There have been growing interests in leveraging experimental measurements to
discover the underlying partial differential equations (PDEs) that govern
complex physical phenomena. Although past research attempts have achieved great
success in data-driven PDE discovery, the robustness of the existing methods
cannot be guaranteed when dealing with low-quality measurement data. To
overcome this challenge, we propose a novel physics-encoded discrete learning
framework for discovering spatiotemporal PDEs from scarce and noisy data. The
general idea is to (1) firstly introduce a novel deep convolutional-recurrent
network, which can encode prior physics knowledge (e.g., known PDE terms,
assumed PDE structure, initial/boundary conditions, etc.) while remaining
flexible on representation capability, to accurately reconstruct high-fidelity
data, and (2) perform sparse regression with the reconstructed data to identify
the explicit form of the governing PDEs. We validate our method on three
nonlinear PDE systems. The effectiveness and superiority of the proposed method
over baseline models are demonstrated.

本文提出一种新的基于物理编码离散学习框架，用于从稀缺且有噪声的数据中发现时空偏微分方程，通过引入基于深度卷积 - 循环网络进行先前的物理知识编码，并利用重构数据的稀疏回归来识别控制 PDEs 的显式形式。作者在三个非线性 PDE 系统上进行了验证，展示了该方法的有效性和优越性。