Abstract
This paper addresses the limitations of current neural ordinary differential equations (NODEs) in modeling and predicting complex dynamics by introducing a novel framework called higher-order-derivative-supervised (HiDeS) NODE. This method extends traditional NODE frameworks by incorporating higher-order derivatives and their interactions into the modeling process, thereby enabling the capture of intricate system behaviors. In addition, the HiDeS NODE employs both the state vector and its higher-order derivatives as supervised signals, which is different from conventional NODEs that utilize only the state vector as a supervised signal. This approach is designed to enhance the predicting capability of NODEs. Through extensive experiments in the complex fields of multi-robot systems and opinion dynamics, the HiDeS NODE demonstrates improved modeling and predicting capabilities over existing models. This research not only proposes an expressive and predictive framework for dynamic systems but also marks the first application of NODEs to the fields of multi-robot systems and opinion dynamics, suggesting broad potential for future interdisciplinary work. The code is available at https://github.com/MengLi-Thea/HiDeS-A-Higher-Order-Derivative-Supervised-Neural-Ordinary-Differential-Equation.