用拉普拉斯变换提升非线性系统建模精度，尤其适合有噪声或间断的复杂动态系统。
LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systems
- 将时域数据转到拉普拉斯域，用积分技巧更准估算导数和间断项。
- 在多个微分方程测试中，准确率与简洁性均优于现有方法。
- 适合需要高鲁棒性建模的物理、工程等领域的研究人员。
稀疏非线性动力系统识别（SINDy）是一种强大的数据驱动发现控制方程的方法。然而，在涉及高阶导数或不连续性的复杂动力系统中，尤其在噪声环境下，其性能受限。为此，我们提出拉普拉斯增强稀疏非线性动力系统识别（LES-SINDy）。通过使用拉普拉斯变换将时间序列测量从时域转换至拉普拉斯域，并结合分部积分法，该方法可更精确地近似导数与不连续项，有效处理无界增长函数及拉普拉斯域中的累积数值误差，从而克服识别过程中的挑战。模型评估阶段从多个候选系统中选出最准确且最简洁的动力系统。在多种常微分方程与偏微分方程上的实验结果表明，相比现有方法，LES-SINDy 在鲁棒性、准确性和简洁性方面均表现更优。
原文摘要 · Abstract (English)
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.
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