# Understanding Mass Transfer Directions via Data-Driven Models with Application to Mobile Phone Data

## TL;DR

This paper presents a data-driven method combining Dynamic Mode Decomposition and Wasserstein distance to infer unknown mass transfer velocity fields from snapshots of mass distribution, applied to mobile phone data to analyze human travel flows.

## Contribution

It introduces a novel algorithm that integrates DMD and Wasserstein distance to solve inverse mass transfer problems using real-world mobile phone data.

## Key findings

- Successfully reconstructed human travel flows from mobile phone density data.
- Demonstrated the effectiveness of the method in large populated areas.
- Provided insights into mass transfer dynamics using real telecommunication data.

## Abstract

The aim of this paper is to solve an inverse problem which regards a mass moving in a bounded domain. We assume that the mass moves following an unknown velocity field and that the evolution of the mass density can be described by partial differential equations (PDEs), which is also unknown. The input data of the problems are given by some snapshots of the mass distribution at certain times, while the sought output is the velocity field that drives the mass along its displacement. To this aim, we put in place an algorithm based on the combination of two methods: first, we use the Dynamic Mode Decomposition to create a mathematical model describing the mass transfer; second, we use the notion of Wasserstein distance (also known as earth mover's distance) to reconstruct the underlying velocity field that is responsible for the displacement. Finally, we consider a real-life application: the algorithm is employed to study the travel flows of people in large populated areas using, as input data, density profiles (i.e. the spatial distribution) of people in given areas at different time instances. This kind of data are provided by the Italian telecommunication company TIM and are derived by mobile phone usage.