Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning

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.