The high computational requirements of nonlinear model predictive control (NMPC) are a long-standing issue and, among other methods, learning the control policy with machine learning (ML) methods has been proposed in order to improve computational tractability. However, these methods typically do not explicitly consider constraint satisfaction. We propose two methods based on learning the optimal control policy by an artificial neural network (ANN) and using this for initialization to accelerate computations while meeting constraints and achieving good objective function value. In the first, the ANN prediction serves as the initial guess for the solution of the optimal control problem (OCP) solved in NMPC. In the second, the ANN prediction is improved by solving a single quadratic program (QP). We compare the performance of the two proposed strategies against two benchmarks representing the extreme cases of (i) solving the NMPC problem to convergence using the shift-initialization strategy and (ii) implementing the controls predicted by the ANN prediction without further correction to reduce the computational delay. We find that the proposed ANN initialization strategy mostly results in the same control policy as the shift-initialization strategy. The computational times are on average similar to 45% longer but the maximum time is similar to 42% smaller and the distribution is tighter, thus more predictable. The proposed QP-based method yields a good compromise between finding the optimal control policy and solution time. Closed-loop infeasibilities are negligible and the objective function is typically greatly improved as compared to benchmark (ii). The computational time required for the necessary second-order sensitivity integration is typically an order of magnitude smaller than for solving the NMPC problem to convergence. (C) 2020 Elsevier Ltd. All rights reserved.