Generalizable Neural Downscaling of Earth System Model Wind Fields via Continuous Dynamics Modeling

Chenxi Yu; Jianan Wei; Hanlin Kong; Hao Sun; Bian He; Wenguan Wang

Figure 1: Overview of our work. (i) Climate downscaling is formulated as a mapping between continuous state fields

\bm{a}

and

\bm{b}

defined over the domain

\Omega

, such that

\bm{b}=G(\bm{a})

, where

G

denotes the theoretical downscaling operator. (ii) A neural operator

G_{\boldsymbol{\theta}}

is trained to approximate the underlying downscaling operator

G

. The HR gridded fields are first progressively coarsened to construct a hierarchy of LR gridded fields, which are used to represent the LR state field

\bm{a}

during training; the neural operator then learns the LR-to-HR function mapping such that

\bm{b}(\bm{x})\approx(G_{\boldsymbol{\theta}}(\bm{a}))(\bm{x})

, and the predicted HR state field is discretized on the target HR grids to compute the loss against the HR reference values for optimizing

G_{\boldsymbol{\theta}}

. (iii) When applying the pretrained neural operator

G_{\boldsymbol{\theta}^{*}}

to bias-corrected LR outputs

\bm{a(x_{\textbf{LR}})}

from ESMs, which are interpreted as representation of the underlying LR state field, it directly generates HR gridded fields

\bm{b(x_{\textbf{HR}})}

discretized on the target HR grids.

Methods	Scale factor	Input grid	MAE		RMSE		95 ${}^{\text{th}}$ PE		5 ${}^{\text{th}}$ PE
Methods	Scale factor	Input grid	$UV$	Speed	$UV$	Speed	$UV$	Speed	$UV$
SD (Interp.)	$\times$ 3	$0.75^{\circ}$	0.142	0.161	0.313	0.367	0.137	0.190	0.140
SD (Clim.)			0.113	0.126	0.244	0.290	0.107	0.137	0.111
Ours			0.046	0.048	0.079	0.083	0.028	0.032	0.028
SD (Interp.)	$\times$ 4	$1^{\circ}$	0.210	0.237	0.411	0.475	0.200	0.279	0.207
SD (Clim.)			0.169	0.190	0.322	0.377	0.158	0.204	0.165

Table 1: Continuous state field mapping performance comparison across the neural downscaling framework and SD baselines.

Methods	MAE (pool)	PCC		ME				95 ${}^{\text{th}}$ PE				5 ${}^{\text{th}}$ PE
Methods	$UV$	$UV$	Speed	$UV$	%	Speed	%	$UV$	%	Speed	%	$UV$	%
BCSD (Interp.)	0.114	0.993	0.985	0.249	55.50	0.306	55.24	0.432	54.98	0.523	56.74	0.422	55.10
BCSD (Clim.)	0.099	0.997	0.992	0.192	65.77	0.254	62.87	0.396	58.70	0.453	62.51	0.386	58.96
Ours	0.048	0.996	0.996	0.195	65.15	0.184	73.08	0.322	66.47	0.309	74.42	0.296	68.48

Table 2: Statistical comparison in CMIP6 [ 9 ] modeling of ERA5 performance across the neural downscaling framework and BCSD baselines.

Figure 2: Qualitative comparison of daily near-surface wind fields. a , Daily near-surface wind field from raw GFDL-ESM4 [ 47 ] simulation. b , Bias-corrected (BC) wind field on the native resolution. c , Near-surface wind field from the target ERA5 reanalysis. d,e,f , Downscaled fields from our neural operator ( d ), BCSD (Clim.) ( e ), and BCSD (Interp.) ( f ), respectively. The local regions highlighted by red boxes and labeled A and B in these panels are enlarged and shown in g and h , respectively.

Figure 3: RAPSD comparison of neural downscaling framework and BCSD baselines. Radial averaged power spectral density (RAPSD) of the zonal wind component

U

, meridional wind component

V

, and wind speed

S

from the downscaled fields. The lower panel shows the enlarged views of the small-scale regime (spatial scales

<100

km). The shaded region denotes

\pm 1

standard deviation over all time steps during the evaluation period.

