We train a deep convolutional neural network to predict hydrodynamic results for flow coefficients, average transverse momenta, and charged particle multiplicities in ultrarelativistic heavy-ion collisions from the initial energy density profiles. We show that the neural network can be trained accurately enough so that it can reliably predict the hydrodynamic results for the flow coefficients and, remarkably, also their correlations like normalized symmetric cumulants, mixed harmonic cumulants, and flow-transverse-momentum correlations. At the same time the required computational time decreases by several orders of magnitude. To demonstrate the advantage of the significantly reduced computation time, we generate 107 initial energy density profiles from which we predict the flow observables using the neural network, which is trained using 5×103, and validated using 9×104 events per collision energy. We then show that increasing the number of collision events from 9×104 to 107 can have significant effects on certain statistics-expensive flow correlations, which should be taken into account when using these correlators as constraints in the determination of the quantum chromodynamics matter properties.