We present a strategy for learning Deep-Neural-Network (DNN)-based Acoustic-to-Articulatory Mapping (AAM) functions where the contribution of an articulatory feature (AF) to the global reconstruction error is weighted by its relevance. We first empirically show that when an articulator is more crucial for the production of a given phone it is less variable, confirming previous findings. We then compute the relevance of an articulatory feature as a function of its frame-wise variance dependent on the acoustic evidence which is estimated through a Mixture Density Network (MDN). Finally we combine acoustic and recovered articulatory features in a hybrid DNN-HMM phone recognizer. Tested on the MOCHA-TIMIT corpus, articulatory features reconstructed by a standardly trained DNN lead to a 8.4% relative phone error reduction (w.r.t. a recognizer that only uses MFCCs), whereas when the articulatory features are reconstructed taking into account their relevance the relative phone error reduction increased to 10.9%