Unsupervised Learning of Dimensionless Groups and Minimally Parametrized Equations

We address the fundamental problem of developing algorithms that learn interpretable physical models from measurement data. In particular, we focus on the data-driven discovery of non-dimensional numbers. An encoder layer constrained by the Buckingham Pi theorem (BuckiNet) and embedded in a deep network that fits input-output measurements, is designed to discover the set of dimensionless numbers that best collapses the data to a low dimensional parameter space and reveals dominant bifurcation parameters. In addition, we develop a SINDy-based algorithm that constrains the dimensionless numbers to be part of a sparse parametric differential equation, thus directly relating the Pi groups to the underlying dynamics that describe the system. We test our methods on nonlinear systems, including the rotating hoop, the boundary layer flow, and Rayleigh-Benard problems.