Symmetric Composition of Anisotropic Operators for Global Subseasonal-to-Seasonal Climate Forecasting

Ziyu Zhou; Yuchen Fang; Weilin Ruan; Tian Zhou; Shiyu Wang; James Kwok; Yuxuan Liang

physics.ao-phFeb 4, 2026

Symmetric Composition of Anisotropic Operators for Global Subseasonal-to-Seasonal Climate Forecasting

Authors: Ziyu Zhou, Yuchen Fang, Weilin Ruan, Tian Zhou, Shiyu Wang, James Kwok, Yuxuan Liang

Organizations: The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China · DAMO Academy, Alibaba Group, Hangzhou, China · The Hong Kong University of Science and Technology, Hong Kong SAR, China

Abstract

Accurate global Subseasonal-to-Seasonal (S2S) climate forecasting is critical for disaster preparedness and resource management, yet it remains challenging due to chaotic atmospheric dynamics. Despite advances in geometry-aware representations, existing methods do not specify how the zonal and meridional interactions that govern real atmospheric dynamics should be explicitly coordinated and composed, leaving an important architectural gap for S2S forecasting. In this paper, we propose AnisoCast, which explicitly models anisotropic global atmospheric dynamics for accurate S2S climate forecasting. It couples: (1) an Anisotropic Embedding strategy that tokenizes the global grid into latitudinal rings, preserving the integrity of zonal periodic structures; and (2) stacked Aniso Blocks that arrange latent Zonal and Meridional Operators in a weight-shared palindromic composition inspired by symmetric operator splitting. Extensive experiments on the ERA5 reanalysis dataset demonstrate that AnisoCast establishes a new state-of-the-art, significantly outperforming existing methods in both forecasting accuracy and computational efficiency.

Figures & tables

Figure 1 : The anisotropic nature of global atmospheric dynamics. Global atmospheric circulation is governed by the coupling of two orthogonal physical processes: (i) Zonal dynamics, characterized by east–west wave propagation, and (ii) meridional transport, driven by the north-south exchange of heat and momentum.

Figure 2 : Main architecture of the proposed AnisoCast. The Anisotropic Embedding projects atmospheric initial states into latitudinal ring tokens by compressing the longitudinal dimension into feature channels. These tokens are then processed through a backbone of several stacked Aniso Blocks. Specifically, each block sequentially applies two weight-shared Zonal Operators and a Meridional Operator to separately process zonal- and meridional-oriented latent interactions. Finally, a Decoder maps the evolved features to bi-weekly forecast fields.

	Model	RMSE ( $\downarrow$ )							ACC ( $\uparrow$ )
		Pressure Level				Single Level			Pressure Level				Single Level
		$z500$	$z850$	$t500$	$t850$	$t2m$	$u10$	$v10$	$z500$	$z850$	$t500$	$t850$	$t2m$	$u10$	$v10$
Weeks 3–4 (Days 15–28)	CMA	69.833	47.056	2.598	2.918	4.221	N/A	N/A	0.968	0.919	0.975	0.974	0.960	N/A	N/A
	NCEP	68.441	44.787	2.225	2.742	4.028	N/A	N/A	0.969	0.921	0.979	0.977	0.966	N/A	N/A
	ECMWF	63.674	42.349	2.067	2.391	3.232	N/A	N/A	0.972	0.929	0.982	0.983	0.971	N/A	N/A
[1.2pt/2.5pt]	Pangu	66.224	37.959	2.271	2.569	N/A	2.431	1.984	0.963	0.926	0.966	0.963	N/A	0.812	0.686

Table 1 : Forecasting results for 14-day mean targets at weeks 3–4 and weeks 5–6. The best result is in bold , and the second-best is underlined . Results for the operational NWP systems are obtained from ( Lyu et al., 2026 ) , whereas we reproduce other data-driven models and report their results. Pressure-level variables

z500

(gpm),

z850

(gpm) and

t500

(K),

t850

(K) are at 500 and 850 hPa, while single-level variables

t2m

(K),

u10

(m/s), and

v10

(m/s) are measured at 2m and 10m above the surface, respectively. N/A indicates the baseline does not predict the variables ( Liu et al., 2025b ) .

