As a main task of inverse problem, model updating has received more and more attention in the area of inspection, sensing, and monitoring technologies during the recent decades, where the estimation of posterior probability density function (PDF) of unknown model parameters is still challenging for expensive-to-evaluate models of interest. In this paper, a novel variational Bayesian inference method is proposed to approximate the real posterior PDF of unknown model parameters by using Gaussian mixture model and measurement responses. A Gaussian process regression model is first trained for approximating the logarithm of the product of likelihood function and prior PDF, with which, another Gaussian process model is induced for approximating the expensive evidence lower bound (ELBO). Then, two Bayesian numerical methods, i.e., Bayesian optimization and Bayesian quadrature, are combined sequentially as a novel Bayesian active learning method for searching the global optima of the parameters of the variational posterior density. The proposed method inherits the advantages of both Bayesian numerical methods, which includes good global convergence, much less number of simulator calls, etc. Three examples, including the dynamic model of a two degrees of freedom structures, the lubrication model of a hybrid journal bearing, and the dynamic model of an airplane structure, are introduced for demonstrating the relative merits of the proposed method. Results show that, given desired requirement of numerical accuracy, the proposed method is more efficient than the parallel methods.

中文翻译：

---

通过贝叶斯主动学习进行高效的变分贝叶斯模型更新


作为逆问题的主要任务，近几十年来，模型更新在检测、传感和监测技术领域受到越来越多的关注，其中未知模型参数的后验概率密度函数 （PDF） 的估计对于成本高昂的感兴趣模型仍然具有挑战性。在本文中，提出了一种新的变分贝叶斯推理方法，通过使用高斯混合模型和测量响应来近似未知模型参数的真实后验 PDF。首先训练一个高斯过程回归模型来近似似然函数和先验 PDF 的乘积的对数，然后用它引入另一个高斯过程模型来近似昂贵的证据下限 （ELBO）。然后，将两种贝叶斯数值方法，即贝叶斯优化和贝叶斯求积，依次组合为一种新的贝叶斯主动学习方法，用于搜索变分后密度参数的全局最优值。所提出的方法继承了两种贝叶斯数值方法的优点，包括良好的全局收敛性、更少的模拟器调用次数等。介绍了三个示例，包括两个自由度结构的动力学模型、混合轴颈轴承的润滑模型和飞机结构的动力学模型，以证明所提出的方法的相对优点。结果表明，在满足数值精度要求的情况下，所提方法比并行方法更有效。