基于流形上概率学习的PBX9502冲击起爆试验不确定度量化

梁霄; 赵延菲; 王瑞利

doi:10.11883/bzycj-2025-0042

基于流形上概率学习的PBX9502冲击起爆试验不确定度量化

doi: 10.11883/bzycj-2025-0042

梁霄^1,,
赵延菲¹,
王瑞利^{2, 3}

1.
山东科技大学数学与系统科学学院，山东青岛 266590
2.
河南省科学院数学研究所，河南郑州 450052
3.
北京应用物理与计算数学研究所，北京 100094

基金项目: 国家自然科学基金（12271054）；国家自然科学-中国工程物理研究院联合基金（U2230208）；挑战计划（TZ2025006）

详细信息

作者简介:
梁　霄（1984－　），男，博士，副教授，mathlx@163.com

中图分类号: O381; TJ55
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- 被引次数: 0
出版历程
- 收稿日期: 2025-02-14
- 修回日期: 2025-04-01
- 网络出版日期: 2025-04-08

Uncertainty quantification of shock to detonation experiment of PBX 9502 based on probability learning on manifold

1.
School of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
2.
Institute of Mathematics, Henan Academy of Sciences, Zhengzhou 450052, Henan, China
3.
Beijing Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

摘要

摘要: 为了解决样本容量稀疏和不确定度对多物理属性爆轰试验研究造成的障碍，采用流形上的概率学习（probability learning on manifold, PLoM）方法通过结合耗散映射与Itô投影采样技术，生成满足爆轰机理的丰富样本，进而实现试验不确定度量化。首先，对具有多物理属性的高能钝感炸药PBX9502的试验样本进行尺度变换。接着利用主成分分析对尺度矩阵规范化处理，构造训练集。然后，采用改进的多维Gauss核密度估计法，标定训练集所对应随机矩阵的概率测度。同时，利用耗散映射提取基于训练集的非线性流形。Wiener过程驱动的耗散Hamilton系统定义的Itô-MCMC随机生成器用于在流形上采样。最后，使用逆变换导出学习集的样本。结果表明，PLoM生成的随机数的Gauss统计量与文献标定的PBX9502的密度的统计信息相吻合。此外，该方法成功导出爆轰距离和爆轰时间与冲击应力服从双对数模型关系，曲线拟合的精度与文献成果相当，而成本可以忽略不计。PLoM通过对已有试验数据的学习与处理，获得更高精度的数字试验结果。PLoM方法泛化能力强，可推广到其他类型炸药的爆轰试验。
- 流形上的概率学习 /
- 不确定度量化 /
- 小样本 /
- 爆轰 /
- 试验不确定度
Abstract: To address the challenges posed by insufficient statistical sampling density and inherent irreducible uncertainties in multi-physical property detonation experiments, probability learning on manifold (PLoM) involving diffusion map and Itô projection sampling are used to generate sufficient dataset satisfying the detonation physical mechanism and therefore, to fulfill the uncertainty quantification of experiment. Firstly, scale transformation is implemented on the experimental data with multi-physical asset of insensitive high explosive PBX 9502. The training set is obtained through the normalization of the scale matrix by means of principal component analysis. Secondly, an improved high-dimensional Gaussian kernel density estimation is utilized to calibrate the probability measure of the random matrix associated with the training dataset. Diffusion map is used to deduce the nonlinear manifold based on the training dataset. Sampling on the manifold is fulfilled through Itô-MCMC generator defined by a dissipative Hamilton system driven by the Wiener process. Finally, the learning set is obtained via inverse transformation. The result shows that the Gaussian statistics obtained from random numbers generated by PLoM coincide with the statistical information of density of PBX 9502 calibrated in the literature. Furthermore, the double logarithm model related to the distance to detonation and initial impact stress is constructed through the data generated. It also holds for the relationship between the time of detonation and initial shock stress. Fitting precision of the curve is almost equivalent to the accuracy of result in literature, however the cost is negligible. More accurate digital test result is obtained through the learning and processing of existing experimental data via PLoM. The PLoM method demonstrates strong generalization capability, enabling its extension to detonation experiments with various types of explosive.
- probability learning on manifold /
- uncertainty quantification /
- small sample size /
- detonation /
- experimental uncertainty

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图 1 协方差矩阵C的特征值

Figure 1. Eigenvalue of covariance matrix C

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图 2 转移矩阵的特征值的递减排列

Figure 2. Descending eigenvalue of transition matrix

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图 3 PLoM生成的样本点

Figure 3. Samples generated from PLoM

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图 4 物理量的概率密度函数

Figure 4. Probability density function of physical quantity

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图 5 爆轰距离和冲击应力的关系

Figure 5. The relationship between distance to detonation and initial shock stress

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图 6 爆轰时间和冲击应力的关系

Figure 6. The relationship between time to detonation and initial shock stress

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表 1 PBX9502起爆试验数据^[11]

Table 1. Initiation experimental data of PBX9502^[11]

ρ/(g·cm⁻³)	u/(km·s⁻¹)	p/GPa	x^*/mm	t^*/μs	ρ/(g·cm⁻³)	u/(km·s⁻¹)	p/GPa	x^*/mm	t^*/μs
1.889	2.349	10.65	11.82	2.22	1.890	2.426	11.16	11.72	2.18
1.889	2.493	11.62	9.17	1.67	1.893	2.474	11.49	11.31	2.10
1.888	2.766	13.55	6.01	1.04	1.890	2.546	11.98	9.44	1.75
1.887	2.860	14.24	4.19	0.74	1.891	2.755	13.47	6.88	1.22
1.886	3.118	16.22	3.68	0.61	1.889	2.759	13.50	7.58	1.36
1.886	2.334	10.55	12.41	2.38	1.889	2.351	10.67	12.78	2.45
1.888	2.599	12.35	8.28	1.48	1.891	2.636	12.61	8.30	1.48
1.887	2.798	13.78	6.26	1.10	1.889	2.756	13.47	6.36	1.12
1.887	2.960	15.00	4.53	0.77	1.889	2.959	14.99	4.77	0.81
1.890	2.394	10.95	14.50	2.75
注：ρ为密度，u为冲击速度，p为冲击应力，x^为爆轰距离，t^为爆轰时间。

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表 2 物理量的Gauss统计信息

Table 2. Gaussian statistical information of physical quantity

物理量	期望	标准差	置信区间下限	置信区间上限
ρ/(g·cm⁻³)	1.8888	0.0018	1.8858	1.8930
u/(km·s⁻¹)	2.6520	0.2344	2.2668	3.1446
p/GPa	12.7741	1.6866	10.1659	16.1521
x^*/mm	8.4581	3.3519	3.3437	15.2677
t^*/μs	1.5467	0.6725	0.5546	2.9207

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表 3 PBX9502的初始密度的Gauss统计量

Table 3. Gaussian statistics of initial density of PBX9502 (g/cm⁻³)

项目	本文结果	LANL标定^[11]	孙承纬标定^[1]
期望	1.8888	1.890	1.895
标准差	0.0018	0.005	<0.01

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