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A Simple Approximate Bayesian Inference Neural Surrogate for Stochastic Petri Net Models

Bright Kwaku Manu, Trevor Reckell, Beckett Sterner, Petar Jevtic

TL;DR

This work tackles parameter recovery in Stochastic Petri Net models with covariate-driven rates where explicit likelihoods are unavailable. It introduces a lightweight 1D-ResNet neural surrogate trained on Gillespie-SPN trajectories to map partial observations to coefficients of covariate-dependent rate functions, with Monte Carlo dropout furnishing calibrated uncertainty. The approach achieves low RMSE (0.043) on synthetic data with 10% missing events and substantially outpaces traditional Bayesian methods in speed, while uncertainty recalibration (STD scaling) improves interval reliability. By integrating environmental covariates and demonstrating real-time capable inference, the method offers a practical pathway for data-driven, likelihood-free SPN parameter estimation in epidemiology and related discrete-event systems.

Abstract

Stochastic Petri Nets (SPNs) are an increasingly popular tool of choice for modeling discrete-event dynamics in areas such as epidemiology and systems biology, yet their parameter estimation remains challenging in general and in particular when transition rates depend on external covariates and explicit likelihoods are unavailable. We introduce a neural-surrogate (neural-network-based approximation of the posterior distribution) framework that predicts the coefficients of known covariate-dependent rate functions directly from noisy, partially observed token trajectories. Our model employs a lightweight 1D Convolutional Residual Network trained end-to-end on Gillespie-simulated SPN realizations, learning to invert system dynamics under realistic conditions of event dropout. During inference, Monte Carlo dropout provides calibrated uncertainty bounds together with point estimates. On synthetic SPNs with $10\%$ missing events, our surrogate recovers rate-function coefficients with an $RMSE = 0.043$ and substantially runs faster than traditional Bayesian approaches. These results demonstrate that data-driven, likelihood-free surrogates can enable accurate, robust, and real-time parameter recovery in complex, partially observed discrete-event systems.

A Simple Approximate Bayesian Inference Neural Surrogate for Stochastic Petri Net Models

TL;DR

This work tackles parameter recovery in Stochastic Petri Net models with covariate-driven rates where explicit likelihoods are unavailable. It introduces a lightweight 1D-ResNet neural surrogate trained on Gillespie-SPN trajectories to map partial observations to coefficients of covariate-dependent rate functions, with Monte Carlo dropout furnishing calibrated uncertainty. The approach achieves low RMSE (0.043) on synthetic data with 10% missing events and substantially outpaces traditional Bayesian methods in speed, while uncertainty recalibration (STD scaling) improves interval reliability. By integrating environmental covariates and demonstrating real-time capable inference, the method offers a practical pathway for data-driven, likelihood-free SPN parameter estimation in epidemiology and related discrete-event systems.

Abstract

Stochastic Petri Nets (SPNs) are an increasingly popular tool of choice for modeling discrete-event dynamics in areas such as epidemiology and systems biology, yet their parameter estimation remains challenging in general and in particular when transition rates depend on external covariates and explicit likelihoods are unavailable. We introduce a neural-surrogate (neural-network-based approximation of the posterior distribution) framework that predicts the coefficients of known covariate-dependent rate functions directly from noisy, partially observed token trajectories. Our model employs a lightweight 1D Convolutional Residual Network trained end-to-end on Gillespie-simulated SPN realizations, learning to invert system dynamics under realistic conditions of event dropout. During inference, Monte Carlo dropout provides calibrated uncertainty bounds together with point estimates. On synthetic SPNs with missing events, our surrogate recovers rate-function coefficients with an and substantially runs faster than traditional Bayesian approaches. These results demonstrate that data-driven, likelihood-free surrogates can enable accurate, robust, and real-time parameter recovery in complex, partially observed discrete-event systems.

Paper Structure

This paper contains 24 sections, 17 equations, 14 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Petri Nets: An overview
  4. Petri Nets in Epidemiology
  5. Simulation-Based Inference
  6. Method
  7. Problem Setting
  8. Forward Model (Petri Net Model)
  9. Information Layers Integration
  10. Mathematical Formulations
  11. Parametrization and Sampling
  12. The Inverse Model (Neural Network)
  13. Monte Carlo Dropout (MC Dropout)
  14. Metrics
  15. Predictive Uncertainty Recalibration
  16. ...and 9 more sections

Figures (14)

  • Figure 1: A visual representation of a simple Petri Net. $P_i's$ represent the places (circles) which contains the markings (dark dots). $t_i's$ denotes the transitions (rectangles). The arrows are directed arcs between a place and a transition, which can be assigned specific weights
  • Figure 2: Method Pipeline
  • Figure 3: (a) Represents within and between patch infection, recovery and mortality dynamics (b) Represents between patch migration dynamics (Note: (a) and (b) together form one model). Double arcs indicate tokens to and from a place and transition
  • Figure 4: (a) True parameter mean compared to overall predicted parameter mean with $\pm 1\sigma$ (b) Distribution of true values across test samples compared to distribution of predicted values across test samples
  • Figure 5: True and predicted sample parameters. The blue points represent the predicted sample parameters with $\pm 1\sigma$ error bars
  • ...and 9 more figures