Data-driven discovery of partial differential equations

Fluid dynamics is inherently governed by spatial-temporal interactions which can be characterized by partial differential equations (PDEs). Emerging sensor and measurement technologies allowing for rich, time-series data collection motivate new data-driven methods for discovering governing equations. We present a novel computational technique for discovering governing PDEs from time series measurements. A library of candidate terms for the PDE including nonlinearities and partial derivatives is computed and sparse regression is then used to identify a subset which accurately reflects the measured dynamics. Measurements may be taken either in a Eulerian framework to discover field equations or in a Lagrangian framework to study a single stochastic trajectory. The method is shown to be robust, efficient, and to work on a variety of canonical equations. Data collected from a simulation of a flow field around a cylinder is used to accurately identify the Navier-Stokes vorticity equation and the Reynolds number to within 1\%. A single trace of Brownian motion is also used to identify the diffusion equation. Our method provides a novel approach towards data enabled science where spatial-temporal information bolsters classical machine learning techniques to identify physical laws.