Machine learning of partial differential equations from noise data

Machine learning of partial differential equations (PDEs) from data is a potential breakthrough for addressing the lack of physical equations in complex dynamic systems. Recently, sparse regression has emerged as an attractive approach. However, noise presents the biggest challenge in sparse regression for identifying equations, as it relies on local derivative evaluations of noisy data. This study proposes a simple and general approach that significantly improves noise robustness by projecting the evaluated time derivative and partial differential term into a subspace with less noise. This method enables accurate reconstruction of PDEs involving high-order derivatives, even from data with considerable noise. Additionally, we discuss and compare the effects of the proposed method based on Fourier subspace and POD (proper orthogonal decomposition) subspace. Generally, the latter yields better results since it preserves the maximum amount of information.