Proper Scoring Rule-based Diffusion for Probabilistic Weather Forecasting

Joonhyeong Park; Giung Nam; Hyungi Lee; Kyunghyun Cho; Byoungwoo Park; Juho Lee

cs.LGSep 29, 2026

Proper Scoring Rule-based Diffusion for Probabilistic Weather Forecasting

Authors: Joonhyeong Park, Giung Nam, Hyungi Lee, Kyunghyun Cho, Byoungwoo Park, Juho Lee

Organizations: KAIST · Kookmin University · New York University

Abstract

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.

Figures & tables

Figure 1: Comparison of probabilistic weather forecasting methods . Diffusion-based models (left) train a deterministic denoiser, while CRPS-based models (middle) train a stochastic predictor to generate ensembles. Our method (right) retains the stochastic predictor and adds a sampled corrupted future and its noise level as conditioning inputs ( purple ). These inputs introduce auxiliary conditional denoising tasks learned through shared parameters, while preserving one-step ensemble generation.

Figure 2: FCN3 ( Bonev et al., 2025 ) train and validation normalized CRPS (nCRPS) across forecasting horizons.

Figure 3: Schematic comparison of the context-only forecasting task and auxiliary conditional denoising tasks.

Figure 4: Effect of conditional denoising tasks on one-step forecasting . For 3-day-ahead weather forecasting, corrupted futures with noise levels

t\in[t_{\min},1]

serve as additional training inputs. Panels show validation nCRPS (left), spread–skill ratio (SSR, middle), and test nCRPS change (%) relative to the CRPS-trained baseline (right). Endpoint-only training (

t_{\min}=1

) closely matches the CRPS baseline. Lowering

t_{\min}

broadens the range of auxiliary tasks, mitigating performance degradation.

	CRPS $\downarrow$									RMSE $\downarrow$
	1d			3d			10d			1d			3d			10d
Method	Z500	T850	Q700	Z500	T850	Q700	Z500	T850	Q700	Z500	T850	Q700	Z500	T850	Q700	Z500	T850	Q700
IFS-ENS $\dagger$	24.1	0.370	0.283	59.3	0.550	0.427	263.8	1.343	0.725	45.8	0.718	0.611	133.4	1.105	0.902	623.1	2.819	1.440
GenCast $\dagger$	20.2	0.269	0.222	54.3	0.459	0.367	253.6	1.281	0.676	39.3	0.542	0.498	123.5	0.955	0.807	606.5	2.737	1.375
NeuralGCM $\dagger$	22.9	0.333	0.263	54.9	0.497	0.380	253.4	1.293	0.679	44.0	0.658	0.543	126.2	1.027	0.812	606.8	2.756	1.374
MOSAIC	22.7	0.300	0.239	58.3	0.480	0.372	261.0	1.303	0.680	44.2	0.601	0.525	133.2	1.001	0.814	619.9	2.781	1.383

Table 3: CRPS and ensemble-mean RMSE across representative variables and forecast lead times 3 3 3 Results marked with † are provided for reference, as their configurations differ from our controlled experimental setup. Our primary comparisons evaluate standard CRPS and DDM training under matched settings. .

Figure 7: Relative CRPS difference between DDM- and CRPS-trained MOSAIC across all 84 forecast channels over 10-day rollouts. Blue indicates better performance of DDM.

Figure 8: Ensemble calibration analysis. (Top left) SSR versus forecast lead time for Z500, T850, and Q700. (Bottom left) SSR distributions across all 84 forecast channels at 1-, 3-, and 10-day lead times. (Right) Binned spread–skill plots for the same variables at 1-, 3-, and 10-day lead times.

Objective	Relative (%)	Relative (%)	Absolute
	nCRPS $\downarrow$	nRMSE $\downarrow$	SSR dev. $\downarrow$
CRPS	$+3.10$	$+2.68$	$+0.017$
DDM	$+2.60$	$+2.24$	$+0.007$

Table 4: Changes in performance under the ERA5-to-HRES distribution shift.

Figure 9: (Left) Relative CRPS differences from IFS-ENS for ERA5-trained MOSAIC evaluated on ERA5 and HRES-fc0. (Right) Ensemble track forecasts and IBTrACS best track for Hurricane Ian.

Appendix figures & tables28 assets

Supplementary material from the paper’s appendix.

