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Continuous Ensemble Weather Forecasting with Diffusion models

Martin Andrae, Tomas Landelius, Joel Oskarsson, Fredrik Lindsten

TL;DR

This paper introduces Continuous Ensemble Forecasting, a diffusion-model framework that samples temporally consistent ensemble trajectories in parallel, conditioned on lead times to produce probabilistic weather forecasts with arbitrary temporal granularity. By fixing the driving noise across lead times or using autocorrelated noise (e.g., Ornstein–Uhlenbeck processes), the method generates coherent trajectories without relying on iterative autoregressive forecasting, while still allowing ARCI—where autoregression plus continuous interpolation refines long horizons. Empirical results on ERA5 WeatherBench data show that ARCI-24/6h achieves competitive RMSE and CRPS across up to 10 days at 6h and 1h resolutions, outperforming several baselines and matching or exceeding existing diffusion-based approaches when using the same backbone architecture. The work highlights the potential of time-continuous diffusion sampling for high-temporal-resolution probabilistic weather forecasting and outlines directions for scaling, refining temporal coherence, and extending to other spatio-temporal forecasting domains.

Abstract

Weather forecasting has seen a shift in methods from numerical simulations to data-driven systems. While initial research in the area focused on deterministic forecasting, recent works have used diffusion models to produce skillful ensemble forecasts. These models are trained on a single forecasting step and rolled out autoregressively. However, they are computationally expensive and accumulate errors for high temporal resolution due to the many rollout steps. We address these limitations with Continuous Ensemble Forecasting, a novel and flexible method for sampling ensemble forecasts in diffusion models. The method can generate temporally consistent ensemble trajectories completely in parallel, with no autoregressive steps. Continuous Ensemble Forecasting can also be combined with autoregressive rollouts to yield forecasts at an arbitrary fine temporal resolution without sacrificing accuracy. We demonstrate that the method achieves competitive results for global weather forecasting with good probabilistic properties.

Continuous Ensemble Weather Forecasting with Diffusion models

TL;DR

This paper introduces Continuous Ensemble Forecasting, a diffusion-model framework that samples temporally consistent ensemble trajectories in parallel, conditioned on lead times to produce probabilistic weather forecasts with arbitrary temporal granularity. By fixing the driving noise across lead times or using autocorrelated noise (e.g., Ornstein–Uhlenbeck processes), the method generates coherent trajectories without relying on iterative autoregressive forecasting, while still allowing ARCI—where autoregression plus continuous interpolation refines long horizons. Empirical results on ERA5 WeatherBench data show that ARCI-24/6h achieves competitive RMSE and CRPS across up to 10 days at 6h and 1h resolutions, outperforming several baselines and matching or exceeding existing diffusion-based approaches when using the same backbone architecture. The work highlights the potential of time-continuous diffusion sampling for high-temporal-resolution probabilistic weather forecasting and outlines directions for scaling, refining temporal coherence, and extending to other spatio-temporal forecasting domains.

Abstract

Weather forecasting has seen a shift in methods from numerical simulations to data-driven systems. While initial research in the area focused on deterministic forecasting, recent works have used diffusion models to produce skillful ensemble forecasts. These models are trained on a single forecasting step and rolled out autoregressively. However, they are computationally expensive and accumulate errors for high temporal resolution due to the many rollout steps. We address these limitations with Continuous Ensemble Forecasting, a novel and flexible method for sampling ensemble forecasts in diffusion models. The method can generate temporally consistent ensemble trajectories completely in parallel, with no autoregressive steps. Continuous Ensemble Forecasting can also be combined with autoregressive rollouts to yield forecasts at an arbitrary fine temporal resolution without sacrificing accuracy. We demonstrate that the method achieves competitive results for global weather forecasting with good probabilistic properties.
Paper Structure (41 sections, 1 theorem, 16 equations, 17 figures, 7 tables, 4 algorithms)

This paper contains 41 sections, 1 theorem, 16 equations, 17 figures, 7 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Work
  4. Ensembles from perturbations.
  5. Generative models.
  6. Temporal resolution.
  7. Spatio-temporal forecasting with diffusion models.
  8. Background
  9. Problem statement.
  10. Autoregressive forecasting.
  11. Conditional Diffusion Models.
  12. Continuous Ensemble Forecasting
  13. Mathematical motivation
  14. Autocorrelated noise
  15. Autoregressive roll-outs with continuous interpolation
  16. ...and 26 more sections

Key Result

Theorem 1

Assume that $f$ satisfies assumption assumption and consider the initial value problem Then, for every $(s_0,z_0)\in D$ and $t\in[0,T]$, there is a unique solution $z(s, s_0,z_0, t)$ with $z(s_0,s_0, z_0, t)=z_0$, and this solution is continuous in $(s,s_0,z_0,t)$ within its domain of definition.

Figures (17)

  • Figure 1: Our proposed framework, Continuous Ensemble Forecasting, generates ensemble weather forecasts using a conditional diffusion model. The model takes lead time as input and forecasts the future weather state in a single step, e.g. the forecast at 24 hours is generated directly from the initial condition, without seeing the intermediate predictions. To ensure temporal consistency, we correlate the driving noises for the different lead times. This can be done by fixing the noise, or by defining a stochastic noise process. Repeating the procedure for different starting noise gives an ensemble of forecasts. Using this framework we can generate 10 day forecasts with 1 hour resolution without sacrificing performance.
  • Figure 2: Diagram depicting the connection between the latent noise space and the solution space.
  • Figure 3: Example forecasts for temperature at 850hPa (t850) at lead time 10 days. The forecasts are generated using ARCI-24/6h except for Deterministic which is sampled using the deterministic model. The bottom row shows 4 ensemble members, randomly chosen out of the 50.
  • Figure 4: RMSE, CRPS and SSR for temperature at 850 hPa (t850).
  • Figure 5: Temporal difference for temperature on 850 hPa (t850) for different values of $\rho$ in algorithm \ref{['alg:ar-noise']}. Choosing $\rho=0$ fixes the noise, $\rho=\text{ln}10$ allows it to vary and $\rho\to\infty$ gives completely uncorrelated noise. The black line refers to the temporal difference of the data.
  • ...and 12 more figures

Theorems & Definitions (1)

  • Theorem 1: Theorem 3.2 in Hale_2009