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.

翻译：本文提出一种基于边界积分与神经算子的高效网格变形方法，将问题表述为线性弹性边界值问题。为克服传统有限元方法的高计算成本以及现有神经算子在处理矢量场狄利克雷边界条件时的局限性，我们引入基于狄利克雷型格林张量的直接边界积分表示。该表述将内部位移场纯粹表示为边界位移的函数，无需求解未知面力。在此基础上，我们设计了基于边界积分的神经算子，用于学习几何与材料感知的格林面力核函数。本框架的关键技术优势在于通过几何描述符实现了物理积分过程与几何表示的数学解耦。本研究虽主要展示了模型在不同边界条件下的强泛化能力，但其架构本质上具备跨几何适应的潜力。数值实验（包括柔性梁的大变形与NACA翼型的刚体运动）验证了模型的高精度及其对线性与叠加原理的严格遵循。结果表明，所提框架在保证网格质量与计算效率的同时，为工程参数化网格生成与形状优化提供了可靠的新范式。