Regime-adaptive partial differential equations for interpretable multi-ethnic urban modeling

Machine learning models for societal applications often sacrifice interpretability for accuracy. We present the Multi-Ethnic Spatial Mixture of Experts (MESMoE), an interpretable framework integrating physics-informed modeling with specialized neural experts to predict urban population dynamics across ethnic groups. MESMoE addresses the challenge of capturing heterogeneous mechanisms that vary by ethnicity, spatial context, and temporal period through a learnable router that directs predictions to specialized experts for distinct demographic regimes (colonization, jump process, decline, and PDE-based diffusion). This approach achieves robust performance (R² values of 0.76-0.81 for 5-year forecasts, 0.71 for 10-year forecasts) while substantially outperforming seven baseline models (45-70% improvement in R²) and maintaining full interpretability through physics-informed parameterization. Using Toronto census data spanning two decades, our model reveals previously undetectable patterns including cross-ethnic influence networks and systematic differences in settlement strategies across ethnic groups. Our findings demonstrate that incorporating domain knowledge through regime-specific, physics-informed modeling can simultaneously enhance predictive accuracy and interpretability-challenging the perceived trade-off in machine learning for complex social systems.