Wasserstein-Fisher-Rao Flow Matching: Resolving Jacobian Stiffness via Dual-Action Probability Transport

Deterministic continuous generative models, including Flow Matching and Continuous Normalizing Flows, restrict probability density transport to mass-conserving kinematic trajectories. When bridging disconnected topological supports, this strict conservation physicalizes as local velocity divergence singularities ($\nabla \cdot {\mathbf{v}}_{\theta }\to -\infty$), which induces severe spatial Jacobian stiffness and collapses explicit ODE solver step sizes. To resolve this geometric bottleneck, we introduce Wasserstein-Fisher-Rao Flow Matching (WFR-FM), a dual-action continuous generative framework parameterized by an unbalanced continuity equation. WFR-FM decouples spatial advection from topological probability weighting via an independent scalar reaction field, allowing trajectories to traverse zero-density corridors by dynamically suppressing and reconstituting mass rather than forcing extreme spatial curvature. Theoretically, we prove that the simulation-free WFR-FM regression strictly upper-bounds the macroscopic Unbalanced Hellinger Divergence. Furthermore, the decoupled objective satisfies Polyak-Łojasiewicz (PŁ) optimization bounds, ensuring exponential convergence and finite gradient variance up to high dimensions ($D=512$), even under discrete minibatch sampling approximations. Empirically, across disconnected 2D manifolds, WFR-FM strictly bounds continuous integration stiffness, reducing inference Number of Function Evaluations (NFE) by up to $236\times$ compared to state-of-the-art mass-conserving baselines. On high-dimensional empirical benchmarks exhibiting intrinsic non-equilibrium dynamics, the decoupled scalar field physically isolates local density variations, achieving a Relative Mass Error (RME) of 0.0016 without adjoint simulations.