Bayesian optimal experimental design (OED) seeks experiments that maximize the expected information gain (EIG) in model parameters. Directly estimating the EIG using nested Monte Carlo is computationally expensive and requires an explicit likelihood. Variational OED (vOED), in contrast, estimates a lower bound of the EIG without likelihood evaluations by approximating the posterior distributions with variational forms, and then tightens the bound by optimizing its variational parameters. We introduce the use of normalizing flows (NFs) for representing variational distributions in vOED; we call this approach vOED-NFs. Specifically, we adopt NFs with a conditional invertible neural network architecture built from compositions of coupling layers, and enhanced with a summary network for data dimension reduction. We present Monte Carlo estimators to the lower bound along with gradient expressions to enable a gradient-based simultaneous optimization of the variational parameters and the design variables. The vOED-NFs algorithm is then validated in two benchmark problems, and demonstrated on a partial differential equation-governed application of cathodic electrophoretic deposition and an implicit likelihood case with stochastic modeling of aphid population. The findings suggest that a composition of 4--5 coupling layers is able to achieve lower EIG estimation bias, under a fixed budget of forward model runs, compared to previous approaches. The resulting NFs produce approximate posteriors that agree well with the true posteriors, able to capture non-Gaussian and multi-modal features effectively.

翻译：贝叶斯最优实验设计（OED）旨在寻找能最大化模型参数期望信息增益（EIG）的实验。直接使用嵌套蒙特卡洛方法估计EIG计算成本高昂且需显式似然函数。相比之下，变分OED（vOED）通过变分形式近似后验分布，无需似然评估即可估计EIG下界，并通过优化变分参数收紧该下界。我们提出在vOED中使用归一化流（NFs）表示变分分布，并将该方法称为vOED-NFs。具体而言，我们采用基于耦合层组合构建的条件可逆神经网络架构的NFs，并引入用于数据降维的摘要网络进行增强。我们给出了下界的蒙特卡洛估计量及其梯度表达式，以实现变分参数与设计变量的梯度同步优化。随后，vOED-NFs算法在两个基准问题中得到验证，并应用于阴极电泳沉积这一偏微分方程控制问题，以及蚜虫种群随机建模的隐式似然案例。研究结果表明，在固定前向模型运行预算下，相较于先前方法，由4-5个耦合层组成的网络能够实现更低的EIG估计偏差。最终生成的NFs所得到的近似后验与真实后验高度吻合，并能有效捕捉非高斯和多模态特征。