PT-Flow: One-Step Schrödinger-Bridge Generation by Proximal Tilting

One-step generative models usually compress a many-step integration of a velocity field into a single step, using distillation, consistency conditions, trajectory straightening, or averaged fields. This compression tends to give up the optimality of the learned map or an explicit transport objective. PT-Flow takes a different route. We choose the governing law of the field (a viscous potential flow, i.e. the Schr\"odinger bridge) so that its time evolution has an exact closed-form solution through the Cole–Hopf transform. The single step is then not a shortened trajectory but the closed-form solution of the flow. The one hard part is a single Gaussian expectation whose naive estimator has variance that grows sharply as the flow becomes sharp. We remove this obstruction with a prox-tilted importance estimator: we center the proposal at the proximal point and scale it by the local curvature, which cancels the leading variation and makes the remaining variance shrink as the viscosity decreases. The estimator therefore becomes more accurate exactly where generation is sharp, and the same network that builds the proposal is the one-step generator itself, so sampling is a single network evaluation with no teacher, no adversarial training, and no ODE solver at inference. The learned scalar potential also defines a normalized Schr\"odinger-bridge marginal. On ImageNet $256\times 256$, PT-Flow reaches an FID of 1.54 with one function evaluation. It combines one-step sampling with teacher-free, adversarial-free training and optimal-transport consistency at convergence.