||Recent advances in data acquisition and computational power herald a new era of breakthrough innovation but also pose several challenges. The unprecedented volumes of measurements from remote sensing and geophysics enhance our understanding of natural systems while processing big environmental data requires significant computational effort. Large-scale simulation tools in high performance computing environments enable one to simulate complex natural processes at multiple spatio-temporal scales, but the number of unknown parameters in such models can be on the order of several millions or billions. With these advances and challenges, new theoretical and computational approaches have been introduced to improve and guarantee the accuracy and computational scalability. In parallel with a popularity of big data analytics and open-source machine learning software packages, data-driven geoscience modeling became an active area of research. In this talk, I will discuss the use of deep learning to construct reduced-order models of PDE equations arising from geoscience applications. Physics-informed or purely data-driven neural networks have shown promising results for the approximation of complex, nonlinear governing equations. By constraining such networks on a nonlinear manifold of low dimensionality and also utilizing automatic differentiation implemented in the open-source software packages, data assimilation and uncertainty quantification can be accelerated significantly compared to traditional approaches. The improvement and performance of these methodologies are illustrated by applying them to several geoscience applications such as subsurface multiphase flow and shallow water equations.