Fourier Features for Identifying Differential Equations (FourierIdent)

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.