Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling

Learned proposals can help explore multimodal, unnormalized Boltzmann targets, but their endpoint densities may be unavailable for statistical correction. We introduce Neural Non-Equilibrium Hamiltonian Monte Carlo (NHMC), which combines learned forward and reverse conditional momentum distributions with reversible, volume-preserving dynamics. Recorded path densities and target energies yield a computable nonequilibrium work for training, importance weighting, and estimation of normalizing constants and free-energy differences, without evaluating the endpoint proposal density. We also introduce a configuration-space round-trip Metropolis–Hastings update: a reverse path takes the current configuration to an intermediate configuration, and a fresh forward path proposes a new configuration from there. We prove that this update is reversible with respect to the target distribution and that, conditional on the intermediate configuration, the update of the auxiliary path record is an independence Metropolis–Hastings step. We derive lower bounds on accepted inter-region probability flow at stationarity, quantifying the effects of conditional path mismatch and dependence between endpoint and intermediate configurations. The path-space KL divergence controls the mismatch term, but high acceptance alone does not ensure transitions between regions. Across analytically tractable many-well targets, lattice ${\varphi }^4$, and two-dimensional compact $U(1)$, $SU(2)$, and $SU(3)$ gauge theories, corrected estimates agree with analytic or independent references. Controlled ${\varphi }^4$ experiments show that sector-probability estimates can remain consistent with reference values while autocorrelation times vary substantially. In compact $U(1)$, NHMC with standard path-space independence Metropolis–Hastings changes topological sectors in the tested regimes where the HMC reference chains remain frozen.