A new idea is presented for efficiently training a low-dimensional, neural network (NN)-based surrogate of a high-fidelity, high-dimensional computational model (HDM). It consists in training the NN by adaptively sampling the parameter space of interest using the acquisition function of a Gaussian Process and exercising the HDM at the sampled parameter points. This approach, which can be described as an active learning approach, is explained and illustrated with numerical experiments associated with the prediction of the lift-over-drag ratio of a cambered NACA airfoil in a large, five-dimensional parameter space of flight conditions and shape design variables. The obtained numerical results demonstrate the superior efficiency as well as accuracy delivered by the proposed training over standard alternatives based on uniform and random parameter samplings. The surrogate models constructed and trained using the proposed approach are suitable for time-critical applications such as design optimization and uncertainty quantification.