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Han T, Ye J, Yan L S, et al. Adaptive mesh partitioning for graph attention Transformer networks[J]. Opto-Electron Eng, 2025, 52(4): 250058. doi: 10.12086/oee.2025.250058
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Adaptive mesh partitioning for graph attention Transformer networks
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Abstract
To address the challenge of balancing the computational accuracy and efficiency in adaptive finite element meshing, this study proposes a GTF-Net model based on the attention fusion mechanism. The model combines the graph attention network with the Transformer architecture, dynamically couples local geometric features with the global physical field through a multi-head cross-attention module, and enhances the representation of singular fields and complex boundaries. The verification of two case studies of waveguide transmission and Bessel equation shows that compared with the traditional Scikit-FEM (skFem) method, GTF-Net improves computational efficiency while reducing the standard deviation of gradient error by 85.9% and 23.8%, respectively. The results show that the model significantly improves the fit between mesh distribution and physical field changes through nonlinear feature mapping, providing a novel deep learning solution for adaptive mesh optimization in engineering calculations. -
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References
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Overview
In the fields of engineering and computational science, finite element analysis is a commonly used numerical simulation tool. Its accuracy and efficiency directly affect the reliability and computational cost of engineering design. However, traditional adaptive mesh refinement technology faces certain challenges in the pursuit of high-precision and high-efficiency collaborative optimization. Especially when dealing with engineering problems with singular fields or complex boundary conditions, traditional iterative methods often show the problems such as uneven gradient error distribution and slow convergence. These limitations not only affect the accuracy of the calculation results, but also limit the application of finite element analysis in complex problems. To address the above problems, this study proposes an adaptive mesh partitioning framework based on the attention fusion mechanism, namely GATv2-Transformer fusion network (GTF-Net). This method transforms the mesh partitioning problem into a node classification problem. Each mesh node is regarded as a node in the graph, and the edges between nodes represent the relationship between units. The relationship between mesh units is modeled using graph neural networks, thereby achieving adaptive adjustment of mesh partitioning. Graph neural networks automatically adjust the mesh structure by learning these relationships. The network innovatively combines the graph attention mechanism with the Transformer architecture, and realizes the dynamic coupling of local geometric features and global physical fields through multi-head cross attention modules, effectively improving the representation ability of complex environments. The analytical solution of multiple equations is introduced into the network training, and a multi-task learning objective is constructed to ensure the generalization performance of the model under different physical field characteristics. The typical optical waveguide transmission problem example and the solution results of the first-kind Bessel function show that compared with the traditional skFem method, GTF-Net has improved the calculation speed while reducing the standard deviation of the gradient error by more than 20% (the Bessel function case is reduced by 23.8%, and the optical waveguide case is reduced by 85.9%). The experimental results show that the network significantly improves the matching degree between the grid density distribution and the physical field changes through nonlinear mapping of the feature space, and the method has a certain generalization ability and can adapt to different types of problems and application scenarios. This method provides a new deep learning solution for adaptive finite element analysis in engineering calculations, and also opens up a new technical path for the development of data-driven intelligent finite element analysis.
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Han T, Ye J, Yan L S, et al. Adaptive mesh partitioning for graph attention Transformer networks[J]. Opto-Electron Eng, 2025, 52(4): 250058. doi: 10.12086/oee.2025.250058
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Figure 1.
Process of adaptive mesh refinement under node classification problem
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Figure 2.
GTF-Net structure diagram
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Figure 3.
The training and validation loss of GTF-Net
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Figure 4.
Bessel's equations. (a) Original mesh; (b) Solution field (original mesh); (c) Gradient error distribution (original mesh); (d) GTF-Net mesh; (e) Solution field (GTF-Net mesh); (f) Error gradient distribution (GTF-Net mesh); (g) skFem mesh; (h) Solution field (skFem mesh); (i) Gradient error distribution (skFem mesh)
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Figure 5.
Solving optical waveguide. (a) Original mesh; (b) Solution field (original mesh); (c) Gradient error distribution (original mesh); (d) GTF-Net mesh; (e) Solution field (GTF-Net mesh); (f) Error gradient distribution (GTF-Net mesh); (g) skFem mesh; (h) Solution field (skFem mesh); (i) Gradient error distribution (skFem mesh)
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Figure 6.
Quality distribution of different mesh cells. (a) SICN value distribution of original mesh cells; (b) SICN value distribution of GTF-Net mesh cells; (c) SICN value distribution of skFem mesh cells; (d) Gamma value distribution of original mesh cells; (e) Gamma value distribution of GTF-Net mesh cells; (f) Gamma value distribution of skFem mesh cells; (g) SIGE value distribution of original mesh cells; (h) SIGE value distribution of GTF-Net mesh cells; (i) SIGE value distribution of skFem mesh cells
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