A Boundary Integral-based Neural Operator for Mesh Deformation

2026-02-27

AI Summary

This work addresses the high computational cost of traditional finite element methods in mesh deformation and the difficulty of existing neural operators in effectively enforcing vector-field Dirichlet boundary conditions. The authors formulate mesh deformation as a linear elastic boundary value problem and introduce a boundary integral representation based on a Dirichlet-type Green’s tensor, which enables exact reconstruction of the interior displacement field from boundary displacements alone. By integrating this boundary integral formulation with neural operators, they develop a geometry- and material-aware Green traction kernel learning framework that decouples physical constraints from geometric representation. The method rigorously satisfies linearity and superposition principles, demonstrating high accuracy, strong generalization across boundary conditions, adaptability to diverse geometries—including large deformations of flexible beams and rigid-body motions of NACA airfoils—while preserving mesh quality and computational efficiency.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Manipulation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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.

Problem

• mesh deformation
• boundary conditions
• computational cost
• neural operators
• linear elasticity

Innovation

• Boundary Integral
• Neural Operator
• Green's Tensor
• Mesh Deformation
• Dirichlet Boundary Condition