Various real-world scientific applications involve the mathematical modeling of complex uncertain systems with numerous unknown parameters. Accurate parameter estimation is often practically infeasible in such systems, as the available training data may be insufficient and the cost of acquiring additional data may be high. In such cases, based on a Bayesian paradigm, we can design robust operators retaining the best overall performance across all possible models and design optimal experiments that can effectively reduce uncertainty to enhance the performance of such operators maximally. While objective-based uncertainty quantification (objective-UQ) based on MOCU (mean objective cost of uncertainty) provides an effective means for quantifying uncertainty in complex systems, the high computational cost of estimating MOCU has been a challenge in applying it to real-world scientific/engineering problems. In this work, we propose a novel scheme to reduce the computational cost for objective-UQ via MOCU based on a data-driven approach. We adopt a neural message-passing model for surrogate modeling, incorporating a novel axiomatic constraint loss that penalizes an increase in the estimated system uncertainty. As an illustrative example, we consider the optimal experimental design (OED) problem for uncertain Kuramoto models, where the goal is to predict the experiments that can most effectively enhance robust synchronization performance through uncertainty reduction. We show that our proposed approach can accelerate MOCU-based OED by four to five orders of magnitude, without any visible performance loss compared to the state-of-the-art. The proposed approach applies to general OED tasks, beyond the Kuramoto model.

翻译：各种现实世界中的科学应用涉及对包含大量未知参数的复杂不确定系统进行数学建模。在这类系统中，精确参数估计往往在实践上不可行，因为可用训练数据可能不足，且获取额外数据的成本可能很高。在此类情况下，基于贝叶斯范式，我们可以设计出能跨所有可能模型保持最佳整体性能的鲁棒算子，并设计能有效降低不确定性以最大限度提升这类算子性能的最优实验。虽然基于MOCU（平均目标不确定性成本）的目标导向不确定性量化（objective-UQ）为复杂系统中的不确定性量化提供了有效手段，但估算MOCU的高计算成本一直是在将其应用于现实科学/工程问题时的挑战。本文提出了一种新颖方案，通过数据驱动方法降低基于MOCU的目标导向不确定性量化计算成本。我们采用神经消息传递模型进行代理建模，并融入一种新颖的公理约束损失函数，可对估计系统不确定性的增加施加惩罚。作为示例，我们考虑了不确定Kuramoto模型的最优实验设计（OED）问题，其目标是预测能通过不确定性降低最有效提升鲁棒同步性能的实验。实验表明，与最先进方法相比，本方案可将基于MOCU的最优实验设计加速四到五个数量级，且未出现任何可见的性能损失。该方案适用于超出Kuramoto模型的通用最优实验设计任务。