Source-linked AI summary
Data-driven discovery of dimensionless numbers and scaling laws from experimental measurements
Xiaoyu Xie, Wing Kam Liu, Zhengtao Gan
TL;DR
Classical dimensional analysis does not uniquely identify dominant dimensionless numbers or their scaling laws, complicating analysis of complex systems with incomplete governing equations. Dimensionless learning embeds dimensional invariance in a two-level machine-learning scheme to discover these relationships from noisy experimental data. The demonstrations include recovery of the Rayleigh number and dimensionless differential equations, with compact dimensionless descriptions as a practical consequence.
Problem
Classical dimensional analysis does not uniquely identify dominant dimensionless groups or their scaling laws, making expert selection and trial-and-error fitting necessary for complex systems with incomplete governing equations.
Source: §1 · p. 2Method
Dimensionless learning embeds dimensional invariance in a two-level machine-learning scheme to discover dominant dimensionless numbers and scaling laws from noisy experimental measurements.
Source: §1 · p. 2Results
The method recovers the classical Rayleigh number from experimental data and demonstrates applications to several engineering problems and dimensionless differential equations.
Source: §1 · p. 2, §2.1 · p. 5Takeaways & Limitations
Dimensionless learning can reduce high-dimensional problems to low-dimensional, interpretable descriptions that support qualitative and quantitative system analysis.
Source: §3 · p. 11Identifying the right dimensionless groups depends on a sufficient parameter list; omitting an important quantity can prevent high R2 and recovery of the correct form.
Source: §2.2 · p. 6
Abstract
from arXiv · showhide
Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a highly-multivariable system with incomplete governing equations. In this study, we embed the principle of dimensional invariance into a two-level machine learning scheme to automatically discover dominant and unique dimensionless numbers and scaling laws from data. The proposed methodology, called dimensionless learning, can reduce high-dimensional parametric spaces into descriptions involving just a few physically-interpretable dimensionless parameters, which significantly simplifies the process design and optimization of the system. We demonstrate the algorithm by solving several challenging engineering problems with noisy experimental measurements (not synthetic data) collected from the literature. The examples include turbulent Rayleigh-Benard convection, vapor depression dynamics in laser melting of metals, and porosity formation in 3D printing. We also show that the proposed approach can identify dimensionally-homogeneous differential equations with minimal parameters by leveraging sparsity-promoting techniques.
1 Introduction
Dimensionless numbers compactly describe physical systems and support scale comparisons, but discovering dominant groups and their scaling laws remains difficult when governing equations are incomplete. Dimensionless learning embeds dimensional invariance in machine learning to discover such groups and laws from noisy experimental measurements.
1 Introduction
- Dimensionless numbers reduce the variables needed to describe a process, can reduce experimental or simulation demands, and provide physically interpretable insights into system behavior.Their scale invariance also underpins similitude between small-scale experiments and full-scale applications.Source: §1 · pp. 1–2
- Classical dimensional analysis yields non-unique dimensionless groups and does not determine their scaling laws, leaving group selection and fitting dependent on expert judgment and trial and error.The challenge is especially acute for complex systems without complete governing equations.Source: §1 · p. 2
- Prior data-driven approaches either assume power-law relationships or rely on idealized experiments with negligible input and output noise.SLAW combines dimensional analysis with multivariate linear regression, while active-subspace methods estimate unique, relevant groups under idealized conditions.Source: §1 · p. 2
- Dimensionless learning embeds dimensional invariance in a two-level machine-learning scheme to discover dominant dimensionless numbers and scaling laws from noisy experimental measurements.The study demonstrates the approach on turbulent Rayleigh-Bénard convection, vapor-depression dynamics, and 3D-printing porosity, and explores identifying dimensionless differential equations with sparsity-promoting methods.Source: §1 · p. 2
2 Results
The results apply dimensionless learning to convection, laser-metal keyhole dynamics, and 3D-printing porosity. Across these cases, the method identifies reduced scaling relations from experimental measurements, while the results also expose limits tied to parameter selection.
2.1 Turbulent Rayleigh-Bénard convection
- R²=0.999 identifies the classical Rayleigh number as the unique best-fit dimensionless number, with the Nu–Ra measurements collapsing onto one curve.The identified coefficients are γ1=γ2=γ3=1.Source: Figure 1 · p. 4, §2.1 · p. 5
- Using heat flux q as the output yields a new set of dimensionless numbers with R²=0.999 and an almost linear log-log relation between them.This expands the dimensionless space beyond the earlier Nu-output example.Source: §2.1 · pp. 5–6
2.2 Vapor depression dynamics in laser-metal interaction
- At the global optimum (γ2=γ3=1), the dimensionless number scores 0.98 R2, has the form of keyhole number Ke, and supports a keyhole-aspect-ratio scaling law.The local maximum’s datapoints are scattered, indicating its dimensionless number is not a good scaling parameter.Source: §2.2 · p. 6
- Adding irrelevant or unimportant quantities can produce additional dimensionless numbers with R² as high as the keyhole number, requiring more experiments to distinguish them.The keyhole number remains identifiable in this scenario.Source: §2.2 · p. 7
2.3 Porosity formation in 3D printing of metals.
- R²=0.13 for a polynomial model using energy density shows that this combined parameter alone does not provide a scale-free relation across materials and processing conditions.The relation must be modified when experimental energy scales vary.Source: §2.3 · p. 8
- Two dimensionless numbers represent porosity data: normalized energy density relates to lack-of-fusion porosity, while a modified normalized enthalpy product relates to keyhole porosity.The two mechanisms are linked to insufficient energy to melt powder and gas bubbles trapped beneath the surface during keyhole fluctuation, respectively.Source: §2.3 · p. 8
- R²=0.95 is reported for another porosity scaling relation based on NED and NEP with different modification terms.The modified term Hd accounts for multiple-track scanning.Source: §2.3 · p. 8, Figure 3 · p. 9
3 Discussions
Dimensionless learning combines dimensional invariance with flexible representation-learning methods, including SINDy, to discover compact, interpretable laws. In the spring-mass-damper example, the dimensionless equation reduces six variables to three.
3 Discussions
- The SINDy example reduces the governing equation from six variables to three, while recovering physical parameters including natural time and length scales and dimensionless damping.Figure 4 outlines the workflow: parameter-combination datasets produce a dimension matrix, followed by sequential optimization and model selection using R2 and BIC.Source: §3 · p. 9, Figure 4 · p. 10
- The first optimization level guarantees dimensional invariance, while the second can use different representation-learning methods to capture scale-free relationships.The paper demonstrates polynomial and tree-based XGBoost methods and identifies deep neural networks, symbolic regression, and Bayesian machine learning as possible extensions.Source: §3 · p. 9
- Embedding dimensional invariance as a constraint reduces the learning space by eliminating strong dependence on variable dimensions.Source: §3 · p. 9
- Dimensionless features yield low-dimensional patterns insensitive to units and scales, making the learned representations physically interpretable.The paper says these representations support qualitative and quantitative analysis of the systems of interest.Source: §3 · p. 11
- The approach targets scale-free laws from high-dimensional data and, in material science, compact expressions that translate optimal process parameters across materials.The discussion also identifies potential applications in manufacturing, microfluidics, solar energy, and financial market analysis.Source: §3 · p. 11