Existing machine learning weather forecasting models typically generate forecasts through autoregressive rollouts at a fixed temporal resolution. While highly efficient for long-range prediction, this formulation can suffer from severe error accumulation when used with shorter time steps and does not explicitly encode the locality and temporal continuity of atmospheric dynamics. To address these limitations, we introduce SDECast, a Neural Stochastic Differential Equation (SDE) framework for continuous-time probabilistic weather forecasting. SDECast extends SDE Matching to learn stochastic dynamics directly in physical space, without requiring repeated SDE simulation during training. On a simulated geophysical flow, we show that SDECast recovers meaningful drift dynamics and faithfully reproduces the underlying continuous-time behavior. We then demonstrate its scalability to global weather forecasting at hourly resolution, where SDECast produces skillful probabilistic forecasts for lead times of up to five days.
Figures & tables
Figure 1: The RMSE ( ↓ ), CRPS ( ↓ ), SSR for experiments on SQG.
Figure 2: Comparison of the true drift, SDECast drift, and the six-hour finite-difference estimate.
Figure 3: The RMSE ( ↓ ), CRPS ( ↓ ), SSR of t850 for experiments on ERA5.
Figure 4: Continuity of dynamics in Northwest Europe ERA5 square. Mean and standard deviation (denoted σ ) are computed from a 24-member rollout.
Appendix figures & tables11 assets
Supplementary material from the paper’s appendix.
Appendix
Component
Value
Grid (lat × lon)
32×64
State variables C
5 ( z500,t850,t2m,u10,v10 )
Conditioning channels
12 ( 6 spatial +2 static +4 temporal)
Padding
circular in longitude only
Posterior interpolant
learnable
Volatility gθ
state-independent, scalar per variable and grid cell
Appendix
Table 1: ERA5 model configuration ( 5.625∘ , 32×64 grid).
Component
Value
Grid
64×64 (doubly periodic)
State channels C
2 (surfaces z=0,z=H )
Conditioning
none
Padding
circular in both axes
Posterior interpolant
linear / learnable
Volatility gθ
state-independent, one scalar
Appendix
Table 2: SQG model configuration ( 64×64 doubly-periodic grid).
Setting
ERA5
SQG
Optimiser
AdamW
AdamW
Learning rate
5×10−4
5×10−4
Weight decay
1×10−4
1×10−4
LR warmup
500 steps, linear
500 steps, linear
LR schedule
cosine →10−5
cosine →10−5
Epochs
50
20
Appendix
Table 3: Training and optimisation settings.
Figure 5: Predicted and ground truth change in u10 and z500 for the following lead times (from top to bottom): 1, 2, 3, 6, 12, 24 and 48 hours. Predicted changes were obtained from a single trajectory.
Figure 7: Latitude-weighted RMSE, CRPS and SSR. RMSE and CRPS are given in physical units; z500 in m 2 /s 2 , t850/t2m in K, u10/v10 in m/s) at 1, 6 and 24 h lead times. ClimODE paper does not report its CRPS and SSR so we omit it from the corresponding plots.
Variable
Time (h)
SDECast
Persistence
ARCI-24-1h
AR-1h
DYffusion
ClimODE
z500
1
24.8
50.9
29.0
20.0
65.8
–
6
66.7
235.3
61.7
68.5
93.9
102.9
24
172.0
603.1
142.9
206.7
186.5
193.4
t850
1
0.32
0.40
0.34
0.26
0.39
–
6
0.94
1.59
0.89
0.93
0.96
1.16
24
1.44
3.09
1.26
1.69
1.43
1.55
Appendix
Table 4: Latitude-weighted RMSE ( ↓ ) on ERA5 in physical units (z500 in m 2 /s 2 , t850/t2m in K, u10/v10 in m/s) at 1, 6 and 24 h lead times.
Variable
Time (h)
SDECast
Persistence
ARCI-24-1h
AR-1h
DYffusion
z500
1
13.4
38.6
16.8
12.7
40.7
6
34.0
169.2
33.1
40.6
53.5
24
86.1
352.6
70.6
121.7
96.7
t850
1
0.16
0.26
0.18
0.13
0.22
6
0.48
1.06
0.47
0.52
0.58
24
0.73
2.02
0.66
0.98
0.79
Appendix
Table 5: Latitude-weighted CRPS ( ↓ ) on ERA5 in physical units (z500 in m 2 /s 2 , t850/t2m in K, u10/v10 in m/s) at 1, 6 and 24 h lead times. ClimODE is deterministic and is therefore not evaluated here.
RMSE ↓
Model
Lead time
1 h
3 h
6 h
12 h
24 h
48 h
96 h
Persistence
–
1.635
3.803
5.319
6.756
7.967
8.925
9.665
CFM
6 h
–
–
0.832
1.107
1.664
3.004
5.301
SDECast (linear)
6 h
1.494
2.733
2.284
2.873
3.918
5.535
7.726
SDECast
6 h
0.561
1.055
1.280
1.584
2.277
3.803
5.863
Appendix
Table 6: Root-mean-square error (RMSE) by forecast lead time, on SQG in physical (denormalized) units. Values are the mean over 24 test trajectories.