Figure 3 : RMSE variation w.r.t. different latitude bands, comparing AnisoCast with baseline models.

Figure 4 : RMSE comparison on variable $z$ and $t$ across different pressure levels.

Figure 5 : Global RMSE distribution of $t850$ with lead times weeks 3-4 in the testing set.

Figure 6 : Ablation study quantifying the contribution of each AnisoCast component.

Figure 7 : Efficiency analysis. The horizontal axis shows inference latency, the vertical axis shows parameter count, and bubble size depicts FLOPs.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8 : Paradigm-level comparison between isotropic modeling and the proposed anisotropic modeling for S2S forecasting.

Category	Variable Name	Symbol	Levels	Resolution	Lat-Lon Range	Time
Pressure Level	Geopotential	$z$	10	$1.5^{\circ}$	$-90^{\circ}\text{S}\;180^{\circ}\text{W}\sim 90^{\circ}\text{N}\;180^{\circ}\text{E}$	$1979\sim 2018$
	Specific Humidity	$q$	10	$1.5^{\circ}$	$-90^{\circ}\text{S}\;180^{\circ}\text{W}\sim 90^{\circ}\text{N}\;180^{\circ}\text{E}$	$1979\sim 2018$
	Temperature	$t$	10	$1.5^{\circ}$	$-90^{\circ}\text{S}\;180^{\circ}\text{W}\sim 90^{\circ}\text{N}\;180^{\circ}\text{E}$	$1979\sim 2018$
	U Component of Wind	$u$	10	$1.5^{\circ}$	$-90^{\circ}\text{S}\;180^{\circ}\text{W}\sim 90^{\circ}\text{N}\;180^{\circ}\text{E}$	$1979\sim 2018$
	V Component of Wind	$v$	10	$1.5^{\circ}$	$-90^{\circ}\text{S}\;180^{\circ}\text{W}\sim 90^{\circ}\text{N}\;180^{\circ}\text{E}$	$1979\sim 2018$
	Vertical Velocity	$w$	10	$1.5^{\circ}$	$-90^{\circ}\text{S}\;180^{\circ}\text{W}\sim 90^{\circ}\text{N}\;180^{\circ}\text{E}$	$1979\sim 2018$

Appendix

Table 2 : Detailed configuration of the ERA5 dataset used in this work.

Period	$z500$ RMSE	$z850$ RMSE	$u10$ RMSE	$v10$ RMSE	$z500$ ACC	$z850$ ACC	$u10$ ACC	$v10$ ACC
w3–4	$51.993{\pm}0.233$	$33.755{\pm}0.412$	$1.995{\pm}0.016$	$1.719{\pm}0.143$	$0.981{\pm}0.002$	$0.958{\pm}0.002$	$0.871{\pm}0.001$	$0.783{\pm}0.002$
w5–6	$52.518{\pm}0.275$	$33.400{\pm}0.382$	$1.995{\pm}0.090$	$1.671{\pm}0.114$	$0.981{\pm}0.002$	$0.958{\pm}0.002$	$0.872{\pm}0.002$	$0.786{\pm}0.002$

Appendix

Table 3 : Multi-seed robustness of AnisoCast over five random seeds. We report mean

\pm

std for RMSE and ACC on weeks 3–4 and weeks 5–6. Lower RMSE and higher ACC indicate better performance.

	Model	RMSE ( $\downarrow$ )							ACC ( $\uparrow$ )
		Pressure Level				Single Level			Pressure Level				Single Level
		$z500$	$z850$	$t500$	$t850$	$t2m$	$u10$	$v10$	$z500$	$z850$	$t500$	$t850$	$t2m$	$u10$	$v10$
2018–2019	ClimaX	67.287	40.132	2.293	2.692	5.298	3.268	2.665	0.960	0.948	0.970	0.968	0.955	0.806	0.641
	ClimODE	70.326	42.260	2.665	3.166	5.122	4.076	2.892	0.957	0.947	0.968	0.965	0.958	0.828	0.735
	CirT	65.211	38.544	2.302	2.987	4.696	2.451	1.989	0.968	0.948	0.974	0.982	0.973	0.831	0.709
	EIMP	66.724	39.138	2.465	2.489	4.592	2.512	2.332	0.974	0.950	0.975	0.983	0.972	0.798	0.732

