The paper introduces a mesh‑free kernel method for continuum‑marginal optimal transport, aiming to recover the minimum‑energy velocity field that reproduces a continuous family of probability marginals. By embedding the weak continuity equation into a reproducing kernel Hilbert space, the authors obtain a sample‑only objective that eliminates spatial discretization. The velocity is represented via a linear‑in‑parameters dictionary or neural network and optimized with mini‑batch stochastic techniques, achieving accurate drift recovery and marginal consistency in synthetic experiments, and the framework also extends to the Nelson problem of stochastic optimal transport.
arXiv:2512. 19196v4 Announce Type: replace-cross Abstract: Solving high-dimensional Fokker-Planck (FP) equations remains a challenging problem in computational physics and stochastic dynamics, due to the curse of dimensionality, unbounded domains, and complex probability landscapes.
arXiv:2608. 02487v1 Announce Type: cross Abstract: Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.
The paper introduces COFM, a framework for consistent optimal transport flow matching that uses partially input convex neural networks (PICNN) to parameterize the transport potential. By adding a Hamilton‑Jacobi residual to the training objective, COFM enforces dynamical consistency and supports both one‑step transport and multi‑step ODE sampling without costly inner optimization. Experiments on benchmark datasets show that COFM achieves competitive performance while reducing L^2‑UVP by over 2× and cutting computational time by about 9× compared to state‑of‑the‑art models.
arXiv:2509.26364v3 Announce Type: replace Abstract: The Schr\"odinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certa...
arXiv:2505. 07124v3 Announce Type: replace Abstract: We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle.
arXiv:2610.07602v1 Announce Type: cross Abstract: We develop a mesh-free neural JKO scheme for advection-reaction-diffusion equations with a gradient-flow structure in the Hellinger-Kantorovich (HK)...
arXiv:2602. 02241v2 Announce Type: replace Abstract: Entropic optimal transport (EOT) in continuous spaces with quadratic cost is a classical tool for solving the domain translation problem.
The paper investigates the computational and statistical properties of c‑rectified flow, a cost‑aware variant of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. It shows that ordinary rectified flow can fail to recover optimal transport couplings, whereas iterative c‑rectified flow always converges to the optimal coupling under suitable regularity and uniform‑integrability assumptions, and provides quantitative contraction and exponential convergence guarantees. Additionally, the authors derive minimax‑optimal score estimation rates that, when combined with iterative c‑rectified flow, yield rate‑optimal estimators of optimal transport in high dimensions and nearly parametric rates in low dimensions.
The paper introduces a new optimal transport framework that uses the Barron energy—an intrinsic neural network complexity measure—instead of the traditional kinetic energy. By modeling transport as deep residual networks that map a reference distribution to a target distribution, the authors define a new metric on probability measures and analyze its properties for generative modeling. They demonstrate that diffusion generative models are suboptimal in this Barron geometry and explore how adaptivity can improve alternative generative approaches.
Light Entropic Optimal Transport on Riemannian Manifolds introduces ManifoldLightOT, a lightweight method for learning kernel‑induced entropic optimal transport couplings directly on common manifolds such as spheres, tori, SO(3), and SE(3). By constructing geometry‑specific Gibbs kernels and compatible potential parameterizations, the approach yields closed‑form normalization and directly sampleable conditional distributions. The formulation extends naturally to products of manifolds and optimizes potentials from samples using Monte Carlo estimates, demonstrating superior performance over existing manifold OT methods in synthetic and real‑world experiments.
The paper introduces a geometric framework for reinforcement learning that treats policies as mappings into the Wasserstein space of action probabilities. It establishes a Riemannian structure induced by stationary distributions, defines the tangent space of policies, and characterizes geodesics while addressing measurability concerns. The authors formulate a general RL optimization problem, construct a gradient flow via Otto's calculus, compute the gradient and Hessian of the energy, and demonstrate the approach with numerical examples for low‑dimensional problems and neural‑network‑parameterized policies for high‑dimensional settings.
The paper introduces Schr"odinger-Bass Bridge (SBB) barycenter models—SBB-encoder, SBB-decoder, SBB-symmetric, and SBB-Markovian—for measure-valued data, leveraging directional asymmetry and dual drift-diffusion control. It proves existence of SBB-barycenters under mild assumptions, shows that for Gaussian inputs the decoder admits a Gaussian minimizer simplifying computation, and develops a hybrid Wasserstein gradient particle descent for general measures. Experiments on spatial images and single-cell RNA sequencing demonstrate that the SBB-decoder and Markovian barycenter outperform existing models in distributional interpolation and inference.