Imprecise Probabilistic Model Updating Using A Wasserstein Distance-based Uncertainty Quantification Metric
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摘要: 复杂物理系统的数学代理模型往往包含多类不确定性因素。在实际工程问题如机械系统可靠性优化设计中，可结合系统响应的部分实测数据，校准模型关键参数取值，修正模型结构，提高代理模型的保真性。但对于具有混合不确定性的非精确概率模型，传统基于欧式距离的模型修正方法并不适用。针对这一问题，提出一种基于Wasserstein距离测度的模型修正方法，该方法基于Wasserstein距离测度构建核函数，利用p维参数空间中Wasserstein距离的几何性质以量化不同概率分布之间的差异，相较于现有模型修正方法，可校准模型的高阶超参数，显著降低模型结构及参数不确定性。针对工程实际需求，进一步采用近似贝叶斯推理与切片分割技术以降低计算成本。通过受迫振动钢板本构参数校核问题与NASA Langley多学科不确定性量化问题验证了本方法在静力学与动力学等实际工程问题中的有效性。
复杂物理系统的数学代理模型往往包含多类不确定性因素。在实际工程问题如机械系统可靠性优化设计中，可结合系统响应的部分实测数据，校准模型关键参数取值，修正模型结构，提高代理模型的保真性。但对于具有混合不确定性的非精确概率模型，传统基于欧式距离的模型修正方法并不适用。针对这一问题，提出一种基于Wasserstein距离测度的模型修正方法，该方法基于Wasserstein距离测度构建核函数，利用p维参数空间中Wasserstein距离的几何性质以量化不同概率分布之间的差异，相较于现有模型修正方法，可校准模型的高阶超参数，显著降低模型结构及参数不确定性。针对工程实际需求，进一步采用近似贝叶斯推理与切片分割技术以降低计算成本。通过受迫振动钢板本构参数校核问题与NASA Langley多学科不确定性量化问题验证了本方法在静力学与动力学等实际工程问题中的有效性。
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关键词:
- Wasserstein距离 /
- 贝叶斯方法 /
- 非精确概率 /
- 不确定性量化 /
- 近似推理
Abstract: Uncertainty factors are usually contained in the mathematical proxy model of complex physical system. In practical engineering problems such as mechanical system reliability optimization design, the key parameters of the model can be calibrated and the model structure can be modified to improve the fidelity of the proxy model. However, for imprecise probabilistic models with mixed uncertainties, the traditional model updating method based on the Euclidean distance is not applicable. To solve this problem, a new model updating method based on the Wasserstein distance measure is proposed, which builds the kernel function based on the Wasserstein distance measure, and uses the geometric properties of Wasserstein distance in P-dimensional parameter space to quantify the differences between different probability distributions. Compared with the existing model updating methods, high-order hyper-parameters of the model can be calibrated to significantly reduce the uncertainty of model structure and parameters. In order to reduce the calculation cost, the approximate Bayesian inference and sliced segmentation technology is further adopted to meet the engineering requirements. The validity of this method for practical engineering problems, such as statics and dynamics, is verified by the constitutive parameter checking problem of forced vibration steel plate and the multidisciplinary uncertainty quantification problem of NASA Langley.-
Key words:
- Wasserstein distance /
- Bayesian methods /
- imprecise probability /
- model updating /
- approximate reasoning
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表 1 因素水平表
尺寸参数 长/mm 宽/mm 高/mm 数值 600 120 3 本构参数 剪切模量/GPa 弹性模量/GPa 质量密度/(kg·m−3) 数值 83 210 7 860 
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表 2 认知不确定性建模
