SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 2, Page 1372-1391, January 2020.
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 a partial differential equation, 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 instants. These kinds of data are provided by the Italian telecommunication company TIM and are derived by mobile phone usage.

中文翻译：

---

通过数据驱动的模型了解质量转移方向及其在手机数据中的应用

SIAM应用动力系统杂志，第19卷，第2期，第1372-1391页，2020年1月。
本文的目的是解决一个关于质量在有界域中运动的逆问题。我们假设质量遵循未知的速度场运动，并且质量密度的演化可以用偏微分方程描述，这也是未知的。问题的输入数据由某些时间的质量分布快照给出，而所需的输出是驱动质量沿其位移的速度场。为此，我们基于两种方法的组合提出了一种算法：首先，我们使用动态模式分解来创建描述传质的数学模型；其次，我们使用Wasserstein距离（也称为推土机距离）的概念来重构引起位移的基础速度场。最后，我们考虑了一个现实生活中的应用程序：该算法用于研究人口稠密地区的人员出行流量，并使用输入数据在不同时刻在给定区域内人员的密度分布（即空间分布）作为输入数据。这些类型的数据由意大利电信公司TIM提供，并通过手机使用得出。