Table 3 Loss residuals and errors of prediction using different number of collocation points

From: Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics

Type

Collocation points

Lp/np

Lb /nb

Error

Beam

np = 10

1.077 × 10-6

8.426 × 10-8

(2.50 ± 1.5)%

np = 25

1.760 × 10−7

3.749 × 10−9

(0.68 ± 0.3%)

np = 50

2.955 × 10−8

9.299 × 10−10

(0.36 ± 0.2%)

np = 100

6.123 × 10−9

4.170 × 10−13

(0.068 ± 0.03%)

Shell

np = 100; nb = 50

9.005 × 10−5

9.255 × 10−6

(9.01 ± 5)%

np = 500; nb = 50

1.905 × 10−5

8.008 × 10−6

(2.94 ± 1)%

np = 1000; nb = 50

8.760 × 10−6

8.660 × 10−6

(2.03 ± 0.8)%

np = 2000; nb = 50

4.691 × 10−6

9.521 × 10−6

(1.59 ± 0.6)%

  1. Notes: np and nb denote the number of collocation points at the interior zone and the boundaries, respectively. \({{\rm{Error}}}={\Vert {\omega }_{PINN}-{\omega }_{FEM}\Vert }_{2}^{2}/{\Vert {\omega }_{FEM}\Vert }_{2}^{2}\) where ωPINN and ωFEM represent the value of the vertical displacement computed by a multi-level physics-informed neural network (ml-PINN) and finite element method (FEM), respectively.