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.
Flow-based models learn a target distribution by modeling a marginal velocity field, defined as the average of sample-wise velocities connecting each sample from a simple prior to the target data. When sample-wise velocities conflict at the same intermediate state, however, this averaged velocity can misguide samples toward low-density regions, degrading generation quality. To address this issue, we propose the Flow Divergence Sampler (FDS), a training-free framework that refines intermediate states before each solver step. Our key finding reveals that the severity of this misguidance is quantified by the divergence of the marginal velocity field that is readily computable during inference with a well-optimized model. FDS exploits this signal to steer states toward less ambiguous regions. As a plug-and-play framework compatible with standard solvers and off-the-shelf flow backbones, FDS consistently improves fidelity across various generation tasks including text-to-image synthesis, and inverse problems.
Sampling from unnormalized multimodal distributions with limited density evaluations remains a fundamental challenge in machine learning and natural sciences. Successful approaches construct a bridge between a tractable reference and the target distribution. Parallel Tempering (PT) serves as the gold standard, while recent diffusion-based approaches offer a continuous alternative at the cost of neural training. In this work, we introduce Conditional Diffusion Sampling (CDS), a framework that combines these two paradigms. To this end, we derive Conditional Interpolants, a class of stochastic processes whose transport dynamics are governed by an exact, closed-form stochastic differential equation (SDE), requiring no neural approximation. Although these dynamics require sampling from a non-trivial initialization distribution, we show both theoretically and empirically that the cost of this initialization diminishes for sufficiently short diffusion times. CDS leverages this by a two-stage procedure: (1) PT is used to efficiently sample the initial distribution, and then (2) samples are transported via the transport SDE. This combination couples the robust global exploration of PT with efficient local transport. Experiments suggest that CDS has the potential to achieve a superior trade-off between sample quality and density evaluation cost compared to state-of-the-art samplers.
Francisco M. Castro-Macías, Pablo Morales-Álvarez, Saifuddin Syed +3
Many scientific datasets, such as molecular conformational ensembles or single-cell tissue measurements, are naturally modeled as meta-distributions: distributions over probability measures on non-Euclidean domains. Existing generative methods largely assume Euclidean geometry and fail to capture this structure. We introduce Riemannian Wasserstein Entropic Flow Matching (RWEFM), a generative framework on the Wasserstein space P2(M) of a Riemannian manifold (M,g). RWEFM is trained by regressing a neural vector field onto Riemannian optimal transport velocities, using McCann displacement interpolations as conditional paths. We confirm theoretically that this construction leads to a valid flow matching approach on P2(M) and introduce the Riemannian Entropic Map, a GPU-efficient approximation of the optimal transport map on manifolds. Our experiments show that by respecting the intrinsic geometry of the data, RWEFM can generate whole single-cell samples in hyperspherical latent spaces and protein conformational ensembles on the torus. As RWEFM requires only a geodesic distance and a projection operator, it is not restricted to manifolds with closed-form geometry, which we demonstrate by generating distributions on a general triangulated mesh.