Regime-adaptive partial differential equations for interpretable multi-ethnic urban modeling
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".