mlsanityData to 7 Oct 2026

Dimensionally Consistent Learning with Buckingham Pi

arXiv 2202.04643 ↗First posted 9 Feb 2022cs.LG cs.CE physics.comp-phcited by 20

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Abstract (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 this 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: (i) a constrained optimization problem with a non-parametric input-output fitting function, (ii) a deep learning algorithm (BuckiNet) that projects the input parameter space to a lower dimension in the first layer, and (iii) a technique based on sparse identification of nonlinear dynamics (SINDy) to discover dimensionless equations whose coefficients parameterize the dynamics. We explore the accuracy, robustness and computational complexity of these methods as applied to three example problems: a bead on a rotating hoop, a laminar boundary layer, and Rayleigh-B'enard convection.

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 10 of its 53 references that are in the map. First posted on arXiv 9 Feb 2022.

Builds on

10 of 53 references.

References · 10

  1. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  2. 2019AI Feynman: a Physics-Inspired Method for Symbolic Regressioncited by 536
  3. 2019Machine Learning for Fluid Mechanicscited by 380
  4. 2022Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoderscited by 58
  5. 2018Deep Generative Markov State Modelscited by 19
  6. 2021Bridging observation, theory and numerical simulation of the ocean using Machine Learningcited by 15
  7. 2021Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equationscited by 15
  8. 2017Data-driven dimensional analysis: algorithms for unique and relevant dimensionless groupscited by 4
  9. 2021Objective discovery of dominant dynamical processes with intelligible machine learningcited by 2
  10. 2017Lurking Variable Detection via Dimensional Analysiscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 3; 2023: 3; 2024: 3; 2025: 6; 2026: 5.

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  1. 2023Benchmarking sparse system identification with low-dimensional chaoscited by 19
  2. 2022Dimensionless machine learning: Imposing exact units equivariancecited by 16
  3. 2022Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effectscited by 15
  4. 2024Symmetry-Informed Governing Equation Discoverycited by 14
  5. 2022Machine learning and invariant theorycited by 13
  6. 2026Think like a Scientist: Physics-guided LLM Agent for Equation Discoverycited by 13
  7. 2023Towards fully covariant machine learningcited by 10
  8. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  9. 2025Improving Controller Generalization with Dimensionless Markov Decision Processescited by 2
  10. 2023Dimensionless Policies based on the Buckingham π Theorem: Is This a Good Way to Generalize Numerical Results?cited by 1
  11. 2024Enhancing Symbolic Regression and Universal Physics-Informed Neural Networks with Dimensional Analysiscited by 1
  12. 2024Similarity Learning with neural networkscited by 0
8 more
  1. 2025Symbolic identification of tensor equations in multidimensional physical fieldscited by 0
  2. 2025Hierarchical Dimensionless Learning (Hi-π): A physics-data hybrid-driven approach for discovering dimensionless parameter combinationscited by 0
  3. 2025Direct transfer of optimized controllers to similar systems using dimensionless MPCcited by 0
  4. 2025A Tutorial on Dimensionless Learning: Geometric Interpretation and the Effect of Noisecited by 0
  5. 2026Data-driven Progressive Discovery of Physical Lawscited by 0
  6. 2026Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equationscited by 0
  7. 2026The Algebra of Units: From Buckingham's Pi-grec Theorem to Latent-Variable Learningcited by 0
  8. 2026Dimensionally consistent surrogate modelling through dimensional analysis and harmonic expansionscited by 0