Equilibrium Flow Maps: Compactified Generative Dynamics And Terminal Projection

Flow matching has emerged as a powerful framework for generative modeling, but sampling typically requires numerically integrating a learned time-dependent vector field, requiring many network evaluations. Flow maps accelerate generation by learning the ODE solution operator directly, replacing many small numerical integration steps with a few large advances along the generative trajectory. Separately, recent methods such as Equilibrium Matching replace the time-dependent dynamics of flow matching with an autonomous vector field with data as equilibria, yielding dynamics with a natural infinite time horizon. These two directions suggest complementary goals: efficient generation with few network evaluations and principled iterative refinement as additional inference-time compute becomes available. To address these two goals simultaneously, we introduce Equilibrium Flow Maps (EqFM), which combine flow map amortization with autonomous equilibrium dynamics in a single model. EqFM compactifies the infinite-horizon autonomous dynamics onto a bounded domain, so the same network can advance a sample by any duration along the flow, from short refinement steps to a direct jump to equilibrium. We construct an autonomous field through infinite-time stochastic interpolants and derive complementary Lagrangian, Eulerian, and semigroup characterizations that lead to practical flow-map objectives for both direct training and distillation from an autonomous teacher. EqFM supports inference from one-step generation to iterative refinement without changing the underlying model. On ImageNet-256, EqFM improves from FID $4.32$ at 1 NFE to $2.09$ at 8 NFE, approaching its 500 NFE teacher at $2.00$.