Low-Rank Clebsch--Gordan Coupling for Equivariant Spherical Forecasting

Yaojun Li; Zhixuan Zhao; Dongzhe Zheng; Christine Allen-Blanchette

cs.LGNov 2, 2026

Low-Rank Clebsch--Gordan Coupling for Equivariant Spherical Forecasting

Authors: Yaojun Li, Zhixuan Zhao, Dongzhe Zheng, Christine Allen-Blanchette

Organizations: Princeton University

Abstract

Spherical-harmonic neural operators provide an efficient representation for global forecasting, but their discretized pointwise nonlinearities can break rotational equivariance through aliasing. Clebsch-Gordan (CG) tensor products provide exactly equivariant interactions between spherical-harmonic representations, but their computational cost grows rapidly with spectral bandwidth and has limited their use at forecasting scale. We introduce LCG-SFNO, which combines low-rank CG interactions within a controlled low-degree band with coefficient-space residual propagation and an exactly projected quadratic nonlinearity, preserving SO(3) equivariance throughout the learned operator. To make the CG interaction practical, we further develop SSHCG, a sparse GPU implementation with a factorized alternative for wider coupled bands. On spherical shallow-water dynamics and the ERA5-based WeatherBench benchmark, LCG-SFNO reduces rotational error to numerical precision (more than three orders of magnitude below SFNO on shallow water and five orders of magnitude below it on WeatherBench), while matching SFNO on shallow-water forecasting and substantially improving forecast skill relative to SFNO on WeatherBench. Matched ablations show that CG coupling accelerates convergence, while SSHCG keeps the coupling to about 10 percent of a training step at the operating bandwidth, making exact equivariant coupling practical at forecasting scale.

Figures & tables

Figure 1: Overview of LCG-SFNO. (a) Input fields are encoded into spherical-harmonic coefficients, processed by coefficient-space residual blocks, and decoded to produce the forecast. (b) Each block combines an equivariant degree-wise spectral map, a low-rank Clebsch–Gordan correction evaluated with SSHCG, and a quadratic nonlinearity with exact spectral projection. The residual state remains in coefficient space between blocks.

	Shallow water	WeatherBench $5.625^{\circ}$
Model	$256\times 512$	1 day	3 days	5 days
ClimaX	$8.5\times 10^{-1}$	$7.9\times 10^{-1}$	$1.0$	$1.0$
FourCastNet	$6.9\times 10^{-1}$	$7.6\times 10^{-1}$	$8.3\times 10^{-1}$	$8.3\times 10^{-1}$
SHNet (GSNO)	$1.1\times 10^{-1}$	$5.6\times 10^{-1}$	$7.2\times 10^{-1}$	$6.7\times 10^{-1}$
SFNO	$9.5\times 10^{-4}$	$9.5\times 10^{-2}$	$1.5\times 10^{-1}$	$1.1\times 10^{-1}$
LCG-SFNO (ours)	$\mathbf{8.2\times 10^{-7}}$	$\mathbf{6.6\times 10^{-7}}$	$\mathbf{7.1\times 10^{-7}}$	$\mathbf{7.2\times 10^{-7}}$

Table 1: Rotational equivariance. Relative equivariance error under joint rotations of the state and auxiliary spatial fields, averaged over eight rotations and sixteen test states. LCG-SFNO reduces SFNO’s error by over three orders of magnitude on SSWE and approximately five on WeatherBench, reaching numerical precision. Best/second-best values are bold/underlined.

Figure 2: Spatial structure of rotational equivariance error. WeatherBench results for one test state $\mathbf{x}$ and one random rotation $R$ . (a) One-day 2 m temperature forecast; (b,c) equivariance residual $R^{-1}\mathcal{G}(R\mathbf{x},R\mathbf{c})-\mathcal{G}(\mathbf{x},\mathbf{c})$ for 2 m temperature and 10 m zonal wind. Rows correspond to SFNO and LCG-SFNO, with shared color scales within each column. The structured residual visible for SFNO is absent at the displayed scale for LCG-SFNO.

