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.

翻译：在本研究中，我们考虑基于能量函数或未归一化密度评估来训练生成器的问题。这是概率推断中的一个基础性问题，对于学习分子三维坐标分布等科学应用至关重要。为解决该问题，我们提出了基于能量的迭代流匹配（iEFM），这是首个基于未归一化密度训练连续归一化流（CNF）模型的离策略方法。我们引入了免仿真的基于能量流匹配目标函数，该函数通过已知能量函数构建的边际向量场的蒙特卡洛估计来训练模型进行预测。我们的框架具有普适性，可扩展至方差爆炸（VE）和最优传输（OT）条件概率路径。我们在二维高斯混合模型（GMM）和八维四粒子双势阱（DW-4）能量函数上评估了iEFM方法。实验结果表明，iEFM优于现有方法，展现了其在复杂高维系统中实现高效可扩展概率建模的潜力。