We propose a neural-network variational quantum algorithm to simulate the time evolution of quantum many-body systems. Based on a modified restricted Boltzmann machine (RBM) wave function ansatz, the proposed algorithm can be efficiently implemented in near-term quantum computers with low measurement cost. Using a qubit recycling strategy, only one ancilla qubit is required to represent all the hidden spins in an RBM architecture. The variational algorithm is extended to open quantum systems by employing a stochastic Schrödinger equation approach. Numerical simulations of spin-lattice models demonstrate that our algorithm is capable of capturing the dynamics of closed and open quantum many-body systems with high accuracy without suffering from the vanishing gradient (or “barren plateau”) issue for the considered system sizes.