A method for adaptive generation of curved surface mesh based on degree of protection calculation geometry and reinforcement learning

Abstract

This invention discloses an adaptive surface mesh generation method based on degree-preserving computational geometry and reinforcement learning. The main steps are as follows: First, the input background triangular mesh surface is parameterized with degree preservation (preserving angles, area, distance, etc.), mapping the surface to a planar parameter domain, and a target size field in the parameter domain is constructed based on the size scaling ratio and Gaussian curvature. Second, the planar mesh generation process is modeled as a Markov decision process, and a reinforcement learning framework based on the front-end advancement method is designed. A near-end policy optimization algorithm combined with curriculum learning and random network distillation mechanisms is used for policy training to achieve adaptive mesh generation that satisfies the target size field within the parameter domain. Finally, the parameter domain mesh is mapped back to the original surface through degree-preserving inverse mapping, resulting in a high-quality adaptive surface mesh. This invention utilizes the autonomous decision-making capability of reinforcement learning, avoiding the quality degradation problem of traditional front-end advancement methods in complex regions, and is applicable to surfaces of any topological type.

Description

Technical Field

[0001] This invention relates to multiple fields such as computer-aided engineering, computational mathematics, artificial intelligence, and computer graphics, and specifically to an adaptive method for generating surface meshes based on degree-preserving computational geometry and reinforcement learning. Background Technology

[0002] In numerical simulation fields such as computer-aided engineering, finite element analysis, and computational fluid dynamics, high-quality surface mesh generation is a crucial prerequisite for ensuring the accuracy and convergence of numerical calculations. Traditional mesh generation methods, such as Delaunay triangulation and the advance-front method, perform well on planar regions but face numerous challenges when dealing with complex surfaces. In recent years, deep reinforcement learning has modeled the mesh generation process as a Markov decision process, using reinforcement learning agents to replace traditional manual heuristic rules. Through extensive trial and error, it learns a advancement strategy that balances local geometric quality and global topological constraints, potentially fundamentally addressing the inherent shortcomings of traditional methods. For surface mesh scenarios, degree-preserving computational geometry, by maintaining intrinsic features such as angles, areas, or distances while parameterizing the surface, can transform the complex surface mesh generation problem into a relatively simple planar mesh generation problem. Further combining this with reinforcement learning methods significantly reduces solution complexity and improves mesh quality. Summary of the Invention

[0003] To address the shortcomings of existing technologies, this invention aims to provide an adaptive surface mesh generation method based on degree-preserving computational geometry and reinforcement learning.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] An adaptive surface mesh generation method based on degree-preserving computational geometry and reinforcement learning is proposed for applications in computer-aided engineering, finite element analysis, computational fluid dynamics, and computer graphics. This method generates high-quality surface meshes that satisfy target size field constraints. Specifically, it maps the surface to a planar parameter domain through degree-preserving parameterization, generates a planar mesh within the parameter domain using a policy network trained by reinforcement learning, and then inversely maps it back to the surface to obtain an adaptive surface mesh. The method includes the following steps:

[0006] S1: For the input triangular mesh surface Perform coordinate-preserving parameterization to map the surface to the planar parameter domain. Based on local scaling ratio and Gaussian curvature Constructing the target size field in the parameter domain This ensures that the size field is proportional to the scaling ratio and inversely proportional to the Gaussian curvature, and smooths the target size field to guarantee the continuity of size changes.

[0007] S1.1: For triangular mesh surfaces Perform degree-preserving parameterization The planar parameter domain is obtained. For each triangular element, calculate the values ​​of each triangle before and after the mapping. The area ratio is used to obtain the local scaling ratio. ,in Represents the area of ​​a unit;

[0008] S1.2: Based on the user-preset target surface size By combining the scaling relationship, the target size of each triangular unit in the parameter domain is calculated. ; Calculate the discrete Gaussian curvature of each vertex of the surface. The target size is adjusted according to the curvature to achieve a smaller target size in areas with high curvature.

[0009] S1.3: Smooth the constructed target size field to suppress discrete noise and limit abrupt size changes between adjacent vertices, thus obtaining the final target size field. .

[0010] S2: The mesh generation process in the planar parameter domain is modeled as a Markov decision process. A reinforcement learning mesh generation framework based on the frontier advancement method is constructed, and the state space, action space and reward function of the agent are designed. The state space integrates local normalized geometric features and target size ratio features. The action space includes two basic operations: closing connection and inserting new points. The reward function comprehensively considers the effectiveness of the cell, the quality of the cell shape and the matching of the side length and size.

