Free Energy Surface Sampling via Reduced Flow Matching

Path sampling methods generate ensembles of reactive trajectories connecting metastable states, but extracting mechanistic insight from these data remains nontrivial. We introduce Flux Matching, a framework that learns two complementary objects directly from reactive trajectory data: a current velocity $u(z)$, whose streamlines trace the dominant reaction pathways, and a scalar potential $h(z)$, obtained from a weighted Helmholtz-Hodge decomposition of the reactive current, that serves as a data-driven reaction coordinate. Both minimize quadratic functionals over the reactive path ensemble, analogous to the flow matching loss in generative modeling, and require no knowledge of the underlying dynamics or stationary distribution. Unlike committor-based methods, $u$ and $h$ remain well-defined under projection onto non-Markovian collective variables, and their level sets in turn provide adaptive interfaces for improved sampling with enhanced sampling methods. Flux Matching is validated through the generation of current velocity trajectories and rate constant calculations on molecular systems.

The rare-event sampling problem has long been the central limiting factor in molecular dynamics (MD), especially in biomolecular simulation. Recently, diffusion models such as BioEmu have emerged as powerful equilibrium samplers that generate independent samples from complex molecular distributions, eliminating the cost of sampling rare transition events. However, a sampling problem remains when computing observables that rely on states which are rare in equilibrium, for example folding free energies. Here, we introduce enhanced diffusion sampling, enabling efficient exploration of rare-event regions while preserving unbiased thermodynamic estimators. The key idea is to perform quantitatively accurate steering protocols to generate biased ensembles and subsequently recover equilibrium statistics via exact reweighting. We instantiate our framework in three algorithms: UmbrellaDiff (umbrella sampling with diffusion models), MetaDiff (a batchwise analogue for metadynamics), and $Δ$G-Diff (free-energy differences via tilted ensembles). Across toy systems, protein folding landscapes and folding free energies, our methods achieve fast, accurate, and scalable estimation of equilibrium properties within GPU-minutes to hours per system-closing the rare-event sampling gap that remained after the advent of diffusion-model equilibrium samplers.

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.