Update: I did some searching and found some interesting demos of a hybrid of neural nets and more conventional DSP called LPCNet:

https://people.xiph.org/~jm/demo/lpcnet_codec/

Sure enough, it was discussed on HN when it came out last year. I think I missed it then.

For those who didn't catch this from the URL, this is by Jean-Marc Valin, of Speex and Opus fame.