  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 variables---yield 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 well-studied example of viscous pipe flow---both turbulent
and laminar cases. The results include a new set of two dimensionless groups
for turbulent pipe flow that are ordered by relevance to the system; the
precise notion of relevance is closely tied to the derivative based global
sensitivity metric from Sobol' and Kucherenko.
