Neural Ordinary Differential Equation With Grünwald–Letnikov Fractional Momentum

Abstract

Forecasting short annual time series is challenging because limited observations increase the risks of unstable optimization, overfitting, and information leakage. This study proposes a standard first-order neural ordinary differential equation trained with finite-memory Grünwald–Letnikov fractional momentum. The fractional operator acts on the parameter update sequence rather than on the latent-state dynamics. The empirical evaluation combines a descriptive comparison on medium-length regional annual energy-consumption series with a 12-year holdout and a reproducible experiment on five extremely short provincial annual series, with the final two observations reserved for strictly recursive testing. The fractional order is selected by particle swarm optimization using only the training data. GLFNODE is compared with a NODE of the same architecture trained using Adam, together with MLP, ARIMA, SARIMA, ETS, and LSTM baselines. Across the five reported series, GLFNODE achieves test MAPE values ranging from 1.848% to 6.300%. Compared with the Adam-trained NODE, GLFNODE reduces the mean MAPE, MAE, and RMSE by 17.56%, 17.14%, and 15.91%, respectively. These results provide reproducible evidence that fractional-memory optimization can improve a compact neural ODE under extremely short-record conditions, particularly relative to the same architecture trained with Adam.