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Probabilistic Weather Forecasting with Hierarchical Graph Neural Networks
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Graph-EFM is a latent-variable probabilistic weather forecasting model on a hierarchical mesh GNN that samples spatially coherent ensemble members with one forward pass per time step.
Problem
Deterministic machine-learning weather models ignore forecast uncertainty, while existing probabilistic approaches are either ad-hoc perturbation ensembles or computationally expensive diffusion samplers.
Approach
Combine a conditional VAE-style latent variable over the top level of a hierarchical mesh graph with a deterministic GNN predictor, so spatial coherence arises from propagating top-level latent noise down the hierarchy; train with a variational ELBO plus CRPS fine-tuning and use Propagation Networks for better information flow.
Key results
- Global z500 at 10 days: Graph-EFM RMSE 695 and CRPS 299 versus GraphCast* RMSE 808 and CRPS 498, with SpSkR 1.15.
- LAM z500 at 57 h: Graph-EFM RMSE 219 and CRPS 115 with SpSkR 0.75, while GraphCast* RMSE 201 is slightly better.
- Sampling 80 global ensemble members takes about 200 s on a single GPU (2.5 s per member).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a hierarchical mesh instead of a multi-scale mesh (ablation) | Graph-EFM · ERA5 global, z500, lead 5 days | CRPS: 211 -> 169 | Helped |
| Use a hierarchical mesh instead of a multi-scale mesh (ablation) | Graph-EFM · ERA5 global, 2t, lead 10 days | RMSE: 2.55 -> 2.32 | Helped |
| Use a hierarchical mesh instead of a multi-scale mesh (ablation) | Graph-EFM · MEPS Nordic LAM, z500, lead 57 h | CRPS: 470 -> 115 | Helped |
| Use a hierarchical mesh instead of a multi-scale mesh (ablation) | Graph-EFM · MEPS Nordic LAM, wvint, lead 57 h | RMSE: 3.51 -> 2.08 | Helped |
| Add a CRPS loss term during fine-tuning (ablation) | Graph-EFM · ensemble calibration | calibration | Helped |
| Replace the conditioning-dependent latent map with a static distribution (ablation) | Graph-EFM · global forecasting | forecast quality | Hurt |
| Use a diagonal Gaussian predictor and sample its observation noise (ablation) | Graph-EFM · forecast rollout | forecast quality | Hurt |
| Use hierarchical rather than multi-scale graph construction (ablation) | Graph-EFM · MEPS Nordic LAM, ensemble spatial coherence | spatial coherency | Helped |
| Use probabilistic rather than deterministic forecasting | Graph-EFM · MEPS Nordic LAM, short lead times | RMSE | No clear effect |
| Use Graph-EFM rather than deterministic models | global ERA5 forecasting · longer lead times | RMSE | Helped |
| Use Graph-EFM rather than other ensemble models | global ERA5 forecasting · global ensemble forecasts | CRPS | Helped |
| Apply SWAG to GraphCast* to create an ensemble | GraphCast* · global and limited-area forecasting | forecast errors | Hurt |
Surprising findings
- At short lead times in the LAM setting, the deterministic GraphCast* gives lower RMSE than the probabilistic Graph-EFM, so probabilistic modeling brings no clear RMSE benefit there.
- GraphCast*+SWAG ensembles are poorly calibrated (SpSkR about 0.07) and do not improve forecast errors.
Limitations
- Choosing KL and CRPS weighting hyperparameters and the training schedule is complicated; high CRPS weight introduces visual artifacts.
- Forecasts remain somewhat blurry, trading visual fidelity for sampling efficiency relative to diffusion models.
- LAM models are under-dispersed (SpSkR below 1), attributed to boundary forcing constraining the ensemble.
Experimental setup
Datasets: ERA5 WeatherBench 2 MEPS
Benchmarks: ERA5 global 1.5 degree test set (2020) MEPS limited area test set (Jul 2022 - Mar 2023)
Metrics: RMSE CRPS SpSkR
Baselines: GraphCast* GraphCast*+SWAG Graph-EFM (multi-scale) Graph-FM
Builds on: GraphCast Interaction Networks SWAG variational autoencoder
Closest papers by meaning
- Tyche: One Step Flow for Efficient Probabilistic Weather Forecasting (NeurIPS 2026 · similarity 0.86)
- Continuous Ensemble Weather Forecasting with Diffusion models (ICLR 2025 · similarity 0.86)
- Njord: A Probabilistic Graph Neural Network for Ensemble Ocean Forecasting (NeurIPS 2026 · similarity 0.86)
- OneForecast: A Universal Framework for Global and Regional Weather Forecasting (ICML 2025 · similarity 0.85)
- EWMoE: An Effective Model for Global Weather Forecasting with Mixture-of-Experts (AAAI 2025 · similarity 0.84)
- Explanation of Dynamic Physical Field Predictions using WassersteinGrad: Application to Autoregressive Weather Forecasting (NeurIPS 2026 · similarity 0.84)
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