Sampling from an unnormalized probability distribution is a fundamental problem in machine learning with applications including Bayesian learning, latent factor inference, and energy-based model training. After decades of research, variations of MCMC remain the default approach to sampling despite slow convergence. 

We propose a fundamentally different approach to this problem via a new Hamiltonian dynamics with a non-Newtonian momentum. In contrast to MCMC approaches like Hamiltonian Monte Carlo, no stochastic step is required. Instead, the proposed deterministic dynamics in an extended state space exactly sample the target distribution, specified by an energy function, under an assumption of ergodicity. 

Energy Sampling Hamiltonian (ESH) dynamics converge faster than their MCMC competitors enabling faster, more stable training of neural network energy models. 

For the non-Newtonian momentum used in Energy Sampling Hamiltonian (ESH) dynamics, when momentum is large, the changes in position appear in slow motion, and when momentum is small it goes into fast-forward. The outcome is that the amount of time spent in each region is exactly proportional to the density exp(-E(x))/Z.