Simulation-Free Flow Map Alignment for High-Fidelity Any-Step Constrained Generation

Flow-based generative models for science and engineering should satisfy physical and operational constraints while preserving distributional fidelity. Inference-time correction improves constraint satisfaction but adds per-sample cost and can distort the generated distribution. Training-time alignment amortizes this cost, but propagating terminal feedback to the local dynamics typically requires costly ODE simulation. Flow map models avoid simulation by evaluating terminal losses directly on learned finite-time transitions. Yet endpoint training alone neither yields the local-velocity update implied by the ODE objective nor ensures that alignment persists when shorter transitions are composed. We introduce simulation-free terminal alignment for any-step flow map generation. A map-based adjoint uses learned intermediate states and the sensitivity of the remaining-time map to transport terminal feedback to the local velocity, approximating the gradient of the ODE terminal objective. Combined with endpoint training and map–velocity consistency, this update aligns a single model across sampling schedules. Our analysis characterizes the map-adjoint approximation error and shows how velocity and consistency errors affect distributional fidelity and cross-schedule agreement. Experiments on constrained distributions, robot trajectories, and physical fields show that the aligned model retains constraint improvements across multi-step schedules while maintaining competitive distributional quality.