In Flow Matching, the transition is defined by the ODE: $dx = v_t(x)dt$. This vector field $v_t$ defines the velocity at each point in space-time to push samples from $p_0$ to $p_1$.

The forward diffusion process adds noise $q(x_t | x_{t-1}) = \mathcal{N}(x_t; \sqrt{1-\beta_t}x_{t-1}, \beta_t I)$. As $t$ increases, the original structure of the image is lost to Gaussian randomness.

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