Surrogate models provide a quick-to-evaluate approximation to complex computational models and are essential for multi-query problems like design optimisation. The inputs of current computational models are usually high-dimensional and uncertain. We consider Bayesian inference for constructing statistical surrogates with input uncertainties and intrinsic dimensionality reduction. The surrogates are trained by fitting to data from prevalent deterministic computational models. The assumed prior probability density of the surrogate is a Gaussian process. We determine the respective posterior probability density and parameters of the posited statistical model using variational Bayes. The non-Gaussian posterior is approximated by a simpler trial density with free variational parameters and the discrepancy between them is measured using the Kullback-Leibler (KL) divergence. We employ the stochastic gradient method to compute the variational parameters and other statistical model parameters by minimising the KL divergence. We demonstrate the accuracy and versatility of the proposed reduced dimension variational Gaussian process (RDVGP) surrogate on illustrative and robust structural optimisation problems with cost functions depending on a weighted sum of the mean and standard deviation of model outputs.

翻译：代理模型能快速评估复杂计算模型的近似结果，对于多查询问题（如设计优化）至关重要。当前计算模型的输入通常具有高维性和不确定性。我们考虑采用贝叶斯推断方法构建具有输入不确定性和内在降维特性的统计代理模型。这些代理模型通过拟合主流确定性计算模型生成的数据进行训练。代理模型的先验概率密度假设为高斯过程，并利用变分贝叶斯方法确定相应的后验概率密度及所设统计模型参数。非高斯后验分布通过包含自由变分参数的更简单试验密度进行近似，两者差异通过Kullback-Leibler（KL）散度衡量。我们采用随机梯度法最小化KL散度，从而计算变分参数及其他统计模型参数。最后，通过依赖模型输出均值与标准差加权和的代价函数，在示例性和鲁棒结构优化问题中验证了所提出的降维变分高斯过程（RDVGP）代理模型的精度与通用性。