Since you mention VAEs: you might be interested in Active Inference, which is built on minimizing variational free energy. Same E−TS form as the thermodynamic kind (Helmholtz free energy), so it does fit your thermo/info/NN question. And the negative free energy is equivalent to the evidence lower bound used in generative models like VAEs. Lots of really fascinating connections between all of these topics :)
There's a free 2022 textbook: https://direct.mit.edu/books/oa-monograph/5299/Active-Infere...
And a brand new one (though it's a bit pricey): https://mitpress.mit.edu/9780262050951/fundamentals-of-activ...