Dimensionally consistent learning with Buckingham Pi.

Area of Science:

• Physics
• Applied Mathematics
• Data Science

Background:

• Dimensional analysis is crucial for understanding physical systems, especially when governing equations are unknown.
• The Buckingham Pi theorem offers a method for finding dimensionless groups but doesn't guarantee uniqueness or optimal data collapse.
• Existing methods often require prior knowledge of system dynamics or governing equations.

Purpose of the Study:

• To develop automated, data-driven techniques for discovering optimal dimensionless groups.
• To leverage measurement data's inherent symmetries for identifying these groups.
• To reduce the dimensionality of complex physical system data effectively.

Main Methods:

• Proposed three novel data-driven techniques constrained by the Buckingham Pi theorem.
• Method 1: Constrained optimization with non-parametric fitting.
• Method 2: A deep learning approach (BuckiNet) for parameter space projection.
• Method 3: Sparse identification of nonlinear dynamics for discovering dimensionless equations.

Main Results:

• Successfully identified key dimensionless groups in three benchmark problems: a bead on a rotating hoop, laminar boundary layer, and Rayleigh-Bénard convection.
• Demonstrated the accuracy, robustness, and computational efficiency of the developed methods.
• Showcased the ability of these techniques to collapse data effectively into a lower-dimensional space.

Conclusions:

• The proposed data-driven methods offer a powerful, automated approach to dimensional analysis.
• These techniques enhance the discovery of physical symmetries and insights from measurement data.
• The methods provide a robust alternative for analyzing systems lacking explicit governing equations.