Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation

Generative models excel at synthesizing high-fidelity samples from complex data distributions, but they often violate hard constraints arising from physical laws or task specifications. A common remedy is to project intermediate samples onto the feasible set at each sampling step. However, repeated projection can disrupt the learned sampling dynamics because this feasible set is defined for ''clean'' samples rather than noisy intermediate states. Thus, recent studies first estimate ''clean'' samples and then project them to the feasible set, but this increases algorithmic complexity and accumulates errors across steps. To address these issues, this paper proposes Chance-constrained Flow Matching (CCFM), a training-free method that formulates constraint enforcement during sampling as stochastic optimization while retaining the hard-constraint feasibility guarantee of projection-based methods. By avoiding direct projection of noisy intermediate states onto the clean-sample feasible set, CCFM mitigates the distributional distortion. Unlike approaches that first estimate and project ''clean'' samples, CCFM avoids complex multi-stage procedures. Experiments show that CCFM outperforms current state-of-the-art constrained generative models in modeling complex physical systems governed by partial differential equations, molecular docking problems, and physics-informed motion generation, delivering superior feasibility and fidelity.