# Three-Dimensional Hyperbolic Mesh Generation Method Based on the Neural Network
## 1 Introduction  
The hyperbolic structured mesh generation method is capable of producing mesh grids with good orthogonality, which has been widely applied in flow field simulations, such as missiles [1], aircraft [2], terrains [3], et al. It is a bottom-up mesh generation approach, where the grids are extrapolated layer by layer from an initial boundary. Since [PERSON] [4] proposed the two-dimensional hyperbolic mesh generation algorithm, its principles have been rapidly applied to three-dimensional models [5; 6; 7]. Under the same constraints, i.e., grid orthogonality and grid size, the governing equations for three-dimensional surface hyperbolic mesh generation are different from the three-dimensional solids [8; 9]. The solution scheme of three-dimensional hyperbolic mesh generation is to locally linearize the governing equations around the initial status, then the governing equations are transformed into a system of equations that are defined on the regular computational space. However, this scheme will lead to issues such as boundary singularities propagating into the interior, instability during the solution process, limited types of boundary conditions along the advancing direction, and the inability to prescribe outer boundary conditions [10; 11; 12]. To address the issue of singularities, researchers have proposed various solutions, such as incorporating spatial variable smoothing coefficients, clustering, matrix correction, and local treatment of the concave-convex region [9; 13]. For solution stability, the introduction of dissipation terms remains a common approach [14; 15]. These techniques effectively ensure the practical applicability of the hyperbolic grid generation method. Consequently,subsequent research has increasingly focused on the application and improving grid generation efficiency [16; 17]. Although the aforementioned studies have enabled the hyperbolic grid generation method to stably produce high-quality grids with different configurations, there has been relatively little focus on the research of boundary control. Consequently, the application of hyperbolic grids in flow field simulation only lies in the boundary layer mesh generation and physical models utilizing far-field boundary conditions [18; 19]. In three-dimensional models, there are only a few boundary control strategies available, such as periodicity, constant plane, and floating edge [8; 13; 20]. These strategies can only satisfy the boundary conditions along the advancing direction under certain circumstances. Moreover, due to the potentially more complex topology of three-dimensional models, special numerical boundary treatments are required at the boundaries, making the imposition of these boundary conditions more complex. Additionally, the current hyperbolic grid generation methods do not offer effective solutions for outer boundary control. Some studies suggested that combining hyperbolic grid generation with the other methods may enable control of the outer boundary. [PERSON] et al. [21] combined the numerical schemes of hyperbolic mesh with parabolic mesh using a weighting coefficient to achieve outer boundary control during the mesh generation process. [PERSON] et al. [22]. achieved outer boundary control by generating hyperbolic mesh near the object surface and regenerating elliptical grids between the hyperbolic mesh boundaries and the outer boundaries. Similarly, [PERSON] et al. [23]. also combined hyperbolic and elliptical grids; in addition to using the elliptical method for outer boundary control, they smoothed the regions of poor-quality grids generated by the hyperbolic method. It is worth noting that, in the case of combining numerical schemes, the influence of outer boundaries on the interior grids is retained, whereas generating elliptical grids in certain regions cannot ensure grid consistency or requires a balance between mesh uniformity and orthogonality.