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Matter evolved under the influence of gravity from minuscule density fluctuations. Nonperturbative structure formed hierarchically over all scales and developed non-Gaussian features in the Universe, known as the cosmic web. To fully understand the structure formation of the Universe is one of the holy grails of modern astrophysics. Astrophysicists survey large volumes of the Universe and use a large ensemble of computer simulations to compare with the observed data to extract the full information of our own Universe. However, to evolve billions of particles over billions of years, even with the simplest physics, is a daunting task. We build a deep neural network, the Deep Density Displacement Model (D3M), which learns from a set of prerun numerical simulations, to predict the nonlinear large-scale structure of the Universe with the Zel'dovich Approximation (ZA), an analytical approximation based on perturbation theory, as the input. Our extensive analysis demonstrates that D3M outperforms the second-order perturbation theory (2LPT), the commonly used fast-approximate simulation method, in predicting cosmic structure in the nonlinear regime. We also show that D3M is able to accurately extrapolate far beyond its training data and predict structure formation for significantly different cosmological parameters. Our study proves that deep learning is a practical and accurate alternative to approximate 3D simulations of the gravitational structure formation of the Universe. [ABSTRACT FROM AUTHOR]