Sampling Boltzmann distributions via normalizing flow approximation of transport maps

Generating independent, equilibrium samples of molecular systems at scale remains a central obstacle in computational statistical mechanics. Boltzmann Generators address this by pairing a generative model with importance sampling to obtain consistent samples from the target distribution. We introduce Normalizing Flow Flow Maps (NF$^2$M), which combines the strengths of recent stochastic flow maps with the tractability of classic normalizing flows to build a Boltzmann Generator. Unlike most methods, which correct the generative model only at the end, NF$^2$M reweighs each denoising transition as generation proceeds, avoiding wasted compute on trajectories that are ultimately discarded. At each denoising step, a conditional normalizing flow proposes clean configurations given the current noisy state (a simpler task than sampling directly from the target) and its exact likelihood enables correcting each proposal toward the true denoising transition of the target Boltzmann distribution. This is in contrast to most existing methods, whose likelihoods are approximate or expensive to evaluate, undermining the statistical reliability of the correction. We establish consistency of the corrected transitions and bound how approximation errors propagate through the sampling chain. We evaluate NF$^2$M on peptide systems, demonstrating improved sampling efficiency and sample quality.

We propose a neural algorithm for sampling from distributions specified by unnormalized Boltzmann densities. Our approach is based on the Jordan--Kinderlehrer--Otto scheme for the Kullback--Leibler divergence in the Wasserstein--Fisher--Rao geometry (WFR JKO scheme). Our contributions are twofold. First, we prove that, for any fixed step size, the exact WFR JKO iterates converge exponentially fast to the target as the number of iterations tends to infinity. Notably, this result requires no structural assumptions on the target, such as log-concavity or a logarithmic Sobolev inequality. Second, we develop a neural implementation of the WFR JKO scheme that parametrizes its transport and reaction components using reweighted normalizing flows. Numerical experiments on challenging multimodal targets demonstrate the promising performance of the proposed method.

Sampling from Boltzmann distributions of molecular systems is an inference problem that has seen significant recent developments fuelled by advances in neural density estimation. We develop a novel sampling algorithm, Neural Transport Nested Sampling (NTNS), which combines the classical strengths of nested sampling with modern neural flow-based methods. NTNS uses a flow matching velocity as the drift in a Metropolis--Hastings corrected Langevin kernel inside a nested sampling outer loop, requiring only evaluations of the target energy function and providing scalable estimation of the full partition function of high-dimensional particle systems. We benchmark NTNS on challenging molecular sampling benchmarks, scaling up to Lennard--Jones clusters of 55 interacting particles, where it reduces both interatomic distance and energy Wasserstein errors to reference MCMC by over an order of magnitude relative to the strongest neural baselines at lower wall-clock cost. To our knowledge, NTNS is also the first neural sampler to return a calibrated, temperature resolved partition function estimate at this scale, recovering the phase structure across temperature from a single run.