Sparse Identification of Nonlinear Dynamical Systems (SINDy) is a powerful tool for the data-driven discovery of governing equations. However, it encounters challenges when modeling complex dynamical systems involving high-order derivatives or discontinuities, particularly in the presence of noise. These limitations restrict its applicability across various fields in applied mathematics and physics. To mitigate these, we propose Laplace-Enhanced SparSe Identification of Nonlinear Dynamical Systems (LES-SINDy). By transforming time-series measurements from the time domain to the Laplace domain using the Laplace transform and integration by parts, LES-SINDy enables more accurate approximations of derivatives and discontinuous terms. It also effectively handles unbounded growth functions and accumulated numerical errors in the Laplace domain, thereby overcoming challenges in the identification process. The model evaluation process selects the most accurate and parsimonious dynamical systems from multiple candidates. Experimental results across diverse ordinary and partial differential equations show that LES-SINDy achieves superior robustness, accuracy, and parsimony compared to existing methods.

翻译：非线性动力系统稀疏辨识（SINDy）是一种用于数据驱动发现控制方程的有力工具。然而，在对涉及高阶导数或不连续性的复杂动力系统进行建模时，尤其是在存在噪声的情况下，该方法面临挑战。这些局限性限制了其在应用数学和物理学各个领域的适用性。为了缓解这些问题，我们提出了拉普拉斯增强非线性动力系统稀疏辨识（LES-SINDy）。通过利用拉普拉斯变换和分部积分法将时间序列测量值从时域转换到拉普拉斯域，LES-SINDy能够实现对导数项和不连续项更精确的近似。该方法还能有效处理拉普拉斯域中的无界增长函数和累积数值误差，从而克服辨识过程中的挑战。模型评估过程从多个候选模型中选出最精确且最简约的动力系统。在多种常微分方程和偏微分方程上的实验结果表明，与现有方法相比，LES-SINDy在鲁棒性、精确性和简约性方面均表现出更优的性能。