Appendix

Table 12

Model	Target	Ensemble randomness	CRPS	DDM	$\Delta$
			# Parameters
Swin	residual	latent vector ( $d{=}32$ )	26.82M	26.84M	$+0.09\%$
U-Net	residual	latent vector ( $d{=}32$ )	22.75M	22.76M	$+0.05\%$
FGN	residual	latent vector ( $d{=}32$ )	40.71M	40.80M	$+0.21\%$
FCN3	state	internal multi-scale noise	17.84M	17.85M	$+0.03\%$

Appendix

Table 7: Backbones used in the

2.8125^{\circ}

pilot study.

Setting	Value
Optimizer	AdamW
Learning rate	$1\times 10^{-4}$
Weight decay	$1\times 10^{-4}$
Epochs	50
Batch size	16
Gradient clipping	1.0

Appendix

Table 8: Optimization settings, shared by every

2.8125^{\circ}

pilot run.

Model	Target	Ensemble randomness	CRPS	DDM	$\bm{\Delta}$
			# Parameters
MOSAIC	state	latent vector ( $d{=}32$ )	214.19M	221.11M	$+3.23\%$
U-Cast	residual	Monte Carlo dropout	895.43M	904.96M	$+1.06\%$

Appendix

Table 9: Backbones used in the

1.5^{\circ}

experiments.

	MOSAIC	U-Cast
Training budget	$250{,}000$ steps	$100$ pretraining / $8$ fine-tuning epochs
Effective batch size	$16$	$48$
Optimizer	Muon	Muon / AdamW
Peak learning rate	$10^{-3}$	$7\times 10^{-3}\,/\,7\times 10^{-5}$
Learning-rate schedule	cosine	linear warmup + cosine
Final learning rate	$10^{-6}$	$10^{-8}$

Appendix

Table 10: Training configurations at

1.5^{\circ}

Figure 10: Training loss (solid, right axis) and validation nCRPS (dashed, left axis) of CRPS-trained FGN, FCN3, Swin, and U-Net across forecast horizons at

2.8125^{\circ}

. Training loss uses residual normalization for FGN, Swin, and U-Net and state normalization for FCN3, whereas validation nCRPS uses state normalization for all models. Training and validation losses therefore share the same normalization only for FCN3. Training configurations are provided in Section C.4 .

Figure 11: Effect of conditional denoising tasks on FGN one-step forecasting. FGN is trained for 100 epochs for 3-day-ahead forecasting at

2.8125^{\circ}

. Panels show validation nCRPS (left), validation SSR (middle), and test nCRPS changes (%) relative to the CRPS baseline (right). Test results use checkpoints selected by validation nCRPS, with

\mathrm{NFE}=1

for all DDM configurations.

Objective	Training $t$	nCRPS $\downarrow$	nRMSE $\downarrow$
CRPS	–	0.1915	0.3920
DDM	$t=1$	0.1906 $(-0.47\%)$	0.3906 $(-0.36\%)$
	$t\sim\mathcal{U}[0.75,1]$	0.1853 $(-3.21\%)$	0.3822 $(-2.52\%)$
	$t\sim\mathcal{U}[0.50,1]$	0.1809 $(-5.53\%)$	0.3749 $(-4.37\%)$
	$t\sim\mathcal{U}[0.25,1]$	0.1780 $(-7.03\%)$	0.3699 $(-5.65\%)$
	$t\sim\mathcal{U}[0,1]$	0.1759 $(-8.10\%)$	0.3664 $(-6.55\%)$

Appendix

Table 11: Effect of

t_{\min}

on FCN3- 3d one-step forecasting performance at

2.8125^{\circ}

used in Figure 4 .

