基于贝叶斯主动学习的自适应离散非参数化概率盒可靠性分析方法

doi:10.7527/S1000-6893.2026.33836

Abstract

Abstract:

Focusing on reliability analysis of nonparametric probability boxes （p-boxes）， an adaptive discrete cumulative distribution function method based on Bayesian active learning is proposed to efficiently estimate failure probability bounds. First， a Bayesian active learning framework is employed to train a Gaussian Process surrogate model， replacing direct functional evaluations in the probability integration process. Second， an explicit expression for the error induced by discretization grid spacing is derived， establishing a quantitative estimation of the numerical integration error bound. Third， an adaptive discretization criterion based on a comprehensive error estimator is developed； this criterion allocates new nodes to subintervals that most significantly impact the accuracy of failure probability bounds， achieving a locally optimal allocation of computational resources. Finally， under a convergence criterion controlled by the upper bound of integration error， convergent upper and lower bounds of the failure probability are obtained. Numerical examples demonstrate the effectiveness and efficiency of the proposed method.

Key words: reliability analysis, nonparametric probability box, adaptive discretization, Bayesian active learning, failure probability boundary

CLC Number:

Yangfan DENG, Jiachang TANG, Zhe ZHANG, Qishui YAO, Wei PAN, Hairui ZHANG. Bayesian active learning based adaptive discretization method for nonparametric probability box reliability analysis[J]. Acta Aeronautica et Astronautica Sinica, 2026, 47(19): 633836.

Figures/Tables 33

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Table 1

Characteristic parameters of nonparametric p-box variables in Example 1

变量	$μ ̲$	$μ ¯$	$σ ̲$	$σ ¯$
$Y 1$	1.76	1.84	0.3	0.3
$Y 2$	1.67	1.73	0.25	0.25

Table 1

Fig.6

Fig.7

Fig.8

Fig.9

Table 2

Reliability analysis results for Example 1

方法	N	$P f L$	$ϵ P f ¯$ /%	$P f U$	$ϵ P f ¯$ /%	N_call
IMC		0.061 9		0.119 6		1×10⁷
IMC-BPNN		0.062 8	1.45	0.115 5	3.43	400
DL-DLSA^［48］		0.060 6	2.10	0.122 6	2.51	64
CCDF-FORM	50	0.056 5	8.70	0.110 2	7.82	1 992
	100	0.059 2	4.32	0.114 7	4.03	3 938
	200	0.060 6	2.03	0.117 1	2.08	7 826
	250	0.060 7	1.94	0.117 6	1.67	9 368
BAAD	44	0.060 1	2.91	0.117 5	1.75	14

Table 2

Fig.10

Table 3

Characteristic parameters of nonparametric p-box variables in Example 2

变量	$μ ̲$	$μ ¯$	$σ ̲$	$σ ¯$
c₁/（ $N ⋅ m - 1$ ）	1.20	1.22	0.10	0.11
c₂/（ $N ⋅ m - 1$ ）	0.50	0.52	0.06	0.08
r/m	0.48	0.51	0.02	0.02
t/s	1.58	1.60	0.18	0.18
P/N	1.18	1.20	0.10	0.11

Table 3

Table 4

Reliability analysis results for Example 2

方法	N	$P f L$	$ϵ P f ¯$ /%	$P f U$	$ϵ P f ¯$ /%	N_call
IMC		0.029 3		0.132 2		1×10⁷
IMC-BPNN		0.028 9	1.37	0.130 2	1.51	2 000
CCDF-FORM	50	0.032 3	10.24	0.139 2	5.30	17 440
	100	0.032 7	11.60	0.140 8	6.51	30 415
	150	0.032 9	12.29	0.140 1	5.96	44 305
BAAD	72	0.028 4	3.07	0.129 5	2.04	33

Table 4

Fig.11

Fig.12

Fig.13

Fig.14

Table 5

Characteristic parameters of nonparametric p-box variables in Example 3

变量	$μ ̲$	$μ ¯$	$σ ̲$	$σ ¯$
$X 1 / m m$	177.8	178.8	0.62	0.62
$X 2 / m m$	4.58	4.63	0.07	0.07
$X 3 / m m$	6.58	6.60	0.08	0.08
$X 4 / ℃$	1 480	1 500	10	10

