# Dimensionally Consistent Learning with Buckingham Pi

arXiv: https://arxiv.org/abs/2202.04643 · HTML: https://mlsanity.com/paper/2202.04643/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 53 references that are in the map.
- First posted on arXiv 9 Feb 2022.

## Builds on (most cited first, 10 of 10)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Deep Generative Markov State Models](https://mlsanity.com/paper/1805.07601/) (2018)
- [Bridging observation, theory and numerical simulation of the ocean using Machine Learning](https://mlsanity.com/paper/2104.12506/) (2021)
- [Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equations](https://mlsanity.com/paper/2106.05102/) (2021)
- [Data-driven dimensional analysis: algorithms for unique and relevant dimensionless groups](https://mlsanity.com/paper/1708.04303/) (2017)
- [Objective discovery of dominant dynamical processes with intelligible machine learning](https://mlsanity.com/paper/2106.12963/) (2021)
- [Lurking Variable Detection via Dimensional Analysis](https://mlsanity.com/paper/1711.03918/) (2017)

## Built on by (most cited first, 22 of 22)

- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Dimensionless machine learning: Imposing exact units equivariance](https://mlsanity.com/paper/2204.00887/) (2022)
- [Think like a Scientist: Physics-guided LLM Agent for Equation Discovery](https://mlsanity.com/paper/2602.12259/) (2026)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [Symmetry-Informed Governing Equation Discovery](https://mlsanity.com/paper/2405.16756/) (2024)
- [Machine learning and invariant theory](https://mlsanity.com/paper/2209.14991/) (2022)
- [Towards fully covariant machine learning](https://mlsanity.com/paper/2301.13724/) (2023)
- [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025)
- [Improving Controller Generalization with Dimensionless Markov Decision Processes](https://mlsanity.com/paper/2504.10006/) (2025)
- [Dimensionless Policies based on the Buckingham π Theorem: Is This a Good Way to Generalize Numerical Results?](https://mlsanity.com/paper/2307.15852/) (2023)
- [Enhancing Symbolic Regression and Universal Physics-Informed Neural Networks with Dimensional Analysis](https://mlsanity.com/paper/2411.15919/) (2024)
- [Similarity Learning with neural networks](https://mlsanity.com/paper/2410.07214/) (2024)
- [Symbolic identification of tensor equations in multidimensional physical fields](https://mlsanity.com/paper/2507.01466/) (2025)
- [Hierarchical Dimensionless Learning (Hi-π): A physics-data hybrid-driven approach for discovering dimensionless parameter combinations](https://mlsanity.com/paper/2507.18332/) (2025)
- [Direct transfer of optimized controllers to similar systems using dimensionless MPC](https://mlsanity.com/paper/2512.08667/) (2025)
- [A Tutorial on Dimensionless Learning: Geometric Interpretation and the Effect of Noise](https://mlsanity.com/paper/2512.15760/) (2025)
- [Data-driven Progressive Discovery of Physical Laws](https://mlsanity.com/paper/2603.13727/) (2026)
- [Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equations](https://mlsanity.com/paper/2605.03511/) (2026)
- [The Algebra of Units: From Buckingham's Pi-grec Theorem to Latent-Variable Learning](https://mlsanity.com/paper/2606.16737/) (2026)
- [Dimensionally consistent surrogate modelling through dimensional analysis and harmonic expansions](https://mlsanity.com/paper/2609.38094/) (2026)
- [pi-Net: Empirical Latent Representations for Differentiable Physics Heads in Data-Scarce Systems](https://mlsanity.com/paper/2610.16504/) (2026)
- [Cepheid: Decoupling Scale from Structure for Cross-Family Motion Completion](https://mlsanity.com/paper/2610.18871/) (2026)

## 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.