Coupling of Lagrangian Mechanics and Physics-Informed Neural Networks for the Identification of Migration Dynamics

An essential aspect of any government in a smart city is to examine the issues of internal and external migration. Migration is a complex phenomenon. In order to effectively manage it, it is not only necessary to be able to accurately predict migration patterns but also to understand which factors influence these patterns. Current approaches to the development of migration models rely on macroeconomic indicators without considering the specificities of intraregional interactions among individuals. In this paper, we propose a method for determining the dynamics of migration balance based on Lagrangian mechanics. We derive and interpret the potential energy of a migration network by introducing specific functions that determine migration patterns. The solution of the migration equations and selection of parameters, as well as external forces, are achieved through the use of physics-informed neural networks. We also use external factors to explain the non-homogeneity in the dynamic equation through the use of a regression model. We analyze settlement priorities using transfer operator theory and invariant density. The findings obtained enable the assessment of migration flows and analysis of external migration factors.