# Three-Dimensional Hyperbolic Mesh Generation Method Based on the Neural Network
## 3 Results
### Terrain Surface  
We selected a terrain surface with numerous features within a one-square-kilometer area. The projection of this surface onto the xy-plane is rectangular. Considering that the spatial range of this surface is the same in both directions, the surface is divided into four parts for sequential mesh generation. After determining the initial boundary, we used a neural network with four hidden layers, each containing 64 nodes, to fit the solution of the governing equations defined on each divided surface, and the value of the Jacobian determinant is set to 0.01. Starting from the second solving process, the last layer boundary generated from the previous solution was used as the new initial boundary. In the final solving stage, the outer boundary constraint was introduced into the loss function to achieve outer boundary control. When it comes to the grid generation step, the distribution of the required \(\xi\) and \(\eta\) values is predicted by the neural network using the uniformly selected points on the red initial boundary and the left constraint boundary. The mesh generation process is shown in Figure 3.  
In Figure 3, the red curve represents the selected initial boundary, the green curve represents the constraint boundary along the advancing direction, and the saffron yellow curve represents the outer boundary. As we can see from Figure 3b, the outer boundary was effectively restricted in the neural-based method. In addition to the surface mesh generation method proposed in this paper, conventional surface meshes are typically generated using linear interpolation or projection methods. To compare the mesh quality generated by these three methods, we choose orthogonality and skewness as mesh quality evaluation criteria. Orthogonality is defined as one minus the maximum absolute value of the dot product of unit vectors on each pair of adjacent edges in each grid, i.e.,  
\[\max\Bigl{\{}1-\Bigl{|}\overset{\rightarrow}{t_{i}}\cdot\overset{\rightarrow} {t_{j}}\Bigr{|}\Bigr{\}}. \tag{16}\]  
Skewness is defined as one minus the ratio of the lengths of the two diagonals of the grid, i.e.,  
\[1-\frac{L_{i}}{L_{j}}. \tag{17}\]  
From the definitions, it can be seen that the larger the value of orthogonality or the smaller the value of skewness, the better the grid quality. The comparison of the mesh quality generated by different methods is shown in Figure 4.  
In Figure 4, the left side shows a comparison of the orthogonality results, while the right side presents a comparison of the skewness results. It can be seen that the grid generated by the neural network has a more uniform distribution in terms of orthogonality and skewness. The grid generated by the other two methods shows a significant decline in quality in areas where the surface changes more drastically. To quantitatively analyze the differences in meshes generated by different methods, we compared the minimum orthogonality value and the maximum skewness value. Additionally, we perform a deeper analysis using mesh smoothness in different directions. Smoothness is defined as the rotation angle between three adjacent nodes in a given direction, which is a commonly used criterion for mesh quality assessment. The smoothness value ranges from 0 to 1, with larger values indicating greater rotation angles, meaning the mesh is less smooth. So, we also compare the minimum value of smoothness in different directions. Given that larger smoothness or orthogonality values mean better mesh quality, we use one minus the maximum skewness value to ensure that each parameter indicates  
Figure 3: Hyperbolic mesh generation of three-dimensional surface: (**a**) intermediate mesh results without outer boundary constraint; (**b**) final mesh results with outer boundary constraint. better quality when the corresponding value increases. The comparison results are shown in Figure 5.  
As shown in Figure 5, except for the smoothness in the I-direction, the surface meshes generated by the neural network exhibit superior quality compared to other methods.