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Elucidated Rolling Diffusion Models for Probabilistic Forecasting of Complex Dynamics
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Proposes Elucidated Rolling Diffusion Models (ERDM), which port EDM's noise schedule, preconditioning, loss weighting and Heun sampler to rolling progressive-noise forecasting, with an external-forecaster initialization and a hybrid 2D+temporal-attention U-Net, outperforming conditional EDM on Navier-Stokes and ERA5.
Problem
Probabilistic forecasting of complex dynamical systems with diffusion models: next-step or uniform-noise window models struggle with temporal dependencies and the progressive growth of uncertainty over lead time, and existing rolling diffusion models are built on DDPM rather than EDM's design princi
Approach
Adapt EDM to rolling windows with snapshot-dependent noise levels (progressive schedule with rho=-10), preconditioning applied per snapshot, a loss weighting that upweights mid-range noise levels, initialization of the first window from an external EDM forecaster, and a 2D U-Net augmented with causal temporal attention.
Key results
- On Navier-Stokes, ERDM achieves about 50% better CRPS at the end of the 64-step rollout than the best baseline (EDM with W=4).
- On ERA5, ERDM improves relative CRPS over single-step EDM by up to 10% across most variables and lead times, and is competitive with IFS ENS and NeuralGCM ENS at long lead times.
- ERDM uses 120 NFEs vs 600 for EDM for a 15-day forecast on a single A100, with similar inference time (209 s vs 237 s) but about 2.3x higher GPU memory.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use progressive noise with rho=-10 rather than the EDM default rho=7 (ablation) | ERDM · Navier-Stokes | CRPS: 2× worse with rho=7 | Helped |
| randomize the noise schedule during training instead of using a fixed optimized schedule (ablation) | ERDM · Navier-Stokes | CRPS: nearly 2× | Hurt |
| add lognormal noise-level loss weighting f(σ) (ablation) | ERDM · Navier-Stokes | performance: >2× | Helped |
| use a dedicated spatiotemporal architecture rather than stacking time into 2D channels (ablation) | ERDM · Navier-Stokes | performance: 4× | Helped |
| use ERDM rather than EDM with window size W=4 | probabilistic forecasting · Navier-Stokes test rollout, 64 time steps | CRPS: 50% better | Helped |
| use ERDM rather than the single-step EDM baseline | probabilistic forecasting · ERA5, selected variables and lead times up to 15 days | relative CRPS: up to 10% | Helped |
| use the rolling window rather than step-by-step EDM generation | forecast generation · ERA5, 15-day forecast, 5-member ensemble | Inference NFEs: 600 -> 120 | Helped |
| use the ERDM hybrid denoiser rather than the EDM 2D architecture | forecast generation · ERA5, 15-day forecast, 5-member ensemble | Inference time (s): 237 -> 209 | Helped |
| use the ERDM hybrid denoiser rather than the EDM 2D architecture | forecast generation · ERA5, 15-day forecast, 5-member ensemble | Inference GPU memory (GB): 21 -> 49 | Hurt |
| use second-order Heun sampling rather than first-order Euler-only sampling (ablation) | ERDM · Navier-Stokes | forecast performance | Helped |
| set churn rate to approximately 0.1 instead of disabling sampling stochasticity (ablation) | ERDM · forecast ensembles | CRPS | No clear effect |
| use progressive temporally correlated noise rather than independent noise (ablation) | ERDM · long-range forecasts | forecast performance | Helped |
Surprising findings
- Randomizing the noise schedule during training degrades performance by nearly 2x, contrary to the randomized training of prior completely randomized schedules.
- EDM-based baselines and PDE-Refiner beat ERDM for the first few Navier-Stokes timesteps, and ERDM's short-range ERA5 forecasts lag IFS ENS.
Limitations
- 3D/temporal-attention denoiser is memory-intensive, limiting scaling to higher resolutions.
- Relies on an external EDM forecaster for first-window initialization and is weaker than IFS ENS on short-range ERA5 forecasts.
- Explicit hand-designed loss weighting may be suboptimal compared to importance sampling.
Experimental setup
Datasets: ERA5 reanalysis (1.5 degree) Navier-Stokes fluid dynamics benchmark (from DYffusion/PDE-Refiner setup)
Benchmarks: ERA5 weather forecasting at 1.5 degree (2021 initial conditions, 64 samples) Navier-Stokes 64-step rollouts
Metrics: CRPS spread-skill ratio power spectra NFEs inference time GPU memory
Baselines: EDM (conditional autoregressive, W=1 to 6) DYffusion PDE-Refiner IFS ENS NeuralGCM ENS Graph-EFM
Builds on: EDM Rolling Sequence Diffusion Models DYffusion GenCast ADM U-Net
Closest papers by meaning
- DiffPTS: Rethinking Diffusion ELBO for Probabilistic Time Series Forecasting (NeurIPS 2026 · similarity 0.90)
- Rolling Diffusion Models (ICML 2024 · similarity 0.90)
- Non-stationary Diffusion For Probabilistic Time Series Forecasting (ICML 2025 · similarity 0.90)
- Auto-Regressive Moving Diffusion Models for Time Series Forecasting (AAAI 2025 · similarity 0.88)
- Continuous Ensemble Weather Forecasting with Diffusion models (ICLR 2025 · similarity 0.88)
- Predicting partially observable dynamical systems via diffusion models with a multiscale inference scheme (NeurIPS 2025 · similarity 0.88)
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