Manifold Attracting Flow Matching: Physics-Constrained Sampling via Continuous Geometric Dynamics

Generating scientific data under previously unseen equality constraints requires satisfying the prescribed constraints without destroying the physical structure learned from data. Existing inference-time approaches often enforce physical constraints by repeatedly correcting predicted endpoints, but such discrete corrections can distort learned generative structure and introduce unstable feedback. In this work, we propose Manifold Attracting Flow Matching (MAFM), which continuously evolves a persistent anchor through an ODE whose dynamics drive it analytically toward the constraint manifold. Specifically, its geometric correction preserves model-guided tangent motion while prescribing the normal motion to contract constraint violations toward the manifold. This construction separates model tracking from constraint attraction and provides theoretical guarantees for residual contraction and local feedback stabilization. Experiments across geometric and partial differential equation benchmarks demonstrate improved generation quality over variant constraint-guided baselines while maintaining low constraint residuals. MAFM provides a flexible framework for coordinating learned physical structure and explicit constraints through controllable sampling dynamics, without retraining the underlying generative model.