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STeP-Diff: Spatio-Temporal Physics-Informed Diffusion Models for Mobile Fine-Grained Pollution Forecasting

Nan Zhou, Weijie Hong, Huandong Wang, Jianfeng Zheng, Qiuhua Wang, Yali Song, Xiao-Ping Zhang, Yong Li, Xinlei Chen

TL;DR

STeP-Diff addresses the challenge of forecasting fine-grained urban air pollution from incomplete mobile-sensor data by marrying conditional diffusion models with DeepONet and a convection-diffusion PDE prior. The framework learns the conditional distribution of the spatio-temporal field while enforcing physics-informed regularization, yielding physically plausible predictions even in data-sparse settings. Empirical results from a 14-day deployment in two cities show substantial improvements over state-of-the-art baselines in MAE, RMSE, and MAPE, and demonstrate strong robustness to irregular data coverage. The work highlights the practicality of physics-guided diffusion for mobile sensing scenarios and points to future enhancements in real-time inference and multi-city data fusion.

Abstract

Fine-grained air pollution forecasting is crucial for urban management and the development of healthy buildings. Deploying portable sensors on mobile platforms such as cars and buses offers a low-cost, easy-to-maintain, and wide-coverage data collection solution. However, due to the random and uncontrollable movement patterns of these non-dedicated mobile platforms, the resulting sensor data are often incomplete and temporally inconsistent. By exploring potential training patterns in the reverse process of diffusion models, we propose Spatio-Temporal Physics-Informed Diffusion Models (STeP-Diff). STeP-Diff leverages DeepONet to model the spatial sequence of measurements along with a PDE-informed diffusion model to forecast the spatio-temporal field from incomplete and time-varying data. Through a PDE-constrained regularization framework, the denoising process asymptotically converges to the convection-diffusion dynamics, ensuring that predictions are both grounded in real-world measurements and aligned with the fundamental physics governing pollution dispersion. To assess the performance of the system, we deployed 59 self-designed portable sensing devices in two cities, operating for 14 days to collect air pollution data. Compared to the second-best performing algorithm, our model achieved improvements of up to 89.12% in MAE, 82.30% in RMSE, and 25.00% in MAPE, with extensive evaluations demonstrating that STeP-Diff effectively captures the spatio-temporal dependencies in air pollution fields.

STeP-Diff: Spatio-Temporal Physics-Informed Diffusion Models for Mobile Fine-Grained Pollution Forecasting

TL;DR

STeP-Diff addresses the challenge of forecasting fine-grained urban air pollution from incomplete mobile-sensor data by marrying conditional diffusion models with DeepONet and a convection-diffusion PDE prior. The framework learns the conditional distribution of the spatio-temporal field while enforcing physics-informed regularization, yielding physically plausible predictions even in data-sparse settings. Empirical results from a 14-day deployment in two cities show substantial improvements over state-of-the-art baselines in MAE, RMSE, and MAPE, and demonstrate strong robustness to irregular data coverage. The work highlights the practicality of physics-guided diffusion for mobile sensing scenarios and points to future enhancements in real-time inference and multi-city data fusion.

Abstract

Fine-grained air pollution forecasting is crucial for urban management and the development of healthy buildings. Deploying portable sensors on mobile platforms such as cars and buses offers a low-cost, easy-to-maintain, and wide-coverage data collection solution. However, due to the random and uncontrollable movement patterns of these non-dedicated mobile platforms, the resulting sensor data are often incomplete and temporally inconsistent. By exploring potential training patterns in the reverse process of diffusion models, we propose Spatio-Temporal Physics-Informed Diffusion Models (STeP-Diff). STeP-Diff leverages DeepONet to model the spatial sequence of measurements along with a PDE-informed diffusion model to forecast the spatio-temporal field from incomplete and time-varying data. Through a PDE-constrained regularization framework, the denoising process asymptotically converges to the convection-diffusion dynamics, ensuring that predictions are both grounded in real-world measurements and aligned with the fundamental physics governing pollution dispersion. To assess the performance of the system, we deployed 59 self-designed portable sensing devices in two cities, operating for 14 days to collect air pollution data. Compared to the second-best performing algorithm, our model achieved improvements of up to 89.12% in MAE, 82.30% in RMSE, and 25.00% in MAPE, with extensive evaluations demonstrating that STeP-Diff effectively captures the spatio-temporal dependencies in air pollution fields.

Paper Structure

This paper contains 53 sections, 2 theorems, 26 equations, 11 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and system overview
  3. Mathematical Setup
  4. Problem Description
  5. System Overview
  6. Methodology
  7. Forecasting with Diffusion Models
  8. Denoising Diffusion Probabilistic Models (DDPM)
  9. Forecasting with DDPM
  10. Spatio-Temporal Physics-Informed Diffusion Models (STeP-Diff)
  11. DeepONet
  12. PDE
  13. PDE-Informed Diffusion Models
  14. Implementation of STeP-Diff for Forecasting
  15. The Architecture of $\epsilon_{\theta}$
  16. ...and 38 more sections

Key Result

Theorem 1

Define the noise term as: where $G(\cdot)$ is the PDE evolution operator defined in Eq. (PDE-l). Then, the pollutant concentration satisfies:

Figures (11)

  • Figure 1: Our air pollution forecasting framework, STeP-Diff, consists of two primary components: Air Pollution Mobile Sensing and the Air Pollution Forecasting Model. The Mobile Sensing component transmits observed measurement data and forecast targets to the Forecasting Model. The Forecasting Model then leverages two core components—DeepONet and PDE-informed Diffusion—to generate accurate predictions.
  • Figure 2: Core components of STeP-Diff
  • Figure 3: Principle of the self-designed portable sensing devices and their installation on mobile platforms.
  • Figure 4: The variation in PM$_{2.5}$ concentration measurements within the selected sub-region of cities over the course of a day is displayed every 4 hours.
  • Figure 5: Visualization of model reconstruction error over a day in selected subregions.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 1
  • proof