For scientists to understand the results of experiments and scientific measurements they often need to have an understanding of how a specific input changes the behavior and output of a system. Finding the underlying connection between the inputs and outputs is crucial to make statements and predictions. In many cases, the exact state and behavior of the system is not known but can be approximated probabilistically. Hence, one wants to look at how those two distributions are connected.

One way to approach this problem is by applying the concept of transportation theory. Given a source density function f and a target g, the aim is to find the most efficient way of pushing the function f to g. The efficiency is governed by a cost function that defines an optimal transport. Typically, this cost function is based on the distance how far a point from f has to be moved to g.

This process of finding the optimal transport reveals a connection between the inputs and outputs. Given that we can find the original transportation map, we know the actual relation and can better understand our system. One of the major problems with current optimal transportation approaches is that they cannot find folded maps (or more generally, non-bijective maps) that occur with discontinuous densities. This means that in most cases the true underlying map cannot be determined. Moosmüller, Dietrich, and Kevrekidis describe a novel approach in [1] on how to find the actual transport map by incorporating observational, time-delayed data.

The aim of this thesis is to implement a neural network based on this concept for transporting densities. While there already is a lot of research concerning neural networks that transport densities, those typically rely on optimal transport and do not find the "correct" solution. With this neural network, the concept can be applied to a wider variety of datasets and especially larger ones.

[1] C. Moosmüller, F. Dietrich, and I. G. Kevrekidis, "A geometric approach to the transport of discontinuous densities," SIAM/ASA Journal on Uncertainty Quantification, vol. 8, no. 3, pp. 1012–1035, 2020. doi: 10.1137/19M1275760.