Robust Identification of Differential Equations by Numerical Techniques from a Single Set of Noisy Observation

We propose robust methods to identify the underlying Partial Differential Equation (PDE) from a given single set of noisy time-dependent data. We assume that the governing equation of the PDE 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 robust methods against a high level of noise and 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. Second, we present two algorithms for PDE identification: Subspace pursuit Time evolution (ST) error 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 a multishooting 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 various numerical experiments to validate our methods. Both methods are efficient and robust to noise.