The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry ($\textit{kinetic energy}$), and the regularization of density paths ($\textit{potential energy}$). These combinations yield different variational problems ($\textit{Lagrangians}$), encompassing many variations of the optimal transport problem such as the Schr\"odinger bridge, unbalanced optimal transport, and optimal transport with physical constraints, among others. In general, the optimal density path is unknown, and solving these variational problems can be computationally challenging. Leveraging the dual formulation of the Lagrangians, we propose a novel deep learning based framework approaching all of these problems from a unified perspective. Our method does not require simulating or backpropagating through the trajectories of the learned dynamics, and does not need access to optimal couplings. We showcase the versatility of the proposed framework by outperforming previous approaches for the single-cell trajectory inference, where incorporating prior knowledge into the dynamics is crucial for correct predictions.

翻译：最优输运的动态形式可通过基础几何（$\textit{动能}$）的不同选择以及密度路径的正则化（$\textit{势能}$）进行扩展。这些组合衍生出不同的变分问题（$\textit{拉格朗日量}$），涵盖了最优输运问题的诸多变体，例如薛定谔桥、非平衡最优输运及带物理约束的最优输运等。通常，最优密度路径是未知的，求解这些变分问题在计算上颇具挑战。利用拉格朗日量的对偶形式，我们提出了一种基于深度学习的新型框架，从统一视角处理所有上述问题。我们的方法无需模拟或反向传播学习动力学的轨迹，也无需访问最优耦合。通过在前沿的单细胞轨迹推断任务中超越先前方法——该任务中，将先验知识融入动力学对正确预测至关重要——我们展示了所提框架的多功能性。