SDECast: Probabilistic Weather Forecasting in Continuous Time with Neural SDEs

1 Introduction

• Autoregressive weather models can accumulate severe errors at shorter timesteps and do not explicitly encode the locality and temporal continuity of atmospheric dynamics.The introduction contrasts these models with differential-equation approaches that evolve the atmosphere through PDEs.Source: §1 · p. 1
• SDECast adapts SDE Matching to learn stochastic dynamics directly in physical space without repeated SDE simulation during training.Neural SDEs add a diffusion term, supporting probabilistic forecasting and ensemble generation.Source: §1 · p. 2, §1.1 · p. 2

1.1 Contributions

• Continuous-time forecasting uses stochastic trajectories from one learned dynamical model to generate ensembles at arbitrary lead times.This capability is presented as a contribution of SDECast.Source: §1.1 · p. 2
• Explicit drift and diffusion functions encode locality and temporal continuity in interpretable physical-space dynamics.These properties are described as inherent to differential equations.Source: §1.1 · p. 2
• Simulation-free training scales Neural SDE learning to high-dimensional weather forecasting without repeated SDE simulation during optimization.The introduction identifies repeated simulation during training as a constraint on applying Neural SDEs to high-dimensional systems.Source: §1.1 · p. 2, §1 · p. 2

2.1 Simulation-free SDE Learning

• The posterior SDE conditions its drift on the observed trajectory while sharing the prior SDE’s diffusion coefficient.The variational objective penalizes the control while reconstructing observations.Source: §2.1 · p. 2
• SDECast evaluates the SDE Matching objective by sampling time and noise and evaluating the posterior parameterization, without integrating the posterior SDE.This analytical drift expression assumes the parameterization is differentiable in time and noise and invertible in noise.Source: §2.1 · pp. 2–3

2.2 Physical-space SDE

• The physical-space variational posterior uses neighboring observations within each interval and matches the physical state at observation times.This Markovian restriction avoids conditioning on the complete trajectory and permits independent evaluation on each observation interval.Source: §2.2 · p. 3
• At inference, the prior SDE starts from the latest observed physical state and is simulated forward to produce forecasts at arbitrary lead times.Independent Wiener paths produce probabilistic forecast ensembles.Source: §2.1 · p. 2, §2.2 · p. 3

3.1 Surface Quasi-Geostrophic Dynamics

• The SQG experiments compare SDECast with autoregressive conditional flow matching and a fixed-linear-posterior ablation, using an autonomous prior with constant, state-independent diffusion.SQG provides a known simulator for assessing both forecast quality and recovery of continuous-time dynamics.Source: §3.1 · pp. 3–4
• Trained on six-hour SQG snapshots, SDECast forecasts hourly with almost ideal SSR, though its RMSE and CRPS are higher than CFM’s; a fixed posterior substantially degrades performance.CFM is restricted to its trained temporal resolution, whereas SDECast generates forecasts at a finer resolution.Source: §3.1 · p. 4, Figure 1 · p. 4
• The learned SQG drift resembles the true drift more faithfully than a six-hour finite-difference estimate, but underrepresents the highest spatial frequencies.The authors attribute some unresolved high-frequency variability to stochasticity; removing diffusion substantially degrades forecasts and produces overly smooth, mean-like trajectories.Source: §3.1 · p. 4, Figure 2 · p. 4

3.2 Global Weather Forecasting with ERA5

• Despite being trained on the same hourly data, SDECast substantially outperforms the one-hour autoregressive baseline at longer lead times, improves on ClimODE, is comparable to DYffusion, and remains behind ARCI-24-1h.Figure 3 reports t850 RMSE, CRPS, and SSR for experiments on ERA5.Source: §3.2 · p. 5, Figure 3 · p. 5
• SDECast trajectories follow the ground-truth evolution while showing ensemble variation in a 24-member rollout over Northwest Europe.The visualization reports the mean and standard deviation of the ensemble.Source: §3.2 · p. 5, Figure 4 · p. 5

4 Conclusions

• SDECast learns explicit physical-space drift and diffusion from discrete trajectories and generates forecast ensembles at arbitrary temporal resolutions.The framework adapts SDE Matching for simulation-free continuous-time probabilistic forecasting.Source: §4 · p. 5
• Experiments on SQG and ERA5 demonstrate recovery of continuous-time dynamics and scaling to global forecasting, but the posterior and diffusion parameterizations and experimental scope remain limited.Source: §4 · p. 6