Model	ACC	MSE	Epochs	Equivariance
A SFNO reference	$84.0\pm 0.0$	$187.7\pm 0.2$	never	$9.5\times 10^{-2}$
B + full-band backbone	$93.4\pm 0.1$	$81.5\pm 0.6$	39	$6.7\times 10^{-2}$
C + CG coupling	$93.7\pm 0.0$	$77.9\pm 0.1$	13	$6.8\times 10^{-2}$
D + oversampled nonlinear pathway	$93.4$	$80.7$	21	$7.2\times 10^{-3}$

Table 2: Incremental component ablation on one-day WeatherBench. Rows progressively add the full-band backbone, CG coupling, and oversampled nonlinear pathway. “Epochs” denotes the number of epochs required to reach row B’s best validation error. Rows A–C report mean

\pm

standard deviation over three seeds; row D is a single-seed diagnostic.

Figure 3: Training efficiency. (a,b) SSWE one-step validation error and epochs to reach each error target over five seeds. (c,d) Corresponding results for one-day WeatherBench over three seeds. Curves terminate when a model fails to reach a stricter target or early stopping occurs. LCG-SFNO matches SFNO’s convergence on SSWE, reaches the intermediate WeatherBench targets in fewer epochs, and matches the best model at the strictest target.

Figure 4: Computational cost versus coupled bandwidth. (a) CG-block forward-and-backward time; (b) end-to-end training-step time, both versus input cutoff $L_{c}$ . The grid route remains nearly constant with increasing bandwidth.

Band $L_{c}/L_{o}$	Triples	Dense	Fused	Tiled	Grid route
4/8	85	3.2	3.2	3.2	3.2
8/16	489	4.5	4.2	3.6	3.6
12/24	1,469	16.6	6.5	4.6	4.1
16/31	3,280	53.2	13.3	7.2	4.7
24/31	9,456	325	51.9	17.4	5.1
31/31	16,400	855 †	152	35.6	5.3

Table 3: CG-block cost versus coupled bandwidth. Forward-and-backward time (ms) for one WeatherBench CG block (186 channels, rank

r=4

, batch 64, single precision). LCG-SFNO uses the

12/24

band. Direct implementations become more expensive with bandwidth, whereas the grid route remains nearly constant. † The

31/31

dense product exceeds 96 GB at batch 64 and is timed using four micro-batches of 16.

Model	Params	1 h	5 h	10 h
ClimaX	1.96M	421 $\pm$ 0	474 $\pm$ 5	535 $\pm$ 10
FourCastNet	1.66M	285 $\pm$ 2	687 $\pm$ 40	1309 $\pm$ 110
SHNet (GSNO)	0.87M	142 $\pm$ 6	391 $\pm$ 16	625 $\pm$ 189
SFNO	0.28M	58.0 $\pm$ 1.2	121 $\pm$ 6	219 $\pm$ 22
LCG-SFNO (ours)	0.27M	57.4 $\pm$ 1.4	119 $\pm$ 7	211 $\pm$ 17

Table 4: Shallow-water forecasting. Relative

L^{2}

rollout error (

\times 10^{-3}

) over 50 test initial conditions, reported as mean

\pm

standard deviation over five seeds. LCG-SFNO matches SFNO within seed variation across all horizons while using the fewest parameters.

		1 day		3 days		5 days
Model	Params	ACC	MSE	ACC	MSE	ACC	MSE
ClimaX	5.40M	93.2 $\pm$ 0.3	76.8 $\pm$ 0.5	78.3 $\pm$ 0.4	242.4 $\pm$ 2.5	53.1 $\pm$ 0.8	435.0 $\pm$ 4.5
FourCastNet	5.60M	92.8 $\pm$ 0.1	82.8 $\pm$ 0.3	69.3 $\pm$ 0.1	325.1 $\pm$ 0.9	50.5 $\pm$ 0.1	440.1 $\pm$ 0.1
SHNet (GSNO)	4.03M	92.4 $\pm$ 0.0	92.7 $\pm$ 0.1	74.2 $\pm$ 0.1	281.6 $\pm$ 0.9	54.9 $\pm$ 0.1	420.1 $\pm$ 1.1
SFNO	5.26M	84.0 $\pm$ 0.0	187.7 $\pm$ 0.2	65.4 $\pm$ 0.2	350.8 $\pm$ 1.4	48.6 $\pm$ 0.1	451.1 $\pm$ 0.6
LCG-SFNO (ours)	5.24M	93.3 $\pm$ 0.1	82.1 $\pm$ 1.4	73.5 $\pm$ 0.1	286.5 $\pm$ 0.9	51.7 $\pm$ 0.2	430.4 $\pm$ 1.4