Objective	Training $t$	nCRPS $\downarrow$	nRMSE $\downarrow$
CRPS	–	0.1868	0.3847
DDM	$t=1$	0.1861 $(-0.41\%)$	0.3828 $(-0.49\%)$
	$t\sim\mathcal{U}[0.75,1]$	0.1781 $(-4.67\%)$	0.3685 $(-4.20\%)$
	$t\sim\mathcal{U}[0.50,1]$	0.1736 $(-7.11\%)$	0.3611 $(-6.13\%)$
	$t\sim\mathcal{U}[0.25,1]$	0.1687 $(-9.73\%)$	0.3528 $(-8.27\%)$
	$t\sim\mathcal{U}[0,1]$	0.1682 $(-9.96\%)$	0.3517 $(-8.55\%)$

Appendix

Table 12: Effect of

t_{\min}

on FGN- 3d -100ep one-step forecasting at

2.8125^{\circ}

used in Figure 11 . Note that the best checkpoint of the CRPS baseline is the same as FGN- 3d -50ep due to overfitting.

		nCRPS $\downarrow$				nRMSE $\downarrow$
Model	Objective	12h	1d	3d	5d	12h	1d	3d	5d
Swin	CRPS	0.0491	0.0712	0.1839	0.2498	0.0966	0.1429	0.3789	0.5151
	DDM	0.0480	0.0698	0.1628	0.2264	0.0947	0.1406	0.3409	0.4739
	$\Delta$ (%)	-2.1	-2.0	-11.5	-9.4	-1.9	-1.6	-10.0	-8.0
U-Net	CRPS	0.0547	0.0799	0.1759	0.2333	0.1072	0.1581	0.3608	0.4782
	DDM	0.0543	0.0789	0.1687	0.2226	0.1064	0.1579	0.3507	0.4632

Appendix

Table 13: Direct forecasting performance at

2.8125^{\circ}

across four architectures.

Model	RI	Objective	24h	72h	120h	240h	288h	360h
FGN	12h	CRPS	0.0747	0.1420	0.1928	0.2442	0.2514	0.2571
		DDM	0.0743	0.1419	0.1926	0.2443	0.2514	0.2570
		$\Delta$ (%)	-0.5	-0.1	-0.1	+0.0	+0.0	-0.0
	1d	CRPS	0.0792	0.1434	0.1934	0.2444	0.2509	0.2566
		DDM	0.0778	0.1412	0.1910	0.2428	0.2499	0.2559
		$\Delta$ (%)	-1.8	-1.6	-1.2	-0.7	-0.4	-0.3

Appendix

Table 14: Autoregressive rollout nCRPS at

2.8125^{\circ}

Model	RI	Objective	24h	72h	120h	240h	288h	360h
FGN	12h	CRPS	0.1497	0.2986	0.4060	0.5082	0.5220	0.5332
		DDM	0.1490	0.2985	0.4056	0.5085	0.5223	0.5329
		$\Delta$ (%)	-0.4	-0.0	-0.1	+0.1	+0.1	-0.0
	1d	CRPS	0.1586	0.3012	0.4066	0.5083	0.5210	0.5321
		DDM	0.1559	0.2978	0.4031	0.5062	0.5199	0.5314
		$\Delta$ (%)	-1.7	-1.1	-0.9	-0.4	-0.2	-0.1

Appendix

Table 15: Autoregressive rollout nRMSE at

2.8125^{\circ}

. The same runs and conventions as Table 14 .

Figure 12: Effective noise levels by harmonic degree for

\kappa_{0}=0.01

(top) and

\kappa_{0}=0.05

(bottom).

Model	RI	baseline	grid	SpG	HK $\kappa_{0}{=}0.01$	HK $\kappa_{0}{=}0.05$	$\Delta$ (%)
		CRPS	DDM
Swin	12h	0.0491	0.0489	0.0496	0.0484	0.0480	-2.1
	1d	0.0712	0.0712	0.0708	0.0698	0.0704	-2.0
	3d	0.1839	0.1644	0.1628	0.1645	0.1705	-11.5
	5d	0.2498	0.2264	0.2268	0.2297	0.2358	-9.4
U-Net	12h	0.0547	0.0553	0.0561	0.0548	0.0543	-0.7

Appendix

Table 16: Noise scheduling for each backbone and RI at

2.8125^{\circ}

, with the single-step nCRPS. Values are test nCRPS. Shaded cells denote the configuration selected by the lowest validation nCRPS.

Figure 13: Relative performance differences in CRPS (left) and ensemble-mean RMSE (right) between DDM and CRPS training for MOSAIC (top) and U-Cast (bottom). Negative values (blue) indicate that DDM shows better performance than CRPS.

Figure 14: CRPS over 10-day autoregressive rollouts in ERA5 2020 .

Figure 15: Ensemble-mean RMSE over 10-day autoregressive rollouts in ERA5 2020 .

