基于物理信息神经网络的通用网格生成方法

doi:10.11996/JG.j.2095-302X.2025050990

图学学报 ›› 2025, Vol. 46 ›› Issue (5): 990-997.DOI: 10.11996/JG.j.2095-302X.2025050990

• 图像处理与计算机视觉 • 上一篇下一篇

基于物理信息神经网络的通用网格生成方法

张浩轩¹^,²^,³(), 李海生¹^,²^,³(), 王敏¹^,²^,³, 李楠¹^,²^,³

¹ 北京工商大学计算机与人工智能学院，北京 100048
² 食品安全大数据技术北京市重点实验室，北京 100048
³ 农产品质量安全追溯技术及应用国家工程实验室，北京 100048

收稿日期:2024-11-12 接受日期:2024-12-31 出版日期:2025-10-30 发布日期:2025-09-10
通讯作者:李海生(1974-)，男，教授，博士。主要研究方向为计算机图形学、网格生成、数字几何处理等。E-mail：lihsh@th.btbu.edu.cn
第一作者:张浩轩(2000-)，男，硕士研究生。主要研究方向为网格生成、数字几何处理。E-mail：zhxggg613@126.com
基金资助:
北京市自然科学基金(L233026);国家自然科学基金(62277001);国家自然科学基金(62272014)

Universal mesh generation method based on physics-informed neural network

ZHANG Haoxuan¹^,²^,³(), LI Haisheng¹^,²^,³(), WANG Min¹^,²^,³, LI Nan¹^,²^,³

¹ School of Computing and Artificial Intelligence, Beijing Technology and Business University, Beijing 100048, China
² Beijing Key Laboratory of Big Data Technology for Food Safety, Beijing 100048, China
³ National Engineering Laboratory for Agri-product Quality Traceability Beijing, Beijing 100048, China

Received:2024-11-12 Accepted:2024-12-31 Published:2025-10-30 Online:2025-09-10
First author：ZHANG Haoxuan (2000-), master student. His main research interests cover mesh generation and digital geometry processing. E-mail：zhxggg613@126.com
Supported by:
Beijing Natural Science Foundation(L233026);National Natural Science Foundation of China(62277001);National Natural Science Foundation of China(62272014)

摘要/Abstract

摘要：

在数值模拟中，结构网格的生成往往需要大量的时间和人力投入。结构网格生成的一般方案是寻找计算域和物理域之间的映射，这种映射可以通过求解偏微分方程获得。然而，现有的结构网格生成方法难以同时保证效率和网格质量。针对上述问题，提出了一种基于物理信息神经网络的通用网格生成模型(UMG-PINN)。该模型将网格生成任务作为一个从计算域到物理域的网格变形问题，以边界曲线为输入，利用注意力网络捕捉计算域与物理域之间的潜在映射，为输入的物理域生成结构网格。UMG-PINN在损失函数中引入线性弹性力学中的Navier-Lamé方程作为底层控制方程，确保神经网络在优化损失值时符合弹性体变形规律。由于该模型是自监督的，所以不需要先验知识或数据集，减少了以往制作结构网格数据集的工作量。实验结果表明，UMG-PINN相比传统的超限插值法能够生成更高质量的结构网格。此外，UMG-PINN在物理信息的约束下，也可以应用于非结构网格生成。

关键词: 网格生成, 物理信息神经网络, Navier-Lamé方程, 自监督学习

Abstract:

Structured mesh generation in numerical simulation often requires a lot of time and manpower. Traditionally, the process relies on establishing a mapping between the computational domain and the physical domain, which is typically obtained by solving partial differential equations. However, existing structured mesh generation methods struggle to simultaneously achieve both high efficiency and superior mesh quality. To address this issue, we propose a universal mesh generation model based on physics-informed neural network (UMG-PINN). This model formulates mesh generation task as a mesh deformation problem from the computational domain to the physical domain. By taking the boundary curve as input and leveraging an attention network, UMG-PINN captures the potential mapping between the computational and physical domains, thereby generating high-quality structured mesh. To enforce physical constraints, the model incorporates the Navier-Lamé equation from linear elasticity into the loss function, ensuring that the neural network conforms to the principles of elastic body deformation during optimization. A key advantage of UMG-PINN is its fully self-supervised training process, which eliminates the need for prior knowledge or pre-existing datasets, greatly reducing the effort required for structured mesh dataset construction. Experimental results show that UMG-PINN outperforms traditional transfinite interpolation methods by generating higher quality structured meshes. In addition, UMG-PINN can also be extended to unstructured mesh generation under the constraints of physical information.

