Deep Learning-Based Residual Augmentation of Neural ODE Approximations: Rollout Error Propagation, Contraction Diagnostics, and CRN Case Study

Neural ordinary differential equations (NODEs) have emerged as an effective methodology in artificial neural networks (ANNs) and deep learning for capturing unknown or unmodeled dynamics in compartmental and dynamical mathematical models arising from real-life applications, particularly under limited-data conditions, through learned data-driven corrections. Nevertheless, accurate one-step prediction errors do not necessarily guarantee reliable long-horizon rollouts. In this work, we study residual Neural ODE models of the form f^=f+hθ and derive a priori rollout-error estimates showing that long-time prediction behavior is generated by the incremental stability structure of the learned dynamics. Contracting regimes produce uniformly controlled rollout errors, whereas weakly contractive or expansive regimes can amplify persistent approximation errors over long time horizons. The analysis is illustrated on a flow-reactor chemical reaction network (CRN), where the washout parameter controls rollout reliability on the data-supported region. Numerical experiments further demonstrate that models with comparable empirical one-step prediction losses may exhibit substantially different multi-step behaviors. Rollout-error analysis and projected-gradient-descent (PGD) sensitivity directions additionally reveal that locally expansive regions align with worst-case perturbation amplification.