New Neural HMC Method Improves Boltzmann Sampling Efficiency

This paper introduces Neural Non-Equilibrium Hamiltonian Monte Carlo (NHMC), a novel approach to sampling from complex probability distributions, specifically unnormalized Boltzmann densities. The core idea involves training a model to learn stochastic Hamiltonian-style paths from a simpler base distribution towards the target distribution. Once trained, these learned paths are used to generate configurations, which are then statistically corrected using a measure of non-equilibrium work. The statistical correction relies on comparing the probability of a forward proposal path with a reverse reference path. This "dimensionless generalized work" is minimized during training to improve path-space alignment and is later used for importance sampling, estimating normalizing constants, or as an acceptance ratio in Metropolis-Hastings. The framework also defines a Metropolis kernel for direct configuration updates. Empirical tests on double-well and lattice $\phi^4$ targets show that NHMC provides corrected estimates when there's sufficient path overlap. However, issues like weight degeneracy and low acceptance can arise if the overlap is poor, indicating proposal failure. A feasibility study in molecular dynamics also suggests its potential.