# Data-driven dimensional analysis: algorithms for unique and relevant dimensionless groups

arXiv: https://arxiv.org/abs/1708.04303 · HTML: https://mlsanity.com/paper/1708.04303/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2017.

## Built on by (most cited first, 4 of 4)

- [Dimensionally Consistent Learning with Buckingham Pi](https://mlsanity.com/paper/2202.04643/) (2022)
- [Dimensionless machine learning: Imposing exact units equivariance](https://mlsanity.com/paper/2204.00887/) (2022)
- [Hierarchical Dimensionless Learning (Hi-π): A physics-data hybrid-driven approach for discovering dimensionless parameter combinations](https://mlsanity.com/paper/2507.18332/) (2025)
- [The Algebra of Units: From Buckingham's Pi-grec Theorem to Latent-Variable Learning](https://mlsanity.com/paper/2606.16737/) (2026)

## Abstract (arXiv)

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.