[0011] S2.1: The state space design is specifically based on a reference point. Construct a local observation neighborhood centered on the region, and extract data to the left and right of the undefined region boundary. Adjacent boundary vertices and And select the g interior points closest to the reference point to form a partial observation set. ;Based on the benchmark Construct a local orthogonal coordinate system for the origin. , where the tangent direction , for Rotate 90 degrees counterclockwise; normalize the representation of neighboring points. , The neighborhood points are used to eliminate the effects of translation, rotation, and scale changes; the target size ratio of the neighborhood points is also calculated. To reflect the trend of size change; the final state is represented as ;

[0012] S2.2: The specific definition of action space is: action space The decision-making process involves two layers: the first layer is the selection of the reference point, which prioritizes risk and selects the boundary vertex with the smallest average interior angle across multiple neighborhoods as the reference point; the second layer is the local generation strategy, which includes two operation types—Type 1 is closed connection, which directly connects the left and right adjacent boundary points of the reference point to form a new triangular unit without introducing new vertices; Type 2 is inserting new points, which is based on the relative step size multiplier output by the strategy network. and local orientation angle According to the formula Calculate the location of the new point and construct a new triangle, then update the boundary of the undivided region;

[0013] S2.3: The reward function is designed as follows: The reward function adopts a hierarchical structure and includes the following components: (1) Validity detection: If the action generates an illegal cell or overlaps with the boundary, a negative reward is given; (2) Completion reward: If the mesh generation is completed, a high positive reward is given; (3) Size matching reward The closer the grid size is to the target size, the higher the reward value; (4) Unit quality reward The quality index is based on the shape of a curved triangle; the higher the unit quality, the higher the reward value.

[0014] S3: The near-end policy optimization algorithm is adopted as the core optimization framework. A course learning mechanism is introduced to enable the agent to gradually transition from simple geometric environment to complex geometric environment. At the same time, a curiosity-driven exploration mechanism based on random network distillation is introduced to enhance the policy exploration capability. The resulting policy network can generate a high-quality planar mesh that satisfies the target size field constraint.

[0015] S3.1: Specific implementation of the course learning mechanism: using the normalized total absolute rotation angle of the boundary. As a measure of geometric complexity, among To vary the orientation angle at each boundary vertex; according to The empirical distribution divides the training sample pool into three course phases: Phase I is the regular plane ( Phase II (approaching 1) is used to learn basic boundary advancement and connection strategies; Phase II is an irregular plane ( Medium), used to improve adaptive partitioning capabilities; Stage III is a complex plane ( Larger (larger) samples are used to train robustness and generalization performance under complex geometric structures; during training, the agent learns first in stage I, and after the average reward or success rate reaches a stable threshold, samples from stage II and stage III are gradually introduced to achieve policy transfer from easy to difficult.

[0016] S3.2: The random network distillation mechanism specifically involves introducing two networks to perform feature mapping on the same state: one is a randomly initialized and fixed target network. The other is a trainable prediction network. ; for a given environment observation Both output feature vectors. and The square of the Euclidean distance between the two is used as the intrinsic reward. The comprehensive reward is defined as follows: ,in This represents the extrinsic reward constituted by the reward function in S2.3. This is the balancing coefficient. By predicting that the network will produce a larger prediction error for unfamiliar states, thus providing a higher intrinsic reward, the agent is encouraged to fully explore the unvisited state space in the early stages of training, and gradually focus on the stable convergence of the optimal strategy in the later stages.

[0017] S4: Use the trained policy network to perform mesh generation in the parameter domain to obtain a high-quality planar mesh that satisfies the target size field constraint. Map the planar parameter domain mesh back to the original surface through degree-preserving inverse mapping to obtain the surface mesh.

[0018] Beneficial effects:

[0019] 1. This invention utilizes the autonomous decision-making ability of reinforcement learning to realize the grid generation process, enabling the agent to learn a promotion strategy that takes into account both local geometric quality and global topological constraints, effectively avoiding the local optimum trap and quality degradation problem of traditional methods.

[0020] 2. This invention constructs a unified target size field in the parameter domain that integrates local scaling compensation and curvature perception through the joint design of degree-preserving parameterization and adaptive size field, so that the surface mesh has curvature adaptive characteristics.

[0021] 3. This invention introduces a joint training strategy of course learning mechanism and random network distillation mechanism, which significantly improves the training convergence speed and stability as well as the agent's ability to explore complex state spaces.

