Thanks! You may find DeepProbLog by Manhaeve et al. interesting, which brings together logic programming, probabilistic programming and gradient descent/neural networks. Also, more generally, I believe in the field of program synthesis there is some research on deriving programs with gradient descent. However, as also pointed out in the comment below, gradient descent may not always be the best approach to such problems (e.g., https://arxiv.org/abs/1608.04428).