Figure 16: SSR over 10-day autoregressive rollouts in ERA5 2020 .

Figure 17: Effect of NFE on MOSAIC forecasting performance. Relative CRPS differences between DDM and CRPS-trained MOSAIC across 84 forecast channels over 10-day autoregressive rollouts. The same DDM checkpoint is evaluated with

\text{NFE}=1

(left) and

\text{NFE}=2

(right) in ERA5 2020 .

Figure 18: CRPS over 10-day autoregressive rollouts in HRES-fc0 2020 .

Figure 19: Ensemble-mean RMSE over 10-day autoregressive rollouts in HRES-fc0 2020 .

Figure 20: SSR over 10-day autoregressive rollouts in HRES-fc0 2020 .

Figure 21: Hurricane Michael track forecasts from 50-member MOSAIC (top) and U-Cast (bottom) ensembles, comparing CRPS (blue, left) and DDM (purple, right). Michael occurred in 2018, within the validation period, and is included as an illustrative case study.

		Hurricane Ian						Hurricane Michael
		Track Error $\downarrow$			Track CRPS $\downarrow$			Track Error $\downarrow$				Track CRPS $\downarrow$
Method	Objective	1d	3d	5d	1d	3d	5d	1d	2d	3d	4d	1d	2d	3d	4d
MOSAIC	CRPS	73	89	189	51	68	125	42	81	156	231	30	54	110	153
MOSAIC	DDM	38	44	55	26	39	63	25	17	68	138	17	17	47	86
U-Cast	CRPS	39	59	245	30	43	148	36	75	150	308	25	51	112	231
U-Cast	DDM	35	37	210	24	38	123	35	62	107	210	24	40	72	141

Appendix

Table 17: Tropical-cyclone ensemble-mean track error and track CRPS (km).

Method	Objective	Lead time	150 km	200 km	300 km
MOSAIC	CRPS	48 h	11/50 (22%)	31/50 (62%)	48/50 (96%)
MOSAIC	DDM	48 h	23/50 (46%)	47/50 (94%)	50/50 (100%)
U-Cast	CRPS	120 h	13/50 (26%)	18/50 (36%)	28/50 (56%)
U-Cast	DDM	120 h	16/50 (32%)	23/50 (46%)	31/50 (62%)

Appendix

Table 18: Number and percentage of ensemble members within 150, 200, and 300 km of the time-matched IBTrACS center for Hurricane Ian. MOSAIC is evaluated at 48 h and U-Cast at 120 h, corresponding to the inset lead times as shown in Figures 22 – 24 .

Figure 22: 50-member ensemble track forecasts for Hurricane Ian initialized at 12 UTC on 23 September 2022, from MOSAIC (top) and U-Cast (bottom), comparing CRPS (blue, left) and DDM (purple, right). Insets show the 48 h MOSAIC and 120 h U-Cast forecast positions. Dashed circles mark a 150 km radius around the time-matched IBTrACS center.

Figure 23: Same as Figure 22 , but with a 200 km radius.

Figure 24: Same as Figure 22 , but with a 300 km radius.

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Tyche: One Step Flow for Efficient Probabilistic Weather Forecasting

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Probabilistic weather forecasting requires not only accurate trajectories, but calibrated distributions over plausible atmospheric futures. Recent data-driven systems have achieved remarkable deterministic skill, and diffusion-based ensemble forecasters have substantially improved sample realism and uncertainty quantification. However, their inference cost scales with forecast horizon, ensemble size, and the number of denoising steps required for each transition, making large operational ensembles expensive. To address this, we present Tyche, a one-step conditional flow model for efficient probabilistic weather forecasting. Tyche models the conditional forecast distribution with a destination-aware average-velocity flow that maps Gaussian noise directly to future weather states in a single function evaluation (1-NFE). To make this one-step transport learnable in high-dimensional geophysical fields, we derive a JVP-regularized rectification objective that enforces temporal self-consistency across source and destination flow timesteps without explicitly forming Jacobians. The transport field is parameterized by an isotropic Swin-style transformer that preserves fine-scale spatial structure while remaining scalable on global grids. To improve ensemble reliability under autoregressive forecasting, we further introduce a rollout-based finetuning stage with curriculum CRPS calibration supervision. Experiments on ERA5 at 1.5

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