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Generating Origin-Destination Matrices in Neural Spatial Interaction Models

Ioannis Zachos, Mark Girolami, Theodoros Damoulas

TL;DR

Addresses reconstructing a discrete $I\times J$ origin-destination matrix $T_{ij}$ from partial statistics for high-resolution ABMs, avoiding discretisation errors from continuous relaxations. Proposes GeNSIT, a neural-physics framework that jointly calibrates a continuous SIM via a neural differential equation and samples the discrete ODM space constrained by ${\mathcal{C}}$, with $O(IJ)$-scaling. Contributions include a joint/disjoint sampling framework with Markov Basis MCMC for discrete tables, neural calibration of SIM parameters $\boldsymbol{\theta}=(\alpha,\beta)$ driving $\Lambda_{ij}$ via Harris-Wilson dynamics, and validation on Cambridge ($I=69$, $J=13$) and Washington, DC ($I=J=179$) showing improved SRMSE and 99% CP at substantially lower compute. This framework extends to other contingency-table inference problems and integrates physics-based dynamics with neural calibration, while discussing limitations and social impacts.

Abstract

Agent-based models (ABMs) are proliferating as decision-making tools across policy areas in transportation, economics, and epidemiology. In these models, a central object of interest is the discrete origin-destination matrix which captures spatial interactions and agent trip counts between locations. Existing approaches resort to continuous approximations of this matrix and subsequent ad-hoc discretisations in order to perform ABM simulation and calibration. This impedes conditioning on partially observed summary statistics, fails to explore the multimodal matrix distribution over a discrete combinatorial support, and incurs discretisation errors. To address these challenges, we introduce a computationally efficient framework that scales linearly with the number of origin-destination pairs, operates directly on the discrete combinatorial space, and learns the agents' trip intensity through a neural differential equation that embeds spatial interactions. Our approach outperforms the prior art in terms of reconstruction error and ground truth matrix coverage, at a fraction of the computational cost. We demonstrate these benefits in large-scale spatial mobility ABMs in Cambridge, UK and Washington, DC, USA.

Generating Origin-Destination Matrices in Neural Spatial Interaction Models

TL;DR

Addresses reconstructing a discrete origin-destination matrix from partial statistics for high-resolution ABMs, avoiding discretisation errors from continuous relaxations. Proposes GeNSIT, a neural-physics framework that jointly calibrates a continuous SIM via a neural differential equation and samples the discrete ODM space constrained by , with -scaling. Contributions include a joint/disjoint sampling framework with Markov Basis MCMC for discrete tables, neural calibration of SIM parameters driving via Harris-Wilson dynamics, and validation on Cambridge (, ) and Washington, DC () showing improved SRMSE and 99% CP at substantially lower compute. This framework extends to other contingency-table inference problems and integrates physics-based dynamics with neural calibration, while discussing limitations and social impacts.

Abstract

Agent-based models (ABMs) are proliferating as decision-making tools across policy areas in transportation, economics, and epidemiology. In these models, a central object of interest is the discrete origin-destination matrix which captures spatial interactions and agent trip counts between locations. Existing approaches resort to continuous approximations of this matrix and subsequent ad-hoc discretisations in order to perform ABM simulation and calibration. This impedes conditioning on partially observed summary statistics, fails to explore the multimodal matrix distribution over a discrete combinatorial support, and incurs discretisation errors. To address these challenges, we introduce a computationally efficient framework that scales linearly with the number of origin-destination pairs, operates directly on the discrete combinatorial space, and learns the agents' trip intensity through a neural differential equation that embeds spatial interactions. Our approach outperforms the prior art in terms of reconstruction error and ground truth matrix coverage, at a fraction of the computational cost. We demonstrate these benefits in large-scale spatial mobility ABMs in Cambridge, UK and Washington, DC, USA.

Paper Structure

This paper contains 34 sections, 1 theorem, 20 equations, 10 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Spatial Interaction Intensities and Contingency Tables
  3. Neural Calibration of Spatial Interaction Models
  4. Constrained Table Sampling
  5. Markov Basis MCMC
  6. Experimental Results
  7. Synthetic ODM Scalability
  8. Real-world ODM Reconstructions and Predictions
  9. Cambridge, UK
  10. Washington, DC, USA
  11. Concluding Remarks
  12. Nomenclature
  13. Theory and Methodology Foundations
  14. Continuous ODM
  15. Spatial Interaction Modelling Choice
  16. ...and 19 more sections

Key Result

Proposition 3.1

(Adapted from diaconis1998): Let $\mu$ be a probability measure on $\mathcal{T}_{{\textcolor{#E20000}{\mathcal{C}}}}$. Given a Markov basis $\mathcal{M}$ that satisfies def:markov_basis, generate a Markov chain in $\mathcal{T}_{{\textcolor{#E20000}{\mathcal{C}}}}$ by sampling $l$ uniformly at rando and move to $\mathbf{T}' = \mathbf{T} + \eta \mathbf{f}_l$ for the choice of $\eta$. An aperiodic,

