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.