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

Area of Science:

• Generative Modeling
• Optimal Control
• Machine Learning

Background:

• Diffusion and flow-matching models excel at capturing complex data distributions.
• Downstream applications often require hard constraints on generated samples, such as obstacle avoidance in robotics.
• Existing projection-based methods are overly restrictive and degrade sample quality.

Purpose of the Study:

• To develop a novel framework for hard-constrained sampling in generative models.
• To reformulate constrained sampling as a tractable trajectory optimization problem.
• To enhance sample quality and constraint satisfaction beyond existing methods.

Main Methods:

• Leveraging numerical optimal control to steer sampling trajectories.
• Exploiting flow-matching model structure and model predictive control techniques.
• Transforming constrained optimization into an efficient and effective tractable surrogate.

Main Results:

• The HardFlow algorithm demonstrates superior constraint satisfaction and sample quality.
• Achieved significant performance improvements across robotics, PDE boundary control, and text-guided image editing.
• Provided a control-theoretic analysis establishing approximation error bounds.

Conclusions:

• HardFlow offers a flexible and effective approach to hard-constrained generative modeling.
• The trajectory optimization perspective enables inclusion of integral costs and terminal objectives.
• The method significantly advances generative modeling for applications with strict constraints.