用二次嵌入方法从数据中自动发现非线性系统的真实方程
Data-driven system identification using quadratic embeddings of nonlinear dynamics
- 将复杂非线性系统映射到高维空间，使其动态变为二次形式
- 在多个基准测试中表现优于SINDy和深度学习方法，准确识别方程
- 适合需要精确建模的物理系统研究者，如流体、机械系统
我们提出一种新型数据驱动方法QENDy（非线性动力学二次嵌入），不仅能学习高度非线性动力系统的二次表示，还能识别其控制方程。该方法基于将系统嵌入更高维特征空间，使动态呈现二次特性。与SINDy类似，QENDy需轨迹数据、时间导数（可用有限差分估计）及预选基函数集合（字典）。通过多个基准问题验证其有效性与精度，并与SINDy及深度学习方法对比。进一步分析了QENDy与SINDy在无限数据极限下的收敛性，揭示二者异同，并比较了二次嵌入与基于Koopman算子的线性化技术。
原文摘要 · Abstract (English)
We propose a novel data-driven method called QENDy (Quadratic Embedding of Nonlinear Dynamics) that not only allows us to learn quadratic representations of highly nonlinear dynamical systems, but also to identify the governing equations. The approach is based on an embedding of the system into a higher-dimensional feature space in which the dynamics become quadratic. Just like SINDy (Sparse Identification of Nonlinear Dynamics), our method requires trajectory data, time derivatives for the training data points, which can also be estimated using finite difference approximations, and a set of preselected basis functions, called dictionary. We illustrate the efficacy and accuracy of QENDy with the aid of various benchmark problems and compare its performance with SINDy and a deep learning method for identifying quadratic embeddings. Furthermore, we analyze the convergence of QENDy and SINDy in the infinite data limit, highlight their similarities and main differences, and compare the quadratic embedding with linearization techniques based on the Koopman operator.
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