The transfer of facial expressions from people to 3D face models is a classic computer graphics problem. In this paper, we present a novel, learning-based approach to transferring facial expressions and head movements from images and videos to a biomechanical model of the face-head-neck musculoskeletal complex. Specifically, leveraging the Facial Action Coding System (FACS) as an intermediate representation of the expression space, we train a deep neural network to take in FACS Action Units (AUs) and output suitable facial muscle and jaw activation signals for the biomechanical model. Through biomechanical simulation, the activations deform the face, thereby transferring the expression to the model. Our approach has advantages over previous approaches. First, the facial expressions are anatomically consistent as our biomechanical model emulates the relevant anatomy of the face, head, and neck. Second, by training the neural network using data generated from the biomechanical model itself, we eliminate the manual effort of data collection for expression transfer. The success of our approach is demonstrated through experiments involving the transfer of head poses and facial expressions from a range of facial images and videos onto our biomechanical face-head-neck model.