Fig. 2: Multi-level physics-informed neural network (ml-PINN) for 1D beam.

a Diagrammatic sketch of beam structure. b Physical relationship between the relevant variables that transforms the uniform load q into the vertical displacement ω by utilizing the three typical equations. M represents bending moment. EI is bending stiffness. The intermediate variables including shear force F_Q, curvature κ, rotation angle θ, and displacement ω are outputted from each neural network. c Schematics of ml-PINN framework for beam structure. L_b() and L_p() represent the loss functions from boundary constraints and loss functions from PDE, respectively. L_p(ω’, θ), L_p(θ’, κ), L_p(M’, F_Q) and L_p(F_Q’, q) are loss functions corresponding to each differential equation. L_b(ω), L_b(θ) and L_b(M) are the loss functions from boundary constraints corresponding to the vertical displacement, section turning angle, and bending moment respectively. {W, b}_i presents the model parameter of neural networks (NNs).