Conditional Flow Matching

The CFM loss scales like standard supervised learning and has produced strong results in image synthesis, audio, video, and molecular design. Conditional flow matching is presented as a fast alternative to diffusion models because it samples through ODE integration. Early continuous normalizing flow training required differentiating through an ODE solver via the adjoint method, which was prohibitively expensive. Flow Matching replaced the expensive adjoint-method training of continuous normalizing flows with a simulation-free regression objective using linear interpolants between independently paired samples. When source and target are paired independently, conflicting trajectory directions at intermediate states produce a curved marginal velocity field requiring many small Euler steps. Conditional flow matching defines a probability path between Gaussian noise and high-resolution precipitation sampled from the data distribution. The learned velocity network minimizes mean squared error between predicted and target velocities while masking missing precipitation pixels. Sampling uses the trained velocity network as an ODE right-hand side from Gaussian noise to the generated field.