A Fourier-based technique for efficient discovery of linear PDEs

Machine learning techniques have been used to identify the form and coefficients of partial differential equations (PDEs). However, these methods require large amounts of data in the presence of noise. We present a new method based in Fourier analysis for determining the coefficients of a PDE when its form is already known. In contrast to a traditional PDE discovery method, our approach presents two major advantages. Firstly, it is robust against noise as it relies on the large Fourier modes that are insensitive to noise. Secondly, it is data-efficient since those lower modes can be reconstructed using a few data points. We demonstrate these advantages by testing our method on the heat and the wave equation. Our goal is to fully integrate this method into a machine learning framework to be able to tackle the sparse regression problem for more general PDEs while requiring only small amounts of data.