[0022] 4. The method of the present invention is universal and applicable to surface models of various topological types, such as open / closed surfaces of different genera. Attached Figure Description

[0023] Figure 1 This is the overall flowchart of the reinforcement learning-based adaptive generation method for surface meshes described in this invention.

[0024] Figure 2 It is the background mesh of the curved surface model in Implementation Case 1.

[0025] Figure 3 This shows the conformal parameterization result of the surface in Implementation Case 1, along with a local magnification. The left image shows the conformal parameterization result, and the right image shows a local magnification.

[0026] Figure 4 This is a magnified view of the adaptive generation result of the planar parametric domain mesh in Implementation Case 1. The left image shows the shape parameterization result, and the right image shows a magnified view of the local area.

[0027] Figure 5 It is the triangular mesh mapped back to the surface in Specific Implementation Case 1.

[0028] Figure 6 It refers to the background mesh of the curved surface model in Implementation Case 2, as well as the segmented cut lines, ring handles, and tunnel rings.

[0029] Figure 7 This is the conformal parameterization result of the surface in Implementation Case 2, along with a local magnification. The left image shows the conformal parameterization result, and the right image shows a local magnification.

[0030] Figure 8 This is a detailed implementation example 2, showing the adaptive generation result of the planar parametric domain mesh and a local magnification. The left image shows the shape parameterization result, and the right image is a local magnification.

[0031] Figure 9 It is the triangular mesh mapped back to the surface in specific implementation case 2. Detailed Implementation

[0032] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention.

[0033] like Figure 1 As shown, this invention provides an adaptive surface mesh generation method based on degree-preserving computational geometry and reinforcement learning. The specific steps are as follows:

[0034] S1: For the input triangular mesh surface Perform coordinate-preserving parameterization to map the surface to the planar parameter domain. Based on local scaling ratio and Gaussian curvature Constructing the target size field in the parameter domain This ensures that the size field is proportional to the scaling ratio and inversely proportional to the Gaussian curvature, and smooths the target size field to guarantee the continuity of size changes.

[0035] S1.1: For triangular mesh surfaces Perform degree-preserving parameterization The planar parameter domain is obtained. For each triangular element, calculate the values ​​of each triangle before and after the mapping. The area ratio is used to obtain the local scaling ratio. ,in Represents the area of ​​a unit;

[0036] S1.2: Based on the user-preset target surface size By combining the scaling relationship, the target size of each triangular unit in the parameter domain is calculated. ; Calculate the discrete Gaussian curvature of each vertex of the surface. Introducing curvature control factor The target size is modified to achieve a smaller target size in high curvature regions, where This is the curvature sensitivity coefficient;

[0037] S1.3: Smooth the constructed target size field to suppress discrete noise and limit abrupt size changes between adjacent vertices, thus obtaining the final target size field. .

[0038] S2: The mesh generation process in the planar parameter domain is modeled as a Markov decision process. A reinforcement learning mesh generation framework based on the frontier advancement method is constructed, and the state space, action space and reward function of the agent are designed. The state space integrates local normalized geometric features and target size ratio features. The action space includes two basic operations: closing connection and inserting new points. The reward function comprehensively considers the effectiveness of the cell, the quality of the cell shape and the matching of the side length and size.

[0039] S2.1: The state space design is specifically based on a reference point. Construct a local observation neighborhood centered on the region, and extract data to the left and right of the undefined region boundary. Adjacent boundary vertices and And select the g interior points closest to the reference point to form a partial observation set. ;Based on the benchmark Construct a local orthogonal coordinate system for the origin. , where the tangent direction , for Rotate 90 degrees counterclockwise; normalize the representation of neighboring points. , The neighborhood points are used to eliminate the effects of translation, rotation, and scale changes; the target size ratio of the neighborhood points is also calculated. To reflect the trend of size change; the final state is represented as ;

[0040] S2.2: The specific definition of action space is: action space The decision-making process involves two layers: the first layer is the selection of the reference point, which prioritizes risk and selects the boundary vertex with the smallest average interior angle across multiple neighborhoods as the reference point; the second layer is the local generation strategy, which includes two operation types—Type 1 is closed connection, which directly connects the left and right adjacent boundary points of the reference point to form a new triangular unit without introducing new vertices; Type 2 is inserting new points, which is based on the relative step size multiplier output by the strategy network. and local orientation angle According to the formula Calculate the location of the new point and construct a new triangle, then update the boundary of the undivided region;

[0041] S2.3: The reward function is designed as follows: The reward function adopts a hierarchical structure and includes the following components: (1) Validity detection: If the action generates an illegal cell or overlaps with the boundary, a negative reward is given; (2) Completion reward: If the mesh generation is completed, a high positive reward is given; (3) Size matching reward ,in This refers to relative dimensional error. Let the target side length be at the centroid of the triangle. For the edge Length, (3) Size matching tolerance; (4) Unit quality bonus Quality indicators for curved triangular shapes, such as aspect ratio and stretchability, are adopted.

