Fractional-grid forecasts split in two: learned frequency prediction, separately reconstructed memory for the error bound

Frequency forecasting for a fractional grid is split into two non-substitutable tasks: one model only predicts future frequency accurately, and another only reconstructs the memory state preceding the observation window, which serves as the reference for an error bound. The prior practice let a single memory estimate both serve frequency prediction and anchor a physical error bound, even though the observation window starts later than the physical history and the in-window phase and frequency cannot determine earlier motion. At observation-noise standard deviation 0.002, force-coordinate neural refinement lowers frequency RMSE in the six-node, ten-cycle, ten-mesh and twenty-mesh groups from the response ridge's 1.600, 1.662, 2.027 and 2.059 to 1.506, 1.558, 1.952 and 1.985, and all eight paired confidence intervals exclude zero.

Once the two tasks are separated, a prediction gain is no longer read as evidence of more accurate memory reconstruction, because part of the learned correction offsets observation noise. Replacing the noisy origin states with exact ones reverses the neural refinement's advantage over the linear predictor, showing that the gain and the memory reconstruction error can grow together. At noise 0.002 the force refinement's force error is 0.000331, above the response ridge's 0.000288; with exact origin states its frequency error is 1.797 against the response ridge's 1.669.

The separately reconstructed force feeds a moving-reference energy inequality, and adding the numerical propagation defect and the reference-to-point-forecast difference lets frequency envelopes be issued where the point predictor alone cannot support them. With the point predictor itself as reference, none of the 64 high-noise twenty-mesh events is issued; with the separate memory reconstruction as reference that group still issues none, but the six-node, ten-cycle and ten-mesh groups begin to pass the test. At high noise the neural reference issues 76 to 82 of 96 six-node events, 24 to 34 of 64 ten-cycle events and 22 to 30 of 64 ten-mesh events, while the direct reference issues all events in all four groups; no issued physical bound fails at the stored times.

The demonstrated scope is simulated transients of the specified finite relaxation model with known parameters, all-node phase and frequency observations, a common fractional order and a common kernel, on six-node, ten-cycle, ten-mesh and twenty-mesh networks that stay connected after a line outage; the authors propose extending the probability statement to field frequency measurements, multiscale data and varied dispatch scenarios, which requires calibration events representative of the new operating conditions.

The issuance guarantee is a marginal joint-failure probability statement, not conditional coverage among issued bounds; calibration is separate per network and noise group, and topology transfer alone provides no coverage guarantee. In the high-noise twenty-mesh group the critical radii fall below the calibrated radii, so every event abstains, which the authors describe as consumed angle margin rather than dynamical instability. The analytic guarantees concern the stated finite equations in real arithmetic, and the double-precision evaluation includes an arithmetic guard but is not a directed-rounding computer-assisted proof.