  Machine learning of partial differential equations from data is a potential
breakthrough to solve the lack of physical equations in complex dynamic
systems, but because numerical differentiation is ill-posed to noise data,
noise has become the biggest obstacle in the application of partial
differential equation identification method. To overcome this problem, we
propose Frequency Domain Identification method based on Fourier transforms,
which effectively eliminates the influence of noise by using the low frequency
component of frequency domain data to identify partial differential equations
in frequency domain. We also propose a new sparse identification criterion,
which can accurately identify the terms in the equation from low
signal-to-noise ratio data. Through identifying a variety of canonical
equations spanning a number of scientific domains, the proposed method is
proved to have high accuracy and robustness for equation structure and
parameters identification for low signal-to-noise ratio data. The method
provides a promising technique to discover potential partial differential
equations from noisy experimental data.
