Thermodynamic-Inspired Explainable GeoAI: Uncovering Regime-Dependent Mechanisms in Heterogeneous Spatial Systems

Thermodynamic-Inspired Explainable GeoAI: A Regime-Aware Framework for Heterogeneous Spatial Systems

Motivation and Limitations of Existing Methods

The modeling of spatial heterogeneity and critical transitions in complex geographic systems presents nontrivial challenges, particularly in the presence of nonlinear, regime-dependent mechanisms. Conventional approaches, such as Geographically Weighted Regression (GWR), account for spatial nonstationarity using location-specific coefficients but make restrictive linearity and smoothness assumptions. Deep learning and GeoAI models achieve state-of-the-art predictive performance but largely rely on uninterpretable latent representations, which impedes mechanistic understanding and policy-relevant diagnosis. Post-hoc explainability tools (e.g., SHAP, LIME) are insufficient as they provide only attribution, rather than explicit decomposition of opposing functional pathways or regime-conditioned role reversals—phenomena widely observed in spatial social, ecological, and physical systems.

ZeGNN: Thermodynamic-Inspired Model Structure

The proposed Zentropy-enhanced Graph Neural Network (ZeGNN) explicitly incorporates thermodynamic principles into the learning process by conceptualizing spatial outcomes as an emergent balance between latent Burden ($E$)—representing disorder-inducing/stressor pathways—and Capacity ($S$)—capturing order-maintaining/resilience mechanisms. Predictions in ZeGNN are structured as a free-energy functional:

$F(\mathbf{x}) = E(\mathbf{x}) - T S(\mathbf{x}),$

where $T$ is an adaptive, learnable effective temperature reflecting regime-specific stochasticity.

This framework partitions predictors into burden and capacity feature blocks using domain knowledge, but allows for soft inductive bias rather than a hard constraint: during training, the model can flexibly diagnose and recover regime-dependent role reversals (e.g., sign changes, cross-channel effects). Regime structure is modeled via a spatial graph-based gating network, constructing a thermodynamic mixture-of-regimes formulation where each spatial location can express a probabilistic mixture over possible regimes. This enables ZeGNN to provide both high-fidelity predictions and interpretable latent decompositions, essential for diagnosing where, why, and how spatial processes shift between contrastive mechanisms.

Simulation Studies and Mechanism Recovery

Rigorous controlled simulations demonstrated the identifiability and faithfulness of the ZeGNN structural prior. In settings where ground-truth burden/capacity fields, regime topology, and variable role reversals are known, ZeGNN recovers large-scale spatial structure, regime boundaries, variable sensitivities, and sharp sign transitions. Gradient-matching experiments confirmed that the derivative-based diagnostics produced by the model closely align with true generative mechanisms, unlike conventional models which systematically fail in the presence of nonlinear and regime-dependent effects.

Empirical Results: Socioeconomic, Health, and Environmental Domains

ZeGNN was benchmarked against OLS, GWR, tree ensembles, and state-of-the-art neural baselines on three distinct real-world cases: King County housing markets, U.S. mental health prevalence by neighborhood, and PM2.5 anomalies during the 2023 Canadian wildfire smoke event. Across all domains, ZeGNN achieved the highest spatial transferability as measured by spatial block cross-validation $R^2$ and RMSE. In the housing case, ZeGNN identified spatially consistent burden/capacity-driven submarkets and revealed how variables such as physical size (burden) and construction grade (capacity) locally dominate the valuation mechanism in qualitatively distinct spatial clusters.

In the mental health domain, ZeGNN uncovered regime-conditioned roles for community resources and stressors operating at regional scales, with spatially coherent uncertainty peaking at interface zones—indicating regime mixtures where attribution is less stable and interventions may have greater leverage. For PM2.5 during the wildfire episode, ZeGNN diagnosed a phase transition into a burden-dominated regime, exposing a regime-conditioned surge in stressors (e.g., mean sea-level pressure) that classic models misattribute as outliers or local anomalies. Residual spatial autocorrelation (Moran's $I$) was sharply attenuated, indicating the model’s ability to internalize structured spatial dependence rather than processing it as residual noise.

Structure and Interpretation of Regimes and Uncertainty

A key contribution of ZeGNN is the soft, interpretable regime assignment—spatial units maintain probabilistic membership in regimes, and gating entropy highlights transition zones where multiple mechanisms plausibly explain the data. These transition corridors spatially organize ambiguity, supporting mechanistic hypotheses regarding process instability and system criticality. Furthermore, the burden/capacity decomposition is robust to initial feature partitioning, as the model learns data-driven role reversals and composes localized mechanistic explanations based on fitted sensitivities rather than fixed assumptions.

Methodological and Theoretical Implications

This thermodynamic perspective places ZeGNN between purely statistical spatial models and strictly physics-constrained ML: it encodes physically and socially interpretable priors (as in physics-informed ML) while allowing for soft, nonparametric adaptation to empirical complexity—a crucial advantage for geographic systems often characterized by uncertain or heterogeneous process laws. The regime-mixing structure also provides new routes for diagnosing boundaries of mechanistic attribution and potential vulnerability to small covariate perturbations, which is essential for identifying critical leverage points in policy and intervention.

Limitations and Future Directions

Interpretation of local gradients remains associational, not causal, and the practical value of the burden/capacity partition depends on domain-specific expertise. Evaluation is sensitive to how spatial folds are constructed in CV; model selection via information criteria is nontrivial for flexible neural estimators. Opportunities for future work include extension to spatiotemporal graphs (where latent regimes and sensitivities accommodate temporal evolution), incorporation of stronger physical/institutional constraints, and harmonization with other energy-based models emerging in theoretical GeoAI.

Conclusion

ZeGNN is a regime-aware, thermodynamic-inspired spatial AI framework that couples the predictive strength of neural architectures with structured, interpretable mechanism decomposition. It advances GeoAI by embedding mechanistic transparency into spatial prediction models, enabling the explicit discovery of regime-dependent role reversals, spatially organized uncertainty, and the structural disentanglement of stressor and buffering dynamics in complex, heterogeneous systems. This approach yields actionable insights for theory-driven hypothesis generation and targeted interventions in geography, environmental science, and allied domains.