Quantum control is a ubiquitous research field that has enabled physicists to delve into the dynamics and features of quantum systems. In addition to steering the system, quantum control has delivered powerful applications for various atomic, optical, mechanical, and solid-state systems. In recent years, traditional control techniques based on optimization processes have been translated into efficient artificial intelligence algorithms. Here, we introduce a computational method for optimal quantum control problems via physics-informed neural networks (PINNs). We apply our methodology to open quantum systems by efficiently solving the state-to-state transfer problem with high probabilities, short-time evolution, and minimizing the power of the control. Furthermore, we illustrate the flexibility of PINNs to solve the same problem under changes in parameters and initial conditions, showing advantages in comparison with standard control techniques.