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In this paper, the generalized regression neural network is used to predict the GNSS position time series. Using the IGS 24-hour final solution data for Bad Hamburg permanent GNSS station in Germany, it is shown that the larger the training of the network, the higher the accuracy is, regardless of the time span of the time series. In order to analyze the performance of the neural network in various conditions, 14 permanent stations are used in different countries, namely, Spain, France, Romania, Poland, Russian Federation, United Kingdom, Czech Republic, Sweden, Ukraine, Italy, Finland, Slovak Republic, Cyprus, and Greece. The performance analysis is divided into two parts, continuous data-without gaps-and discontinuous ones-having intervals of gaps with no data available. Three measure of error are presented, namely, symmetric mean absolute percentage error, standard deviation, and mean of absolute errors. It is shown that for discontinuous data the position can be predicted with an accuracy of up to 6 centimeters, while the continuous data positions present a higher prediction accuracy, as high as 3 centimeters. In order to compare the results of this machine learning algorithm with the traditional statistical approaches, the Theta method is used, which is well-established for high-accuracy time series prediction. The comparison shows that the generalized regression neural network machine learning algorithm presents better accuracy than the Theta method, possibly up to 250 times. In addition, it is approximately 4.6 times faster.