Constrained Flow Matching via Lagrangian Dual Flows

What the math gives to ML

The paper transfers continuous-time primal-dual dynamics into flow-based generative sampling: a generated state and a Lagrange multiplier are integrated jointly rather than repeatedly projecting samples onto a nonlinear constraint manifold. The transferable asset is that constraint gradients enter the sample velocity through a dual co-state, while the dual state accumulates constraint violation, avoiding inner optimization, pseudoinverses, and explicit projection. A practical neural implementation is a constrained flow-matching sampler that augments the learned velocity field with a Jacobian-transpose dual correction and integrates the primal and dual ODEs with the same adaptive solver.