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<p>This paper presents the variational physics-informed neural network (VPINN) as an effective tool for static structural analyses. One key innovation includes the construction of the neural network solution as an admissible function of the boundary-value problem (BVP), which satisfies all geometrical boundary conditions. We then prove that the admissible neural network solution also satisfies natural boundary conditions, and therefore all boundary conditions, when the stationarity condition of the variational principle is met. Numerical examples are presented to show the advantages and effectiveness of the VPINN in comparison with the physics-informed neural network (PINN). Another contribution of the work is the introduction of Gaussian approximation of the Dirac delta function, which significantly enhances the ability of neural networks to handle singularities, as demonstrated by the examples with concentrated support conditions and loadings. It is hoped that these structural examples are so convincing that engineers would adopt the VPINN method in their structural design practice.</p>

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Neural network solution based on the minimum potential energy principle for static problems of structural mechanics

  • 期刊论文

  • 发表日期：

    2025年6月2日

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  • Jiamin Qian,
  • Lincong Chen,
  • J. Q. Sun

摘要

This paper presents the variational physics-informed neural network (VPINN) as an effective tool for static structural analyses. One key innovation includes the construction of the neural network solution as an admissible function of the boundary-value problem (BVP), which satisfies all geometrical boundary conditions. We then prove that the admissible neural network solution also satisfies natural boundary conditions, and therefore all boundary conditions, when the stationarity condition of the variational principle is met. Numerical examples are presented to show the advantages and effectiveness of the VPINN in comparison with the physics-informed neural network (PINN). Another contribution of the work is the introduction of Gaussian approximation of the Dirac delta function, which significantly enhances the ability of neural networks to handle singularities, as demonstrated by the examples with concentrated support conditions and loadings. It is hoped that these structural examples are so convincing that engineers would adopt the VPINN method in their structural design practice.