  The dynamical formulation of the optimal transport can be extended through
various choices of the underlying geometry (kinetic energy), and the
regularization of density paths (potential energy). These combinations yield
different variational problems (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. We propose a novel
deep learning based framework approaching all of these problems from a unified
perspective. Leveraging the dual formulation of the Lagrangians, 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.
