We describe a new technique for performing Hamiltonian Monte Carlo (HMC) simulations using an alternative leapfrog integrator that is parameterized by weights in a neural network. We look at applying this technique to a two-dimensional Gaussian Mixture Model and a two-dimensional U(1) lattice gauge theory, and compare the results against traditional HMC. Ongoing issues and potential areas for improvement are discussed, particularly within the context of HPC and long-term goals of the lattice QCD community.