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.

翻译：自然系统的连续动态已通过神经常微分方程（Neural ODE）得到有效建模。然而，为实现准确且有意义的预测，模型必须遵循支配这些系统的基本规则或定律。本文提出一种针对Neural ODE的自适应惩罚算法，以支持对受约束自然系统的建模。该自适应惩罚函数能够动态调整惩罚参数，且显式引入先验知识有助于增强基于Neural ODE模型的可解释性。我们通过建模三个具有先验知识约束的自然系统（种群增长、化学反应演化及阻尼谐振子运动）验证了所提方法。数值实验及与其他惩罚型Neural ODE方法及普通（vanilla）Neural ODE的对比表明，所提自适应惩罚算法在建模受约束自然系统中的有效性。此外，该自适应惩罚方法提供了更准确、稳健的模型，其预测结果可靠且具有意义。