Displacement Reconstruction Based on Physics-Informed DeepONet Regularizing Geometric Differential Equations of Beam or Plate

Physics-informed DeepONet (PIDeepONet) is utilized for the reconstruction task of structural displacement based on measured strain. For beam and plate structures, the PIDeepONet is built by regularizing the strain–displacement relation and boundary conditions, referred to as geometric differential equations (GDEs) in this paper, and the training datasets are constructed by modeling strain functions with mean-zero Gaussian random fields. For the GDEs with more than one Neumann boundary condition, an algorithm is proposed to balance the interplay between different loss terms. The algorithm updates the weight of each loss term adaptively using the back-propagated gradient statistics during the training process. The trained network essentially serves as a solution operator of GDEs, which directly maps the strain function to the displacement function. We demonstrate the application of the proposed method in the displacement reconstruction of Euler–Bernoulli beams and Kirchhoff plates, without any paired strain–displacement observations. The PIDeepONet exhibits remarkable precision in the displacement reconstruction, with the reconstructed results achieving a close proximity, surpassing 99%, to the finite element calculations.