Machine learning of partial differential equations from noise data

Machine learning of partial differential equations from data is a potential breakthrough to solve the lack of physical equations in complex dynamic systems, and sparse regression is an attractive approach recently emerged. Noise is the biggest challenge for sparse regression to identify equations because sparse regression relies on local derivative evaluation of noisy data. This study proposes a simple and general approach which greatly improves the noise robustness by projecting the evaluated time derivative and partial differential term into a subspace with less noise. This approach allows accurate reconstruction of PDEs involving high-order derivatives from data with a considerable amount of noise. In addition, we discuss and compare the effects of the proposed method based on Fourier subspace and POD subspace, and the latter usually have better results since it preserves the maximum amount of information.