  We propose robust methods to identify underlying Partial Differential
Equation (PDE) from a given set of noisy time dependent data. We assume that
the governing equation is a linear combination of a few linear and nonlinear
differential terms in a prescribed dictionary. Noisy data make such
identification particularly challenging. Our objective is to develop methods
which are robust against a high level of noise, and to approximate the
underlying noise-free dynamics well. We first introduce a Successively Denoised
Differentiation (SDD) scheme to stabilize the amplified noise in numerical
differentiation. SDD effectively denoises the given data and the corresponding
derivatives. Secondly, we present two algorithms for PDE identification:
Subspace pursuit Time evolution error (ST) and Subspace pursuit
Cross-validation (SC). Our general strategy is to first find a candidate set
using the Subspace Pursuit (SP) greedy algorithm, then choose the best one via
time evolution or cross validation. ST uses multi-shooting numerical time
evolution and selects the PDE which yields the least evolution error. SC
evaluates the cross-validation error in the least squares fitting and picks the
PDE that gives the smallest validation error. We present a unified notion of
PDE identification error to compare the objectives of related approaches. We
present various numerical experiments to validate our methods. Both methods are
efficient and robust to noise.
