You can. I don't know ML or math much, but I think this is how ML works, or at least some of the ML problem? Give some matching input and output date sample, you find the "best" mapping.

For typical optimization problems, people want to find the best input X that results in minimum energy, but your case is a little special, which you want to find the best mapping of X. So your energy term is the sum of loss for all data samples, and the X is the parameters inside your NN. NN is differentiable and L is differentiable, so it is totally doable. I have also done this, e.g., to find the best trajectory that minimize the energy consuption/impact force, or to position objects so their silouette matches a given one, though I think it is more a pure non-linear optimization problem where you use an NN rather than a ML problem.