非线性动力学端到端建模的神经动态模式分解
Neural Dynamic Mode Decomposition for End-to-End Modeling of Nonlinear Dynamics
摘要：
Koopman谱分析在理解非线性动力系统方面引起了人们的注意，通过该分析，我们可以通过使用非线性函数提升观测来分析具有线性区域的非线性动力学。为了进行分析，我们需要找到一个合适的提升函数。虽然已经提出了几种基于神经网络的提升函数估计方法，但是现有的方法训练神经网络而不进行谱分析。在本文中，我们提出了神经动态模式分解，其中神经网络进行训练，使得预测误差最小化时，动态建模的基础上，在提升空间的谱分解。利用我们提出的方法，预测误差通过神经网络和谱分解反向传播，从而实现Koopman谱分析的端到端学习。当动态的频率或增长率的信息可用时，所提出的方法可以利用它作为训练的正则化器。我们还提出了一个扩展我们的方法时，观察外生控制时间序列的影响。我们的实验证明了我们提出的方法的有效性，在特征值估计和预测性能。
英文摘要：
Koopman spectral analysis has attracted attention for understanding nonlinear dynamical systems by which we can analyze nonlinear dynamics with a linear regime by lifting observations using a nonlinear function. For analysis, we need to find an appropriate lift function. Although several methods have been proposed for estimating a lift function based on neural networks, the existing methods train neural networks without spectral analysis. In this paper, we propose neural dynamic mode decomposition, in which neural networks are trained such that the forecast error is minimized when the dynamics is modeled based on spectral decomposition in the lifted space. With our proposed method, the forecast error is backpropagated through the neural networks and the spectral decomposition, enabling end-to-end learning of Koopman spectral analysis. When information is available on the frequencies or the growth rates of the dynamics, the proposed method can exploit it as regularizers for training. We also propose an extension of our approach when observations are influenced by exogenous control time-series. Our experiments demonstrate the effectiveness of our proposed method in terms of eigenvalue estimation and forecast performance.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
