Structural health monitoring (SHM) based on global navigation satellite systems (GNSS) is an interesting solution to provide absolute positions at different locations of a structure in a global reference frame. In particular, low-cost GNSS stations for large-scale bridge monitoring have gained increasing attention these last years because recent experiments showed the ability to achieve a subcentimeter accuracy for continuous monitoring with adequate combinations of antennas and receivers. Technical solutions now allow displacement monitoring of long bridges with a cost-effective deployment of GNSS sensing networks. In particular, the redundancy of observations within the GNSS network with various levels of correlations between the GNSS time series makes such monitoring solution a good candidate for anomaly detection based on machine learning models, using several predictive models for each sensor (based on environmental conditions, or other sensors as input data). This strategy is investigated in this paper based on GNSS time series, and an anomaly indicator is proposed to detect and locate anomalous structural behavior. The proposed concepts are applied to a cable-stayed bridge for illustration, and the comparison between multiple tools highlights recurrent neural networks (RNN) as an effective regression tool. Coupling this tool with the proposed anomaly detection strategy enables one to identify and localize both real and simulated anomalies in the considered data set.