Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition
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发表时间：
2017-10
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通讯作者：
Naoya Takeishi;Y. Kawahara;T. Yairi
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作者：
Naoya Takeishi;Y. Kawahara;T. Yairi
库普曼算子的谱分解作为非线性动力系统分析的工具而引起人们的关注。动态模式分解是库普曼谱分析的一种流行数值算法；然而，我们经常需要根据潜在的动力学手动准备非线性可观测量，这并不总是可能的，因为我们可能没有任何关于它们的先验知识。在本文中，我们基于从观测数据中学习库普曼不变子空间的原理，提出了一种完全数据驱动的库普曼谱分析方法。为此，我们提出最小化线性最小二乘回归的残差平方和来估计一组函数，将数据转换为线性回归拟合良好的形式。我们介绍了神经网络的实现，并使用非线性动力系统和应用程序根据经验评估性能。
Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral analysis; however, we often need to prepare nonlinear observables manually according to the underlying dynamics, which is not always possible since we may not have any a priori knowledge about them. In this paper, we propose a fully data-driven method for Koopman spectral analysis based on the principle of learning Koopman invariant subspaces from observed data. To this end, we propose minimization of the residual sum of squares of linear least-squares regression to estimate a set of functions that transforms data into a form in which the linear regression fits well. We introduce an implementation with neural networks and evaluate performance empirically using nonlinear dynamical systems and applications.