Investigations of a Traffic Flow Model which is based on a Non-Linear Conservation Law
Abstract
Nonlinear conservation laws play a pivotal role in various scientific fields, including engineering, fluid dynamics, physics, and biology, as they describe a wide range of wave propagation and transport phenomena. This thesis centers on hyperbolic conservation laws, which are integral to numerous scientific and engineering disciplines. A main challenge in these equations is that the solution typically creates shocks, i.e., one or several jumps moving in space and possibly changing over time. Given this complexity, explicit solutions are often unattainable, necessitating the development of numerical methods to approximate or simulate their solutions. To design efficient numerical methods, it is crucial to understand the analytical structure of conservation law solutions. Therefore, the theoretical properties of these solutions, which are pertinent to the design and analysis of numerical schemes, are briefly discussed. This thesis addresses the design and implementation of such numerical methods, specifically through solving the Lighthill-Whitham-Richards (LWR) traffic flow model.Recent advancements in artificial intelligence have introduced innovative approaches to solving partial differential equations (PDEs) using machine learning tools. In the context of conservation laws, deep learning has significantly contributed to extracting solutions, with neural networks being utilized to solve inverse problems. This study explores a novel neural network, the symbolic multilayer neural network (S-Net), designed for learning various types of nonlinear functions. Inspired by techniques in recent literature, S-Net is employed to learn the unknown flux function in nonlinear conservation laws using observational data. A framework combining an entropy-satisfying numerical scheme with S-Net has been studied. This integrated approach leverages S-Net and the numerical scheme to determine hidden flux functions in nonlinear conservation laws. This method has been applied to a real-life problem, identifying the flux of vehicles using S-Net with observational data obtained from the Rusanov scheme.