Discovering Governing Spatial Interaction Mechanisms in Dynamic Urban Systems

Abstract

Governing equations are fundamental for describing and predicting dynamic urban geographic systems. Unlike physical systems guided by first principles, urban spatiotemporal phenomena emerge from coupled geographic processes that lack deterministic theoretical foundations, making the discovery of governing equations elusive and largely heuristic. Spatiotemporal dynamics in urban systems are often observed as sequential snapshot data of spatial distribution, while the cause of such dynamics is often implied or unknown. In this study, we propose a unified differential equation formalism that decomposes urban dynamics into a time-invariant spatial interaction process and a self-dynamic component. Building on this formalism, we introduce the Urban Discovery Framework (U-Discovery), which integrates hypothesis generation, neural fitting, and governing equation identification for the discovery of governing spatial interaction laws. U-Discovery leverages Large Language Models and literature-based reasoning to propose differential equation candidates. Each candidate was calibrated from the observed spatiotemporal dynamics using a neural fitting method. The candidates are evaluated and ranked based on the fitting error and mathematical complexity. Our synthetic experiments prove that U-Discovery can find the sole governing equation from the simulated dynamics. Empirical experiments in Hennepin County, Minnesota, further demonstrate the potential of U-Discovery in identifying optimal governing laws from real-world human activity dynamics.

Figures (8)

• Figure 1: The Urban Discovery framework (U-Discovery). In Hypothesis Generation (HG), GraphRAG is utilized to construct a knowledge graph from existing literature, extracting scientific priors as context for a Large Language Model (LLM) to propose equation candidates for spatial evolution. In Neural Fitting Examination (NFE), the Urban Differential Equation Network (UrbanDE-Net) evaluates these candidates against observed dynamics by fitting the differential terms. In Governing Equation Identification (GEI), Candidate equations are ranked based on fitting error and complexity. The equation that optimally balances these two metrics is identified as the potential governing equation from the candidate set.
• Figure 2: Workflow and prompts for extracting scientific priors and generating equation candidates via two LLM-based agents: a Scientific Priori Extractor and an Equation Discovery Assistant. Scientific Priori Extractor and Equation Discovery Assistant are two LLM-based agents used in this workflow. (a) The Extractor retrieves knowledge from a literature-based knowledge graph via GraphRAG. (b) The Assistant synthesizes the extracted priors to propose mathematical equations for the spatial evolution term. (c) The structure of the retrieval query prompt, which enforces a strict "no invention" rule to distill raw literature into scientific priors. (d) The structure of the equation generation prompt, which operates under constrained modes, No invention and Partially invention, to translate the extracted priors into equation candidates.
• Figure 3: Model architecture of the Urban Differential Equation Network (UrbanDE-Net). The Urban Dynamics Decoupler takes a geospatial network $G$ and a time window of snapshots $P_{w:w+q-1}$ as input to first generate the system dynamics embedding $H^w$ as a deep representation of the underlying mechanisms. The SE and TD learners then decouple this embedding to estimate the parameters, $\hat{\theta}^{se}$ and $\hat{\theta}^{td}$, for a set of equation candidates. Finally, the Urban Dynamics Simulator integrates the equation candidates proposed by the HG to reconstruct the snapshots. The reconstruction error is then backpropagated to update the weights in the Decoupler, optimizing the parameter estimation process.
• Figure 4: Synthetic spatiotemporal system and dynamics. (a) The ground truth governing equation. (b) A 20-node fully connected spatial network with initial node state $p_{i,0} \sim U(20, 70)$ and edge weight $d_{ij} \sim U(3, 30)$. We colored the lines in (c) and (d) based on the node color in (b) for tracing the dynamics from different initial node states. (c) Additive components of the governing equation driving temporal change: a spatial evolution term with the single-constrained spatial gravity mechanism, a sine function as uniform periodic forcing, and Gaussian noise. (d) The resulting synthetic time-series dynamics.
• Figure 5: Ranking of equation candidates and parameter convergence. (a) Heatmap of equation candidates ranked by $nMSE_\Delta$. The three columns represent $nMSE_\Delta$, the rank of the $nMSE$, and equation complexity. The red star indicates the optimal candidate, the Single-constrained gravity model (POW). The Single-constrained gravity model (POW), marked with a red star, is identified as the optimal candidate. (b) Error frontiers showing $nMSE_\Delta$ and $nMSE$ across different complexity. The red stars denote the minimum error achieved by the top-ranked equation. (c) Training convergence of the parameters for the ground truth equation during optimization. Blue dots represent the initial parameter values, which converge toward the final estimated values (orange dots) near the target ground truth (red dashed lines).