Physical layer message authentication in underwater acoustic networks (UWANs) leverages the characteristics of the underwater acoustic channel (UWAC) as a fingerprint of the transmitting device. However, as the device moves its UWAC changes, and the authentication mechanism must track such variations. In this paper, we propose a context-based authentication mechanism operating in two steps: first, we estimate the position of the underwater device, then we predict its future position based on the previously estimated ones. To check the authenticity of the transmission, we compare the estimated and the predicted position. The location is estimated using a convolutional neural network taking as input the sample covariance matrix of the estimated UWACs. The prediction uses either a Kalman filter or a recurrent neural network (RNN). The authentication check is performed on the squared error between the predicted and estimated positions. The solution based on the Kalman filter outperforms that built on the RNN when the device moves according to a correlated Gauss-Markov mobility model, which reproduces a typical underwater motion.