The Algebra of Units: From Buckingham's Pi-grec Theorem to Latent-Variable Learning

2026-06-15

AI Summary

Traditional dimensional analysis relies heavily on expert knowledge, hindering its automation and scalability. This work proposes a data-driven approach that, for the first time, unifies classical dimensional analysis with modern machine learning within a common algebraic framework, enabling the automatic discovery of physically interpretable dimensionless groups without requiring prior physical knowledge. By applying a logarithmic transformation to map measurement data onto a low-dimensional manifold, the method integrates singular value decomposition with an integer exponent combination search and incorporates a repeating-variable filtering mechanism. Evaluated on a compressor dataset comprising 16,000 samples, the approach accurately recovers canonical dimensionless parameters—including flow coefficient, head coefficient, and Mach number—with performance map reconstruction errors below 0.01%.

Abstract

Engineers often measure many quantities-speed, pressure, temperature, length-expressed in different physical units. The Buckingham Pi-grec theorem states that these variables can always be combined into a smaller set of dimensionless numbers whose values fully determine the system's behaviour. Identifying the appropriate dimensionless groups has traditionally required expert knowledge and physical insight. This paper shows that they can instead be discovered automatically from data, without prior knowledge of the governing physics. The key observation is that, after logarithmic transformation, measurements collected under different scalings of the same system lie on a low-dimensional manifold whose geometry is determined by the underlying dimensionless groups. Singular value decomposition (SVD) identifies this manifold directly from data. A subsequent search over integer-exponent combinations recovers candidate dimensionless quantities, while a repeating-variable filter retains only those constructed from the machine's characteristic scales. This procedure recovers familiar engineering groups, including the flow coefficient, head coefficient, and Mach number, while excluding equivalent but less interpretable alternatives. The method is demonstrated on a synthetic compressor dataset containing 16,000 measurements. Starting from raw dimensional variables and no physics input, it recovers the correct dimensionless groups to numerical precision and reproduces the compressor performance map with an error below 0.01%. More broadly, the work reveals a close connection between classical dimensional analysis and modern data-driven learning. Both rely on the same underlying algebraic structure, suggesting new approaches for building physical models that are simultaneously interpretable, scalable, and data-efficient.

Problem

• dimensionless groups
• Buckingham Pi theorem
• data-driven discovery
• dimensional analysis
• latent-variable learning

Innovation

• dimensionless groups
• Buckingham Pi theorem
• singular value decomposition
• data-driven discovery
• physical interpretability