[0042] S3: The near-end policy optimization algorithm is adopted as the core optimization framework. A course learning mechanism is introduced to enable the agent to gradually transition from simple geometric environment to complex geometric environment. At the same time, a curiosity-driven exploration mechanism based on random network distillation is introduced to enhance the policy exploration capability. The resulting policy network can generate a high-quality planar mesh that satisfies the target size field constraint.

[0043] S3.1: Specific implementation of the course learning mechanism: using the normalized total absolute rotation angle of the boundary. As a measure of geometric complexity, among To vary the orientation angle at each boundary vertex; according to The empirical distribution divides the training sample pool into three course phases: Phase I is the regular plane ( Phase II (approaching 1) is used to learn basic boundary advancement and connection strategies; Phase II is an irregular plane ( Medium), used to improve adaptive partitioning capabilities; Stage III is a complex plane ( Larger (larger) samples are used to train robustness and generalization performance under complex geometric structures; during training, the agent learns first in stage I, and after the average reward or success rate reaches a stable threshold, samples from stage II and stage III are gradually introduced to achieve policy transfer from easy to difficult.

[0044] S3.2: The random network distillation mechanism specifically involves introducing two networks to perform feature mapping on the same state: one is a randomly initialized and fixed target network. The other is a trainable prediction network. ; for a given environment observation Both output feature vectors. and The square of the Euclidean distance between the two is used as the intrinsic reward. The comprehensive reward is defined as follows: ,in This represents the extrinsic reward constituted by the reward function in S2.3. This is the balancing coefficient. By predicting that the network will produce a larger prediction error for unfamiliar states, thus providing a higher intrinsic reward, the agent is encouraged to fully explore the unvisited state space in the early stages of training, and gradually focus on the stable convergence of the optimal strategy in the later stages.

[0045] S4: Use the trained policy network to perform mesh generation in the parameter domain to obtain a high-quality planar mesh that satisfies the target size field constraint. Map the planar parameter domain mesh back to the original surface through degree-preserving inverse mapping to obtain the surface mesh.

[0046] Example 1

[0047] This embodiment uses a zero-genus open surface (Chinese Lion model) as an example to further describe the invention in detail. The specific steps are as follows:

[0048] S1: For the ChineseLion triangular mesh surface (like Figure 2 (As shown) Conformal parameterization is performed to map the surface to the planar parameter domain. (like Figure 3 (As shown). Calculate the local scaling ratio for each triangular unit. According to the preset target side length Calculate the target side length of the parameter domain The initial target size of the vertex is obtained by weighting the area of ​​the parameter domain. Calculate the discrete Gaussian curvature of each vertex. Introducing a curvature control factor (taking the curvature sensitivity coefficient) After correction, size field smoothing is performed to obtain the final target size field. .

[0049] S2: Construct a reinforcement learning mesh generation framework. Set local observation parameters n=2, g=2, that is, take two adjacent boundary vertices on the left and right sides of the boundary, and select the two interior points closest to the reference point. For neighboring points... Perform a local coordinate transformation for the normalization factor and calculate the size ratio feature. In the action space, the two types of connections—closed connections and insertions of new points—are determined by the discrete actions output by the policy network. The position of the new point is determined by the relative step size multiplier. and direction angle Determined. Size matching tolerance of the reward function. The shape quality index is set to the stretch ratio. ,in It is the radius of the circumcircle of the triangle. It is the length of the longest side of the triangle.

[0050] S3: Training is performed using a proximal policy optimization algorithm. Both the policy network and the value network employ a three-layer fully connected network structure (256-256-256), with rectified linear units as the activation function. A course learning mechanism is introduced, dividing the 300 training boundaries into... The metric is divided into three stages: Stage I (100 regular planar boundaries) is used to learn the basic policy, while Stage II (100 irregular planar boundaries) and Stage III (100 complex planar boundaries) progressively increase the complexity. A random distillation mechanism and an intrinsic reward balancing coefficient are also introduced. Take 0.01. The total number of training steps is approximately Step. The optimizer uses Adam with a learning rate of [missing information]. .

