You're asking for online learning. This is certainly possible with modern techniques - define an unsupervised loss and update the weights with gradients as new

You're asking for online learning. This is certainly possible with modern techniques - define an unsupervised loss and update the weights with gradients as new data is encountered. Federated learning also works along these lines. 'Raw' online learning is unpopular because you have no garauntee that the network won't do something funky in the field. That said, I think there are production systems in the world which learn on a day by day basis. Eg, take all of the logs from the last day and use them to update the production model for tomorrow. Then there's enough data that you don't risk a bad step. Think of it as learning while dreaming...