A Visual Introduction to Flow-based Generative Models

Flow-based generative models have become one of the dominant paradigms in modern generative modeling, powering state-of-the-art results in image and video generation. We present a visual survey tracing their evolution from normalizing flows, which transform simple distributions into complex ones through invertible mappings and the change-of-variables formula, to continuous normalizing flows, which replace discrete transformations with ordinary differential equations and unlock more flexible architectures. We then cover flow matching and stochastic interpolants, which enable efficient simulation-free training by reducing the problem to a simple velocity regression objective. A key challenge that remains is that learned flow trajectories tend to be curved, requiring many neural network evaluations at generation time and incurring high computational cost. Rectified flows address this by straightening trajectories through an iterative reflow procedure, enabling faster sampling with fewer integration steps. Throughout, 2D visualizations and animations — built using trained neural networks on toy distributions — illustrate probability paths, velocity fields, Euler integration, trajectory curvature, and the straightening effect of rectified flows, making the geometric intuitions behind these methods tangible and accessible.