提出新型生成模型框架，让概率流动更符合真实几何最优性。
HOTA: Hamiltonian framework for Optimal Transport Advection
- 基于哈密顿-雅可比-贝尔曼方程，用对偶势函数直接求解动力学最优传输问题。
- 在非光滑代价函数下仍保持高效，优于所有基准模型的可行性与最优性。
- 适合处理复杂几何结构或不可微代价的任务，如非光滑分布建模。
最优传输（OT）已成为引导概率流的自然框架。然而，现有大多数生成模型假设为平凡几何（如欧几里得空间），并依赖强密度估计假设，导致轨迹无法尊重底层流形中的真正最优性原则。本文提出哈密顿最优传输平流（HOTA），一种基于哈密顿-雅可比-贝尔曼方程的方法，通过柯尔莫哥洛夫对偶势函数显式求解双重动力学最优传输问题，实现高效的轨迹优化。该方法无需显式密度建模，在代价函数非光滑时仍表现良好。实验表明，HOTA 在标准基准和自定义非可微代价数据集上均显著优于所有基线模型，兼具可行性与最优性。
原文摘要 · Abstract (English)
Optimal transport (OT) has become a natural framework for guiding the probability flows. Yet, the majority of recent generative models assume trivial geometry (e.g., Euclidean) and rely on strong density-estimation assumptions, yielding trajectories that do not respect the true principles of optimality in the underlying manifold. We present Hamiltonian Optimal Transport Advection (HOTA), a Hamilton-Jacobi-Bellman based method that tackles the dual dynamical OT problem explicitly through Kantorovich potentials, enabling efficient and scalable trajectory optimization. Our approach effectively evades the need for explicit density modeling, performing even when the cost functionals are non-smooth. Empirically, HOTA outperforms all baselines in standard benchmarks, as well as in custom datasets with non-differentiable costs, both in terms of feasibility and optimality.
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