Mixed-Field Matching: Time-Conditioned Transport with Energy-Based Refinement

We introduce Mixed-Field Matching (MFM), which trains one network whose time input follows the noise level up to a threshold and stays fixed above it. Diffusion and flow-matching models give the network the noise level at every step of generation, while recent equilibrium models remove it entirely. We argue that the need for the noise level changes during generation. Far from the data, one noisy sample is consistent with many noise levels and many clean images, and the noise level reduces this uncertainty about where the sample should move. Close to the data, the manifold hypothesis suggests that the sample itself carries this information. MFM sampling first transports noise toward the data with the time-conditioned field and then refines each sample using a single time-independent field. A penalty that sets the field to zero on clean training examples keeps finished samples in place, so extra refinement steps do not degrade them. On a two-dimensional example where equilibrium matching (EqM) moves every sample onto the data but gives the modes the wrong proportions, MFM assigns samples to modes about as accurately as flow matching (FM). On ImageNet-256 with DiT-XL/2, at the same training and sampling cost as the baselines, MFM lowers FID by 15% relative to EqM and 28% relative to FM without guidance, and by 14% and 33% at a shared classifier-free guidance scale.