We introduce Perron--Frobenius Operator Matching (PFOM), a generative framework that matches density evolution via the integral PF operator, subsuming flow, diffusion, and jump models. We prove that among Bregman divergences, only Kullback--Leibler divergence preserves equality between density-level and sample-conditioned objectives, yielding a practical loss equivalent to Koopman path matching. We further develop Nesterov-accelerated training and sampling that stabilize discretization and accelerate convergence. %On Gaussian mixtures and two-moons, PFOM achieves faster KL/$W_2$/MMD decrease and improved wall-clock efficiency with empirical validation. PFOM unifies operator-theoretic identification with modern generative modeling and opens paths to adaptive dictionaries and high-dimensional applications.

翻译：我们提出Perron--Frobenius算子匹配(PFOM)框架，通过积分PF算子实现密度演化匹配，统一了流模型、扩散模型和跳跃模型。我们证明在Bregman散度中，仅Kullback--Leibler散度能保持密度级目标与样本条件目标之间的等价性，从而得到等价于Koopman路径匹配的实用损失函数。我们进一步开发了Nesterov加速的训练与采样方法，有效稳定离散化过程并加速收敛。PFOM算子理论识别与现代生成建模的统一框架，为自适应字典学习及高维应用开辟了新路径。