SDECast: Probabilistic Weather Forecasting in Continuous Time with Neural SDEs
Maria Marchenko, Martin Andrae, Fredrik Lindsten, Christian A. Naesseth
NeurIPS 2026
stat.ML, cs.LG, physics.ao-ph
2026-10-02
SDECast trains a neural SDE without simulating trajectories, giving hourly probabilistic weather forecasts out to five days and beating autoregressive baselines at long leads.
Machine-learning weather models learn a one-step predictor at a fixed time resolution and roll it out autoregressively. At 6-hour steps this is efficient. At shorter steps the errors compound fast, and the setup ignores two inductive biases that differential equations provide for free: temporal continuity and spatial locality. Neural ODEs recover continuity, but existing weather applications are deterministic, and training them with a simulation-based loss is expensive. Neural SDEs learn both a drift and a diffusion term, which makes probabilistic ensembles natural, yet classic training needs finely sampled trajectories or repeated SDE simulation during optimization. Neither scales to global weather. SDECast makes Neural SDE training simulation-free and pushes it to global forecasting.
Two layers: the simulation-free SDE Matching objective below, and a physical-space modification on top.
The object being learned is a prior SDE, dzt = hθ(zt, t) dt + gθ(zt, t) dWt, where the drift hθ is the deterministic tendency and the diffusion gθ sets the noise magnitude. Training brings in a posterior SDE that shares the diffusion but conditions its drift on the observed trajectory, acting as a control that bends prior paths toward the observations; the loss combines the control norm with a reconstruction term. Evaluating this naively means integrating the posterior SDE over and over. SDE Matching sidesteps the solver by parameterizing the posterior marginals directly, zt = Fφ(ε, t, X): with smoothness in t and ε and invertibility in ε, the posterior drift has a closed form, so the loss reduces to sampling (t, ε) and evaluating Fφ.
Vanilla SDE Matching is a latent-variable model; zt has no reason to match the physical state. SDECast forces the two together:
Plugged back in, the reconstruction loss becomes a constant and drops out of the objective entirely. At inference, the prior SDE is initialized at the latest observation and integrated forward with any step size; read it out at whatever lead time you want, and fresh Wiener paths give new ensemble members. On ERA5 the drift and diffusion are also conditioned on time of day and year to absorb diurnal and seasonal cycles, and the diffusion varies by location; the learned uncertainty magnitude visibly separates land from ocean.
On SQG, an idealized geophysical turbulence model with a known simulator, every model trains on 6-hour snapshots and is evaluated at hourly resolution. Against an autoregressive conditional flow-matching baseline, SDECast has higher (worse) RMSE and CRPS but a spread-skill ratio close to the ideal value of 1, and it keeps producing 1-hour forecasts the baseline cannot emit at all. Swapping the learned posterior for fixed linear interpolation degrades results sharply. With ground truth available, the learned drift matches the true drift closely and is more faithful to the instantaneous dynamics than a 6-hour finite-difference estimate, though it underrepresents the highest spatial frequencies and pushes that fast variability into the noise term. Removing the diffusion at inference hurts badly and yields oversmooth, mean-like trajectories. The stochasticity is doing real work.
On ERA5, downsampled to 5.625° with five variables (z500, t850, t2m, u10, v10), trained hourly and evaluated out to five days:
| Baseline | Relative result |
| AR-1h, hourly autoregressive diffusion | SDECast clearly better at longer lead times |
| ClimODE, physics-informed Neural ODE (comparable from 6 h) | SDECast better |
| DYffusion, trained across 1-24 h lead times | comparable |
| ARCI-24-1h, trained across 1-24 h lead times | SDECast behind |
The paper attributes the gap to inputs: ARCI-24-1h conditions on the two previous states, while SDECast and DYffusion propagate a single state under a Markov assumption. Exact metric values are plotted in the figures; the text reports no tabulated numbers.
This is not a leaderboard result, and the paper does not sell it as one. The value is architectural: one dynamical model covers arbitrary lead times and resolutions without per-resolution retraining, ensembles come from resampling Wiener paths rather than rerunning a generative model per step, and the drift and diffusion are explicit functions whose learned magnitudes show where the model expects uncertainty. For practitioners the honest framing is a workshop-scale proof of concept: five variables on a coarse grid, against research baselines in the same family, with no comparison against operational state-of-the-art systems.
The authors flag simplified posterior and diffusion parameterizations (diagonal, with constant diffusion in the SQG setup) and a limited experimental scope. Reading closely adds more. The learned drift covers only coarse time scales, with fine-scale dynamics absorbed into noise, so claims about recovering the physics come with that caveat. The deterministic variant collapses, so the point-forecast story is weak and the case rests on ensembles. Trailing ARCI-24-1h suggests the single-state Markov assumption leaves cheap temporal signal unused. And while training is simulation-free, inference still numerically integrates the SDE at whatever resolution you target; compute cost at scale goes undiscussed.