Hankel Residual Observer

Integrate the observer after a neural forecaster's multi-step output head and before inference-time prediction is returned or before the forecasting loss is evaluated. Let the network output $\hat y_{t+1:t+H}\in\mathbb{R}^{pH}$. When targets become available, compute the one-step residual $r_t=y_t-\hat y_t$ and append it to a bounded FIFO buffer. Once at least $T_{ini_d}+H$ residual vectors are available, construct dictionary columns from consecutive windows: $D_p[:,j]=[r_j;\ldots;r_{j+T_{ini_d}-1}]$ and $D_f[:,j]=[r_{j+T_{ini_d}};\ldots;r_{j+T_{ini_d}+H-1}]$. Retain the latest $K$ columns and rebuild or rank-update the Gram matrix. At prediction time, form $d_{ini}$ from the latest $T_{ini_d}$ residuals and solve $g=(D_p^TD_p+\lambda_gI)^{-1}D_p^Td_{ini}$ with Cholesky. Compute $\hat d=D_fg$. Define normalized mismatch $q=\|D_pg-d_{ini}\|_2/(\|d_{ini}\|_2+10^{-8})$ and set $\gamma_i=\operatorname{clip}(1-q/\tau,0,1)$, optionally multiplied by a learned horizon decay. Return $\hat y_{corr}=\hat y+\Gamma\hat d$. The Hankel construction and ridge solve come directly from the mathematical method; select $K$, $T_{ini_d}$, $\lambda_g$, $\epsilon$, and $\tau$ on a validation stream. First test a small GRU or TCN on synthetic forecasting with additive sinusoidal disturbances, then Electricity or Traffic. Compare the baseline at equal neural FLOPs and report multi-step RMSE, long-horizon error, observer CPU latency, and memory. The expected signal is lower error on persistent disturbances without degradation on clean data, with corrections automatically suppressed when dictionary mismatch is high.