  This paper concerns the study of optimal (supremum and infimum) uncertainty
bounds for systems where the input (or prior) probability measure is only
partially/imperfectly known (e.g., with only statistical moments and/or on a
coarse topology) rather than fully specified. Such partial knowledge provides
constraints on the input probability measures. The theory of Optimal
Uncertainty Quantification allows us to convert the task into a constraint
optimization problem where one seeks to compute the least upper/greatest lower
bound of the system's output uncertainties by finding the extremal probability
measure of the input. Such optimization requires repeated evaluation of the
system's performance indicator (input to performance map) and is
high-dimensional and non-convex by nature. Therefore, it is difficult to find
the optimal uncertainty bounds in practice. In this paper, we examine the use
of machine learning, especially deep neural networks, to address the challenge.
We achieve this by introducing a neural network classifier to approximate the
performance indicator combined with the stochastic gradient descent method to
solve the optimization problem. We demonstrate the learning based framework on
the uncertainty quantification of the impact of magnesium alloys, which are
promising light-weight structural and protective materials. Finally, we show
that the approach can be used to construct maps for the performance certificate
and safety design in engineering practice.
