Note: This paper is part of the JCP Special Topic on Machine Learning Meets Chemical Physics.

Aircraft designers are legally required to treat manufacturing variability so as to minimize the probability of structural failure, pursuant to Code of Federal Regulations (CFR) 25.613. The usual design strategy employed for compliance with the regulations is the use of allowable values. However, allowables are not guaranteed to satisfy the regulations as written. In this work, we thoroughly investigate the use of allowables via a formal reliability analysis using quantile evaluation. We find that allowables are provably conservative for simple settings but admit anticonservative structural analysis under nonmonotone structural response, strongly nonnormal distributions, correlated material properties, and high-dimensional material property spaces such as those arising in laminate composites. This analysis develops intuition for when allowables are a conservative approximation of a true reliability analysis. We introduce an inexpensive procedure to detect analysis pathologies and illustrate its use on a set of aerospace-relevant structural analysis problems.

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Classical dimensional analysis has two limitations: (i) the computed dimensionless groups are not unique, and (ii) the analysis does not measure relative importance of the dimensionless groups. We propose two algorithms for estimating unique and relevant dimensionless groups assuming the experimenter can control the system's independent variables and evaluate the corresponding dependent variable; e.g., computer experiments provide such a setting. The first algorithm is based on a response surface constructed from a set of experiments.The second algorithm uses many experiments to estimate finite differences over a range of the independent variables. Both algorithms are semi-empirical because they use experimental data to complement the dimensional analysis. We derive the algorithms by combining classical semi-empirical modeling with active subspaces, which-given a probability density on the independent variablesyield unique and relevant dimensionless groups. The connection between active subspaces and dimensional analysis also reveals that all empirical models are ridge functions, which are functions that are constant along low-dimensional subspaces in its domain. We demonstrate the proposed algorithms on the wellstudied example of viscous pipe flow-both turbulent and laminar cases. The results include a new set of two dimensionless groups for turbulent pipe flow

Lurking variables represent hidden information, and preclude a full understanding of phenomena of interest. Detection is usually based on serendipity -visual detection of unexplained, systematic variation. However, these approaches are doomed to fail if the lurking variables do not vary. In this article, we address these challenges by introducing formal hypothesis tests for the presence of lurking variables, based on Dimensional Analysis. These procedures utilize a modified form of the Buckingham π theorem to provide structure for a suitable null hypothesis. We present analytic tools for reasoning about lurking variables in physical phenomena, construct procedures to handle cases of increasing complexity, and present examples of their application to engineering problems. The results of this work enable algorithm-driven lurking variable detection, complementing a traditionally inspection-based approach.