参数 不确定性模型 认知不确定性建模 E 弹性模量$ E\sim N\left({\mu }_{E}, {\sigma }_{E}^{2}\right) $ $ {{{\theta }} }_{1}={\mu }_{E}\sim U\left[180, 230\right] $ 均值${\mu _E} \in \left[ {180, 230} \right]$ $ {{{\theta }} }_{2}={\sigma }_{E}^{2}\sim U\left[0, 10\right] $ 方差$\sigma _E^2 \in \left[ {0, 10} \right]$ G 剪切模量$ G\sim N\left({\mu }_{G}, {\sigma }_{E}^{2}\right) $ $ {{{\theta }} }_{3}={\mu }_{G}\sim U\left[60, 100\right] $ 均值${\mu _G} \in \left[ {60, 100} \right]$ $ {{{\theta }} }_{4}={\sigma }_{G}^{2}\sim U\left[0, 10\right] $ 方差$\sigma _G^2 \in \left[ {0, 10} \right]$
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表 3 设定认知不确定性参数的名义值
不确定性参数 名义值 $ E\sim N\left({\mu }_{E}, {\sigma }_{E}^{2}\right) $ ${\mu _{_E}} = 200.5, \sigma _E^2 = 2.0$ $ G\sim N\left({\mu }_{G}, {\sigma }_{G}^{2}\right) $ ${\mu _G} = 81.4, \sigma _G^2 = 1.5$
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表 4 后验结果与真值比较数据
参数 真值/GPa W距离极大似然估计/GPa W距离修正95%置信区间 E距离极大似然估计/GPa E距离修正95%置信区间 ${\mu _E}$ 200.5 200.261 1 [200.257 5, 200.264 7] 200.302 2 [200.298 3, 200.306 0] $\sigma _E^2$ 2.0 2.078 1 [2.064 5, 2.091 8] — — ${\mu _G}$ 81.4 81.567 7 [81.565 2, 81.570 1] 81.553 6 [81.550 5, 81.556 8] $\sigma _G^2$ 1.5 1.408 8 [1.402 8, 1.414 9] — —
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表 5 认知不确定性建模
参数 类别 不确定性模型 认知不确定性建模 ${p_1}$ Ⅲ 单峰Beta分布，${\mu _1} \in \left[ {0.6, 0.8} \right], \sigma _1^2 \in \left[ {0.02, 0.04} \right]$ ${{{\theta }} _1} = {\mu _1}, {{{\theta }} _2} = \sigma _1^2$ ${p_2}$ Ⅱ 常数，${p_2} \in \left[ {0.0, 1.0} \right]$ ${{{\theta }} _3} = {p_2}$ ${p_3}$ Ⅰ 均匀分布，$ {p}_{3}\sim U\left(0.0, 1.0\right) $ - ${p_4}, {p_5}$ Ⅲ 二元正态分布，${\mu _i} \in \left[ { - 5.0, 5.0} \right], \sigma _i^2 \in \left[ {0.002\;5, 4.0} \right], \rho \in \left[ { - 1.0, 1.0} \right], i = 4, 5$ ${{{\theta }} _4} = {\mu _4}, {{{\theta }} _5} = {\mu _5}, {{{\theta }} _6} = \sigma _4^2, {{{\theta }} _7} = \sigma _5^2, {{{\theta }} _8} = \rho $
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表 6 不同近似贝叶斯推理结果比较
修正参数 95%置信区间 极大似然估计 真值 分位数 E距离 W距离 分位数 E距离 W距离 ${{{\theta }} _1}$ [0.702 2, 0.709 1] [0.700 6, 0.702 9] [0.683 4, 0.684 4] 0.705 7 0.701 8 0.683 9 0.636 4 ${{{\theta }} _2}$ [0.027 5, 0.028 2] [0.029 9, 0.030 6] [0.031 9, 0.032 3] 0.027 9 0.030 2 0.032 1 0.035 6 ${{{\theta }} _3}$ [0.431 5, 0.474 4] [0.451 1, 0.485 2] [0.297 0, 0.330 8] 0.452 9 0.468 1 0.313 9 1 ${{{\theta }} _4}$ [0.601 5, 0.953 4] [−0.177 8, 0.189 2] [−3.703 6, −3.547 8] 0.777 5 0.005 7 −3.625 7 4 ${{{\theta }} _5}$ [−0.760 7, −0.457 2] [1.685 1, 1.986 3] [−3.771 4, −3.621 4] −0.609 0 1.835 7 −3.696 4 −1.5 ${{{\theta }} _6}$ [1.779 7, 1.923 7] [1.772 8, 1.914 4] [1.819 6, 1.946 5] 1.851 7 1.843 6 1.883 0 0.04 ${{{\theta }} _7}$ [1.959 8, 2.115 1] [1.777 4, 1.921 3] [1.110 9, 1.253 1] 2.037 5 1.849 3 1.182 0 0.36 ${{{\theta }} _8}$ [−0.095 8, −0.026 5] [−0.009 1, 0.059 8] [0.128 6, 0.195 6] −0.061 1 0.025 4 0.162 1 0.5
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