# Inference-time constrained generation

When a pretrained flow-matching model must satisfy an equality constraint supplied only at inference time, common practice repeatedly corrects predicted endpoints, and those discrete corrections can disturb the learned physical structure. MAFM instead evolves a persistent endpoint anchor continuously under an ordinary differential equation, separating model-guided tangential motion from normal attraction toward the constraint. On five PDE benchmarks—Poisson, Burgers, reaction–diffusion, Euler and Navier–Stokes—MAFM meets the prescribed constraint tolerance and achieves lower sample-level relative error than baselines that meet the same tolerance.

When a pretrained flow-matching model must satisfy an equality constraint supplied only at inference time, common practice repeatedly corrects predicted endpoints, and those discrete corrections can disturb the learned physical structure. MAFM instead evolves a persistent endpoint anchor continuously under an ordinary differential equation, separating model-guided tangential motion from normal attraction toward the constraint. On five PDE benchmarks—Poisson, Burgers, reaction–diffusion, Euler and Navier–Stokes—MAFM meets the prescribed constraint tolerance and achieves lower sample-level relative error than baselines that meet the same tolerance.