  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.
