S3N pairs finite-element LL–HP mapping with quad-spiral state-space scanning to cut 10-day forecast nRMSE to 0.5177

S3N achieves lower error at long lead times: nRMSE of 0.3171, 0.4411, and 0.5177 at 4, 7, and 10 days, reducing the 10-day nRMSE by 19.1% relative to the second-best OneForecast (0.6398). Relative to prior MLWP methods that operate mainly on LL grids or process HEALPix pixels in separate patches, this framework unifies an equal-area spherical representation with global cross-face information propagation, so its advantage grows with forecast horizon. On ERA5/WeatherBench 2 with 1979–2017 training, 2018–2019 validation, and 2020 testing, 69 atmospheric fields at 6-hour temporal resolution, with all comparative experiments under the same conditions; Table 2 reports nRMSE per model and lead time.

L2Proj represents LL and HP data as continuous piecewise-linear fields on the unit sphere and solves the mapping through a projection, giving an average bidirectional mapping MSE of 0.000252 versus 0.002153 for bilinear interpolation. Prior LL–HP mapping usually relies on pointwise or learned resampling even though the two grids differ in sampling density and spatial resolution; L2Proj provides a unified finite-element formulation for nonmatching spherical grids, with mass and cross-grid coupling matrices precomputable offline and reusable. Using 2017 and 2020 data, the 69 standardized atmospheric fields are mapped from LL to HP and back to LL, and the reconstructed fields are compared with the original LL fields; Table 1 reports the average MSE over all variables.

The AQSS block uses cross-latitude attention to guide selective state-space updates along four global pole-to-pole spiral paths, letting hidden states propagate directly across HEALPix base-face boundaries without face-wise padding or shifted-window processing. Prior HP methods organize pixels into separate base faces or local windows, so cross-face communication depends on handcrafted boundary handling, rotated or copied padding, shifted windows, masks, or layout-specific indexing; AQSS replaces these with global spiral paths. The method section specifies eight cross-latitude neighbor sampling, multi-head attention, context modulation of the scan input and of the dynamic Mamba parameters, and selective scanning over four RING-ordered spiral paths; ablation shows that removing attention-guided SSM parameter modulation raises 10-day nRMSE from 0.5177 to 2.3103.

Ablation indicates that weighted path fusion mainly benefits short-range accuracy, whereas attention-guided SSM parameter modulation is critical for suppressing autoregressive error accumulation. This decomposition attributes long-range stability to specific components rather than to the architecture as a whole, giving component-level evidence that later designs can test. Table 4 compares the full model against fixed-weight path fusion, no SSM guidance, no attention, and no input fusion across 6-hour to 10-day lead times, with the full model achieving the lowest nRMSE at all lead times.

The result applies to a global deterministic medium-range forecasting setting using ERA5/WeatherBench 2 data at 6-hour temporal resolution with 69 atmospheric fields and an HP representation containing 49,152 equal-area pixels; training is one-step prediction followed by long-range autoregressive fine-tuning, taking less than 12 days on one NVIDIA RTX 6000D GPU with 84 GB memory. Directly reusable elements include the offline-precomputed L2Proj mapping matrices, whose mass and cross-grid coupling matrices depend only on spherical node positions and the resolutions of the two grids and require no mesh-intersection processing or numerical integration at runtime, and the AQSS construction of four pole-to-pole spiral paths in RING order, each visiting every HP pixel exactly once. For researchers working on spherical grid representations, equal-area discretization, or long-range autoregressive stability, the framework offers transferable components; the authors state that future work will improve the mapping resolution between the two grids and short-range forecasting performance.

The mapping evaluation uses 2017 and 2020 data, and Table 1 reports only the average MSE over all variables without per-variable distributions in the text; the main results and ablations rest on the 2020 test period and a single dataset, so behavior across datasets, resolutions, and probabilistic forecasting remains open. In the ablation, the fixed-weight fusion and no-SSM-guidance variants show markedly higher short-range error, indicating strong coupling among components, so replacing any one of them would require recalibration. The authors list improved mapping resolution and short-range forecasting as future work, so the current relative position at 6-hour and 1-day lead times should be read as an observation under this setting rather than a general conclusion.