Decomposing Traffic Flow Patterns Under Extreme Weather: A Novel Field-Theoretic Framework Using Discrete Differential Operators

Extreme weather events increasingly challenge urban transportation systems, requiring advanced methods to understand and predict traffic adaptations. Traditional graph-based approaches to traffic analysis, while effective for static conditions, struggle to capture the continuous and dynamic nature of flow patterns during weather disruptions. Current methods lack the mathematical framework to bridge discrete network structures with continuous flow dynamics, particularly in analyzing how traffic patterns reorganize under stress. This paper presents a novel field-theoretic framework for analyzing traffic flow patterns, integrating Hodge decomposition with continuous field visualization. The framework decomposes traffic flows into three fundamental components: gradient flows representing direct movements, curl flows capturing circulation patterns, and harmonic flows revealing background patterns. By transforming discrete network measurements into continuous fields, our method enables comprehensive analysis of flow pattern evolution under varying conditions. Through both synthetic experiments and a case study of London's central district during rainfall events, we demonstrate our framework's ability to reveal previously unobservable traffic adaptation patterns. The method successfully captures the emergence of alternative routes, identifies critical adaptation zones, and quantifies systematic changes in flow organization. This approach provides valuable insights for urban planners and traffic managers, offering a mathematical foundation for understanding network adaptation mechanisms and informing resilience planning strategies.