Approximating Deep Neural Networks (DNNs) with Binarized Neural Networks (BNNs)
I expect you'll run into two challenges in this line of research: (1) there are no known ways to construct a BNN that exactly matches an ordinary neural network; (2) verifying a BNN with a SAT solver won't scale to the size models needed for most interesting tasks.

Since you asked specifically about (1), no, I don't know of any way to do that.  Probably you won't find exact equivalence.  You might be able to train a BNN that has a very high agreement with the ordinary neural network on the training set; I don't know whether it will have the same adversarial examples or not.