[0051] S4: Utilize the trained and converged policy network to perform mesh generation within the parameter domain, obtaining a planar parameter domain mesh (e.g., ...). Figure 4 As shown), by conformal inverse mapping back to the original ChineseLion surface, the final result is obtained (as shown). Figure 5 (As shown).

[0052] Example 2

[0053] This embodiment uses a genus closed surface (Kitten model) as an example to further describe the invention in detail. The specific steps are as follows:

[0054] S1: For Kitten surfaces (such as...) Figure 6 (As shown) Path cutting and conformal parameterization are performed. First, a pair of ring-handle loops and tunnel loops on the surface are searched and cut along them to obtain a topological disk containing the loop cut boundary. The planar parameter domain is obtained through conformal parameterization. (like Figure 7As shown in the figure, the kerf boundary satisfies the periodic alignment condition in the parameter domain. Then, the target size field in the parameter domain is constructed according to step S1. .

[0055] S2: Construct a reinforcement learning framework similar to Example 1, but additionally consider periodic boundary conditions: During mesh generation, for vertices on periodic boundaries, their paired vertices need to be updated synchronously to ensure a one-to-one correspondence between nodes on both sides of the cut. The design of the state space and action space is the same as in Example 1.

[0056] S3: Training is performed using the same proximal strategy optimization training framework, curriculum learning mechanism, and random distillation mechanism as in Example 1. Since the periodic boundaries of the one-genus surface parameter domain introduce additional structural constraints, it is necessary to ensure that the generated mesh satisfies the sampling consistency condition on the periodic boundaries during training.

[0057] S4: Utilize the trained policy network to generate a mesh that satisfies the target size field within the parameter domain (e.g., ...). Figure 8 As shown), the final result is obtained by mapping the mesh back to the original Kitten surface through conformal inverse mapping (as shown). Figure 9 As shown in the figure, the nodes on both sides of the slit maintain a strict one-to-one correspondence, and the surface mesh is well closed after inverse mapping.

[0058] The technical means disclosed in this invention are not limited to those disclosed in the above embodiments, but also include technical solutions composed of any combination of the above technical features. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications are also considered within the scope of protection of this invention.

Claims

1. A method for adaptive generation of surface meshes based on degree-preserving computational geometry and reinforcement learning, characterized in that, It is used in computer-aided engineering, finite element analysis, computational fluid dynamics or computer graphics to generate high-quality surface meshes that meet the target size field constraints; Specifically, the surface is mapped to the planar parameter domain by preserving the degree of parameterization. A planar mesh is generated in the parameter domain using a policy network trained by reinforcement learning. The mesh is then mapped back to the surface to obtain an adaptive surface mesh. Includes the following steps: S1: For the input triangular mesh surface Perform coordinate-preserving parameterization to map the surface to the planar parameter domain. Based on local scaling ratio and Gaussian curvature Constructing the target size field in the parameter domain This ensures that the size field is directly proportional to the scaling ratio and inversely proportional to the Gaussian curvature, and smooths the target size field to guarantee the continuity of size changes. S2: Model the mesh generation process in the plane parameter domain as a Markov decision process, construct a reinforcement learning mesh generation framework based on the frontier advancement method, and design the agent's state space, action space, and reward function; The state space integrates local normalized geometric features and target size ratio features, the action space includes two basic operations: closed connection and insertion of new points, and the reward function comprehensively considers cell effectiveness, cell shape quality and side length size matching. S3: The near-end policy optimization algorithm is adopted as the core optimization framework. A course learning mechanism is introduced to enable the agent to gradually transition from simple geometric environment to complex geometric environment. At the same time, a curiosity-driven exploration mechanism based on random network distillation is introduced to enhance the policy exploration capability. The resulting policy network can generate a high-quality planar mesh that satisfies the target size field constraint. S4: Use the trained policy network to perform mesh generation in the parameter domain to obtain a high-quality planar mesh that satisfies the target size field constraint. Map the planar parameter domain mesh back to the original surface through degree-preserving inverse mapping to obtain the surface mesh.

