Intelligent surrogate modeling for penetration prediction: solving forward and inverse problems with multi-fidelity data

doi:10.1007/s10483-026-3397-7

Abstract

Abstract:

The analysis of penetration mechanics is critical for the offensive targeting and defensive design of underground facilities. Although computational methods are fundamental to penetration analysis, they are often constrained by a trade-off between accuracy and computational efficiency. Emerging artificial intelligence (AI) methods, with inherent strengths in modeling complex high-dimensional relationships from available data, provide promising alternatives for building intelligent surrogate models. This study proposes a fusion-enhanced radial basis function network (FE-RBFN) for penetration prediction, solving forward and inverse problems with multi-fidelity data. FE-RBFN employs three interconnected subnetworks to extract features and capture nonlinear correlations at varying fidelity levels. To overcome the challenge of data scarcity, FE-RBFN embeds a data fusion strategy to fully leverage multi-fidelity data from multiple sources. The experimental results demonstrate that our network yields rapid and precise predictions, outperforming traditional machine learning methods. Notably, in multi-fidelity scenarios, FE-RBFN exhibits robust prediction accuracy despite the limited availability of high-fidelity data.

Key words: penetration, artificial intelligence (AI), data fusion, multi-fidelity

2010 MSC Number:

O385

Danning JING, Xuguang CHEN, Shuo WANG, Qinglin WANG, Jie LIU, Xinhai CHEN. Intelligent surrogate modeling for penetration prediction: solving forward and inverse problems with multi-fidelity data. Applied Mathematics and Mechanics (English Edition), 2026, 47(6): 1341-1362.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3397-7

https://www.amm.shu.edu.cn/EN/Y2026/V47/I6/1341

Figures/Tables 21

Fig. 1

Fig. 2

Table 1

Comprehensive configuration of experimental scenarios, data sampling, and multi-fidelity task mapping"

Variable	Symbol	Unit	Range	Discrete point (HF/LF)	Total case
CRH	ψ	–	1.75–4.25	Fixed/11
Velocity	V₀	m/s	400–900	21/21	HF: 147
Angle	θ	(°)	0–60	7/13	LF: 3 003
Time	t	ms	0–t_max	32/20
Problem	Physical task		Input	Output	Dataset split (train: test)
Forward 1	Final depth prediction		$ψ, V 0, θ$	P_d	HF
Forward 2	Time-series displacement		$ψ, V 0, θ, t$	X(t)=(X, Y, Z)	40% : 60%
Forward 3	Time-series force response		$ψ, V 0, θ, t$	$F (t) = (F x, F y, F z)$	LF
Inverse	Parameter identification		X_final	$V 0, θ$	80% : 20%

Table 1

Fig. 3

Fig. 4

Table 2

Fig. 5

Fig. 6

Table 3

Fig. 7

Table 4

Fig. 8

Table 5

Fig. 9

Table 6

Definitions of two distinct extrapolation scenarios for generalization"

Scenario	Training region	Testing region
Down-extrapolation	High energy: $v > V th$ and $θ > Θ th$	Low energy: $v ⩽ V th$ or $θ ⩽ Θ th$
Up-extrapolation	Low energy: $v < V th$ and $θ < Θ th$	High energy: $v ⩾ V th$ or $⩾ Θ th$
*Note: V_th and $Θ th$ denote the velocity and impact angle thresholds, respectively

Table 6

Fig. 10

Fig. 11

Fig. 12

Table A1

Projectile properties"

$ρ p / (kg ⋅ m − 3)$	$Y p / MPa$	$r p / mm$	$L / mm$	ψ
7 800	835	13.45	203	3

Table A1

Table A2

Target properties"

$f c / MPa$	$E t / GPa$	$Y t / MPa$	$ρ t / (kg ⋅ m − 3)$	λ	$τ / MPa$	$c fric$
44.74	25	44.74	2 340	1.023	40	0.01

Table A2

Fig. B1

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Method	R²	E_MSE	E_MAPE/%
Linear regression	0.924 3	0.001 0	5.11
Decision tree	0.926 4	0.000 9	5.38
k-nearest neighbor	0.939 9	0.000 8	4.49
XGBoost	0.961 1	0.000 5	4.59
FE-RBFN	0.981 5	0.000 2	2.59

Method	X			Y			Z
Method	R²	E_MSE	E_MAPE/%	R²	E_MSE	E_MAPE/%	R²	E_MSE	E_MAPE/%
	Displacement
Linear regression	0.830 3	0.174 1	137.4	0.626 4	0.356 4	168.0	0.835 6	0.162 1	148.3
Decision tree	0.881 5	0.112 6	231.5	0.693 9	0.305 4	147.3	0.894 6	0.103 1	234.6
k-nearest neighbor	0.993 6	0.006 4	36.73	0.954 9	0.039 7	63.66	0.993 0	0.006 9	33.78
XGBoost	0.994 7	0.004 9	24.04	0.868 9	0.126 3	106.5	0.995 1	0.004 5	28.40
FE-RBFN	0.998 3	0.001 6	10.31	0.921 9	0.081 0	84.91	0.998 1	0.001 8	26.20
	Force component
Linear regression	0.638 9	0.362 5	80.98	0.000 9	1.079 7	171.62	0.814 6	0.201 9	63.61
Decision tree	0.858 5	0.142 1	43.21	-0.061 0	1.146 6	165.68	0.973 0	0.029 4	26.50
k-nearest neighbor	0.927 8	0.072 5	30.48	0.197 7	0.867 1	130.25	0.993 0	0.007 6	16.88
XGBoost	0.923 4	0.076 8	34.88	0.012 3	1.067 4	160.42	0.991 5	0.009 2	20.11
FE-RBFN	0.942 6	0.060 5	25.43	0.194 3	0.655 1	122.78	0.994 0	0.006 5	15.33

Method	Velocity			Angle
Method	R²	E_MSE	E_MAPE/%	R²	E_MSE	E_MAPE/%
Linear regression	0.940 3	0.068 5	51.05	0.875 8	0.117 6	70.05
Decision tree	0.951 9	0.055 3	47.32	0.943 7	0.053 3	17.54
k-nearest neighbor	0.917 8	0.094 4	55.92	0.889 9	0.104 3	91.31
XGBoost	0.968 6	0.036 1	39.19	0.949 0	0.048 3	31.11
FE-RBFN	0.979 7	0.023 4	29.51	0.993 1	0.006 5	22.26

Training data	R²	E_MSE	E_MAPE/%
Only HF data (limited)	0.613	0.005 1	13.92
Multi-fidelity data (LF & limited HF)	0.875	0.001 6	7.34