Accurately modelling the dynamics of complex systems and discovering their governing differential equations are critical tasks for accelerating scientific discovery. Using noisy, synthetic data from two damped oscillatory systems, we explore the extrapolation capabilities of Neural Ordinary Differential Equations (NODEs) and the ability of Symbolic Regression (SR) to recover the underlying equations. Our study yields three key insights. First, we demonstrate that NODEs can extrapolate effectively to new boundary conditions, provided the resulting trajectories share dynamic similarity with the training data. Second, SR successfully recovers the equations from noisy ground-truth data, though its performance is contingent on the correct selection of input variables. Finally, we find that SR recovers two out of the three governing equations, along with a good approximation for the third, when using data generated by a NODE trained on just 10% of the full simulation. While this last finding highlights an area for future work, our results suggest that using NODEs to enrich limited data and enable symbolic regression to infer physical laws represents a promising new approach for scientific discovery.

翻译：准确建模复杂系统的动力学特性并发现其控制微分方程，是加速科学发现的关键任务。利用来自两个阻尼振荡系统的含噪合成数据，我们探究了神经常微分方程（NODEs）的外推能力以及符号回归（SR）恢复底层方程的能力。我们的研究得出三个关键发现。首先，我们证明只要所得轨迹与训练数据具有动力学相似性，NODEs能够有效地外推至新的边界条件。其次，SR能够成功从含噪的真实数据中恢复方程，但其性能取决于输入变量的正确选择。最后，我们发现当使用仅在完整仿真数据10%上训练的NODE所生成的数据时，SR恢复了三个控制方程中的两个，并对第三个方程给出了良好近似。虽然最后一项发现指明了未来工作的方向，但我们的结果表明，利用NODEs扩充有限数据并促使符号回归推断物理规律，为科学发现提供了一种前景广阔的新途径。