Last modified: 7 Oct 2025 This note is organized during the study of the "Introduction to Flow Matching and Diffusion Models" course (https://diffusion.csail.mit.edu/). The content is derived from course notes, lectures and labs. I reorganized the material for my own understanding. The lab parts should be viewed with the source code for a comprehensive understanding since I didn’t show most utility functions.

1️⃣ Introduction

Generation → Sampling: Formalize what it means to “generate” objects

We represent the objects we want to generate as vectors.

How good an image is $\approx$ How likely it is under data distribution.

Distribution of data we want to generate: Probability Density $p_{data}: \mathbb{R}^d \rightarrow \mathbb{R}{\ge0}, \ \ \ z\mapsto p{data}(z)$

• Generation means sampling the data distribution: $z \sim p_{data}$
• Conditional generation means sampling the conditional data distribution: $z \sim p_{data}(\cdot|y)$, $y$ is the condition variable.
• Initial distribution: $p_{init} = N(0, I_d)$

A generative model converts samples from a initial distribution (e.g. Gaussian) into samples from the data distribution.

Flow Models

Diffusion Models

• Trajectory $X: [0, 1] \rightarrow \mathbb{R}^d, \ \ \ t \mapsto X_t$ (a solution to ODE)

• Stochastic Process $(X_t)_{0 \le t \le 1}$ is given by:

  • $X_t:Random\ Variable\ \ \ (0 \le t \le 1)$
  • $X:$ $[0, 1] \rightarrow \mathbb{R}^d, \ \ \ t \mapsto X_t$

• Vector Field $u: \mathbb{R}^d \times [0,1] \rightarrow \mathbb{R}^d, \ \ \ (x,t) \mapsto u_t(x)$ (defines an ODE)

• Vector Field $u: \mathbb{R}^d \times [0,1] \rightarrow \mathbb{R}^d+\sigma, \ \ \ (x,t) \mapsto u_t(x)$

             $\\sigma:[0, 1]\\rightarrow\\mathbb{R}_{\\ge 0},\\ \\ \\ t \\mapsto \\sigma_t\\ \\ \\ (diffusion\\ coefficient)$
  

• Brownian motion:

  • Stochastic process $W = (W_t)_{t \ge 0}$
    • $W_0 = 0$
    • $Gaussian\ increments: W_t - W_s \sim N(0, (t-s)I_d),\ \ \ (0\le s \lt t)$
    • $Independent\ increments: W_{t_1}-W_{t_0},...,W_{t_n}-W_{t_{n-1}}$ are independent

• Ordinary Differential Equation: describe conditions of a trajectory

  • $X_0 = x_0 \ \ \ (initial \ condition)$
  • $\frac d{dt} X_t = u_t(X_t) \ \ \ (ODE)$

  not differentiable…

  • need an equivalent formulation of ODEs not using derivatives：

  

• From ODE to Stochastic Differential Equation (SDE):

  • $X_0 = x_0 \ \ \ (initial \ condition)$
  • $d X_t = u_t(X_t)dt + \sigma_tdW_t \ \ \ (SDE)$

  $\Rightarrow$ extend to SDE to encourage more exploration (the origin!!):