Deep conditional distribution learning via conditional Föllmer flow

We propose a conditioning mechanism for diffusion models based on multi-speed joint diffusion of the target and the condition. The mechanism learns an unconditional joint score network and enforces conditioning at inference via a plug-in correction term. The plug-in term separates the conditioning contribution from the learned unconditional dynamics, offering a transparent view of how the condition steers generation of the target distribution. Building on this, we derive explicit conditional reverse-time SDEs and approximate probability-flow ODEs, enabling principled and directly comparable conditional samplers. To reduce the induced ODE--SDE discrepancy, we introduce a log-Fokker--Planck residual regularization that improves ODE sampling quality. Experiments on conditional image generation tasks demonstrate competitive performance and support the effectiveness of the plug-in conditioning view. Additional ODE--SDE comparison experiments show that the log-Fokker--Planck residual regularization improves deterministic ODE sampling.

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 matching has emerged as a powerful generative family for simulation-based inference (SBI). However, sampling and density evaluation require numerical integration at inference time. This makes posterior sampling in neural posterior estimation costly. More importantly, it can make repeated likelihood evaluation prohibitively expensive and impractical for Markov chain Monte Carlo (MCMC) sampling, which may require thousands of repeated density evaluations. Here, we develop conditional mean flows for posterior and likelihood approximation, combining few-step sampling with tractable density evaluation. We show that the mean flow objective identifies conditional flow maps and that independent masking enables direct estimation of marginal likelihoods. We evaluate both posterior and likelihood estimation across several tasks and show that conditional mean flows retain the flexibility of flow-based density estimators while significantly reducing inference time. In particular, single-step conditional mean flow density evaluation yields estimation error comparable to 200-step flow matching while enabling about 200-fold faster downstream MCMC sampling.