We survey continuous-time generative modeling methods based on transporting a simple reference distribution to a data distribution via stochastic or deterministic dynamics. We present a unified framework in which diffusion models, score-based generative models, and flow matching are instances of learning a time-dependent vector field that induces a family of marginals $(ρ_t)_{t \in [0,1]}$ governed by continuity and Fokker-Planck equations. Such a unified theory is timely because these methods are converging methodologically, yet fragmented notation and competing derivations continue to obscure their shared structure and the practical tradeoffs governing sampling, stability, and computation. Within this framework, we (i) derive reverse-time sampling for diffusion and score-based models as controlled stochastic dynamics, (ii) show that the probability flow ODE yields identical marginals and connects diffusion to likelihood-based normalizing flows, and (iii) interpret flow matching as direct regression of the velocity field under a chosen interpolation, clarifying when it coincides with or differs from score-based training. We compare objectives, sampling schemes, and discretization errors under unified notation, discuss connections to Schrodinger bridges and entropic optimal transport, and summarize theoretical guarantees and open problems on approximation, stability, and scalability.

翻译：我们综述了基于随机或确定性动力学将简单参考分布输送到数据分布的连续时间生成建模方法。我们提出一个统一框架，其中扩散模型、基于得分的生成模型和流匹配均可视为学习一个依赖于时间的向量场，该向量场诱导一族由连续性方程和福克-普朗克方程控制的边际分布$(ρ_t)_{t \in [0,1]}$。这一统一理论恰逢其时，因为这些方法在方法论上趋于收敛，但分散的符号体系和相互竞争的推导方式仍掩盖了其共同结构以及控制采样、稳定性和计算的实践权衡。在此框架内，我们(i)将扩散模型和基于得分模型的反向时间采样推导为受控随机动力学，(ii)表明概率流常微分方程可产生相同的边际分布，并将扩散模型与基于似然的归一化流联系起来，(iii)将流匹配解释为在所选插值下对速度场的直接回归，阐明其何时与基于得分的训练一致或不同。我们在统一符号体系下比较目标函数、采样方案和离散误差，讨论与薛定谔桥和熵最优输运的联系，并总结关于逼近性、稳定性和可扩展性的理论保证与开放问题。