We investigate the optimal transport problem between probability measures
when the underlying cost function is understood to satisfy a least action
principle, also known as a Lagrangian cost. These generalizations are useful
when connecting observations from a physical system where the transport
dynamics are influenced by the geometry of the system, such as obstacles (e.g.,
incorporating barrier functions in the Lagrangian), and allows practitioners to
incorporate a priori knowledge of the underlying system such as non-Euclidean
geometries (e.g., paths must be circular). Our contributions are of
computational interest, where we demonstrate the ability to efficiently compute
geodesics and amortize spline-based paths, which has not been done before, even
in low dimensional problems. Unlike prior work, we also output the resulting
Lagrangian optimal transport map without requiring an ODE solver. We
demonstrate the effectiveness of our formulation on low-dimensional examples
taken from prior work. The source code to reproduce our experiments is
available at this https URL

我们研究了具有一种最小作用原理的拉格朗日成本的概率测度之间的最优传输问题，这些推广在将受系统几何影响的物理系统的观测结果连接起来时非常有用，如障碍物（例如，将屏障函数合并到拉格朗日法中），并允许从先验知识、非欧几里得几何中获得基础系统的知识（例如，路径必须是圆形）。我们的贡献具有计算兴趣，我们展示了有效计算测地线和摊销基于样条的路径的能力，这在低维问题中以前从未做过。与以前的工作不同的是，我们还输出了不需要 ODE 求解器的结果拉格朗日最优传输映射。我们通过来自以往工作的低维示例证明了我们的公式的有效性。可以在此 https URL 上获取重复我们实验的源代码。