Fractional Neural Ordinary Differential Equations for Time-Series Forecasting

Neural ordinary differential equations (Neural ODEs) describe the feature evolution of deep networks by continuous-time dynamical systems and enable end-to-end learning through differentiable numerical solvers. Nevertheless, in closed-loop rolling prediction for small-sample time series, conventional Neural ODEs remain vulnerable to error accumulation and numerical instability. To improve the controllability of long-term evolution, this study proposes a neural ordinary differential equation framework based on fractional-order operators. Rather than directly introducing full-history convolution kernels into the governing dynamics, the proposed approach constructs a fractional effective step size from the closed-form expression of the Riemann–Liouville fractional integral of a constant function and consistently embeds it into all sub-steps of a fourth-order Runge–Kutta solver. In this way, the scale of continuous-depth propagation is regulated by a single tunable parameter. Combined with a residual output structure, the method preserves the interpretability of continuous dynamics while effectively suppressing trajectory drift in closed-loop prediction and improving training stability. To investigate the impact of the fractional-order parameter on fitting and extrapolation, particle swarm optimization is employed to search automatically for the optimal order. Experimental evaluations on the linear spiral system and Lorenz continuous dynamical systems and on a small-sample provincial annual electricity-consumption dataset show that the proposed model achieves lower prediction errors across multiple tasks and exhibits superior trajectory preservation and robustness under long-horizon forecasting.