Figure 4: Bias correction capability comparison of neural downscaling framework and BCSD baselines. a , MAE distribution across speed percentiles for global near-surface wind components and wind speed relative to ERA5, with the shaded region representing the interquartile range of errors (25

{}^{\text{th}}

–75

{}^{\text{th}}

percentile). b , The relative frequency of near-surface wind components and wind speed with respect to ERA5. c , The global distribution of mean EKE bias.

Figure 5: Predictive uncertainty of the neural downscaling framework for daily near-surface wind fields from the GFDL-ESM4 simulations. a,b , The ensemble mean ( a ) and standard deviation ( b ) of the downscaled fields from our framework with stochastic forward passes for the near-surface wind speed, respectively.

Figure 6: Future climate scenario projections downscaled by the neural downscaling framework. a,b , Spatial patterns of future global near-surface wind speed trends produced by our framework ( a ) and the original GFDL-ESM4 [ 47 ] simulations ( b ) under the SSP119, SSP245, and SSP585 scenarios. c , Latitudinally averaged wind speed trends of our framework and the original GFDL-ESM4 simulations over the global domains for the SSP119, SSP245, and SSP585 scenarios, from left to right.

Figure 7: Overview architecture of the downscaling neural operator. a , The overall architecture of the downscaling neural operator. b , The detailed architecture of the Fourier integral operator

F_{\boldsymbol{\theta}_{3}}

, i.e. , the U-shaped FNO, which consists of an encoder-decoder architecture with skip-connections in which convolutional layers are replaced by FNO [ 66 ] blocks. c , The detailed architecture of single FNO block.

Figure S.1: Global MAE distribution comparison for state mapping across the neural downscaling framework and SD baselines. From left to right, MAE of our framework, SD (Clim.), and SD (Interp.), respectively. From top to bottom, MAE of the downscaled fields from coarsened ERA5 reanalysis data with spatial resolution of 1.5

{}^{\circ}\times

1.5 ∘ , 1.25

{}^{\circ}\times

1.25 ∘ , 1

{}^{\circ}\times

1 ∘ , and 0.75

{}^{\circ}\times

0.75 ∘ to HR ERA5 reanalysis with spatial resolution of 0.25 ∘ .

Figure S.2: RAPSD comparison for state mapping across the neural downscaling framework and SD baselines. The RAPSD of the zonal wind component

U

( a ), meridional wind component

V

( b ), and wind speed

S

( c ) from coarsened ERA5 reanalysis data with spatial resolution of 1.5

{}^{\circ}\times

1.5 ∘ , 1.25

{}^{\circ}\times

1.25 ∘ , 1

{}^{\circ}\times

1 ∘ , and 0.75

{}^{\circ}\times

0.75 ∘ to HR ERA5 reanalysis with spatial resolution of 0.25 ∘ for our framework and SD baselines. The lower panel shows the enlarged views of the small-scale regime (spatial scales

<100

km). The shaded region denotes

\pm 1

standard deviation over all time steps during the evaluation period.

Figure S.3: Qualitative comparison of raw ESM simulations and downscaled fields for the same date. a,b , The raw coarse-scale ESM simulations for the MPI-ESM1-2-HR [ 48 ] ( a ), and MRI-ESM2-0 [ 49 ] ( b ) simulations, respectively. c,d , The downscaled fields from our framework for the MPI-ESM1-2-HR ( c ), and MRI-ESM2-0 ( d ) simulations.

Figure S.4: RAPSD comparison of neural downscaling framework and BCSD baselines across different ESMs. a , RAPSD of the zonal wind component

U

, meridional wind component

V

, and wind speed

S

for the MPI-ESM1-2-HR [ 48 ] simulation. b , RAPSD of

U

,

V

, and

S

for the MRI-ESM2-0 [ 49 ] simulation. The lower panel shows the enlarged views of the small-scale regime (spatial scales

<100

km). The shaded region denotes

\pm 1

standard deviation over all time steps during the evaluation period.

Figure S.5: Global distribution of the 95

{}^{\text{th}}

percentile wind speed error relative to ERA5 for the neural downscaling framework, BCSD baselines, and GFDL-ESM4 [ 47 ] simulations. The global distribution of the 95

{}^{\text{th}}

percentile wind speed error relative to ERA5 for our framework ( a ), BCSD (Clim.) ( b ), BCSD (Interp.) ( c ), and GFDL-ESM4 simulations ( d ), respectively.

