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

doi:10.1007/s10483-025-3257-8

Abstract

Abstract:

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.

Key words: physics-informed neural network (PINN), variational physics-informed neural network (VPINN), structural statics, admissible function, Gaussian approximation

2010 MSC Number:

O324

Jiamin QIAN, Lincong CHEN, J. Q. SUN. Neural network solution based on the minimum potential energy principle for static problems of structural mechanics. Applied Mathematics and Mechanics (English Edition), 2025, 46(6): 1125-1142.

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Figures/Tables 18

Fig. 1

Table 1

Comparison of computational accuracy and training time between the VPINN and PINN methods for solving static responses of the cantilever beam with elastic supports"

Method	Iteration	Neural network structure	Sample	CPU time/s	$e L 1$
PINN	40	1-60-1	2 000	34.12	16.120
PINN	200	1-48-48-48-48-1	10 000	298.12	0.931
VPINN	40	1-60-1	2 000	11.25	0.354

Table 1

Table 2

Comparison of the moments and shear forces at discontinuous points of the cantilever beam with elastic supports computed by the VPINN and PINN with the FE solution. M1 is the moment to the left of the concentrated force, and M2 is the moment to the right of the concentrated force. V1 is the shear force to the left of the torsional spring, and V2 is the shear force to the right of the torsional spring"

Method	$M 1$ /( $N ⋅ m$ )	$M 2$ /( $N ⋅ m$ )	$V 1$ /N	$V 2$ /N
FE	176.91	$6.01 × 10 − 5$	1 000	$− 7.69 × 10 − 5$
PINN	165.66	$− 8.73$	979.73	$− 9.94$
VPINN	172.15	$− 1.40$	984.12	$− 2.25$
Method	$e M 1$ /%	$e M 2$ /%	$e V 1$ /%	$e V 2$ /%
PINN	6.36	8.73	2.03	9.94
VPINN	2.69	1.40	1.59	2.25
Note that $e$ represents the percentage error between the approximate and FE solutions

Table 2

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Table 3

Comparison of the solution accuracy and training steps of the VPINN and PINN methods for solving the responses of the cantilever Timoshenko beam with uniform loading"

Method	Iteration	Neural network structure	Sample	CPU time/s	$e L 1$
PINN	200	1-20-20-2	14 000	351.33	4.210 00
PINN	500	1-20-20-2	10 000	298.12	0.763 00
VPINN	200	1-20-20-2	14 000	78.00	0.004 12

Table 3

Table 4

Effect of activation functions of the VPINN on the responses of the cantilever Timoshenko beam"

Iteration	Neural network structure	Activation function	Sample	$e L 1$
200	1-20-20-2	sigmoid	14 000	0.052 40
200	1-20-20-2	ReLU	14 000	1.240 00
200	1-20-20-2	tanh	14 000	0.004 12

Table 4

Table 5

The moment M and shear force V of the beam at x=0"

Method	$M$ /(N·m)	$e M$	$V$ /N	$e V$
Analytical	$− 800.00$	0.000 00	$− 400.00$	0.000 0
PINN	$− 798.06$	0.242 00	$− 393.93$	1.500 0
VPINN	$− 802.14$	0.008 28	$− 405.14$	0.012 8

Table 5

Fig. 6

Fig. 7

Table 6

Comparison of L2 error and training time between the VPINN and PINN methods for solving the displacement fields of the thick-walled cylinder"

Method	Iteration	Neural network structure	CPU time/s	Sample	$e L 2$
PINN	100	2-10-10-2	130.07	3 600	0.037 340
PINN	100	2-20-20-20-2	682.14	3 600	0.079 880
VPINN	100	2-10-10-2	113.10	3 600	0.004 158

Table 6

Table 7

Influence of activation function types on the results of the VPINN method for solving displacements of the thick-walled cylinder"

VPINN iteration	Neural network structure	Activation function	Sample	$e L 2$
100	2-10-10-2	tanh	3 600	0.007 202
100	2-10-10-2	softmax	3 600	0.006 222
100	2-10-10-2	sigmoid	3 600	0.005 291

Table 7

Fig. 8

Fig. 9

Fig. 10

Table 8

Comparison of stresses and errors at x=0, y=1 and x=1, y=0 of the thick-walled cylinder"

Method	$σ r r, max$	$σ r r, min$	$σ θ θ, max$	$σ θ θ, min$	$σ r θ$
Analytical	11.33	-10.00	11.33	-10.00	0.000
VPINN	11.17	-10.18	11.67	$− 9.83$	0.221
PINN	11.97	-10.05	10.81	-10.12	0.207
Method	$e r r, max$	$e r r, min$	$e θ θ, max$	$e θ θ, min$	$e r θ$
VPINN	1.41	1.800	3.00	1.70	2.21
PINN	5.65	0.500	4.56	1.20	2.07

Table 8

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