# Learning to Transport with Neural Networks

## TL;DR

This paper compares various neural network-based methods for learning optimal transport maps between probability distributions, introducing a novel approach using dynamic flows and supervised learning reductions.

## Contribution

It presents a new approach that reduces optimal transport to supervised learning and compares it with heuristic and mathematically justified methods.

## Key findings

- The novel dynamic flow-based method performs competitively.
- Mathematically justified approaches show stronger theoretical guarantees.
- Heuristic methods are faster but less accurate.

## Abstract

We compare several approaches to learn an Optimal Map, represented as a neural network, between probability distributions. The approaches fall into two categories: ``Heuristics'' and approaches with a more sound mathematical justification, motivated by the dual of the Kantorovitch problem. Among the algorithms we consider a novel approach involving dynamic flows and reductions of Optimal Transport to supervised learning.