Flow Matching

Flow matching learns a velocity field that moves samples from a simple source, often Gaussian noise, toward a target data distribution. Conditional flow matching trains the field on paths between source and data samples, while optimal-transport couplings and rectified paths aim to make those trajectories easier to learn and integrate with an ordinary differential equation solver. Extensions handle discrete data, curved manifolds and symmetries, physical constraints, and applications from image and speech generation to robotics, scientific forecasting, and inverse problems. A central challenge is getting high fidelity and diverse, reliable samples with few solver steps; theory on convergence and stability is advancing, but the best paths, source distributions, and schedules depend on the task.