Partial differential equations and machine learning integration for transit-oriented development

The accurate classification of rail transit stations is critical for advancing Transit-Oriented Development (TOD) and promoting sustainable urban growth. This research presents a novel hybrid framework that integrates Partial Differential Equations (PDEs) with Machine Learning (ML) techniques for the classification of rail transit stations. Unlike conventional TOD models, this study applies the heat equation to the Node, Place, Ridership-Time (NPRT) framework, offering a mathematically grounded approach to capture spatial-temporal dynamics in transit systems. This integration represents the first known application of PDE-based physical modeling combined with supervised learning for classifying transit stations within a TOD framework. This approach significantly enhances the model's interpretability while maintaining competitive prediction accuracy. Through extensive case studies and empirical validation, the PDE-NPRT model demonstrates strong performance, with Mean Squared Error (MSE) values ranging from 0.0075 to 0.0222. Although slightly outperformed by enhanced ML models—such as K-Nearest Neighbors (KNN), Deep KNN (DKNN), and Deep Distributed Neural Networks (DDNN)—which achieve MSEs as low as 0.0034, the PDE-NPRT model offers a more interpretable and theoretically robust alternative. Additionally, the study introduces a multi-layer modeling strategy that combines regression analysis, clustering algorithms, PDEs, and neural networks, further enriching the understanding of ridership patterns and congestion mechanisms. Clustering outcomes are validated through external indices, confirming the alignment of model predictions with real-world site characteristics. This work represents a significant advancement in TOD modeling, offering a robust and explainable tool for urban planners and decision-makers. By bridging advanced mathematical modeling with machine learning, it paves the way for more intelligent, data-driven, and sustainable urban mobility strategies.