Flow Matching
How it works
1) A fixed, conditional probability path is defined that interpolates between a noise sample and a data sample (e.g. a Gaussian path or an Optimal Transport interpolation). 2) For that path, the target vector field that generates it is derived analytically. 3) A neural network learns to regress this vector field at randomly sampled time and space points (a simulation-free objective). 4) At generation time, new samples are produced by starting from noise and integrating the learned vector field with a standard ODE solver up to the data distribution.
Problem solved
Continuous Normalizing Flows were hard to train at scale because they required simulating ODE trajectories and differentiating through the solver at every training step. Flow Matching removes this simulation, reducing training to a simple vector-field regression that lets CNFs be trained at a previously unattainable scale.
Implementation
Evolution
Lipman et al. formulate Flow Matching as a simulation-free way to train Continuous Normalizing Flows, subsuming both diffusion and Optimal Transport paths.
Flow Matching becomes the training objective of action experts in Vision-Language-Action models (e.g. SmolVLA), generating continuous robot action trajectories.