optimal transport flow matching

This work addresses the time-parametrization optimization problem for continuous-time transport trajectories in flow-based generative modeling: specifically, how to schedule the time axis to minimize the spatial Lipschitz constant of the velocity field induced by a given transport map—thereby reducing learning error and enhancing model stability. We propose a smooth variational approximation framework grounded in Γ-convergence, unifying optimal transport and dynamical systems theory, and derive, for the first time, a closed-form solution for the optimal time schedule. Theoretically, this solution reduces the Lipschitz constant exponentially compared to conventional constant-speed parametrizations (e.g., Wasserstein geodesics), thereby overcoming the fundamental limitation of zero-acceleration paradigms. Our method yields analytic solutions across broad classes of distribution pairs and transport maps, significantly improving generalization capability and training robustness of flow-based generative models.

Existing optimal transport (OT) mapping estimation theory heavily relies on Brenier’s theorem—which requires quadratic cost and absolutely continuous source distributions—rendering it inadequate for stochastic OT mappings with mass splitting, commonly encountered in real-world settings involving singular, discrete, or corrupted source/target distributions. Method: We propose a novel metric to quantify the quality of stochastic OT mappings and develop the first universal, robust, finite-sample optimal risk bound framework. Our approach integrates generalization error analysis, adversarially robust statistical learning, parameterized stochastic mapping modeling, and regularized empirical risk minimization. Contribution/Results: We derive near-optimal finite-sample risk bounds under minimal distributional assumptions. Experiments demonstrate substantial improvements in transport accuracy over conventional OT methods in challenging non-absolutely-continuous and corrupted-data regimes where standard approaches fail.

This work addresses the problem of efficiently learning Wasserstein geodesics and their associated optimal transport velocity fields directly from samples to model the dynamic transport process between a source and a target distribution. Building upon the dynamical formulation of optimal transport, the constrained optimization problem is reformulated as a minimax game, enabling joint approximation of the geodesic, optimal map, and full velocity field using deep neural networks. The proposed method constitutes the first purely sample-driven neural solver capable of handling general cost functions—including the quadratic cost—without requiring explicit density estimation. Experiments on both synthetic and real-world datasets demonstrate that the approach accurately reconstructs geodesics and velocity fields and enables direct sampling from the target distribution, thereby validating its effectiveness and broad applicability.

This work addresses the problem of constructing an exact one-step generative mapping between a source distribution and a singular target distribution supported on a low-dimensional manifold. To this end, it proposes a flow matching approach based on time-independent (autonomous) velocity fields, which directly learns the desired mapping through a conservation equation and provides a dynamical interpretation of the flux constraint arising in the Beckmann formulation of optimal transport. Theoretically, the paper establishes the equivalence between autonomous flows and one-step mappings, unifying Poisson flow generative models with equilibrium matching under quadratic regression loss, thereby resolving inconsistencies in existing methods. Empirically, the proposed approach demonstrates its effectiveness through high-quality image generation on ImageNet at 256×256 resolution.

This work addresses the optimization of flow architectures for generative modeling in dynamic optimal transport, specifically targeting *straight flows*—stochastic processes whose particle accelerations vanish identically and are thus exactly integrable by first-order numerical schemes. Adopting an Eulerian perspective, we model the velocity field via conditional expectations and derive a novel partial differential equation characterizing straight flows: its solutions must satisfy exact balance between conditional acceleration and the divergence of a weighted covariance tensor. We prove that, under affine time interpolation, straight flows exist if and only if the source and target distributions admit a deterministic endpoint coupling. This result provides the first complete necessary and sufficient geometric and probabilistic characterization of zero-acceleration flows. It establishes a rigorous theoretical foundation and constructive design principles for analytically integrable, computationally efficient normalizing flow models.

This work investigates the statistical limits and sample complexity lower bounds for estimating any valid transport map without relying on optimal transport (OT). Formalizing the estimation task within a minimax framework, the study integrates tools from optimal transport theory, minimax analysis, and stability-based hypothesis testing to establish, for the first time, rigorous statistical lower bounds for non-optimal yet valid transport maps. The results demonstrate that under standard stability conditions, the statistical difficulty of estimating an arbitrary valid transport map is comparable to that of estimating the OT map itself. However, when the stability assumption fails, alternative valid maps can substantially improve estimation accuracy, outperforming the OT-based approach.

This study addresses the challenge of learning the time-evolving probability density dynamics of high-dimensional systems from unlabeled snapshot data alone, without access to true trajectory information. The authors propose a two-stage flow-based method: first, they construct time-parameterized transport maps from a reference distribution to the marginal distributions at each observed time point using conditional flow matching; second, they estimate the physical velocity field by regressing on synthetic trajectories generated from these maps, thereby recovering the full dynamical system. This approach uniquely identifies non-gradient physical dynamics—including rotational and cyclic behaviors—without requiring trajectory data, overcoming limitations of classical optimal transport frameworks. The method accurately reconstructs complex collective dynamics in high-dimensional settings, demonstrating its capability to capture intricate, non-equilibrium processes.

This work proposes a continuous normalizing flow method based on the generalized Benamou–Brenier formulation to solve general $p$-cost optimal transport ($p$-OT) problems. The approach parameterizes the velocity field via the gradient of a scalar potential function and introduces a self-induced matching loss, training the model along the straight-line bridge defined by its own endpoints while leveraging maximum mean discrepancy (MMD) for flexible terminal distribution matching. Under regularity, exact terminal matching, and uniqueness assumptions, the authors theoretically establish that zero-loss solutions strictly satisfy the generalized Benamou–Brenier optimality system, thereby exactly recovering the $p$-OT map and dynamics. Experiments demonstrate that the method accurately reconstructs theoretical $p$-OT maps on synthetic data, achieves strong performance in high-dimensional tabular density modeling, and validates the flexibility of terminal matching in sample-wise color transfer tasks.

This work addresses the challenge of efficient and accurate ensemble forecasting in chaotic, turbulent, and stochastic systems by proposing a trajectory-aware surrogate modeling approach. The method uniquely learns the probability flow velocity directly from trajectory data, enabling modeling of trajectory-dependent dynamical quantities—such as fluxes and circulations—through first-order trajectory matching (FTM), without requiring estimation of conventional drift or diffusion coefficients, score functions, or explicit simulation. By integrating a simulation-free one-step training loss with stability analysis, the approach achieves high-fidelity ensemble predictions at low computational cost across diverse stochastic dynamical systems and partial differential equation benchmarks, significantly enhancing both trajectory resolution and predictive efficiency.

This work addresses the absence of a natural geometric structure on paths of probability measures—constrained by the continuity equation—that is amenable to optimization. The authors propose a novel convection Fisher–Rao metric and, for the first time, establish its connection to measure transport optimization. They reveal its geometric essence through three complementary perspectives: its emergence as a zero-noise limit, its characterization as the expected form of the second variation of a large deviation rate functional, and its identification with the Hessian of the dynamical optimal transport action functional. Integrating tools from information geometry, large deviation theory, and variational analysis, the theoretical development—supported by numerical experiments—demonstrates that this metric effectively enables optimal fitting of probability densities, with the Gauss–Newton method proving particularly well-suited for optimizing the associated velocity fields.