HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization

Recent advances in diffusion and flow-matching models have revolutionized generative AI, enabling realistic image synthesis, video generation, and robotic planning. These models learn a time-dependent velocity or score function to transport simple distributions to complex data distributions without requiring explicit likelihoods. Despite their success, incorporating hard constraints—such as obstacle avoidance in robotics or safety bounds in control—remains challenging. Existing solutions rely on projection methods, which restrict the entire sampling path, often degrading sample quality and exploration. Some recent works explore optimal control principles for guidance, but mainly in unconstrained or softly constrained settings. The gap persists in developing scalable, theoretically grounded methods that enforce strict terminal constraints without sacrificing diversity or efficiency.

The core issue is that current projection-based methods enforce pathwise feasibility, which is overly restrictive and hampers the exploration of high-quality solutions. They often cause a distribution shift away from the target, reducing sample fidelity. Moreover, these methods struggle with high-dimensional problems and complex constraints, leading to computational inefficiencies. The challenge is to develop a method that guarantees the final sample satisfies the constraints while allowing the sampling process enough flexibility to generate diverse, high-quality outputs. Achieving this balance requires rethinking the constraint enforcement paradigm, shifting from pathwise to terminal-only constraints, and integrating control-theoretic insights into the generative process.

The key innovation is formulating the constrained sampling as a trajectory optimization problem, where only the terminal state is constrained. This avoids unnecessary restrictions on the entire path, preserving exploration. The authors employ a reverse reparameterization, transforming the decision variables from the current state to the predicted terminal state, simplifying the feasible set. They incorporate MPC principles to decompose the long-horizon problem into manageable subproblems, enabling efficient, scalable optimization. The framework also includes auxiliary costs for distributional consistency and sample quality, providing a unified approach. Theoretical bounds on approximation errors ensure the method’s reliability, marking a significant departure from prior projection-based techniques.

• �� Start with a pretrained flow-matching model, learning a neural velocity field vθt(x).
• �� Define the hard constraints h(x) ≤ 0 and terminal cost C(x).
• �� Formulate an optimal control problem: minimize terminal cost plus control effort, subject to system dynamics and terminal constraints.
• �� Introduce control inputs ut to perturb the velocity field during sampling.
• �� Use MPC to decompose the long-horizon problem into short-step subproblems, solving iteratively.
• �� Apply reverse reparameterization: instead of optimizing over the entire path, optimize over the terminal state prediction, reducing complexity.
• �� Derive theoretical bounds on the approximation error between the surrogate and original problem.
• �� Implement an efficient solver based on collocation and fixed-point iteration, enabling scalable inference.

The authors evaluate HardFlow on three main tasks: robotic path planning with obstacle avoidance, PDE boundary control with precise boundary condition satisfaction, and text-guided image editing for content preservation. They use datasets like Robotics Dataset, PDE Boundary Data, and Text-Image Dataset. Metrics include constraint satisfaction rate, FID, Inception Score, and sample diversity. Baselines include traditional projection methods, Lagrangian relaxation, and unconstrained flow models. Hyperparameters such as sampling steps, regularization weights, and model architectures are tuned via ablation studies. The experiments demonstrate that HardFlow consistently achieves higher feasibility rates (>98%) and better sample quality, with significant improvements over baselines.

HardFlow outperforms existing methods by achieving 98% constraint satisfaction across tasks, with a 12% increase in sample quality scores. In robotics, obstacle avoidance improves by 10%, while in PDE control, the boundary error drops to 0.01. In image editing, content consistency improves by 20%. Ablation studies reveal that terminal constraints alone yield 95% feasibility; adding integral costs reduces bias by 18%. The results confirm that the proposed approach effectively balances constraint satisfaction, sample diversity, and quality, demonstrating broad applicability and robustness.

This framework is applicable to robotics, PDE boundary control, and creative AI tasks like image editing, where strict constraints are essential. It enables safe, reliable, and high-quality sample generation without extensive retraining. Practical deployment requires pre-trained flow models and computational resources for optimization. The method can be integrated into existing pipelines to enhance safety and fidelity, especially in scenarios demanding real-time constraint enforcement.

Computational complexity remains high in very high-dimensional spaces, limiting real-time applications. Dependence on the accuracy of pre-trained models makes it sensitive to model errors. Strict terminal constraints may reduce sample diversity in some cases, requiring careful cost balancing. Future work should focus on efficiency improvements, adaptive constraint handling, and extension to dynamic, multi-agent environments.