Figure S.6: Relative frequency distributions of near-surface wind fields downscaled by the neural downscaling framework and BCSD baselines across different ESMs. a , The relative frequency distribution of the near-surface wind fields for MPI-ESM1-2-HR [ 48 ] simulations and their downscaled fields with respect to ERA5. b , The relative frequency distribution of the near-surface wind fields for MRI-ESM2-0 [ 49 ] simulations and their downscaled fields.

Figure S.7: Double-ITCZ bias reduction performance of bias-correction preprocessing (BC) and neural downscaling framework for different ESMs. a , The divergence of the near-surface wind field for the ESM simulations (GFDL-ESM4 [ 47 ] , MPI-ESM1-2-HR [ 48 ] , and MRI-ESM2-0 [ 49 ] ). b , The divergence of the wind field for the bias-corrected ESM simulations. c , The divergence of the wind field for the downscaled fields from our framework. d , The divergence of the wind field for the ground truth ERA5 reanalysis data.

ESMs	Methods	MAE (pool)	PCC		ME				95 ${}^{\text{th}}$ PE				5 ${}^{\text{th}}$ PE
ESMs	Methods	$UV$	$UV$	Speed	$UV$	%	Speed	%	$UV$	%	Speed	%	$UV$	%
MPI- ESM1-2- HR	BCSD (Interp.)	0.115	0.993	0.986	0.250	60.13	0.289	60.82	0.421	58.77	0.456	62.47	0.401	56.94
	BCSD (Clim.)	0.097	0.996	0.992	0.198	68.42	0.241	67.27	0.388	61.91	0.396	67.42	0.370	60.36
	Ours	0.041	0.996	0.996	0.202	67.79	0.189	74.34	0.327	67.91	0.294	75.75	0.299	67.96
MRI- ESM2-0	BCSD (Interp.)	0.147	0.991	0.983	0.275	62.00	0.309	69.93	0.440	68.75	0.509	72.12	0.449	65.18
	BCSD (Clim.)	0.126	0.995	0.991	0.214	70.45	0.251	75.55	0.403	71.42	0.435	76.21	0.411	68.14

Table S.1: Statistical comparison in CMIP6 (MPI-ESM1-2-HR [ 48 ] and MRI-ESM2-0 [ 49 ] ) modeling of ERA5 performance across the neural downscaling framework and BCSD baselines.

ESMs	Methods	MAE (MKE)	%	MAE (EKE)	%
GFDL-ESM4	BCSD (Interp.)	1.081	57.05	1.289	55.33
	BCSD (Clim.)	0.855	66.03	1.243	56.92
	Ours	0.856	65.98	0.816	71.73
MPI-ESM1-2-HR	BCSD (Interp.)	1.090	62.85	1.184	58.49
	BCSD (Clim.)	0.885	69.84	1.144	59.89
	Ours	0.894	69.55	0.833	70.81

Table S.2: Statistical comparison in specific kinetic energy decomposition (MKE and EKE) for CMIP6 (GFDL-ESM4 [ 47 ] , MPI-ESM1-2-HR [ 48 ] , and MRI-ESM2-0 [ 49 ] ) modeling of ERA5 across the neural downscaling framework and BCSD baselines.

Figure S.8: Comparison of temporal consistency across the neural downscaling framework and BCSD baselines. a , Temporal autocorrelation and biases with respect to ERA5 of the zonal wind component

U

for the our framework and the BCSD baselines. b , Temporal autocorrelation and biases of the meridional wind component

V

.

Figure S.9: Uncertainty quantification of neural downscaling framework for the downscaled fields from GFDL-ESM4 [ 47 ] simulations. a,b , The ensemble mean ( a ) and standard deviation ( b ) of the downscaled fields from our framework with stochastic forward passes for the wind speed. c , The CRPS for the ensemble of downscaled fields from our framework with stochastic forward passes for the wind speed. d , The CRPSS for the ensemble of downscaled fields from our framework with stochastic forward passes relative to BCSD (Clim.) baseline ensemble.

