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Relativistic hydrodynamics is a powerful tool to simulate the evolution of the quark-gluon plasma in relativistic heavy-ion collisions. Using 10 000 initial and final profiles generated from (2+1)-dimensional relativistic hydrodynamics vish2+1 with Monte Carlo Glauber (MC-Glauber) initial conditions, we train a deep neural network based on the stacked U-net, and use it to predict the final profiles associated with various initial conditions, including MC-Glauber, MC Kharzeev-Levin-Nardi (MC-KLN), a multiphase transport (AMPT) model, and the reduced thickness event-by-event nuclear topology (TRENTo) model. A comparison with the vish2+1 results shows that the network predictions can nicely capture the magnitude and inhomogeneous structures of the final profiles, and creditably describe the related eccentricity distributions P(ɛ_{n}) (n=2, 3, 4). These results indicate that a deep learning technique can capture the main features of the nonlinear evolution of hydrodynamics, showing its potential to largely accelerate the event-by-event simulations of relativistic hydrodynamics.