  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.
