Neural Optimal Transport with Lagrangian Costs

TL;DR

This paper introduces a novel neural optimal transport method that incorporates Lagrangian costs, enabling efficient computation of geodesics and transport maps in systems influenced by physical geometry and constraints.

Contribution

It presents a new computational approach for optimal transport with Lagrangian costs, allowing direct transport map computation without ODE solvers, and demonstrates efficiency in low-dimensional systems.

Findings

Efficient computation of geodesics and spline paths.

Ability to output transport maps without ODE solvers.

Effective in low-dimensional physical system examples.

Abstract

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 https://github.com/facebookresearch/lagrangian-ot.