Appendix

Table 4 : Extended-period evaluation on the 2018–2019 held-out years. Scores are averaged over the weeks 3–4 and weeks 5–6 forecast windows. The best result across all models is in bold , and the best baseline result is underlined . Lower RMSE and higher ACC indicate better performance. ACC is dimensionless. The variable groups follow Table 1 .

Figure 9 : Monthly RMSE breakdown for $z500$ .

Figure 10 : Monthly RMSE breakdown for $z850$ .

Figure 11 : Monthly RMSE breakdown for $t500$ .

Figure 12 : Monthly RMSE breakdown for $t850$ .

Model	TCC W3–4					TCC W5–6					CSI			SEDI
Model	$z500$	$z850$	$t2m$	$v10$	tp	$z500$	$z850$	$t2m$	$v10$	tp	$u10$	$v10$	$t2m$	$u10$	$v10$	$t2m$
CirT	0.420	0.419	0.365	0.301	0.261	0.351	0.350	0.370	0.303	0.169	0.09	0.12	0.22	0.29	0.33	0.40
EIMP	0.412	0.411	0.368	0.292	0.251	0.334	0.331	0.360	0.313	0.177	0.10	0.12	0.22	0.31	0.34	0.39
TelePiT	0.428	0.424	0.370	0.289	0.276	0.362	0.358	0.372	0.312	0.182	0.10	0.11	0.22	0.32	0.32	0.38
AnisoCast	0.442	0.440	0.372	0.313	0.282	0.373	0.370	0.368	0.309	0.196	0.12	0.13	0.24	0.34	0.35	0.42

Appendix

Table 5 : Evaluation under additional criteria. We report TCC for

z500

z850

t2m

v10

, and total precipitation ( tp ). CSI and SEDI are reported for extreme-event evaluation on

u10

v10

, and

t2m

. All metrics are dimensionless, and higher values indicate better performance.

Model	Resolution	$z500$ RMSE	$t2m$ RMSE	$z500$ ACC	$t2m$ ACC
CirT	$5.625^{\circ}$	56.103	5.510	0.978	0.957
EIMP	$5.625^{\circ}$	66.326	4.783	0.980	0.974
TelePiT	$5.625^{\circ}$	59.780	4.121	0.976	0.973
AnisoCast	$5.625^{\circ}$	52.015	3.320	0.981	0.983
CirT	$1.5^{\circ}$	55.505	5.478	0.980	0.958
EIMP	$1.5^{\circ}$	66.498	4.785	0.981	0.974

Appendix

Table 6 : Multi-resolution evaluation on Weeks 3–4. We compare different models under

5.625^{\circ}

1.5^{\circ}

, and

0.25^{\circ}

spatial resolutions. The best results under each resolution are highlighted in bold. Lower RMSE and higher ACC indicate better performance.

Figure 13 : Evaluation over the non-polar latitude band

|\phi|\leq 60^{\circ}

, excluding high-latitude regions.

	Model	RMSE ( $\downarrow$ )							ACC ( $\uparrow$ )
		Pressure Level				Single Level			Pressure Level				Single Level
		$z500(gpm)$	$z850(gpm)$	$t500(K)$	$t850(K)$	$t2m(K)$	$u10(m/s)$	$v10(m/s)$	$z500(gpm)$	$z850(gpm)$	$t500(K)$	$t850(K)$	$t2m(K)$	$u10(m/s)$	$v10(m/s)$
Weeks 3-4	AnisoCast	51.956	33.453	1.824	2.145	3.255	1.994	1.621	0.982	0.958	0.985	0.985	0.982	0.872	0.783
	w/o AE	57.088	38.211	2.481	2.630	6.801	3.012	2.531	0.953	0.952	0.948	0.920	0.957	0.833	0.732
	w/o ZO	61.207	39.118	2.467	2.742	4.061	3.053	3.441	0.970	0.939	0.953	0.934	0.961	0.818	0.721
	w/o MO	64.218	42.507	2.334	3.176	4.119	3.284	3.121	0.955	0.926	0.962	0.908	0.958	0.844	0.764

Appendix

Table 7 : Full ablation study results of AnisoCast on weeks 3–4 and weeks 5–6.

