We explore using Artificial Neural Networks (ANNs) for model parameter identification tasks in high-dimensional problems with multi-dimensional outputs. The calibration of these models involves learning hundreds of parameters from sparse and noisy observations and solving high-dimensional inverse problems. Firstly, we examine ANNs as surrogates of the forward mapping. When applied to a one-dimensional hemodynamics model of the human arterial system, ANNs are excellent surrogates of time-series outputs, with only a 3% prediction error. We also discuss the trade-off between the architectures' complexity and the learning database's size. We utilize the pre-trained forward surrogates to solve the ill-posed inverse problem formulated as a likelihood maximum estimation (MLE). We also explore gradient-based and gradient-free non-linear optimization methods. Finally, we investigate ANNs as direct surrogates of the inverse mapping from observations to model parameters, with comparable parameter identification accuracy to MLE. We also emphasize the importance of restricting informative observations to form the inputs of the inverse ANN surrogate.