Velocity Adaptation for Flow-Matching Models

Flow-Matching models generate samples by learning a continuous-time velocity field that transports an initial distribution to a target data distribution. Although these models achieve strong empirical performance, the learned dynamics often exhibit suboptimal structure that can degrade sample quality as measured by Fréchet Inception Distance (FID). We propose a relatively lightweight post-training velocity adaptation framework for pretrained Flow-Matching models. The adaptation leverages the Helmholtz--Hodge decomposition to decompose the learned velocity field into its conservative and solenoidal components. We introduce a fully convolutional scalar potential network ($\phi_{\mathrm{NET}}$, $\sim$3M parameters), whose gradient defines the conservative component. The adapted velocity is obtained by reweighting the conservative and solenoidal components during inference. Across CIFAR-10, ImageNet-64, ImageNet-32, CelebA-128, and AFHQ datasets, the proposed adaptation consistently improves generative quality and substantially reduces Fréchet Inception Distance (FID) compared to the corresponding pretrained Flow-Matching models. The adaptation also has strong cross-domain transferability: a $\phi_{\mathrm{NET}}$ trained on one dataset effectively adapts velocity fields trained across different datasets and image resolutions. Our results suggest that structured decompositions of learned velocity fields provide an effective mechanism for improving sampling in continuous-time generative models.