A Boundary Integral-based Neural Operator for Mesh Deformation
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Abstract (arXiv)
This paper presents an efficient mesh deformation method based on boundary integration and neural operators, formulating the problem as a linear elasticity boundary value problem (BVP). To overcome the high computational cost of traditional finite element methods and the limitations of existing neural operators in handling Dirichlet boundary conditions for vector fields, we introduce a direct boundary integral representation using a Dirichlet-type Green's tensor. This formulation expresses the internal displacement field solely as a function of boundary displacements, eliminating the need to solve for unknown tractions. Building on this, we design a Boundary-Integral-based Neural Operator (BINO) that learns the geometry- and material-aware Green's traction kernel. A key technical advantage of our framework is the mathematical decoupling of the physical integration process from the geometric representation via geometric descriptors. While this study primarily demonstrates robust generalization across diverse boundary conditions, the architecture inherently possesses potential for cross-geometry adaptation. Numerical experiments, including large deformations of flexible beams and rigid-body motions of NACA airfoils, confirm the model's high accuracy and strict adherence to the principles of linearity and superposition. The results demonstrate that the proposed framework ensures mesh quality and computational efficiency, providing a reliable new paradigm for parametric mesh generation and shape optimization in engineering.
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References · 12
- 2015Deep Residual Learning for Image Recognitioncited by 52,096
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
- 2018On the Spectral Bias of Neural Networkscited by 1,341
- 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
- 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
- 2022Wavelet neural operator: a neural operator for parametric partial differential equationscited by 78
- 2024Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometriescited by 47
- 2022Learning the Solution Operator of Boundary Value Problems using Graph Neural Networkscited by 17
- 2022BI-GreenNet: Learning Green's functions by boundary integral networkcited by 13
- 2023Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometriescited by 13
- 2024Learning the boundary-to-domain mapping using Lifting Product Fourier Neural Operators for partial differential equationscited by 4
- 2020Mesh deformation techniques in fluid-structure interaction: robustness, accumulated distortion and computational efficiencycited by 2
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- 2026Physics-integrated neural differentiable modeling for immersed boundary systems
- 2024A mechanism-driven reinforcement learning framework for shape optimization of airfoils
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