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.

尽管许多最常见的混沌动力系统在时间上是连续的，但对混沌的大部分理解都是通过离散时间映射形成的。Henri Poincaré 首先通过使用低维横向子空间跟踪连续流的连续迭代来建立这种联系。通过流与子空间的连续交叉点迭代动力学的映射现在称为庞加莱映射，它是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很突出。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为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。变量的可逆变化基于自动编码器，它允许降维，并且具有使用拓扑共轭的等价关系对混沌系统进行分类的优点。事实上，拓扑共轭的执行是学习坐标和动力学配对的关键神经网络正则化。我们提供了该方法在低维系统（例如 Rössler 和 Lorenz 系统）上的说明性应用，同时还展示了该方法在无限维系统（例如 Kuramoto-Sivashinsky 方程）上的实用性。