Moxian Qian

Sampling from an unnormalized Boltzmann density requires proposals that move probability mass globally while retaining enough path-probability information for statistical correction. We introduce Neural Non-Equilibrium Hamiltonian Monte Carlo (NHMC), a train-then-correct learned Hamiltonian sampler. Starting from a tractable base distribution, NHMC learns stochastic Hamiltonian-style paths toward the target. Once training is complete, the learned proposal parameters are fixed; the proposal then generates complete paths and endpoint configurations, which are statistically corrected using the recorded non-equilibrium work. This dimensionless generalized work is determined by the probability ratio between the forward proposal path and a reverse reference path. During training, minimizing its mean reduces a path-space KL divergence and controls an upper bound on endpoint mismatch. During evaluation, the same quantity defines weights for self-normalized importance sampling on paths (path-SNIS), estimates normalizing constants or free-energy differences, and gives the acceptance ratio for path-space independent Metropolis-Hastings (path-IMH). The same forward-reverse laws also define a shared-bridge round-trip Metropolis kernel that acts directly on configurations and preserves the Boltzmann target. On double-well and finite-volume lattice phi^4 targets, the NHMC construction gives corrected estimates when path overlap is sufficient; when overlap is poor, weight degeneracy, low acceptance, and long autocorrelation expose proposal failure. We additionally report a molecular internal-coordinate feasibility study using an MD prior and learned-force path proposal.

In lattice field theory, target distributions are known only up to normalization, while the partition function is generally intractable. Markov chain Monte Carlo simulations often become inefficient near phase transitions or the continuum limit due to critical slowing down. We propose a novel sampler based on nonequilibrium thermodynamics, called Stochastic Path Sampler (SPS), which can generate configurations for the unnormalized target distribution without a need for training data. The central idea is to establish a trajectory-level balance for learnable forward and backward stochastic dynamics between two equilibrium states—the prior and target distributions—by minimizing the path-space variational free energy, equivalently an entropy-production upper bound. The learned forward process provides independent proposals, subsequently corrected by an extended-space Independence Metropolis–Hastings step. In two-dimensional phi^4 theory, our neural sampler matches HMC’s sampling quality with a much shorter autocorrelation time in the pseudocritical region. SPS offers a stochastic-quantization-inspired route to data-free proposal construction for lattice field theory.

We analyze a gravitation-induced contribution to the secular shrinkage of Mercury’s orbit, in addition to the standard relativistic perihelion precession.