Characteristic-Steered Flow Matching for Oracle-Defined Sets

Generating samples that satisfy complex physical or structural constraints remains a fundamental challenge when valid regions cannot be expressed in closed form. For instance, finding valid candidates, such as a stable crystal structure or a collision-free robot trajectory, fundamentally relies on evaluating proposals and rejecting failures against a binary verifier. This problem can be expressed as learning distributions over oracle-defined sets, where admissibility is accessible only through pointwise verification. We study how to adapt continuous generative flows in this regime, where certified reference samples seed a base transport, but further constraint satisfaction must be learned from oracle evaluations that provide no gradient feedback. We introduce Characteristic-Steered Flow Matching (CSFM), a method that converts discrete membership evaluations into directional supervision along generative paths. By pairing each inadmissible candidate with a certified feasible anchor, CSFM transports the required terminal correction backward along the generative trajectory via reverse-time linearization, supervising the velocity field to realize first-order endpoint corrections without differentiating the oracle. Across synthetic geometric sets, physics-constrained continuous control, and long-horizon trajectory planning, CSFM achieves favorable validity–coverage and downstream trade-offs relative to existing flow-correction baselines, preserving coverage and distributional fidelity as ambient dimension scales.