Entropy-Controlled Flow Matching

Modern explicit-time generative models, such as Flow Matching [Lipman et al., 2023] and Rectified Flow [Liu et al., 2023], are typically derived top-down via Optimal Transport and the continuity equation. This standard Eulerian approach focuses on the macroscopic transport of probability mass. In this paper, we present an alternative, bottom-up mechanical derivation grounded in a Lagrangian (particle-centric) perspective. By analyzing the local Taylor expansion of a continuous denoiser, we motivate a strict invariance condition required for optimal, singlestep generation: the conservation of target identity. Enforcing this condition yields a governing quasi-linear advection Partial Differential Equation (PDE). We demonstrate that solving this PDE via the Method of Characteristics analytically yields the straight-line trajectories of Flow Matching. This geometric perspective isolates the Jacobian of the denoiser as the primary source of trajectory curvature, providing a direct mathematical explanation for why straight-line flows enable massive step sizes, and why empirical models require distillation to flatten intersecting characteristics.

Flow matching models learn to transport samples from a simple prior distribution to a complex data distribution. When prior-data pairs are coupled via optimal transport (OT), the learned trajectories are straight and non-crossing, enabling fast, even single-step, generation. However, computing the OT coupling in high dimensions is intractable, and existing methods attempt to solve the OT problem, at the cost of persistent bias or significant overhead. Rather than solving for the OT coupling, we reformulate the problem. Once the prior is treated as a design choice rather than a fixed input, the OT coupling between prior and data is no longer unique. Many priors admit an OT-optimal identity coupling to the data, leaving us free to choose one that is also tractable to sample. We identify low-frequency projection of natural images as such a choice. The identity coupling between data and its low-frequency representation is empirically OT-optimal, the prior is structured enough to be sampled by a lightweight model at inference, and the remaining flow-matching task reduces to synthesizing high-frequency detail. Interpolating the prior with Gaussian noise further improves generation quality while preserving the OT coupling. The approach requires no modifications to the flow model itself, and integrates naturally with latent-space models, classifier-free guidance, and one-step generation frameworks. Across all benchmarks, our method reduces trajectory curvature by more than $2\times$ compared to existing flow matching methods, yielding better generation quality in the few-step regime.

The space $\mathcal{P}_2(\mathbb{R}^d$) of probability measures with finite second moment carries a natural geometry: the quadratic Wasserstein distance W_2 makes it a complete metric space and, following Otto, a (formal) Riemannian manifold whose geodesics are the optimal-transport interpolations. On this manifold, the gradient flow of the free energy F(rho) = KL(rho || π) is exactly the Fokker-Planck equation, and its implicit-Euler discretization is the JKO scheme. This is the geometry underlying diffusion models: the forward process descends the free energy, and each denoising step realizes one JKO step, which recovers DDPM, DDIM, NCSN/SMLD, and Energy Matching; this is one scheme, not separate theories. The same manifold supports a second variational principle. Its geodesics - the minimum-action curves of the Benamou-Brenier formula - are precisely the optimal-transport paths that Flow Matching learns. Fixing both endpoints and following the geodesic, generation becomes a deterministic ODE along a straight line, hence far fewer sampling steps. Placing both families of models on one manifold makes their relationship exact: diffusion follows a free-energy gradient flow, an initial-value problem; optimal-transport Flow Matching follows a Wasserstein geodesic, a boundary-value problem. The two reach the same endpoints along different paths.