An early example of artificial self-replication is Von Neumann’s self-reproducing CA which is a cellular automaton.
Self-replication in neural networks can be done with neural network quines (Chang and Lipson 2018).
Chang, Oscar, and Hod Lipson. May 24, 2018. "Neural Network Quine". arXiv:1803.05859 [Cs]. http://arxiv.org/abs/1803.05859.