Abstract
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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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 (https://github.com/josephbakarji/bucki-data)45, 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 (https://github.com/josephbakarji/bucki-data)45.
References
Barenblatt, G. I. Scaling, Self-Similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics 14 (Cambridge Univ. Press, 1996).
Sterrett, S. G. Physically Similar Systems – A History of the Concept, pp 377–411 (Springer, 2017).
Buckingham, E. On physically similar systems; illustrations of the use of dimensional equations. Phys. Rev. 4, 345–376 (1914).
del Rosario, Z., Lee, M. & Iaccarino, G. Lurking variable detection via dimensional analysis. SIAM/ASA J. Uncertain. 7, 232–259 (2019).
Jofre, L., del Rosario, Z. R. & Iaccarino, G. Data-driven dimensional analysis of heat transfer in irradiated particle-laden turbulent flow. Int. J. Multiphase Flow 125, 103198 (2020).
Fukami, K. & Taira, K. Robust machine learning of turbulence through generalized Buckingham pi-inspired pre-processing oftraining data, APS Division of Fluid Dynamics Meeting Abstracts, A31-004 (2021).
Xie, X., Liu, W. K. & Gan, Z. Data-driven discovery of dimensionless numbers and scaling laws from experimental measurements. Preprint at https://arxiv.org/abs/2111.03583 (2021).
Cerda, E. & Mahadevan, L. Geometry and physics of wrinkling. Phys. Rev. Lett. 90, 074302 (2003).
Morris, S. W., Bodenschatz, E., Cannell, D. S. & Ahlers, G. Spiral defect chaos in large aspect ratio Rayleigh–Bénard convection. Phys. Rev. Lett. 71, 2026 (1993).
Shi, X. D., Brenner, M. P. & Nagel, S. R. A cascade of structure in a drop falling from a faucet. Science 265, 219–222 (1994).
Grzybowski, B., Stone, H. A. & Whitesides, G. M. Dynamic self-assembly of magnetized, millimetre-sized objects rotating at a liquid–air interface. Nature 405, 1033–1036 (2000).
Seminara, A. et al. Osmotic spreading of bacillus subtilis biofilms driven by an extracellular matrix. Proc. Natl Acad. Sci. USA 109, 1116–1121 (2012).
Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Ref. Mod. Phys. 65, 851–1112 (1993).
Callaham, J. L., Koch, J. V., Brunton, B. W., Kutz, J. N. & Brunton, S. L. Learning dominant physical processes with data-driven balance models. Nat. Commun. 12, 1–10 (2021).
Holmes, P. & Guckenheimer, J. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42 of Applied Mathematical Sciences (Springer-Verlag, 1983).
Yair, O., Talmon, R., Coifman, R. R. & Kevrekidis, I. G. Reconstruction of normal forms by learning informed observation geometries from data. Proc. Natl. Acad. Sci. 114, E7865–E7874 (2017).
Kalia, M., Brunton, S. L., Meijer, H. G., Brune, C. & Kutz, J. N. Learning normal form autoencoders for data-driven discovery of universal, parameter-dependent governing equations. Preprint at https://arxiv.org/abs/2106.05102 (2021).
Schmidt, M. & Lipson, H. Distilling free-form natural laws from experimental data. Science 324, 81–85 (2009).
Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113, 3932–3937 (2016).
Rudy, S. H., Brunton, S. L., Proctor, J. L. & Kutz, J. N. Data-driven discovery of partial differential equations. Sci. Adv. 3, e1602614 (2017).
Lu, L., Jin, P., Pang, G., Zhang, Z. & Karniadakis, G. E. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mach. Intell. 3, 218–229 (2021).
Raissi, M., Perdikaris, P. & Karniadakis, G. E. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019).
Karniadakis, G. E. et al. Physics-informed machine learning. Nat. Rev. Phys. 3, 422–440 (2021).
Noé, F., Olsson, S., Köhler, J. & Wu, H. Boltzmann generators: sampling equilibrium states of many-body systems with deep learning. Science 365, eaaw1147 (2019).
Brenner, M., Eldredge, J. & Freund, J. Perspective on machine learning for advancing fluid mechanics. Phys. Rev. Fluids 4, 100501 (2019).
Duraisamy, K., Iaccarino, G. & Xiao, H. Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357–377 (2019).
Brunton, S. L., Noack, B. R. & Koumoutsakos, P. Machine learning for fluid mechanics. Ann. Rev. Fluid Mech. 52, 477–508 (2020).
Sonnewald, M. et al. Bridging observations, theory, and numerical simulation of the ocean using machine learning. Environ. Res. Lett. 16, 073008 (2021).
Kaiser, B. E., Saenz, J. A., Sonnewald, M. & Livescu, D. Objective discovery of dominant dynamical processes with intelligible machine learning. Preprint at https://arxiv.org/abs/2106.12963.
Sonnewald, M., Wunsch, C. & Heimbach, P. Unsupervised learning reveals geography of global ocean regimes. Earth Space Sci. 6, 784–794 (2019).
Wu, H., Mardt, A., Pasquali, L. & Noe, F. Deep generative Markov state models in Proc. of 32nd Conference on NeuralInformation Processing Systems (eds Bengio, S. et al.) (2018).
Champion, K., Lusch, B., Kutz, J. N. & Brunton, S. L. Data-driven discovery of coordinates and governing equations. Proc. Natl Acad. Sci. USA 116, 22445–22451 (2019).
Bakarji, J., Champion, K., Kutz, J. N. & Brunton, S. L. Discovering governing equations from partial measurements with deep delay autoencoders. Preprint at https://arxiv.org/abs/2201.05136 (2022).
Constantine, P. G., del Rosario, Z. & Iaccarino, G. Data-driven dimensional analysis: algorithms for unique and relevant dimensionless groups. Preprint at https://arxiv.org/abs/1708.04303 (2017).
Udrescu, S.-M. & Tegmark, M. AI Feynman: a physics-inspired method for symbolic regression. Sci. Adv. 6, eaay2631 (2020).
Gunaratnam, D. J., Degroff, T. & Gero, J. S. Improving neural network models of physical systems through dimensional analysis. Appl. Soft Comput. 2, 283–296 (2003).
Saha, S. et al. Hierarchical deep learning neural network (HiDeNN): an artificial intelligence (AI) framework for computational science and engineering. Comput. Methods Appl. Mech. Eng. 373, 113452 (2021).
Mozaffar, M. et al. Mechanistic artificial intelligence (mechanistic-AI) for modeling, design, and control of advanced manufacturing processes: current state and perspectives. J. Mater. Process. Technol. 302, 117485 (2021) .
Strogatz, S. H. Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering (CRC, 2018).
Lorenz, E. N. Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963).
Pandey, A., Scheel, J. D. & Schumacher, J. Turbulent superstructures in Rayleigh–Bénard convection. Nat. Commun. 9, 2118 (2018) .
Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability (Clarendon, 1961).
Mortensen, M. Shenfun: High performance spectral Galerkin computing platform. J. Open Source Softw. 3, 1071 (2018).
Loiseau, J.-C. & Brunton, S. L. Constrained sparse Galerkin regression. J. Fluid Mech. 838, 42–67 (2018).
Bakarji, J. & Callaham, J. Bucki-Data GitHub repository. Zenodo https://doi.org/10.5281/zenodo.7187741 (2022).
Acknowledgements
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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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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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). https://doi.org/10.1038/s43588-022-00355-5
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DOI: https://doi.org/10.1038/s43588-022-00355-5
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