基于有理多项式混沌展开的贝叶斯优化用于线性结构动力学模型的最大后验估计
Maximum a Posteriori Estimation for Linear Structural Dynamics Models Using Bayesian Optimization with Rational Polynomial Chaos Expansions
AI总结：
本文提出将有理多项式混沌展开代理模型与贝叶斯优化相结合，通过稀疏贝叶斯学习和自适应实验设计降低计算成本，高效求解结构动力学模型的最大后验参数估计问题。
摘要：
贝叶斯分析能够将先验知识与测量数据相结合，以学习模型参数。当只对参数的点估计感兴趣时，通常采用计算最大后验（MAP）估计的方法。我们将MAP估计应用于结构动力学模型中，在该类模型中系统响应可由频率响应函数描述。为了减轻反复调用昂贵模型所带来的高计算负担，我们采用有理多项式混沌展开（RPCE）代理模型，该模型将系统频率响应表示为两个具有复系数多项式的有理式。我们针对RPCE提出了一种对现有稀疏贝叶斯学习方法的扩展，该扩展基于对分母系数后验分布的拉普拉斯近似。此外，我们引入了一种贝叶斯优化方法，该方法允许在MAP估计的整个优化过程中自适应地丰富实验设计。在此过程中，我们利用期望改进采集函数作为识别输入空间中可能与较大目标函数值相关的样本点的手段。该采集函数通过蒙特卡洛采样进行估计，采样基于稀疏贝叶斯学习过程中识别出的展开系数后验分布。通过将稀疏诱导学习过程与序贯实验设计相结合，我们有效减少了MAP估计问题中的模型评估次数。我们在一个代数二自由度系统的参数更新问题以及一个交叉层压木板的有限元模型上，证明了所提出方法的适用性。
英文摘要：
Bayesian analysis enables combining prior knowledge with measurement data to learn model parameters. Commonly, one resorts to computing the maximum a posteriori (MAP) estimate, when only a point estimate of the parameters is of interest. We apply MAP estimation in the context of structural dynamic models, where the system response can be described by the frequency response function. To alleviate high computational demands from repeated expensive model calls, we utilize a rational polynomial chaos expansion (RPCE) surrogate model that expresses the system frequency response as a rational of two polynomials with complex coefficients. We propose an extension to an existing sparse Bayesian learning approach for RPCE based on Laplace's approximation for the posterior distribution of the denominator coefficients. Furthermore, we introduce a Bayesian optimization approach, which allows to adaptively enrich the experimental design throughout the optimization process of MAP estimation. Thereby, we utilize the expected improvement acquisition function as a means to identify sample points in the input space that are possibly associated with large objective function values. The acquisition function is estimated through Monte Carlo sampling based on the posterior distribution of the expansion coefficients identified in the sparse Bayesian learning process. By combining the sparsity-inducing learning procedure with the sequential experimental design, we effectively reduce the number of model evaluations in the MAP estimation problem. We demonstrate the applicability of the presented methods on the parameter updating problem of an algebraic two-degree-of-freedom system and the finite element model of a cross-laminated timber plate.
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