How to retrain the neural network when new data comes in?
Yes, there is some methods to do this based on the library. The proper keyword for this functionality is Checkpoint. Hence, you can check functions to save or restore the checkpoints from hard disk to continue learning or prediction in a new session. For example you can see the following in Tensorflow:


  The tf.train.Saver class provides methods for saving and restoring models. The tf.train.Saver constructor adds save and restore ops to the graph for all, or a specified list, of the variables in the graph. The Saver object provides methods to run these ops, specifying paths for the checkpoint files to write to or read from.


For more details about this in Tensorflow, you can follow this link.

To complete the case you can follow this subject in another library such as CNTK in this link.