We address the computational efficiency in solving the A-optimal Bayesian design of experiments problems for which the observational model is based on partial differential equations and, consequently, is computationally expensive to evaluate. A-optimality is a widely used and easy-to-interpret criterion for the Bayesian design of experiments. The criterion seeks the optimal experiment design by minimizing the expected conditional variance, also known as the expected posterior variance. This work presents a novel likelihood-free method for seeking the A-optimal design of experiments without sampling or integrating the Bayesian posterior distribution. In our approach, the expected conditional variance is obtained via the variance of the conditional expectation using the law of total variance, while we take advantage of the orthogonal projection property to approximate the conditional expectation. Through an asymptotic error estimation, we show that the intractability of the posterior does not affect the performance of our approach. We use an artificial neural network (ANN) to approximate the nonlinear conditional expectation to implement our method. For dealing with continuous experimental design parameters, we integrate the training process of the ANN into minimizing the expected conditional variance. Specifically, we propose a non-local approximation of the conditional expectation and apply transfer learning to reduce the number of evaluations of the observation model. Through numerical experiments, we demonstrate that our method significantly reduces the number of observational model evaluations compared with common importance sampling-based approaches. This reduction is crucial, considering the computationally expensive nature of these models.

翻译：我们解决了基于偏微分方程的观测模型在A-最优贝叶斯实验设计问题中的计算效率问题，这类模型的计算成本通常很高。A-最优性是一种广泛使用且易于解释的贝叶斯实验设计准则，它通过最小化期望条件方差（也称为期望后验方差）来寻找最优实验设计。本文提出了一种新颖的无似然方法，无需对贝叶斯后验分布进行采样或积分即可求解A-最优实验设计。在该方法中，我们利用全方差公式通过条件期望的方差获取期望条件方差，并借助正交投影性质近似条件期望。通过渐近误差估计，我们证明后验分布的难解性不影响方法性能。我们采用人工神经网络（ANN）近似非线性条件期望来实现该方法。为处理连续实验设计参数，我们将ANN的训练过程融入期望条件方差的最小化中。具体而言，我们提出条件期望的非局部近似方法，并应用迁移学习减少观测模型评估次数。数值实验表明，与常见的重要性采样方法相比，本方法显著降低了观测模型评估次数。考虑到这些模型的高计算成本特性，这一降低至关重要。