Figure S.10: Future climate scenario projection trends under the SSP119, SSP245, and SSP585 scenarios over the global, land and ocean domains. a , Latitudinally averaged wind speed trends of the GFDL-ESM4 simulations without bias correction over the global, land, and ocean domains. b , Latitudinally averaged wind speed trends of the downscaled fields obtained with our framework over the global, land, and ocean domains for three different future scenarios.

ESMs	Methods	MAE (pool)	PCC		ME				95 ${}^{\text{th}}$ PE				5 ${}^{\text{th}}$ PE
ESMs	Methods	$UV$	$UV$	Speed	$UV$	%	Speed	%	$UV$	%	Speed	%	$UV$	%
GFDL- ESM4	BCSD (Interp.)	0.114	0.993	0.987	0.254	54.59	0.300	55.57	0.411	57.37	0.510	57.68	0.427	54.87
	BCSD (Clim.)	0.099	0.996	0.994	0.198	64.54	0.246	63.47	0.374	61.26	0.441	63.35	0.391	58.72
	Ours	0.048	0.996	0.997	0.202	63.96	0.180	73.28	0.315	67.39	0.332	72.46	0.318	66.39

Table S.3: Statistical comparison in SSP245 future climate scenario projections across the neural downscaling framework and BCSD baselines from GFDL-ESM4 [ 47 ] simulations.

ESMs	Methods	MAE (MKE)	%	MAE (EKE)	%
GFDL-ESM4	BCSD (Interp.)	1.077	56.23	1.239	58.57
	BCSD (Clim.)	0.848	65.56	1.191	60.17
	Ours	0.846	65.63	0.785	73.74

Table S.4: Statistical comparison in specific kinetic energy decomposition (MKE and EKE) for SSP245 future climate scenario projections across the neural downscaling framework and BCSD baselines from GFDL-ESM4 [ 47 ] simulations.

Model	$s$	$\times$ 3		$\times$ 4		$\times$ 5		$\times$ 6
Model	$s$	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE
SD (Interp.)	-	0.1602	0.3694	0.2359	0.4775	0.2937	0.5618	0.3586	0.6492
SD (Clim.)	-	0.1256	0.2927	0.1883	0.3794	0.2372	0.4519	0.2929	0.5277
Ours	3	0.0376	0.0663	0.1227	0.2442	0.1847	0.3728	0.2480	0.5665
Ours	6	0.1110	0.3077	0.1477	0.3804	0.1819	0.3758	0.1131	0.1794
Ours	$\mathcal{U}(3,6)$	0.0484	0.0840	0.0664	0.1115	0.0936	0.1505	0.1167	0.1859

Table S.5: Ablation study on the downscaling neural operator with different training scale factor strategies.

Model	$\lambda_{2}$	$\times$ 3		$\times$ 4		$\times$ 5		$\times$ 6
Model	$\lambda_{2}$	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE
SD (Interp.)	-	0.1602	0.3694	0.2359	0.4775	0.2937	0.5618	0.3586	0.6492
SD (Clim.) [ 42 ]	-	0.1256	0.2927	0.1883	0.3794	0.2372	0.4519	0.2929	0.5277
CNN [ 67 ]	0	0.0639	0.1289	0.0941	0.1795	0.1301	0.2395	0.1695	0.3050
SRNO [ 68 ]	0	0.0583	0.1155	0.0837	0.1530	0.1150	0.2019	0.1521	0.2631
DFNO [ 40 ]	0	0.0545	0.0946	0.0699	0.1173	0.0968	0.1556	0.1192	0.1899

Table S.6: State mapping capability of our downscaling neural operator with different model architectures and gradient loss weights.

Figure S.11: Comparison of global MAE distribution for state mapping with different training scale factor strategies. From left to right, MAE of our framework with continuous sampling range

s\sim\mathcal{U}(3,6)

, and with discrete sampling range (3 and 6), respectively. From top to bottom, MAE of the downscaled fields from coarsened ERA5 reanalysis data with spatial resolution of 1.5

{}^{\circ}\times

1.5 ∘ , 1.25

{}^{\circ}\times

1.25 ∘ , 1

{}^{\circ}\times

1 ∘ , and 0.75

{}^{\circ}\times

0.75 ∘ to HR ERA5 reanalysis with spatial resolution of 0.25 ∘ .

Generalizable Neural Downscaling of Earth System Model Wind Fields via Continuous Dynamics Modeling

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