Learning fluid physics from highly turbulent data using sparse physics-informed discovery of empirical relations (SPIDER)

We show how a complete mathematical description of a complicated physical phenomenon can be constructed from observational data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality, and symmetry, a weak formulation of differential equations, and sparse regression. To illustrate this, we extract a complete system of governing equations describing flows of incompressible Newtonian fluids -- the Navier-Stokes equation, the continuity equation, and the boundary conditions -- from numerical data describing a highly turbulent channel flow in three dimensions. The hybrid approach is remarkably robust, yielding accurate results for very high noise levels, and should thus work equally well for a wide range of experimental data. In addition, this approach provides easily interpretable information about the relative importance of different physical effects (such as viscosity) as well as useful insight into the quality of the data, making it a useful diagnostic tool.