2. The adaptive generation method for surface meshes based on degree-preserving computational geometry and reinforcement learning according to claim 1, characterized in that, The construction of the target size field in step S1 specifically includes the following steps: S1.1: For triangular mesh surfaces Perform degree-preserving parameterization The planar parameter domain is obtained. For each triangular unit, calculate the values ​​of each triangle before and after the mapping. The area ratio is used to obtain the local scaling ratio. ,in Represents the area of ​​a unit; S1.2: Based on the user-preset target surface size By combining the scaling relationship, the target size of each triangular unit in the parameter domain is calculated. ; Calculate the discrete Gaussian curvature of each vertex of the surface. The target size is adjusted according to the curvature to achieve a smaller target size in areas with high curvature. S1.3: Smooth the constructed target size field to suppress discrete noise and limit abrupt size changes between adjacent vertices, thus obtaining the final target size field. .

3. The adaptive generation method for surface meshes based on degree-preserving computational geometry and reinforcement learning according to claim 2, characterized in that, The design of the state space in step S2 is specifically as follows: based on the reference point Construct a local observation neighborhood centered on the region, and extract data to the left and right of the undefined region boundary. Adjacent boundary vertices and And select the g interior points closest to the reference point to form a partial observation set. ;Based on the benchmark Construct a local orthogonal coordinate system for the origin. , where the tangent direction , for Rotate 90 degrees counterclockwise; normalize the representation of neighboring points. , The neighborhood points are used to eliminate the effects of translation, rotation, and scale changes; the target size ratio of the neighborhood points is also calculated. To reflect the trend of size change; the final state is represented as .

4. The adaptive generation method for surface meshes based on degree-preserving computational geometry and reinforcement learning according to claim 3, characterized in that, The definition of the action space in step S2 is specifically as follows: Action space The decision-making process involves two layers: the first layer is the selection of the reference point, which prioritizes risk and selects the boundary vertex with the smallest average interior angle across multiple neighborhoods as the reference point; the second layer is the local generation strategy, which includes two operation types—Type 1 is closed connection, which directly connects the left and right adjacent boundary points of the reference point to form a new triangular unit without introducing new vertices; Type 2 is inserting new points, which is based on the relative step size multiplier output by the strategy network. and local orientation angle According to the formula Calculate the location of the new point and construct a new triangle, and update the boundary of the undivided region.

5. The adaptive generation method for surface meshes based on degree-preserving computational geometry and reinforcement learning according to claim 4, characterized in that, The reward function designed in step S2 is as follows: The reward function adopts a hierarchical structure and includes the following components: (1) Validity detection: If the action generates an illegal cell or overlaps with the boundary, a negative reward is given; (2) Completion reward: If the mesh generation is completed, a high positive reward is given; (3) Size matching reward The closer the grid size is to the target size, the higher the reward value; (4) Unit quality reward The quality index is based on the shape of a curved triangle; the higher the unit quality, the higher the reward value.

6. The adaptive generation method for surface meshes based on degree-preserving computational geometry and reinforcement learning according to claim 1, characterized in that, The course learning mechanism described in step S3 is specifically as follows: using the normalized total absolute rotation angle of the boundary. As a measure of geometric complexity, To vary the orientation angle at each boundary vertex; according to The empirical distribution divides the training sample pool into three course phases: Phase I is the regular plane, i.e. A value close to 1 is used to learn basic boundary advancement and connection strategies; Phase II is an irregular plane, that is Medium, used to improve adaptive partitioning capabilities; Phase III is a complex plane, that is The larger size is used to train robustness and generalization performance under complex geometric structures. During the training process, the agent learns first in stage I. After the average reward or success rate reaches a stable threshold, samples from stage II and stage III are gradually introduced to achieve policy transfer from easy to difficult.

7. The adaptive generation method for surface meshes based on degree-preserving computational geometry and reinforcement learning according to claim 1, characterized in that, The random network distillation mechanism described in step S3 specifically involves introducing two networks to perform feature mapping on the same state: one is a randomly initialized and fixed target network. The other is a trainable prediction network. ; for a given environment observation Both output feature vectors. and The square of the Euclidean distance between the two is used as the intrinsic reward. The comprehensive reward is defined as follows: ,in This represents the extrinsic reward constituted by the reward function in S2. The balancing coefficient is used to predict the network's large prediction error in unfamiliar states, thereby giving it a higher intrinsic reward. This encourages the agent to fully explore the unvisited state space in the early stages of training, and gradually focus on the stable convergence of the optimal strategy in the later stages.

8. The adaptive generation method for surface meshes based on degree-preserving computational geometry and reinforcement learning according to claim 1, characterized in that, The method is applied to mesh generation tasks in fields such as computer-aided engineering, finite element analysis, computational fluid dynamics, and computer graphics.

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