Real-time prediction of train arrivals is important for proactive traffic control and information provision in passenger rails. Despite many studies in predicting arrival times or delays at stations, they are essentially the next-step time series prediction problem which may limit their applications in practice. For example, passengers on the trains or waiting on platforms may have different destinations and need the predicted train arrival times for any downstream stations rather than only the next station. The paper aims to formulate a real-time train arrival times prediction problem at multiple stations and arbitrary times. We develop multi-output machine learning models and systematically evaluate their performance using train operation data in Sweden. The direct multi-output regression models with different regression functions are tested, including LightGBM, linear regression, random forest regression, and gradient boosting regression models. The hyperparameters are optimized using random grid search and five-fold cross-validation methods. The results show that the Direct Multi-Output LightGBM significantly outperformed other models in terms of accuracy. The predictions at downstream stations improve as the train moves along given more real-time information is observed.