Constrained sampling stops patching endpoints: MAFM evolves a persistent anchor along geometric dynamics and cuts PDE error on five benchmarks
Synopsis
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.
Interpretation
Constrained sampling can now introduce a new equality constraint at sampling time without touching the trained generative model, with the constraint residual decaying continuously at a prescribed rate. Earlier inference-time methods such as PCFM repeatedly correct predicted endpoints and re-interpolate with the original source, and ECI alternates endpoint extrapolation, physical correction and interpolation; such discrete corrections can distort learned generative structure and introduce unstable feedback. On the sparse-observation Poisson task, terminal projection drives observation error to zero but leaves 22.7% PDE relative error; MAFM meets the observation tolerance while lowering the best feasible PDE relative error from 5.55% to 1.70%.
The method maintains an endpoint anchor that persists across sampling steps and serves as the evolving candidate for the final output, rather than attaching constraint control only to transient sampling states. The anchor velocity combines endpoint predictions from the frozen model, admitted through finite-rate tracking into tangential motion, with a Lagrange multiplier that independently prescribes normal residual contraction; the construction separates the two roles. On the mass-constrained Burgers and reaction–diffusion tasks, MAFM reaches numerical feasibility with own-initial-condition rollout relative errors of 0.965% and 2.74%, against next-best feasible errors of 1.03% and 2.96%.
Ablations show that anchor history and input-adjoint guidance carry distinct roles, and removing either degrades the result. Removing endpoint tracking raises Poisson PDE relative error from 1.70% to 29.0%, and removing the input adjoint raises it to 38.0%, while all three Poisson variants meet the same observation tolerance. On Burgers, replacing the anchor's tangential component each step with the latest prediction while keeping the normal update fixed leaves per-step mass residuals differing by at most a tiny amount, yet raises rollout relative error from 0.965% to 7.68%.
Perspective
The demonstrated scope is six physical-space generation tasks on synthetic data: a sphere–plane intersection, partially observed Poisson fields with 64 and 256 sensors, mass-constrained Burgers and reaction–diffusion trajectories, and Euler and controlled Navier–Stokes flows given only a kinetic-energy target, all performed as inference-time sampling on frozen flow-matching models. The intended readers are researchers and engineers working on scientific generative modeling and constrained sampling.
The residual-contraction law describes ideal continuous dynamics with a full-rank constraint Jacobian and exact multiplier solves; Gram regularization, solve error and time discretization in a numerical sampler can alter that law. Guidance quality depends on the frozen endpoint model, whose conditional-mean prediction may average modes or become less reliable away from the training distribution. Empirical evidence is limited to one training seed per task, Burgers samples show mild fluctuation-amplitude contraction, and reaction–diffusion samples retain a small negative-value tail.