Table 5: WeatherBench forecasting. ACC (%) and normalized latitude-weighted MSE (

\times 10^{-3}

), reported as mean

\pm

standard deviation over three seeds. LCG-SFNO substantially improves over SFNO across all horizons and achieves the highest mean one-day ACC; ClimaX and SHNet lead on the remaining aggregate metrics.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Cumulative fraction of WeatherBench variance below each spherical-harmonic degree over 512 test states. The dotted lines mark the input and output cutoffs of the coupled band. The input cutoff covers most of the state variance, while the output cutoff covers nearly all target variance and most one-day tendency variance.

Figure 6: Representation costs of broad-band tensor products. (a) Dense e3nn coefficient domain relative to real-basis nonzeros. (b) Logical path-result volume relative to the final output; the diamond marks the compiled observation. (c) Registered path metadata in the constructed cuEquivariance operator, whose backend path count equals the channel count

C

times the number of angular nonzeros in every tested configuration.

Figure 7: Matched-operator eager forward-plus-backward crossover as bandwidth, batch size, precision, and per-sector CP rank vary. Color represents $\log_{2}(t_{\mathrm{LCG}}/t_{\mathrm{Gaunt}})$ ; blue favors Gaunt, red favors direct CG, and the black contour marks equal latency. We use channel multiplicity eight, batch sizes $B\in\{1,4,8,16,32,64\}$ , and equal even- and odd-sector ranks $K\in\{4,8,16\}$ .

Task	$C_{u}$	Degrees (L/R/O)	e3nn	SSHCG	cuEquivariance
MACE-large	128	0–2 / 0–3 / 0–3	2.02	1.04	2.30
MACE-medium	128	0–1 / 0–3 / 0–2	0.774	0.345	0.399
NequIP-LiPS	32	0–2 / 0–2 / 0–2	1.58	0.665	1.32
NequIP-aspirin	64	0–1 / 0–1 / 0–1	0.516	0.175	0.225
NequIP-toluene	64	0–2 / 0–2 / 0–2	2.18	0.756	2.79
NequIP-benzene	64	0–3 / 0–3 / 0–3	7.94	2.83	17.3

Appendix

Table 6: Task-derived UVU configurations and forward–backward latency (CUDA-event ms), FP32, CUDA Graph, A100 PCIe 40GB,

B=1024

. Degree ranges are inclusive.

Workload	$L/H$	e3nn	SSHCG	cuEquivariance
S1	4	1.47	0.0738	7.59
S1	8	7.47	0.226	180
S1	12	27.3	0.832	TO
S2	8	1.17	0.119	13.6
S2	16	4.55	0.284	129
S2	24	8.74	0.472	TO

Appendix

Table 7: Spectral forward–backward latency (ms), FP32, CUDA Graph, A100 SXM4 80GB,

B=C=8

Benchmark	Model	Params	Architecture (epochs run)
Shallow water	ClimaX	1.96M	patch 4, width 128, depth 6, 8 heads
	FourCastNet	1.66M	patch 4, width 128, depth 8, 8 AFNO blocks
	SHNet (GSNO)	0.87M	width 8, scale factor 4
	SFNO	0.28M	width 28, 4 blocks, $L=85$ , scale factor 2
	LCG-SFNO	0.27M	width 25, 4 blocks, $L=85$ , CG band 12/24 rank 4, quadratic activation, oversampling 1.5
WeatherBench	ClimaX	5.40M	patch 4, width 256, depth 6, 8 heads, decoder depth 2

Appendix

Table 8: Architectures of the compared models at the sizes of the benchmark paper, and the training length of our models (epochs run until early stopping at 1, 3 and 5 days on WeatherBench; fixed budget on the shallow-water task). WeatherBench models use the benchmark optimiser and schedule (AdamW, learning rate

4\times 10^{-4}

, batch 128, warm-up and cosine decay, early-stopping patience 3); shallow-water models use the benchmark recipe (Adam, learning rate

10^{-3}

, batch 16, 50 epochs).

