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爆轰试验由于操作风险高、样品制备和测试成本大等特征，导致试验样本稀疏，给标定待测物理量的概率分布和使用机器学习方法带来巨大的挑战。流形上的概率学习(PLoM,probability learning on manifold)能生成丰富的、符合实用常识的、遵循试验数据物理机理的样本,成为处理小样本的有效工具。首先将炸药PBX9502的含有多物理属性试验数据做缩比变换，然后，使用主成分分析将缩比数据规范化，并将规范化矩阵的分量作为训练集。进而，使用改进的多元Gauss核密度估计法表征训练集的先验概率。紧接着利用耗散映射提炼基于训练集的非线性流形。具体而言，通过转移矩阵的第一个特征值和对应的特征向量构造耗散基函数和耗散映射。其次，将训练集作为Wiener过程驱动的Hamilton系统的初值,先验概率作为不变测度构造Itô-MCMC随机数生成器，Störmer-Verlet格式用以求解随机耗散Hamilton方程。最后，使用反演变换，实现学习集的扩容。结果发现：PLoM能生成符合工业生产和高精度模拟需求的密度和爆速的随机数。利用学习集导出炸药的密度和爆速服从仿射变换，密度和爆压服从非线性关系，密度的微小波动会引起爆速和爆压的剧烈的变化。比较学习集的变异系数，还发现爆压离散程度最高，与孙承纬院士等专家的论断吻合。所用方法具备普适性，能推广到其他的爆轰系统。Detonation test suffers small experimental datasets due to high risk of implementation and substantial cost of samples production and measurement. The major challenges of limited data consist in constructing the probability distribution of physical quantities and application of machine learning. Probability learning on manifold (PLoM) can generate a large number of realizations reconcilable with practical common sense, and the underlying physical mechanism is preserved in these samples generated. So PLoM is viewed as an efficient tool of tackling small samples. To begin with, experimental data is assumed to be concentrated on an unknown subset of Euclidean space and can be treated as the sampling of random vector to be determined. Meanwhile, experimental is solved in the framework of matrix and the scaling transformation is conducted on the datasets of PBX9502 with multi-physics attributes. Then the principal component analysis is utilized to normalize the scaling matrix, and the normalization matrix is labeled as training sets. Moreover, the altered multi-dimensional Gaussian kernel density estimation is utilized for estimating the probability distribution of training set. Furthermore, diffusion map is used to discover and characterize the geometry and structure of dataset. In other words, nonlinear manifold based on the training set is constructed trough diffusion map. To specifically, the first eigenvalue and corresponding eigenvector is related to the construction of diffusion basis and diffusion maps. To make it further, Itô-MCMC sampler is associated with dissipative Hamilton systems driven by Wiener process, for which the initial condition is set to be training set and prior probability is conceived as invariant measure. Störmer-Verlet scheme is used for solving the stochastic dissipative Hamilton equations. At last, additional realizations of learning dataset are fulfilled through the inversion transformation. The result shows that random number generated from PLoM satisfies the requirement of industrial and high fidelity simulation. The 95% confidence interval of density is included in the range calibrated by Los Alamos National Laboratory. And value of detonation velocity calibrated by Prof. Chengwei Sun also falls into 95% confidence interval of detonation velocity generated by PLoM. It is also deduced from the learning set that density and detonation velocity satisfies the affine transformation. Furthermore, detonation pressure has nonlinear relationship with density. Tiny variation of density can lead to magnificent fluctuation of detonation pressure and detonation velocity. Detonation pressure has the largest discreetness among all the physical quantities through the comparison of variation coefficients of learning set, which coincides with the assertion of experts such as Prof. Chengwei Sun. The methodology used is sufficiently general and can be extended to other detonation systems.
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Keywords:
- Probabilistic learning on manifold /
- diffusion map /
- stochastic dissipative Hamilton equations /
- small datasets /
- detonation /
- uncertainty quantification
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