  We propose a sparse regression method capable of discovering the governing
partial differential equation(s) of a given system by time series measurements
in the spatial domain. The regression framework relies on sparsity promoting
techniques to select the nonlinear and partial derivative terms terms of the
governing equations that most accurately represent the data, bypassing a
combinatorially large search through all possible candidate models. The method
balances model complexity and regression accuracy by selecting a parsimonious
model via Pareto analysis. Time series measurements can be made in an Eulerian
framework where the sensors are fixed spatially, or in a Lagrangian framework
where the sensors move with the dynamics. The method is computationally
efficient, robust, and demonstrated to work on a variety of canonical problems
of mathematical physics including Navier-Stokes, the quantum harmonic
oscillator, and the diffusion equation. Moreover, the method is capable of
disambiguating between potentially non-unique dynamical terms by using multiple
time series taken with different initial data. Thus for a traveling wave, the
method can distinguish between a linear wave equation or the Korteweg-deVries
equation, for instance. The method provides a promising new technique for
discovering governing equations and physical laws in parametrized
spatio-temporal systems where first-principles derivations are intractable.
