No, it doesn't. Again, the law is very clear about this. The US Copyright Office guidelines are extremely blunt: machines are not authors. Period. It doesn't matter if you twiddled with the hyperparameters or how much you paid for the electricity. (Similarly, you can spend billions of dollars measuring a physical constant, but as a fact, it is not copyrightable.)
Couldn't this allow the person hosting the model to claim copyright?
I am also thinking about hyper-parameter configuration and selection of training data as creative human input in this process.