Flow Matching with Semidiscrete Couplings

Flow models parameterized as time-dependent velocity fields can generate data from noise by integrating an ODE. These models are often trained using flow matching, i.e. by sampling random pairs of noise and target points (x0,x1)(\mathbf{x}_0, \mathbf{x}_1)(x0​,x1​) and ensuring that the velocity field is aligned, on average, with x1−x0\mathbf{x}_1 - \mathbf{x}_0x1​−x0​ when evaluated along a segment linking x0\mathbf{x}_0x0​ to x1\mathbf{x}_1x1​. While these pairs are sampled independently by default, they can also be selected carefully by matching batches of nnn noise to nnn target.