SA-NODEs Improve Long-Term Dynamical System Approximation.

Approximating complex dynamical systems over long periods using neural ordinary differential equations (NODEs) is challenging, as error bounds typically worsen exponentially with time. This paper presents two new training strategies for Semi-Autonomous NODEs (SA-NODEs) that effectively tackle this problem. The "model predictive strategy" adaptively partitions the time horizon and restarts training from observed data within each window. This ensures uniform error control over the entire duration with a parameter budget that grows linearly with the horizon. The "Floquet strategy" is designed for autonomous systems with stable limit cycles, using a certified contraction of the learned return map to confine errors to linear growth per period, even without deployment data. These methods provide more robust and scalable solutions for long-term trajectory approximation.