The continuous dynamics of natural systems has been effectively modelled using Neural Ordinary Differential Equations (Neural ODEs). However, for accurate and meaningful predictions, it is crucial that the models follow the underlying rules or laws that govern these systems. In this work, we propose a self-adaptive penalty algorithm for Neural ODEs to enable modelling of constrained natural systems. The proposed self-adaptive penalty function can dynamically adjust the penalty parameters. The explicit introduction of prior knowledge helps to increase the interpretability of Neural ODE -based models. We validate the proposed approach by modelling three natural systems with prior knowledge constraints: population growth, chemical reaction evolution, and damped harmonic oscillator motion. The numerical experiments and a comparison with other penalty Neural ODE approaches and \emph{vanilla} Neural ODE, demonstrate the effectiveness of the proposed self-adaptive penalty algorithm for Neural ODEs in modelling constrained natural systems. Moreover, the self-adaptive penalty approach provides more accurate and robust models with reliable and meaningful predictions.

本文提出了一种自适应惩罚算法，用于在神经常微分方程中模拟受限自然系统，并通过在人口增长、化学反应演化和阻尼谐振子运动中对其进行建模进行验证，结果表明自适应惩罚算法在模拟受限自然系统中的神经常微分方程上具有更好的有效性和可靠性。

一种自适应惩罚方法，将先验知识约束融入神经ODE