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Dimensionally consistent learning with Buckingham Pi

A preprint version of the article is available at arXiv.


In the absence of governing equations, dimensional analysis is a robust technique for extracting insights and finding symmetries in physical systems. Given measurement variables and parameters, the Buckingham Pi theorem provides a procedure for finding a set of dimensionless groups that spans the solution space, although this set is not unique. We propose an automated approach using the symmetric and self-similar structure of available measurement data to discover the dimensionless groups that best collapse these data to a lower dimensional space according to an optimal fit. We develop three data-driven techniques that use the Buckingham Pi theorem as a constraint: (1) a constrained optimization problem with a non-parametric input–output fitting function, (2) a deep learning algorithm (BuckiNet) that projects the input parameter space to a lower dimension in the first layer and (3) a technique based on sparse identification of nonlinear dynamics to discover dimensionless equations whose coefficients parameterize the dynamics. We explore the accuracy, robustness and computational complexity of these methods and show that they successfully identify dimensionless groups in three example problems: a bead on a rotating hoop, a laminar boundary layer and Rayleigh–Bénard convection.

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Fig. 1: Illustration of the BuckiNet layer for the rotating hoop problem.
Fig. 2: Dimensionless SINDy applied to the rotating hoop problem, showing the angle as a function of time.
Fig. 3: Summary of the dimensionless SINDy approach applied to the rotating hoop problem.
Fig. 4: Learnt Blasius scaling in laminar boundary layer.
Fig. 5: The dimensionless SINDy method applied to the Rayleigh–Bénard convection problem.

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Data availability

The data that support the findings of this study have been generated by numerically solving the example models proposed in the manuscript. The solvers are available in the GitHub repository (, and the data can be found in ‘data/figure-X’, where X is the number of the figure. Source data for Figs. 2, 4 and 5 are also available with this manuscript.

Code availability

The Bucki-Data package is available in the GitHub repository (


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The authors acknowledge support from the Army Research Office (ARO W911NF-19-1-0045) and the National Science Foundation AI Institute in Dynamic Systems (grant no. 2112085). J.C. acknowledges funding support from the Department of Defense through the National Defense Science and Engineering Graduate Fellowship Program. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

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Authors and Affiliations



J.B, J.C., S.L.B. and J.N.K. designed research. J.B. and J.C. performed research. J.B, J.C., S.L.B. and J.N.K. wrote the paper.

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Correspondence to Joseph Bakarji, Jared Callaham, Steven L. Brunton or J. Nathan Kutz.

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Nature Computational Science thanks Zhengtao Gan and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Handling editor: Jie Pan, in collaboration with the Nature Computational Science team. Peer reviewer reports are available.

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Bakarji, J., Callaham, J., Brunton, S.L. et al. Dimensionally consistent learning with Buckingham Pi. Nat Comput Sci 2, 834–844 (2022).

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