Keywords: Partial Differential Equations, Physics-Guided Learning, Spatial Kernel Estimation.
TL;DR: We for the first time reports the discovery of PDEs with highly nonlinear coefficients from sparse and noisy data
Abstract: Partial differential equations (PDEs) fitting scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects. The data-driven discovery of PDEs from scientific data thrives as a new attempt to model complex phenomena in nature, but the effectiveness of current practice is typically limited by the scarcity of data and the complexity of phenomena. Especially, the discovery of PDEs with highly nonlinear coefficients from low-quality data remains largely under-addressed. To deal with this challenge, we propose a novel physics-guided learning method, which can not only encode observation knowledge such as initial and boundary conditions but also incorporate the basic physical principles and laws to guide the model optimization. We empirically demonstrate that the proposed method is more robust against data noise and sparsity, and can reduce the estimation error by a large margin; moreover, for the first time we are able to discover PDEs with highly nonlinear coefficients.