Flow Sampling: Learning to Sample from Unnormalized Densities via Denoising Conditional Processes

2026-05-05

AI Summary

This work addresses the inefficiency of sampling from density distributions defined by unnormalized energy functions by proposing a data-free sampling framework. The approach redefines the diffusion objective, shifting from data-conditional forward noising to noise-conditional denoising drift, and integrates flow matching with interpolation strategies to substantially reduce the number of energy function evaluations. The method naturally extends to Riemannian manifolds, enabling closed-form conditional drift computations on non-Euclidean geometries such as hyperspheres and hyperbolic spaces. Empirical evaluations demonstrate superior sampling performance and computational efficiency across diverse tasks, including synthetic energy benchmarks, small peptide and large-scale molecular conformation generation, and spherical distribution modeling.

Abstract

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.

Problem

• unnormalized densities
• energy-based sampling
• efficient sampling
• data-free generative modeling
• Riemannian manifolds

Innovation

• Flow Sampling
• unnormalized density sampling
• denoising conditional processes
• diffusion on manifolds
• energy-based models