A comparative study of neural ordinary differential equations and neural operators for modeling temporal dynamics

Abstract Capturing the dynamics of relational systems is a key challenge in the natural sciences, with applications ranging from simulating molecular interactions to analyzing particle mechanics. Machine learning approaches have made significant progress in this area by using graph neural networks to learn and visualize spatial interactions effectively. Neural ordinary differential equations (Neural ODEs) and neural operators (NO) represent two distinct paradigms. However, a clear comparative understanding of when to prefer one over the other is still lacking. To address this gap, we present the first systematic comparison between two representative architectures: EGNO (Equivariant Graph Neural Operator) and SEGNO (Second-order Equivariant Graph Neural Ordinary Differential Equation). Through a series of experiments, we investigate their strengths and limitations in various simulation scenarios in the multi-step trajectory prediction tasks. Specifically, we employ rollout strategies and different input/output configurations, including multiple and irregularly sampled time steps. Our findings highlight a key trade-off between precision and stability that is central to model selection. SEGNO demonstrates superior robustness and stability over long prediction horizons, making it well-suited for tasks requiring reliable long-term forecasting. Conversely, EGNO offers higher precision during early stages of the trajectory and better leverages diverse training configurations, thanks to its discretization-invariant design. In summary, Neural Operators (EGNO) are preferable when short-term accuracy and data efficiency are critical, while Neural ODEs (SEGNO) are advantageous for scenarios demanding stable long-term predictions. This work not only clarifies the practical advantages of each approach but also lays the groundwork for informed model selection and future hybrid strategies in dynamical system modeling.