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<p>This paper presents an efficient mesh deformation framework that integrates boundary integral theory with neural operators, formulating the task as a linear elasticity boundary value problem (BVP). To overcome the high computational costs associated with physics-informed neural networks (PINNs) and the limitations of existing neural operators in handling Dirichlet boundary conditions for vector fields, we introduce a direct boundary integral representation utilizing a Dirichlet-type Green’s tensor. This formulation expresses the internal displacement field solely as a function of boundary displacements, effectively reducing a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d\)</EquationSource> </InlineEquation>-dimensional spatial problem to a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((d-1)\)</EquationSource> </InlineEquation>-dimensional boundary problem. Building upon this mathematical foundation, we propose the Boundary-Integral-based Neural Operator (BINO), which learns a Green’s traction kernel to achieve robust generalization across diverse boundary conditions. Comprehensive numerical experiments—ranging from the deformation of flexible beams to complex 3D manifolds such as multi-connected perforated discs—confirm that the proposed framework maintains high mesh quality while strictly adhering to the physical principles of linearity and superposition.</p>

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A boundary integral-based neural operator for mesh deformation

  • 期刊论文

  • 发表日期：

    2026年7月11日

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  • Zhengyu Wu,
  • Jun Liu,
  • Wei Wang

摘要

This paper presents an efficient mesh deformation framework that integrates boundary integral theory with neural operators, formulating the task as a linear elasticity boundary value problem (BVP). To overcome the high computational costs associated with physics-informed neural networks (PINNs) and the limitations of existing neural operators in handling Dirichlet boundary conditions for vector fields, we introduce a direct boundary integral representation utilizing a Dirichlet-type Green’s tensor. This formulation expresses the internal displacement field solely as a function of boundary displacements, effectively reducing a \(d\) -dimensional spatial problem to a \((d-1)\) -dimensional boundary problem. Building upon this mathematical foundation, we propose the Boundary-Integral-based Neural Operator (BINO), which learns a Green’s traction kernel to achieve robust generalization across diverse boundary conditions. Comprehensive numerical experiments—ranging from the deformation of flexible beams to complex 3D manifolds such as multi-connected perforated discs—confirm that the proposed framework maintains high mesh quality while strictly adhering to the physical principles of linearity and superposition.