Flow and Diffusion Models

We start by defining a trajectory.

Ordinary Differential Equations (ODEs): The mathematical foundation for flow models.

Trajectory: The solution to an ODE. It is a function that maps time $t$ to a specific location in a $d$-dimensional space. - Function: $X : [0, 1] \to \mathbb{R}^d, \quad t \mapsto X_t$

Vector Field ($u$): The rule that defines an ODE. It assigns a specific velocity vector to every point in space at any given time - Function: $u : \mathbb{R}^d \times [0, 1] \to \mathbb{R}^d, \quad (x, t) \mapsto u_t(x)$

The ODE System

An ODE requires that a trajectory $X$ “follows along the lines” of the vector field $u_t$, beginning at a specific starting point.

This is formalized by a system of two equations:

• The ODE (Direction):

  $$\frac{d}{dt}X_t = u_t(X_t)$$

  (Meaning: The derivative/velocity of $X_t$ is determined by the vector field $u_t$.)

• Initial Conditions (Starting Point):

  $$X_0 = x_0$$

  (Meaning: At time $t=0$, we start at location $x_0$.)

3. Flow

The Core Question: If we start at $X_0 = x_0$ at $t = 0$, where exactly are we at time $t$ (what is $X_t$)?

The Answer: A function called the flow ($\psi$). The flow is the mathematical solution to the ODE.

• Flow Function: $\psi : \mathbb{R}^d \times [0, 1] \to \mathbb{R}^d, \quad (x_0, t) \mapsto \psi_t(x_0)$

• Governing Equations for Flow:

  • Flow ODE: $\frac{d}{dt}\psi_t(x_0) = u_t(\psi_t(x_0))$

  • Flow Initial Conditions: $\psi_0(x_0) = x_0$

4. Key Takeaway & Intuition

If you know the initial condition ($X_0 = x_0$), you can find the entire trajectory using the flow: $X_t = \psi_t(X_0)$.

Because of this relationship, you can think of vector fields, ODEs, and flows as three different ways to describe the exact same object:

Golden Rule: Vector fields define ODEs whose solutions are flows.

• unqiue soln to an SDE means the distribution over trajectories is constant with time