Key words: mesh generation, physics-informed neural network, Navier-Lamé equation, self-supervised learning

中图分类号:

TP183

张浩轩, 李海生, 王敏, 李楠. 基于物理信息神经网络的通用网格生成方法[J]. 图学学报, 2025, 46(5): 990-997.

ZHANG Haoxuan, LI Haisheng, WANG Min, LI Nan. Universal mesh generation method based on physics-informed neural network[J]. Journal of Graphics, 2025, 46(5): 990-997.

图/表 12

图1 UMG-PINN整体网络框架

Fig. 1 The overall network architecture of UMG-PINN

图2 模型3上不同结构网格生成方法的结果((a) TFI；(b) MGNet+双曲线方程；(c) MGNet+泊松方程；(d) UMGPINN)

Fig. 2 Results of different structured mesh generation methods on model3 ((a) TFI; (b) MGNet+hyperbola equation; (c) MGNet+poisson equation; (d) UMGPINN)

图3 模型4上不同结构网格生成方法的结果((a) MGNet+双曲线方程；(b) MGNet+泊松方程；(c) UMGPINN)

Fig. 3 Results of different structured mesh generation methods on model4 ((a) MGNet+hyperbola equation;(b) MGNet+poisson equation; (c) UMGPINN)

图4 模型5上不同结构网格生成方法的结果((a) TFI；(b) MGNet+双曲线方程；(c) MGNet+泊松方程；(d) UMGPINN)

Fig. 4 Results of different structured mesh generation methods on model5 ((a) TFI; (b) MGNet+hyperbola equation; (c) MGNet+poisson equation; (d) UMGPINN)

图5 在不同模型下三维结构网格生成的结果((a) 模型1；(b) 模型2；(c) 模型3)

Fig. 5 Results of 3D structured mesh generation on different models ((a) Model 1; (b) Model 2; (c) Model 3)

表1 不同结构网格模型的平均最小/最大夹角的比较/(°)

Table 1 A comparison of average min/max included angle for different models of structured meshes/(°)

模型	TFI	MGNet^[31]+泊松方程	MGNet^[31]+双曲线方程	Ours
模型1	83.66/96.40	85.58/94.42	84.84/95.18	85.75/94.21
模型2	83.82/96.20	84.63/96.37	85.09/95.89	85.40/96.42
模型3	65.08/114.70	69.00/110.82	65.74/114.31	70.11/109.54
模型4	-	79.78/100.53	78.86/101.54	75.13/105.40
模型5	82.94/97.10	83.17/96.91	81.79/98.27	83.32/96.77
模型6	84.14/95.86	86.75/93.25	85.31/94.66	87.91/92.09

表2 不同结构网格模型的网格生成时间的比较/s

Table 2 A comparison of mesh generation time for different models of structured meshes/s

模型	TFI	MGNet^[31]+ 泊松方程	MGNet^[31]+ 双曲线方程	Ours
模型1	0.011	0.004	0.005	0.001
模型2	0.082	0.005	0.005	0.001
模型3	0.015	0.003	0.003	0.003
模型4	-	0.003	0.004	0.003
模型5	0.082	0.005	0.004	0.001
模型6	0.017	0.003	0.005	0.001

图6 模型1上不同非结构网格生成方法的结果

Fig. 6 Results of different unstructured mesh generation methods on model 1 ((a) Delaunay; (b) UMGPINN)

图7 模型2上不同非结构网格生成方法的结果

Fig. 7 Results of different unstructured mesh generation methods on model2 ((a) Delaunay; (b) UMGPINN)

图8 在不同模型下三维非结构网格生成的结果((a) 模型1；(b) 模型2；(c) 模型3)

Fig. 8 Results of 3D unstructured mesh generation on different models ((a) Model 1; (b) Model 2; (c) Model 3)

表3 不同非结构网格模型的平均最小/最大夹角的比较/(°)

Table 3 A comparison of average min/max included angle for different models of unstructured meshes/(°)

模型	Delaunay	Ours
模型1	46.26/77.72	36.22/86.63
模型2	46.13/77.78	42.19/79.94
模型3	46.36/77.53	52.23/68.84
模型4	46.7/77.16	52.98/67.78
模型5	46.50/77.44	52.54/68.62
模型6	46.70/77.18	48.02/74.34

表4 不同非结构网格模型的平均纵横比的比较

Table 4 A comparison of average aspect ratio for different models of unstructured meshes

模型	Delaunay	Ours
模型1	1.359	1.722
模型2	1.362	1.482
模型3	1.356	1.185
模型4	1.346	1.164
模型5	1.352	1.176
模型6	1.346	1.321

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基于物理信息神经网络的通用网格生成方法

Universal mesh generation method based on physics-informed neural network

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