dimensional models. By generating the mesh of three-dimensional surfaces and three-dimensional solid models, the effectiveness of this method was demonstrated. Additionally, a variance-constrained outer boundary control strategy is proposed, enabling the control of the outer boundary of different structures during the hyperbolic mesh generation process. Compared to traditional hyperbolic mesh generation methods, the approach proposed in this paper not only achieves higher quality hyperbolic mesh generation but also allows for the control of arbitrary boundary conditions along the advancing direction and outer boundary conditions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & \multicolumn{3}{c}{**Equiangular Skewness**} & \multicolumn{3}{c}{**Deviation Angle**} \\ \hline  & **Min** & **Max** & **Average** & **Min** & **Max** & **Average** \\ \hline Neural-based & 0.677 & 0.998 & 0.918 & 0.212 & 29.092 & 7.376 \\ Traditional & 0.419 & 0.995 & 0.916 & 0.477 & 52.335 & 7.574 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Extreme and average values of equiangle skewness and deviation angle for different methods concerning the terrain solids mesh. Figure 10: Histogram of the mesh quality distribution of the terrain solids generated by different methods. In traditional hyperbolic mesh generation methods, the mesh extrapolation is based on numerically solving linearized governing equations, simple boundary control along the advancing direction can only be achieved through using different numerical methods, while controlling the outer boundary is not feasible. Therefore, a key innovation of this paper is the introduction of outer boundary control into the loss function. As illustrated by the mapping relationship in Figure 1, the boundaries in the physical space correspond to an extremum iso-surface in the computational space. Consequently, although we cannot determine the exact extremum value corresponding to the outer boundary under a given Jacobian determinant, the outer boundary must be an iso-surface, meaning that the values at points on this surface should be identical. This forms the foundation of our outer boundary control strategy. Furthermore, outer boundary constraints can be omitted if necessary, making the method equally applicable to free outer boundaries. Although the cases presented in this paper do not include highly complex constrained boundaries, the computational process for handling complex boundaries does not differ. Even when the complexity of the solution increases due to the consideration of complex boundaries, we can still reduce the training complexity by dividing the overall structure into multiple regions, ensuring successful mesh generation. As for the training process, in three-dimensional hyperbolic mesh generation, the number of governing equations involved becomes greater compared to the 2D case. Moreover, the number of boundaries also becomes greater. Consequently, the loss function includes more terms. As described in Section 2, the loss function for three-dimensional surface mesh generation consists of six terms, while the three-dimensional solid model includes seven terms. For the neural network, this essentially represents a highly complex multi-task learning problem, which can reduce the convergence rate of the network. Therefore, during the training process, considering there is no conflict between the different loss terms that constrain the governing equations, we sum these parts into a single loss term. Similarly, all loss terms related to boundary control are aggregated into another loss term. Finally, the two summed terms are combined with the loss on the initial boundary to form a total loss function, and the seven tasks are reduced to three tasks, thereby reducing task complexity and enhancing the convergence of the neural network.