We address the computational efficiency in solving the A-optimal Bayesian design of experiments problems for which the observational map 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 Bayesian experimental design. This criterion seeks the optimal experimental design by minimizing the expected conditional variance, which is also known as the expected posterior variance. This study presents a novel likelihood-free approach to the A-optimal experimental design that does not require sampling or integrating the Bayesian posterior distribution. The expected conditional variance is obtained via the variance of the conditional expectation using the law of total variance, and we take advantage of the orthogonal projection property to approximate the conditional expectation. We derive an asymptotic error estimation for the proposed estimator of the expected conditional variance and show that the intractability of the posterior distribution does not affect the performance of our approach. We use an artificial neural network (ANN) to approximate the nonlinear conditional expectation in the implementation of our method. We then extend our approach for dealing with the case that the domain of experimental design parameters is continuous by integrating the training process of the ANN into minimizing the expected conditional variance. Through numerical experiments, we demonstrate that our method greatly reduces the number of observation model evaluations compared with widely used importance sampling-based approaches. This reduction is crucial, considering the high computational cost of the observational models. Code is available at https://github.com/vinh-tr-hoang/DOEviaPACE.

翻译：本文解决了基于偏微分方程观测映射进行A-最优贝叶斯实验设计时的计算效率问题。A-最优性是贝叶斯实验设计中广泛使用且易于解释的准则，通过最小化期望条件方差（也称期望后验方差）来寻求最优实验设计。本研究提出了一种无需采样或积分贝叶斯后验分布的似然无关新方法。利用全方差公式，期望条件方差可通过条件期望的方差求得，我们借助正交投影性质来近似条件期望。推导了期望条件方差估计量的渐近误差估计，证明后验分布的难处理性不影响本方法的性能。在实现过程中，采用人工神经网络(ANN)近似非线性条件期望。进而将方法扩展至实验设计参数域连续的情况，通过将ANN训练过程融入期望条件方差最小化来实现。数值实验表明，与广泛使用的基于重要性采样的方法相比，本方法显著减少了观测模型评估次数。考虑到观测模型的高计算成本，这种缩减至关重要。代码开源于https://github.com/vinh-tr-hoang/DOEviaPACE。