Paradigm	Period	z500 RMSE	z850 RMSE	t2m RMSE	v10 RMSE	z500 ACC	z850 ACC	t2m ACC	v10 ACC
AR	W3–4	56.122	40.112	5.881	2.421	0.952	0.951	0.954	0.730
Direct	W3–4	51.956	33.453	3.255	1.621	0.982	0.958	0.982	0.783
AR	W5–6	57.468	37.011	5.035	2.012	0.960	0.938	0.960	0.725
Direct	W5–6	52.268	33.437	3.231	1.606	0.982	0.958	0.982	0.786

Appendix

Table 8 : Comparison between autoregressive and direct S2S formulations. The direct formulation predicts the bi-weekly mean targets directly from the initial state, while the autoregressive variant recursively rolls out future states before computing the target averages.

Figure 14 : RMSE and ACC variations across different numbers of Aniso Blocks ( $L$ ).

Figure 15 : SpecRes comparison across pressure levels.

Figure 16 : RMSE comparison on variables $u,v$ across different pressure levels.

Figure 17 : RMSE comparison on variables $z$ and $t$ across different pressure levels.

Figure 18 : Daily RMSE and ACC variations across critical variables during weeks 3–4 (Days 15–28).

Figure 19 : Daily RMSE and ACC variations across critical variables during weeks 5–6 (Days 29–42).

Figure 20 : Global RMSE distribution for $t850$ at weeks 5–6 lead time.

Figure 21 : Global RMSE distribution for $t500$ at weeks 3–4 lead time.

Figure 22 : Global RMSE distribution for $t500$ at weeks 5–6 lead time.

Figure 23 : Global RMSE distribution for $z500$ at weeks 3–4 lead time.

Figure 24 : Global RMSE distribution for $z500$ at weeks 5–6 lead time.

Figure 25 : Global RMSE distribution for $z850$ at weeks 3–4 lead time.

Figure 26 : Global RMSE distribution for $z850$ at weeks 5–6 lead time.

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Subseasonal-to-seasonal (S2S) forecasting is societally critical, supporting decision-making in sectors ranging from water and agricultural management to disaster risk reduction, energy planning, and insurance. Achieving reliable predictions at these timescales requires representing the ocean and its dynamics, but traditional physics-based Ocean General Circulation Models (OGCMs), are computationally expensive and difficult to develop and improve because of the code complexity. In this work, we propose Neptune, an end-to-end data-driven framework for global ocean and sea-ice components emulation tailored for S2S timescales, up to 60 days. Neptune combines Convolutional Neural Networks (CNNs) and Spherical Fourier Neural Operators (SFNOs) to effectively capture local features and global cross-scale interactions, thereby obtaining a coherent representation of the ocean state. Forced by prescribed daily atmospheric fields, Neptune emulates ocean state variables, from temperature and salinity, to zonal and meridional currents, from sea surface height to sea ice thickness and concentration, with daily outputs at the ocean surface and through the water column. Specifically, we propose two variants of Neptune, Neptune-1 and Neptune-025, capable of emulating the ocean state at 1° and 0.25° resolution, respectively. Evaluated against a suite of metrics, including statistics (RMSE, CRPS and ACC), physical coherency (Ocean Heat Content, Eddy Kinetic Energy and Ice Brier Score) and climate indices (ENSO and Z20 metric, IOD), Neptune successfully reproduces the spatio-temporal evolution of the oceanic fields up to 60 days, and is stable over long timescales. Neptune provides compelling evidence that end-to-end data-driven ocean emulators can become a powerful component of next-generation S2S forecasting systems, emulating ocean state at high spatio-temporal resolution.

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