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. assume that governing equation PDE is linear combination few and nonlinear differential terms in prescribed dictionary. Noisy data make such identification particularly challenging. Our objective develop against high level noise approximate noise-free dynamics well. first introduce Successively Denoised Differentiation (SDD) scheme stabilize amplified numerical differentiation. SDD effectively denoises corresponding derivatives. Second, we present two algorithms for identification: Subspace pursuit Time evolution (ST) error Cross-validation (SC). general strategy find candidate using Pursuit (SP) greedy algorithm, then choose best one via time or cross-validation. ST uses multishooting selects which yields least error. SC evaluates cross-validation least-squares fitting picks gives smallest validation various experiments validate our methods. Both are efficient noise.