Figures (10)

  • Figure 1: The ground truth discrete ODM (two-way contingency table) can be reconstructed through either multiple expensive ABM simulations anirudh2022 (ABM rectangle) or approximated by a continuous representation $\boldsymbol{\Lambda}$ coupled with the Harris-Wilson SDE (SIM rectangle). In the latter, the ground truth can be reconstructed by sampling in the discrete combinatorial space of constrained ODMs conditioned on $\boldsymbol{\Lambda}$ (GeNSIT rectangle). ABM simulations scale with $\mathcal{O}(M\log(M))$ compared to $\textsc{GeNSIT}$ which scales with $\mathcal{O}(IJ)$, where $M\gg I+J$ is the size of the agent interaction graph.
  • Figure 2: The space $\mathcal{T}_{{\textcolor{#E20000}{\mathcal{C}}}}$ of $3\times3$ discrete ODMs with summary statistics ${\textcolor{#E20000}{\mathcal{C}_{T}}}$. Sampling on the continuous relaxation of $\mathcal{T}_{{\textcolor{#E20000}{\mathcal{C}}}}$ ($\boldsymbol{\Lambda}$ level) with quantisation can lead to either large rejection rates, or poor exploration of the distribution over $\mathcal{T}_{{\textcolor{#E20000}{\mathcal{C}}}}$.
  • Figure 3: GeNSIT : (a) successive iterations of Alg. \ref{['alg:neural_sde_table_inference']} for a given ensemble member, (b) plate diagram for every iteration, ensemble member. We propose two schemes: a Joint and a Disjoint (see App. \ref{['app:disjoint_vs_joint']} for details). Contrary to the latter, the former passes table $\mathbf{T}$ information to the loss $\mathcal{L}$ (see in (b)). We perform an optimisation step in the intensity $\boldsymbol{\Lambda}$ space and a sampling step in $\mathbf{T}$ space, with associated complexities $\mathcal{O}(\tau J+IJ)$ and $\mathcal{O}(IJ)$. The $\boldsymbol{\Lambda}$ arises by the well-known family of SIMs \ref{['eq:totally_constrained_sim']},\ref{['eq:singly_constrained_sim']} coupled with the HW-SDE \ref{['eq:harris_wilson_sde']}. The $\mathbf{T}$ sampling step generates discrete ${\textcolor{#E20000}{\mathcal{C}_{T}}}$-constrained ODMs contrary to ellam2018gaskin2023, which only operate on the continuous mean-field level $\boldsymbol{\Lambda}$.
  • Figure 4: Total computation time (left) of GeNSIT and SIM-NNgaskin2023 versus the number of origin-destination pairs $IJ$ with $(I\times J)$ equal to $100\times100,200\times200,\dots,1000\times1000$. The two algorithms are run for $N=10^3$ iterations with ${\textcolor{#E20000}{\mathcal{C}_{T}}}=\{{\textcolor{#E20000}{\mathbf{T}_{\cdot+}}},{\textcolor{#E20000}{\mathbf{T}_{+\cdot}}},{\textcolor{#E20000}{\mathbf{T}_{\mathcal{X}_{50\%}}}}\}$. Constraint ${\textcolor{#E20000}{\mathbf{T}_{\mathcal{X}_{50\%}}}}$ means that 50% of table cells chosen uniformly at random are fixed. Total computation time is the sum of the $\mathbf{T}$ sampling (middle) and $\boldsymbol{\Lambda}$ learning (right) times. Intensity learning and table sampling times are computed for lines \ref{['alg:line:intensity_start']}-\ref{['alg:line:intensity_end']} and lines \ref{['alg:line:table_start']}-\ref{['alg:line:table_end']} of Alg. \ref{['alg:neural_sde_table_inference']}, respectively. Our framework scales linearly with $IJ$, which is much faster than SIM-MCMC and SIT-MCMC. SIM-NN does not operate at all in the discrete table space, which explains its faster computational speed.
  • Figure 5: SRMSE by total number of agents $A$ (left) and ODM dimension $(I\times J)$ (right) of GeNSIT's discrete ODM sampling (line \ref{['alg:line:table_end']} of Alg. \ref{['alg:neural_sde_table_inference']}) for $N=10^4$ iterations, a fixed intensity $\boldsymbol{\Lambda}$ and ${\textcolor{#E20000}{\mathcal{C}_{T}}}=\{{\textcolor{#E20000}{\mathbf{T}_{\cdot+}}},{\textcolor{#E20000}{\mathbf{T}_{\mathcal{X}_{50\%}}}}\}$. On the left we set $(I\times J)=150\times150$. The reconstruction error (SRMSE) scales linearly in $IJ$ and exponentially in $A={\textcolor{#E20000}{T_{++}}}$.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1