  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.