CRPS ↓
Model
Lead time
1 h
3 h
6 h
12 h
24 h
48 h
96 h
Persistence
–
1.150
2.741
3.915
5.107
6.188
7.077
7.771
CFM
6 h
–
–
0.441
0.591
0.879
1.561
2.822
SDECast (linear)
6 h
0.907
1.729
1.402
1.677
2.172
2.989
4.112
SDECast
6 h
0.303
0.599
0.717
0.868
1.195
1.956
3.163
Appendix
Table 7: Continuous ranked probability score (CRPS) by forecast lead time, on SQG in physical (denormalized) units. Values are the mean over 24 test trajectories.
SSR
Model
Lead time
1 h
3 h
6 h
12 h
24 h
48 h
96 h
Persistence
–
0.000
0.000
0.000
0.000
0.000
0.000
0.000
CFM
6 h
–
–
0.854
0.833
0.810
0.833
0.935
SDECast (linear)
6 h
0.211
0.179
0.334
0.440
0.583
0.817
1.419
SDECast
6 h
0.622
0.552
0.641
0.789
0.945
0.981
0.966
Appendix
Table 8: Spread–skill ratio (SSR) by forecast lead time (dimensionless; SSR=1 indicates a calibrated ensemble). Values are the mean over 24 test trajectories. This is an ensemble-only metric, so the deterministic baseline (Persistence) report 0 and are excluded from highlighting; dashes denote leads the model does not produce.
In precipitation forecasting, not only accuracy but also temporal resolution is critical. However, increasing temporal resolution is constrained by observational limitations and the computational cost of dense discrete modeling. To overcome this limitation, we reformulate precipitation forecasting as a continuous-time dynamical system and propose RainODE, a framework that models precipitation evolution in latent space using a Neural ODE. This formulation enables derivative-consistent temporal dynamics and captures the dominant large-scale advective motion of precipitation systems. Nevertheless, a purely deterministic ODE struggles to represent non-advective intensity changes such as localized growth, decay, and sub-grid variability, often leading to over-smoothed predictions. To address this issue, we introduce a stochastic source modeling module based on a Brownian Bridge formulation, which refines residual intensity variations and restores fine-grained structures while preserving advective consistency. By combining deterministic continuous dynamics with stochastic refinement, RainODE enables arbitrary-time inference while maintaining sharp predictions. Experiments on SEVIR and the newly introduced Radar-based Precipitation Integrated Dataset (RAPID) demonstrate consistent improvements across multiple temporal intervals and precipitation regimes. The code is available at https://github.com/SeongYE/RainODE.
Yeeun Seong, Doyi Kim, Minseok Seo +1
Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea
Multi-station multivariate weather forecasting aims to forecast future weather variables at multiple weather stations from historical surface observations. Existing station forecasting models learn statistical dependencies among discrete stations, but lack explicit physical evolution. Meanwhile, PDE-based weather models provide interpretable physical dynamics, yet require continuous fields and upper-air variables unavailable in surface station data. To bridge this gap, we propose StationPDE, a station-oriented surface PDE learning model. StationPDE constructs a terrain-aware continuous surface field from discrete station observations and decomposes its physical evolution into surface wind transport and upper-air inference. Surface wind transport explicitly evolves observable weather variables, while upper-air inference uses learnable horizontal diffusion to approximate the missing influence of unavailable upper-air variables. A parallel data-driven diffusion branch captures complementary motion patterns, and an adaptive router integrates the two forecasts for station-level multivariate forecasting. Experiments on Weather2K and MeteoNet show that StationPDE consistently outperforms state-of-the-art baselines, reducing MSE by about 9.6% on average compared with the strongest baseline.
Recent probabilistic weather forecasters train stochastic predictors with the continuous ranked probability score (CRPS) to generate each ensemble member in a single forward pass. These models learn the predictive distribution from the forecast context alone, which becomes difficult at longer forecast horizons where uncertainty is high. To learn the predictive distribution more effectively, we introduce auxiliary conditional denoising tasks that predict the same future state from the context and its corrupted version, which provides partial future information that can reduce prediction ambiguity. Building on distributional diffusion models, we learn the conditional distributions of these tasks with a single stochastic predictor by minimizing a proper scoring rule across noise levels. At inference, the predictor can still generate each ensemble member in a single forward pass at the fully corrupted endpoint. Standard CRPS training is recovered as the endpoint-only special case of our formulation, so our framework extends existing CRPS-based forecasters with only additional conditioning inputs. Controlled experiments show that the auxiliary tasks improve one-step forecasting across architectures, with larger gains at longer forecast horizons. The gains extend to high-dimensional global weather forecasting under both training from scratch and fine-tuning, along with improved calibration and potential benefits for generalization under distribution shift.