IDENT Review: Recent Advances in Identification of Differential Equations from Noisy Data

TL;DR

This paper reviews recent methods for identifying differential equations from noisy data, emphasizing linear system formulation, denoising, sparsity, and model selection to improve accuracy in real-world applications.

Contribution

It provides a comprehensive overview and analysis of recent advances in differential equation identification, highlighting key themes and strategies for improved coefficient recovery.

Findings

Proper denoising enhances identification accuracy

Sparsity and model selection are crucial for correct coefficient support

Various approaches improve coefficient recovery from noisy data

Abstract

Differential equations and numerical methods are extensively used to model various real-world phenomena in science and engineering. With modern developments, we aim to find the underlying differential equation from a single observation of time-dependent data. If we assume that the differential equation is a linear combination of various linear and nonlinear differential terms, then the identification problem can be formulated as solving a linear system. The goal then reduces to finding the optimal coefficient vector that best represents the time derivative of the given data. We review some recent works on the identification of differential equations. We find some common themes for the improved accuracy: (i) The formulation of linear system with proper denoising is important, (ii) how to utilize sparsity and model selection to find the correct coefficient support needs careful attention, and (iii) there are ways to improve the coefficient recovery. We present an overview and analysis of these approaches about some recent developments on the topic.