We present a numerical method to simulate the dynamics of continuous variable quantum many-body systems. Our approach is based on custom neural-network many-body quantum states. We focus on dynamics of two-dimensional quantum rotors and simulate large experimentally-relevant systems with using state-of-the-art sampling approaches based on Hamiltonian Monte Carlo. We demonstrate the method can access quantities like the return probability and vorticity oscillations after a quantum quench in two-dimensional systems of up to 64 (8x8) coupled rotors. Our approach can be used to perform previously unexplored non-equilibrium simulations bridging the gap between simulation and experiment.