论文标题
Koopman操作员，几何和学习
Koopman Operator, Geometry, and Learning
论文作者
论文摘要
我们提供了一个框架，以学习植根于表示和库普曼运营商概念的动态系统。这两个之间的相互作用导致可以根据Koopman运算符频谱的属性在有限维度线性地表示系统的完整描述。状态空间的几何形状都连接到表示的概念，无论是在线性情况下与特征函数的关节水平相关的，在非线性表示情况下。如下所示，即使是非线性有限维表示，也可以使用Koopman操作员框架来学习，从而导致一类新的表示特征问题。给出了使用神经网络学习的联系。提供了Koopman操作员理论到不同空间之间“静态”地图的扩展。讨论了Koopman操作员频谱对Mori-Zwanzig类型表示的影响。
We provide a framework for learning of dynamical systems rooted in the concept of representations and Koopman operators. The interplay between the two leads to the full description of systems that can be represented linearly in a finite dimension, based on the properties of the Koopman operator spectrum. The geometry of state space is connected to the notion of representation, both in the linear case - where it is related to joint level sets of eigenfunctions - and in the nonlinear representation case. As shown here, even nonlinear finite-dimensional representations can be learned using the Koopman operator framework, leading to a new class of representation eigenproblems. The connection to learning using neural networks is given. An extension of the Koopman operator theory to "static" maps between different spaces is provided. The effect of the Koopman operator spectrum on Mori-Zwanzig type representations is discussed.