A Details on SDE Matching

• The posterior transport Fφ yields samples and posterior-drift evaluations without numerically integrating the SDE during training.Its differentiability in time and noise, and invertibility in noise, support the posterior SDE representation; when diffusion is state-independent, the divergence term vanishes.Source: A Details on SDE Matching · p. 8

A.1 SDECast algorithm

• On each observation interval, the algorithm uses a Gaussian posterior whose mean interpolates the bracketing observations with a learnable correction and whose scale is δ plus a learnable correction.Both corrections vanish at interval endpoints; fixing them to zero and one, respectively, recovers the linear-interpolant Brownian-bridge variant.Source: §A.1 · p. 8

B.1 ERA5

• ERA5 uses five standardized variables on a 32 × 64 latitude–longitude grid, with longitude-periodic convolutions.Training, validation, and test years are 1979–2015, 2016, and 2018, respectively.Source: §B.1 · p. 9
• For ERA5, the drift network receives five state channels plus 12 conditioning channels, while the interpolant receives two endpoint states plus those conditioning channels.The diffusion field has 5 × 32 × 64 entries, and inference uses 64 Euler–Maruyama steps.Source: §B.1 · p. 9
• Baseline implementation details are referenced to [5].Source: §B.1 · p. 9

B.2 SQG

• The SQG state is scaled surface potential temperature on two bounding surfaces, represented by two channels on a doubly periodic 64 × 64 grid.Its convolutions wrap along both spatial axes; there is no extra static or temporal conditioning, and the diffusion coefficient is a scalar.Source: §B.2 · p. 9
• The SQG experiments use the same backbone and model size as ERA5 for the prior drift and learnable interpolant, and inference uses 200 Euler–Maruyama steps.Both linear and learnable interpolants are trained.Source: §B.2 · p. 9

• Training uses the simulation-free objective in equation 4 and AdamW, with linear learning-rate warmup followed by cosine annealing.Source: C Training and Optimisation · p. 9
• δ is set to 0.001 in both experiments, small enough not to affect results while remaining nonzero to avoid division by zero in equation 13.Source: C Training and Optimisation · p. 9
• Training and optimization settings are summarized in Table 3.Source: C Training and Optimisation · p. 9

• Additional ERA5 visualizations include forecast residuals and learned volatility.Figure 5 compares predicted and ground-truth changes in u10 and z500 at 1, 2, 3, 6, 12, 24, and 48 hours; predictions come from a single trajectory.Source: D Additional results on ERA5 · p. 9, Figure 5 · p. 10
• Table 1 identifies the ERA5 model configuration at 5.625° resolution on a 32 × 64 grid.Source: Table 1 · p. 9
• Table 3 summarizes training and optimization settings.Source: Table 3 · p. 11
• Tables 4 and 5 report latitude-weighted ERA5 RMSE and CRPS, respectively, at 1, 6, and 24-hour lead times in physical units.The captions specify units for z500, t850/t2m, and u10/v10; ClimODE is not evaluated in the CRPS table because it is deterministic.Source: Table 4 · p. 11, Table 5 · p. 13
• Figure 7 presents latitude-weighted RMSE, CRPS, and SSR at 1, 6, and 24 hours, omitting ClimODE from CRPS and SSR plots because those metrics are not reported for it.RMSE and CRPS are reported in physical units for z500, t850/t2m, and u10/v10.Source: Figure 7 · p. 12

• Additional SQG scores are provided in Tables 6 to 8.Source: E Additional results on SQG · p. 14
• Table 6 reports SQG RMSE by forecast lead time in physical, denormalized units, averaged over 24 test trajectories.Source: Table 6 · p. 14
• Table 7 reports SQG CRPS by forecast lead time in physical, denormalized units, averaged over 24 test trajectories.Source: Table 7 · p. 14
• Table 8 reports SQG spread–skill ratio by forecast lead time, averaged over 24 test trajectories; SSR = 1 indicates a calibrated ensemble.Persistence is excluded from highlighting because it is deterministic; dashes denote leads the model does not produce.Source: Table 8 · p. 14