Kinetic Models on Networks with Applications Traffic Flow and Supply Chains

动力学方程可以提供各种物理过程的精细描述。我们对存在额外网络结构的物理系统感兴趣。这些系统只是在交通和生产网络的背景下进行了粗略的研究。粗尺度或宏观模型是基于偏微分方程（偏微分方程）的平均量，而细尺度模型是基于偏微分方程的分布函数。考虑到网络拓扑结构，不同弧线上的许多过程必须通过合适的耦合条件进行耦合。在宏观模型的情况下，它们的详细形式是一个正在进行讨论的问题。在本建议中，我们从玻尔兹曼类型的动力学描述开始，应用于交通流，供应链以及气体动力学。动力学方程在尽可能精细的尺度上描述了弧上的动力学。由于这些方程的线性结构，顶点处动力学的耦合条件自然产生。除了对这些耦合系统的分析外，我们还对利用矩闭关系在顶点引入平均量所得到的耦合条件感兴趣。这引入了可能的非线性宏观耦合条件。我们将这些新得到的条件与已知的宏观模型的耦合条件进行了比较。此外，我们在耦合条件中包含离散决策。这些决策可以建模，例如在不同的操作模式或规划和设计决策之间的瞬时切换。对于由动力学模型、耦合条件和可能的离散决策sj控制的网络的仿真，我们通过离散化和重新公式化引入了适当的混合整数规划问题的公式化。我们给出了交通流和供应链应用的数值结果，并将这些结果与现有的宏观方法进行了比较。特别地，我们讨论了一种多尺度方法，结合了生产过程中具有高度非线性行为的部分和混合整数公式的经典离散化概念。

Kinetic equations can provide a fine-scale description of a variety of physical processes. We are interested in physical systems where an additional network structure is present. Such systems have been studied in the context of traffic and production networks on a coarse-scale only. Coarse scale or macroscopic models are based on partial differential equations (PDEs) for averaged quantities whereas fine scale models are based on PDEs for distribution functions. Considering a network topology many processes on different arcs have to be coupled by suitable coupling conditions. In the case of macroscopic models their detailed form is a point of ongoing discussion. In this proposal we start from a kinetic description of Boltzmann type with applications in traffic flow, supply chains as well as gas dynamics. The kinetic equations describe the dynamics on the arcs on the finest possible scale. Coupling conditions for the dynamics at vertices arise naturally due to the linear structure of these equations. Besides the analysis of these coupled systems we are interested in the coupling conditions obtained by introducing averaged quantities at the vertex using moment closure relations. This introduces possible nonlinear macroscopic coupling conditions. We compare these newly obtained conditions with already known coupling conditions for the macroscopic models. Additionally, we include discrete decisions in the coupling conditions. These decisions can model for example instantaneous switching between different modes of operation or planning and design decisions. For the simulation of networks governed by kinetic models, coupling conditions and possibly discrete decisionSj we introduce an appropriate formulation by discretization and reformulation as mixed-integer programming problems. We give numerical results for traffic flow and supply chain applications and compare these results with existing macroscopic approaches. In particular, we discuss a multi-scale approach combining parts of the production process with highly nonlinear behavior and classical discretization concepts with the mixed-integer formulation.