# Hard-constrained sampling

MIT researchers introduced HardFlow, which recasts hard-constrained generation as a trajectory-optimization problem and enforces safety, physical or task requirements only on the final output rather than on every intermediate sampling step. The common alternative, projection-based sampling, repeatedly pulls intermediate samples back into the feasible region throughout generation, which can keep the model from reaching a better final solution. Across robotic manipulation, maze navigation and text-guided image editing, HardFlow achieved perfect constraint satisfaction and consistently beat baselines on solution quality, with computation time comparable to or lower than most competing methods.

MIT researchers introduced HardFlow, which recasts hard-constrained generation as a trajectory-optimization problem and enforces safety, physical or task requirements only on the final output rather than on every intermediate sampling step. The common alternative, projection-based sampling, repeatedly pulls intermediate samples back into the feasible region throughout generation, which can keep the model from reaching a better final solution. Across robotic manipulation, maze navigation and text-guided image editing, HardFlow achieved perfect constraint satisfaction and consistently beat baselines on solution quality, with computation time comparable to or lower than most competing methods.