用神经微分方程生成数据，再用符号回归发现物理规律
An Empirical Investigation of Neural ODEs and Symbolic Regression for Dynamical Systems
- 先用神经微分方程学习动态，再用符号回归找微分方程
- 符号回归能从噪声数据中恢复出3个方程中的2个，第3个近似良好
- 仅用10%数据训练的NODE可提升符号回归性能，适合小样本研究
准确建模复杂系统的动态并发现其控制微分方程，对加速科学发现至关重要。我们使用来自两个阻尼振荡系统的噪声合成数据，研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。研究得出三个关键发现：第一，只要新轨迹与训练数据具有动态相似性，NODEs 能有效外推至新的边界条件；第二，当输入变量选择正确时，SR 可从带噪声的真实数据中成功恢复方程；第三，当使用仅含10%完整模拟数据训练的NODE生成的数据时，SR能够恢复出三个控制方程中的两个，并对第三个给出良好近似。这一结果表明，利用NODE扩充有限数据，再通过符号回归推导物理定律，是一种有前景的新方法。
原文摘要 · Abstract (English)
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.
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