StationPDE: Station-Oriented Surface PDE Learning for Multi-Station Multivariate Weather Forecasting

2026-08-20

AI Summary

This study addresses the limitations of multi-site weather forecasting, where discrete statistical models lack physical evolution mechanisms and continuous PDE models depend on upper-air observational data. To overcome these bottlenecks, we propose a station-oriented surface PDE learning framework. This method constructs a terrain-aware continuous surface field, decomposing it into wind advection and upper-air inference processes. Crucially, it approximates the influence of missing upper-air variables through learnable horizontal diffusion, integrating parallel data-driven diffusion branches with an adaptive routing mechanism to achieve precise predictions. Extensive evaluations on the Weather2K and MeteoNet benchmarks demonstrate that the proposed approach outperforms existing state-of-the-art baselines, reducing the average mean squared error (MSE) by approximately 9.6%.

Abstract

Multi-station multivariate weather forecasting aims to forecast future weather variables at multiple weather stations from historical surface observations. Existing station forecasting models learn statistical dependencies among discrete stations, but lack explicit physical evolution. Meanwhile, PDE-based weather models provide interpretable physical dynamics, yet require continuous fields and upper-air variables unavailable in surface station data. To bridge this gap, we propose StationPDE, a station-oriented surface PDE learning model. StationPDE constructs a terrain-aware continuous surface field from discrete station observations and decomposes its physical evolution into surface wind transport and upper-air inference. Surface wind transport explicitly evolves observable weather variables, while upper-air inference uses learnable horizontal diffusion to approximate the missing influence of unavailable upper-air variables. A parallel data-driven diffusion branch captures complementary motion patterns, and an adaptive router integrates the two forecasts for station-level multivariate forecasting. Experiments on Weather2K and MeteoNet show that StationPDE consistently outperforms state-of-the-art baselines, reducing MSE by about $9.6\%$ on average compared with the strongest baseline. Code and implementation details are available at https://github.com/hnu-vis/StationPDE.

Problem

• Multi-station weather forecasting
• Multivariate forecasting
• Surface PDE learning
• Physical evolution
• Discrete station observations

Innovation

• Surface PDE Learning
• Terrain-aware Continuous Field
• Upper-air Inference
• Adaptive Router
• Multivariate Weather Forecasting