Interpretable Solar-Cycle Forecasting via Causal TrendPriors and Residual Attention

Forecasting the solar cycle remains challenging because the monthly Sunspot Number (SSN) exhibits irregular amplitudes, variable timings, and quasi-periodic, non-stationary behavior. We present an interpretable hybrid frame-work that decomposes the signal into a smooth causal trend and a short-term residual component. The trend is estimated with a one-sided (causal), robust locally weighted smoothing (LOWESS) filter so that only past data influence each estimate, preventing information leakage. Residual variations are modeled by a lightweight attention network adapted for scientific time-series forecasting. The residual model uses physically and temporally meaningful inputs— solar-cycle phase, odd–even parity, the 10.7 cm solar radio flux (F10.7), the planetary geomagnetic index (Ap), and sunspot-group area. Training with a quantile loss produces predictive quantiles, which are refined by a phase-aware conformal calibration procedure to obtain reliable prediction intervals (PIs). On a strictly chronological hindcast of Solar Cycle 24, the model achieves a median mean absolute error (MAE) of about 14 and a root-mean-square error (RMSE) of about 19, outperforming a persistence-with-drift baseline (MAE ≈65) and a gradient-boosting regressor. Calibrated 80 % PIs attain empirical coverage close to the nominal level. Feature-removal tests show that explicit cycle phase is the main driver of forecast skill. Once the causal trend and phase are included, standard physical covariates add little marginal benefit; removing them leaves accuracy essentially unchanged and can even improve it slightly, yielding a more parsimonious model. Forecasts initialized at the official December 2019 minimum provide plausible trajectories for Solar Cycles 25 and 26 with calibrated uncertainty bands. This causal-trend plus attention-residual framework yields consistent, interpretable, and well-calibrated forecasts, complementing review studies of solar-cycle variability (e.g. Hathaway 2015) and employing established tools for smoothing (Cleveland 1979), attention (Vaswani et al. 2017), and interval calibration (Romano, Patterson, and Candes 2019).