Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspective

Diffusion models have emerged as a dominant framework for generative modeling, but their mathematical foundations are often presented separately through diffusion probabilistic models, score-based modeling, stochastic differential equations, and numerical sampling methods. We write this tutorial to provide a unified and self-contained account of these viewpoints from the perspective of differential equations. Starting from a conditional Gaussian noising process, we derive ordinary differential equation (ODE) and stochastic differential equation (SDE) representations, pass to the corresponding marginal forward dynamics, and then obtain the reverse-time SDE and probability-flow ODE that make generation possible. We show that the central unknown quantity in reverse sampling is the marginal score, explain how score matching becomes the standard denoising objective under a noise-prediction parameterization, and discuss practical reverse-time sampling and guidance. We further place DDPM, DDIM, flow matching, and score-based SDEs in a common framework, and conclude with diffusion language models in continuous embedding space together with a brief discussion of discrete masked-token diffusion. The tutorial is intended as a bridge between the analytical foundations of diffusion processes and the modern generative algorithms built upon them.

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.

This work studies prediction parameterization for stochastic generative dynamics in diffusion models. Existing velocity-based generative models provide the simplicity of learning a single transport field, but their standard formulation is deterministic, whereas stochastic extensions generally require additional score information or an intermediate velocity-to-score reconstruction. To retain single-field prediction while directly supporting stochastic reverse dynamics, this paper introduces Mean Velocity Matching (MVM). MVM constructs a Gaussian perturbation process for which the conditional expectation of a restoration-oriented velocity, $(x_0-x_t)/t$, directly forms the reverse-SDE drift. Consequently, a single learned field is sufficient to parameterize the stochastic reverse process without separately estimating or reconstructing the score. Because direct regression of this velocity becomes unbounded near $t=0$, MVM further introduces a $\sqrt{t}$-scaled parameterization that preserves the reverse dynamics while yielding a bounded training target. The same learned field also induces a deterministic probability-flow ODE, enabling stochastic and deterministic sampling to be studied within a unified formulation. Experiments with Transformer-based generative models achieve an FID of $\MVMImageNetThirtyTwoFID$ at \MVMImageNetThirtyTwoNFE\ NFE on ImageNet $32\times32$ and $\MVMImageNetTwoFiftySixFID$ at \MVMImageNetTwoFiftySixNFE\ NFE on ImageNet $256\times256$. Controlled SDE--ODE comparisons further show that the ODE performs better under very low NFE, whereas the stochastic reverse process achieves lower FID when sufficient function evaluations are available. These results demonstrate that MVM provides a direct single-field parameterization of stochastic reverse dynamics while maintaining competitive generation quality.