This paper explores the critical role of differentiation approaches for
data-driven differential equation discovery. Accurate derivatives of the input
data are essential for reliable algorithmic operation, particularly in
real-world scenarios where measurement quality is inevitably compromised. We
propose alternatives to the commonly used finite differences-based method,
notorious for its instability in the presence of noise, which can exacerbate
random errors in the data. Our analysis covers four distinct methods:
Savitzky-Golay filtering, spectral differentiation, smoothing based on
artificial neural networks, and the regularization of derivative variation. We
evaluate these methods in terms of applicability to problems, similar to the
real ones, and their ability to ensure the convergence of equation discovery
algorithms, providing valuable insights for robust modeling of real-world
processes.

这篇论文探讨了差分方程发现的差异化方法在数据驱动中的关键作用，对于可靠的算法操作而言，输入数据的准确导数是至关重要的，尤其是在不可避免地存在测量质量下降的真实场景中。我们提出了一些替代常用的有限差分方法的方法，这些方法因其在噪声存在下的不稳定性而臭名昭著，噪声可能会加剧数据中的随机误差。我们的分析涵盖了四种不同的方法：Savitzky-Golay 滤波、谱微分、基于人工神经网络的平滑和导数变差的正则化。我们评估了这些方法在类似真实问题上的适用性以及它们保证方程发现算法收敛的能力，为真实世界过程的健壮建模提供了有价值的见解。