https://arxiv.org/abs/1407.2650
https://arxiv.org/abs/1701.01285
http://therisingsea.org/notes/MScThesisJamesClift.pdf
As neel_k notes, a good way to understand this picture is in terms of differential linear logic (a refinement of simply-typed differential lambda calculus). I did not provide references in the talk as unfortunately I did not understand the subject at the time, but the introduction to the second paper above hopefully serves to remedy that omission. Suffice to day that Ehrhard and Regnier discovered something quite profound in differential lambda calculus, the implications of which are still being explored; this discovery should have significant bearing on what people refer to as differentiable programming, I think.
The third link above is the master’s thesis of my talented student James Clift. We have subsequently extended that work and are finishing a paper which explains the computational content of derivatives of Turing machines, according to differential linear logic. This might be of interest to people here. I will post a link when we put it online.