Abstract: We present a simulation method for the dynamics of continuous-variable quantum many-body systems based on neural-network quantum states. The focus is put on dynamics of experimentally relevant two-dimensional quantum rotors. We simulate previously unreachable system sizes and simulation times using a neural-network trial wavefunction in a continuous basis and using modern sampling approaches based on Hamiltonian Monte Carlo. The method is demonstrated to be able to access quantities like the return probability and vorticity oscillations after a quantum quench in two-dimensional systems of up to 64 (8 $\times$ 8) coupled rotors. Our approach can be used for accurate non-equilibrium simulations of continuous systems at previously unexplored system sizes and evolution times, bridging the gap between simulation and experiment.