Iterated Energy-based Flow Matching for Sampling from Boltzmann Densities

In this work, we consider the problem of training a generator from evaluations of energy functions or unnormalized densities. This is a fundamental problem in probabilistic inference, which is crucial for scientific applications such as learning the 3D coordinate distribution of a molecule. To solve this problem, we propose iterated energy-based flow matching (iEFM), the first off-policy approach to train continuous normalizing flow (CNF) models from unnormalized densities. We introduce the simulation-free energy-based flow matching objective, which trains the model to predict the Monte Carlo estimation of the marginal vector field constructed from known energy functions. Our framework is general and can be extended to variance-exploding (VE) and optimal transport (OT) conditional probability paths. We evaluate iEFM on a two-dimensional Gaussian mixture model (GMM) and an eight-dimensional four-particle double-well potential (DW-4) energy function. Our results demonstrate that iEFM outperforms existing methods, showcasing its potential for efficient and scalable probabilistic modeling in complex high-dimensional systems.