Flow Matching: A Visual Introduction (peterroelants.github.io)

This post is a hands‑on, visual introduction to flow matching (FM), demonstrating how to train a generative model that transports samples from a simple noise distribution to a complex target distribution. Using a minimal 1D toy (Gaussian noise π0 → bimodal Gaussian π1) the author implements a PyTorch notebook that (a) defines straight‑line reference paths between independently sampled noise and target points, (b) trains a neural velocity field FMθ(x_t, t) that predicts v = dx_t/dt, and (c) visualizes path densities and a velocity heatmap and animation showing integration from t=0 to t=1. Training minimizes the expected squared error E_{t,x_t} ||FMθ(x_t,t) − v(x_t,t)||^2 with t∼U[0,1], and inference integrates the learned velocity beginning from x0 to produce samples from π1. Significance: flow matching is a practical alternative to score‑ or diffusion‑based generative methods because it directly models a velocity field that can be integrated to generate samples, often requiring fewer steps when trained on straight‑line (linear) conditional paths — a variant equivalent to rectified flow. The notebook highlights key technical choices (independent couplings, linear interpolation x_t=(1−t)x0 + t x1, Euler discretization, path density estimation) and provides intuition on why straight‑line reference paths yield straighter, faster trajectories. Though presented in 1D for clarity, the approach scales conceptually to high‑dimensional domains (images, video, control), making FM a compelling technique for efficient, interpretable generative modeling.