Table 5

Fig.15

Fig.16

Table 6

Reliability analysis results for Example 3

方法	N	$P f L$	$ϵ P f ¯$ /%	$P f U$	$ϵ P f ¯$ /%	时间/s	N_call
IMC		0.007 2		0.053 7			1×10⁷
IMC-BPNN		0.007 9	9.72	0.054 5	1.49		4 000
BAAD（无自适应）	50	0.007 6	5.56	0.053 0	1.30	45.9	28
	100	0.007 5	4.17	0.053 5	0.37	139.3
	150	0.007 3	1.39	0.053 8	0.19	145.8
BAAD	54	0.007 3	1.39	0.053 6	0.19	49.3

Table 6

Table 7

Characteristic parameters of nonparametric p-box variables in Example 4

变量	$μ ̲$	$μ ¯$	$σ ̲$	$σ ¯$
$η / (P a ⋅ s)$	0.023 7	0.024 1	0.002 8	0.003 0
$R x / m m$	2.289	2.302	0.05	0.06
$F m a x / N$	7.225	7.235	0.03	0.04
$σ 0 / (10 - 4 m m)$	2.04	2. 06	0.045	0.055

Table 7

Table 8

Reliability function of bearing lubrication under different rotational speeds

转速/（r∙min^-1）	功能函数 $g (η, R x, F m a x, σ 0)$
1.8×10⁴	$8.759 × 10 - 3 η 0.68 R x 0.393 (F m a x) - 0.073 - 3 σ 0$
1.2×10⁴	$9.001 × 10 - 3 η 0.68 R x 0.393 (F m a x) - 0.073 - 3 σ 0$
8×10³	$8.649 × 10 - 3 η 0.68 R x 0.393 (F m a x) - 0.073 - 3 σ 0$

Table 8

Fig.17

Fig.18

Fig.19

Fig.20

Fig.21

Fig.22

Fig.23

Fig.24

Table 9

Reliability analysis results for Example 4

方法	转速/（r∙min^-1）	N	$P f L$	$ϵ P f ¯$ /%	$P f U$	$ϵ P f ¯$ /%	时间/s	N_call
IMC	1.8×10⁴		9.91×10^-4		4.15×10^-3			1×10⁷
	1.2×10⁴		4.69×10^-4		2.37×10^-3
	8×10³		1.26×10^-3		5.11×10^-3
BAAD（无自适应）	1.8×10⁴	50	1.08×10^-3	8.98	4.33×10^-3	4.34	66.4	37
		75	1.06×10^-3	6.96	4.25×10^-3	2.41	71.3
		100	1.03×10^-3	3.94	4.24×10^-3	2.17	131.6
		125	1.01×10^-3	1.92	4.23×10^-3	1.93	133.7
		150	9.72×10^-4	1.92	4.21×10^-3	1.45	125.4
	1.2×10⁴	50	4.95×10^-4	5.54	2.41×10^-3	1.69	192.5	31
		75	4.83×10^-4	2.99	2.19×10^-3	7.59	257.3
		100	4.47×10^-4	4.69	2.24×10^-3	5.49	341.8
		125	4.41×10^-4	5.97	2.25×10^-3	5.06	668.4
		150	4.43×10^-4	5.54	2.23×10^-3	5.91	607.6
	8×10³	50	1.16×10^-3	7.94	5.03×10^-3	1.57	115.7	33
		75	1.17×10^-3	7.14	5.04×10^-3	1.37	152.9
		100	1.19×10^-3	5.56	4.96×10^-3	2.94	230.9
		125	1.20×10^-3	4.76	4.94×10^-3	3.33	221.8
		150	1.22×10^-3	3.17	5.00×10^-3	2.15	265.2
BAAD	1.8×10⁴	49	1.01×10^-3	1.92	4.18×10^-3	0.72	48.9	37
	1.2×10⁴	45	4.81×10^-4	2.56	2.31×10^-3	2.53	178.1	31
	8×10³	46	1.24×10^-3	1.59	5.19×10^-3	1.57	129.5	33

Table 9

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Bayesian active learning based adaptive discretization method for nonparametric probability box reliability analysis

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