  We investigate the benefits and challenges of utilizing the frequency
information in differential equation identification. Solving differential
equations and Fourier analysis are closely related, yet there is limited work
in exploring this connection in the identification of differential equations.
Given a single realization of the differential equation perturbed by noise, we
aim to identify the underlying differential equation governed by a linear
combination of linear and nonlinear differential and polynomial terms in the
frequency domain. This is challenging due to large magnitudes and sensitivity
to noise. We introduce a Fourier feature denoising, and define the meaningful
data region and the core regions of features to reduce the effect of noise in
the frequency domain. We use Subspace Pursuit on the core region of the time
derivative feature, and introduce a group trimming step to refine the support.
We further introduce a new energy based on the core regions of features for
coefficient identification. Utilizing the core regions of features serves two
critical purposes: eliminating the low-response regions dominated by noise, and
enhancing the accuracy in coefficient identification. The proposed method is
tested on various differential equations with linear, nonlinear, and high-order
derivative feature terms. Our results demonstrate the advantages of the
proposed method, particularly on complex and highly corrupted datasets.