	Model	ACC	MSE	Epochs	Equivariance
A	SFNO, benchmark configuration	84.0 $\pm$ 0.0	187.7 $\pm$ 0.2	never	$9.5\times 10^{-2}$
B	+ full spectral band, two-layer codec, regularization	93.4 $\pm$ 0.1	81.5 $\pm$ 0.6	39	$6.7\times 10^{-2}$
C	+ CG coupling, activations on the data grid	93.7 $\pm$ 0.0	77.9 $\pm$ 0.1	13	$6.8\times 10^{-2}$
D	+ dealiased grid, band-limited residual stream	93.4	80.7	21	$7.2\times 10^{-3}$

Appendix

Table 9: WeatherBench 24-hour ablation from the reference SFNO to LCG-SFNO. ACC and normalized MSE measure forecast skill; “Epochs” is the number of epochs required to reach the best validation error of row B; equivariance is the relative error defined in Appendix D.4 . Rows A–C are mean and standard deviation over three seeds; row D is a single seed. Row D uses the uniform oversampling configuration as a diagnostic ablation; the final block/codec allocation is selected in Appendix E.2 . All rows use the benchmark-scale parameter budget and the GELU activation; the final model replaces it by the quadratic activation (Appendix B.3 ).

Blocks	Codec	Equivariance	Step
$1.5$	$1.5$	$2.5\times 10^{-2}$	$2.57\times$
$\mathbf{1.5}$	$\mathbf{2.0}$	$\mathbf{7.8\times 10^{-3}}$	$\mathbf{2.81\times}$
$1.5$	$3.0$	$4.1\times 10^{-3}$	$3.46\times$
$1.5$	$4.0$	$3.9\times 10^{-3}$	$4.30\times$
$2.0$	$2.0$	$6.7\times 10^{-3}$	$3.66\times$
$2.0$	$3.0$	$1.1\times 10^{-3}$	$4.33\times$

Appendix

Table 10: Initialization-time equivariance error and training-step cost for different oversampling factors in the residual blocks and codec. Step time is normalized by the same architecture without CG coupling or oversampling. The bold row is the configuration used by the final WeatherBench model.

Figure 8: SSWE relative $L^{2}$ rollout error over twenty one-hour steps for all compared models, mean and standard deviation over five seeds with fifty test initial conditions each. The spherical spectral models accumulate error far more slowly than the position-dependent architectures, and LCG-SFNO coincides with SFNO throughout.

	1 h			5 h			10 h
Model	$h$	vorticity	divergence	$h$	vorticity	divergence	$h$	vorticity	divergence
ClimaX	5.2 $\pm$ 0.0	259 $\pm$ 1.0	1000 $\pm$ 0.0	4.4 $\pm$ 0.0	417 $\pm$ 15	1000 $\pm$ 0.1	3.8 $\pm$ 0.0	600 $\pm$ 31	1000 $\pm$ 0.2
FourCastNet	3.0 $\pm$ 0.0	101 $\pm$ 3.1	750 $\pm$ 4.0	6.3 $\pm$ 0.8	635 $\pm$ 70	1418 $\pm$ 125	6.9 $\pm$ 1.1	1927 $\pm$ 398	1992 $\pm$ 245
SHNet (GSNO)	1.4 $\pm$ 0.1	147 $\pm$ 6.4	279 $\pm$ 16	3.9 $\pm$ 0.8	419 $\pm$ 46	750 $\pm$ 19	7.5 $\pm$ 7.2	783 $\pm$ 314	1085 $\pm$ 250
SFNO	0.6 $\pm$ 0.0	116 $\pm$ 1.4	58 $\pm$ 4.2	2.5 $\pm$ 0.2	191 $\pm$ 1.8	169 $\pm$ 16	4.7 $\pm$ 1.7	311 $\pm$ 17	340 $\pm$ 49
LCG-SFNO (ours)	0.6 $\pm$ 0.0	115 $\pm$ 0.5	57 $\pm$ 4.1	2.5 $\pm$ 0.2	189 $\pm$ 3.1	166 $\pm$ 18	4.5 $\pm$ 2.1	289 $\pm$ 6.8	338 $\pm$ 56

