Speaker
Description
This presentation will describe a method to discover the governing equations in physical systems with multiple regimes and lengthscales, using minimum entropy criteria to optimize results. The historically challenging problem of turbulent flow is used as an example, infamous for its half-ordered, half-chaotic behavior across several orders of magnitude. Exact solutions to the Navier-Stokes equations are not known to exist, and the resolution to this problem remains the subject of a Clay Millenium Prize. Accordingly, various approximations have been developed to describe turbulent regimes, including the Reynolds-Averaged Navier-Stokes (RANS) equations that separate velocity and pressure quantities into constant and stochastic terms. However, the RANS equations are nonoptimal and can be improved using information-theoretic techniques from ODE. Two components are used to analyze this problem. First is the observation of invariants, symmetries, and conserved quantities. Invariants are quantities that remain constant when subject to symmetry transformations, and conserved quantities are properties of dynamic systems that remain constant over time. The second component is the Minimum Description Length (MDL) criterion, which provides a mathematically rigorous way to identify the most accurate equations to describe a given dataset. Using a Bayesian selection process, the search space of possible governing equations is navigated to find the optimal expressions for fluid flow. After this step, the MDL criterion is applied again at a larger lengthscale to partition the flow field into distinct regimes and generate higher-level transfer equations. The end result is a more accurate version of the RANS decomposition grounded in information theory, which we call a Kolmogorov decomposition. While the specific fluid mechanics example has a wide range of applications, from propulsion design to weather prediction and oceanography, the mathematical techniques discussed in this presentation are domain-agnostic and can apply to all areas of physics.
