# Planar Delaunay Mesh Smoothing Method Based on Angle and a Deep Q-Network
## 5 Conclusions  
This paper proposes a planar Delaunay mesh smoothing method based on angle and a deep Q-network. Firstly, the central node based on the vertex angle of the bisected polygon is updated, and the convex hull algorithm is used to construct the inner polygon for the updated scattered points. In addition, to address the problem of low generalization of the unconstrained DQN model, an inner polygon is proposed to be iteratively updated with repeated training of reinforcement learning, and the updated inner polygon is used as the basis for the reward function setting during neural network training. The effectiveness of this method was demonstrated through simple examples and meshes with size gradient features. Moreover, a large number of untrained examples were used to  
Figure 32: Example 2: finite element calculation results after smoothing for each optimization method. (**a**) Laplacian; (**b**) angle-based; (**c**) ODT; (**d**) CVT; (**e**) unconstrained DQN; and (**f**) constrained DQN.  
verify the high generalization of the DQN model trained using this method. Additionally, the comparison experiments of this paper's method with Laplacian, angle-based, ODT, CVT, and unconstrained DQN on four examples show that this paper's method can better control the minimum angle, improve the minimum quality, and reduce the number of poor elements. Finally, two numerical examples were used for finite element analysis to verify the improvement effect of the proposed method on the finite element calculation results.  
This study explores the impact of the constructed inner polygon on a deep Q-based network smoothing method and improves the performance of the DQN model. Although some achievements have been made, many problems need to be solved. One problem is that the proposed method only focuses on triangular elements as the research object. However, the element types also include quadrilateral elements, tetrahedron elements, Hexahedron elements, etc. In future work, we plan to expand our research on element types, construct inner polygon or even inner polyhedra, and use reinforcement learning methods to study the mesh. Finally, we will also research the latest deep network structures to shorten training time.  
Conceptualization, F.Z. and G.S.; methodology, F.Z.; software, F.Z.; validation, F.Z. and G.S.; formal analysis, F.Z.; investigation, F.Z.; resources, F.Z.; data curation, F.Z.; writing--original draft preparation, F.Z.; writing--review and editing, G.S.; visualization, F.Z.; supervision, G.S.; project administration, G.S.; funding acquisition, G.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (51878347).  
Not applicable.  
The data used to support the findings of this study are available from the corresponding author upon request.  
This study was financially supported by the National Natural Science Foundation of China (No. 51878347), and the authors acknowledge all the other scholars for their advice in the process of improving this article.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.