共轭映射的深度学习
Deep Learning of Conjugate Mappings
摘要：
尽管许多最常见的混沌动力系统在时间上是连续的，但对混沌的理解是通过离散时间映射形成的。亨利·庞加莱（Henri Poincaré）首先通过跟踪连续流的连续迭代与低维横向子空间建立了这种联系。通过流与子空间的连续交叉迭代动态的映射现在被称为庞加莱映射，它是用于解释和分类混沌动力学的主要方法。不幸的是，除了最简单的系统之外，在所有系统中，这种映射的显式形式仍然是突出的。这项工作提出了一种通过使用深度学习将可逆坐标变换构造成共轭表示来获得显式庞加莱映射的方法，其中动力学由相对简单的混沌映射控制。变量的可逆变化是基于一个自动编码器，它允许降维，并具有利用拓扑共轭的等价关系分类混沌系统的优点。事实上，拓扑共轭的实施是学习坐标和动力学配对的关键神经网络正则化。我们提供的方法，如Rössler和洛伦兹系统的低维系统的暂时应用，同时也展示了无限维系统，如仓本-Sivashinsky方程的方法的实用性。
英文摘要：
Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincaré map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincaré mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the Rössler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto--Sivashinsky equation.
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。
