Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. 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 theoretically show that our proposed method strictly reduces the coefficient estimation error of existing baselines, and is also robust against noise. Extensive experiments show that the proposed method is more robust against data noise, and can reduce the estimation error by a large margin. Moreover, all the PDEs in the experiments are correctly discovered, and for the first time we are able to discover three-dimensional PDEs with highly nonlinear coefficients.

翻译：偏微分方程作为拟合科学数据的数学工具，能够以可解释的机制表征物理定律，广泛应用于物理学、金融学等数学导向学科。基于科学数据发现偏微分方程的数据驱动方法，为建模自然界的复杂现象提供了新思路，但现有方法的有效性通常受限于数据稀疏性和现象复杂性。特别地，从低质量数据中发现具有高非线性系数的偏微分方程仍未得到充分解决。针对这一挑战，我们提出了一种新颖的物理引导学习方法，该方法既能编码初始条件、边界条件等观测知识，又能融合基本物理原理与定律以指导模型优化。我们从理论上证明，所提方法能够严格降低现有基线的系数估计误差，并具有抗噪鲁棒性。大量实验表明，该方法对数据噪声具有更强的鲁棒性，且能大幅降低估计误差。此外，实验中所有偏微分方程均被正确发现，并且我们首次实现了具有高非线性系数的三维偏微分方程的可发现性。