We introduce a method to simulate parametrized quantum circuits, an architecture behind many practical algorithms on near-term hardware, focusing on the Quantum Approximate Optimization Algorithm (QAOA). A neural-network parametrization of the many-qubit wave function is used, reaching 54 qubits at 4 QAOA layers, approximately implementing 324 RZZ gates and 216 RX gates without requiring large-scale computational resources. our approach can be used to provide accurate QAOA simulations at previously unexplored parameter values and to benchmark the next generation of experiments in the Noisy Intermediate-Scale Quantum (NISQ) era.