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

Station weather forecasting is fundamentally shaped by both complex spatial dependencies across stations and strong physical coupling among weather variables. However, existing studies often consider these relationships separately and use different datasets and experimental settings, hindering systematic assessment of their individual and joint contributions. In this paper, we introduce $M^2$Weather, a benchmark for joint multi-station and multi-variable weather forecasting. Through multi-criteria quality control and station stratification, we collect 2,809 high-quality stations with 5 physically coupled weather variables across three spatial scales: France, Europe, and Global. This multi-scale design lets us examine whether conclusions persist from national to global station networks. We also introduce unified training and evaluation protocols to enable fair comparison of different station-variable modeling paradigms. To further examine the benefits of modeling station-variable relationships, we design a lightweight, plug-and-play adapter. With a trained weather forecasting model, this adapter can introduce missing station or variable relationships without retraining the model. This enables fair and efficient investigation of station-variable relationships. Systematic evaluation of 16 representative models shows the benefits of jointly modeling station and variable relationships. Completing missing relationships further reduces MSE for all adapted models on all three datasets. Together, these results identify the complementary information across stations and variables as an important resource for improving station weather forecasting. Our code can be obtained at https://github.com/hnu-vis/M2-Weather.

Near-surface weather varies over tens to hundreds of meters, yet remains unresolved in analyses and forecasts. We test whether this variation can be inferred without resolving atmospheric dynamics. Combining sparse weather stations, high-resolution Earth observation, and coarse atmospheric dynamics, we infer temperature, dewpoint, and wind at 30-m resolution across the contiguous United States. Against measurements held out in space and time, estimates reduce error by 11-28% relative to the strongest baseline. Within held-out $0.25^\circ$ grid cells, we recover more spatial variance than baselines, explaining nearly half of temperature variability in the median cell. The method captures time-varying differences between locations and produces coherent patterns associated with topography and land cover. Beyond weather, our findings illustrate how sparse observations of a dynamical system can be combined with dense observations of persistent environmental structure to recover otherwise unresolved spatial variability.

We present Varda-single-1.0, a medium-range data-driven weather prediction system built for the Alpine domain. It provides hourly deterministic regional forecasts on a mesh of 1 km resolution and global forecasts on a 31 km mesh. The system comprises two independently trained stretched-grid Graph Transformer models with encoder-processor-decoder architecture, developed in the Anemoi framework: a 6-hourly autoregressive forecaster and a temporal downscaler reconstructing hourly forecasts between the forecaster's steps. Its training curriculum includes pre-training on ERA5 reanalysis data, followed by training on a 20-year kilometre-scale regional reanalysis, and finally fine-tuning on operational kilometre-scale analyses. Verified over one year against operational analyses and surface station observations, Varda-single is competitive with or improves on MeteoSwiss' operational numerical weather prediction baselines for most headline scores and variables. It broadly matches the skill of the high-resolution 1 km ICON-CH1-EPS control at lead times up to +33 h and generally outperforms the 2 km ICON-CH2-EPS control at lead times up to +120 h. Despite competitive aggregate scores, Varda-single underestimates some local wind maxima and produces overly smooth convective precipitation fields, consistent with the smoothing associated with squared-error training. To gain insight into the model's behaviour, we investigate three case studies beyond the aggregated headline scores, and find particular weaknesses in Varda-single's representation of local winds over complex terrain. Varda-single represents an important step in the development of high-resolution ML forecasting over complex terrain, in complementing the operational regional numerical weather prediction models of MeteoSwiss with data-driven models and in providing a pretrained model for researchers and user-specific applications.