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

TL;DR

This paper introduces robust numerical methods for identifying partial differential equations from noisy data, utilizing denoising schemes and greedy algorithms to accurately recover underlying dynamics despite high noise levels.

Contribution

The paper develops a Successively Denoised Differentiation scheme and two PDE identification algorithms, ST and SC, which are robust against noise and improve accuracy in PDE discovery.

Findings

Methods are robust to high noise levels.

Algorithms accurately identify PDEs from noisy data.

Numerical experiments validate efficiency and robustness.

Abstract

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.