SIMANF: Sample Free Learning of Unnormalized Distributions via Simulated Annealing in Normalizing Flows

Sampling from unnormalized densities is analogous to the generative modeling problem, but the target distribution is defined by a known energy function instead of data samples. Because evaluating the energy function is often costly, a primary challenge is to learn an efficient sampler. We introduce Flow Sampling, a framework built on diffusion models and flow matching for the data-free setting. Our training objective is conditioned on a noise sample and regresses onto a denoising diffusion drift constructed from the energy function. In contrast, diffusion models' objective is conditioned on a data sample and regresses onto a noising diffusion drift. We utilize the interpolant process to minimize the number of energy function evaluations during training, resulting in an efficient and scalable method for sampling unnormalized densities. Furthermore, our formulation naturally extends to Riemannian manifolds, enabling diffusion-based sampling in geometries beyond Euclidean space. We derive a closed-form formula for the conditional drift on constant curvature manifolds, including hyperspheres and hyperbolic spaces. We evaluate Flow Sampling on synthetic energy benchmarks, small peptides, large-scale amortized molecular conformer generation, and distributions supported on the sphere, demonstrating strong empirical performance.

Efficient sampling from Boltzmann distributions is central to modelling complex physical systems. Markov Chain Monte Carlo (MCMC) methods suffer from critical slowing down, high autocorrelation, and poor mode-mixing, limiting their scalability. Recent advances, like Boltzmann Generators, offer a promising alternative but remain constrained by costly MCMC-based training, inefficient sampling, and poor ergodicity. We introduce an algorithm for learning Boltzmann distributions that does not require any true samples for training. Our approach draws inspiration from flow matching but departs fundamentally from sample-trajectory matching to distribution-trajectory matching. The algorithm iteratively reshapes the target distribution, using model generated samples to guide learning and ensure comprehensive mode coverage. We validate our method on standard benchmarks, including a 2D Gaussian mixture, Many-Well distributions, and high-dimensional scalar $φ^4$ theory. The proposed approach not only improves sampling performance and accuracy over traditional MCMC and flow-based baselines but also establishes a new method for sample-free learning of complex physical distributions.

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.