Deep-Learning Discovers Macroscopic Governing Equations for Viscous Gravity Currents from Microscopic Simulation Data

Although deep-learning has been successfully applied in a variety of science and engineering problems owing to its strong high-dimensional nonlinear mapping capability, it is of limited use in scientific knowledge discovery. In this work, we propose a deep-learning based framework to discover the macroscopic governing equation of viscous gravity current based on high-resolution microscopic simulation data without the need for prior knowledge of underlying terms. For two typical scenarios with different viscosity ratios, the deep-learning based equations exactly capture the same dominated terms as the theoretically derived equations for describing long-term asymptotic behaviors, which validates the proposed framework. Unknown macroscopic equations are then obtained for describing short-term behaviors, and additional deep-learned compensation terms are eventually discovered. Comparison of posterior tests shows that the deep-learning based PDEs actually perform better than the theoretically derived PDEs in predicting evolving viscous gravity currents for both long-term and short-term regimes. Moreover, the proposed framework is proven to be very robust against non-biased data noise for training, which is up to 20%. Consequently, the presented deep-learning framework shows considerable potential for discovering unrevealed intrinsic laws in scientific semantic space from raw experimental or simulation results in data space.