Dimensionally Consistent Learning with Buckingham Pi. (arXiv:2202.04643v1 [cs.LG])

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.

Source: https://arxiv.org/abs/2202.04643

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