Appendix

Table 11: SSWE forecasting per variable: relative

L^{2}

rollout error in physical units (

\times 10^{-3}

) of height

h

, vorticity, and divergence at one, five, and ten hours over the fifty test initial conditions, mean and standard deviation over five seeds. The best value in each column is shown in bold and the second best underlined.

One day
Model	2T	10U	10V	U600	V600	T600	Average
ClimaX	94.3	91.4	91.5	93.3	92.2	96.7	93.2
FourCastNet	94.9	90.8	90.8	92.6	91.3	96.5	92.8
SHNet (GSNO)	97.1	89.5	89.4	91.7	90.2	96.5	92.4
SFNO	95.3	76.5	74.9	83.8	80.1	93.3	84.0
LCG-SFNO	97.4	90.6	90.3	92.9	91.6	97.1	93.3

Appendix

Table 12: WeatherBench

5.625^{\circ}

per variable, anomaly correlation coefficient in percent (higher is better), by lead time. All models trained by us; mean over three seeds. The best value in each column and block is in bold and the second best underlined.

Model	2T	10U	10V	U600	V600	T600	Average
One day
ClimaX	11.8	89.9	136	68.5	144	9.87	76.8
FourCastNet	9.66	96.0	146	75.3	160	10.5	82.8
SHNet (GSNO)	6.21	109	167	84.5	179	10.7	92.7
SFNO	9.93	228	367	158	344	19.8	188
LCG-SFNO	5.55	97.3	154	72.1	155	8.86	82.1

Appendix

Table 13: WeatherBench

5.625^{\circ}

per variable, latitude-weighted MSE of normalised variables (

\times 10^{-3}

, lower is better), by lead time. All models trained by us; mean over three seeds. The best value in each column and block is in bold and the second best underlined.

Figure 9: 24-hour forecasts on WeatherBench $5.625^{\circ}$ for 15 July 2017 at 12 UTC. Columns (a) to (f) show the target, ClimaX, FourCastNet, SHNet, SFNO, and LCG-SFNO, with one colour scale per row. Rows show 2 m temperature, 10 m zonal wind, and 600 hPa temperature. Every model captures the large-scale structure; the differences lie in the finer features that the error maps of Figure 10 isolate.

Figure 10: Forecast errors (forecast minus target) of the models in Figure 9 for the same date. Columns (a) to (e) show ClimaX, FourCastNet, SHNet, SFNO, and LCG-SFNO on one symmetric colour scale per row, set by the pooled errors of all models; each panel states the root-mean-square error of that model and variable on this sample. LCG-SFNO leaves the smallest and least coherent residuals of the spherical models.

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E(3)

-equivariant neural networks have proven to be effective in a wide range of 3D modeling tasks. A fundamental operation of such networks is the tensor product, which allows interaction between different feature types. Because this operation scales poorly, there has been considerable work towards accelerating this interaction. However, recently \citet{xieprice} have pointed out that most speedups come from a reduction in expressivity rather than true algorithmic improvements on computing Clebsch-Gordan tensor products. A modification of Gaunt tensor product \citep{gaunt} can give a true asymptotic speedup but is incomplete and misses many interactions. In this work, we provide the first complete algorithm which truly provides asymptotic benefits Clebsch-Gordan tensor products. For full CGTP, our algorithm brings runtime complexity from the naive

O(L^6)

O(L^4\log^2 L)

, close to the lower bound of

O(L^4)

. We first show how generalizing fast Fourier based convolution naturally leads to the previously proposed Gaunt tensor product \citep{gaunt}. To remedy antisymmetry issues, we generalize from scalar signals to irrep valued signals, giving us tensor spherical harmonics. We prove a generalized Gaunt formula for the tensor harmonics. Finally, we show that we only need up to vector valued signals to recover the missing interactions of Gaunt tensor product.

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