# A Boundary Integral-based Neural Operator for Mesh Deformation

arXiv: https://arxiv.org/abs/2602.23703 · HTML: https://mlsanity.com/paper/2602.23703/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 44 references that are in the map.
- First posted on arXiv 27 Feb 2026.

## Builds on (most cited first, 12 of 12)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks](https://mlsanity.com/paper/2206.14092/) (2022)
- [BI-GreenNet: Learning Green's functions by boundary integral network](https://mlsanity.com/paper/2204.13247/) (2022)
- [Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometries](https://mlsanity.com/paper/2308.12939/) (2023)
- [Learning the boundary-to-domain mapping using Lifting Product Fourier Neural Operators for partial differential equations](https://mlsanity.com/paper/2406.16740/) (2024)
- [Mesh deformation techniques in fluid-structure interaction: robustness, accumulated distortion and computational efficiency](https://mlsanity.com/paper/2006.14051/) (2020)

## 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.