Inductive Logic Progamming (ILP) is a class of machine learning algorithms that learn logic programs from examples and background knowledge (where background knowledge is a logic theory, i.e. another logic program, as ar the examples).

SAT is the boolean satisfiability problem, the problem of finding variable assignments to the variables of a boolean formula that make the formula true.

So the two are not similar and one is not an instance of the other.

The δILP paper describes a differentiable ILP system that learns logic programs with a differentiable logic representation.

Finally, the paper above (SATNet) describes a differentiable satisfiabiilty solver that can be incorporated in a neural network architecture to enable the nerual net to solve satisfiability problems and perform reasoning.