标题： 学习非线性动力学的全局线性表示 显示英文标题

标题： Learning Global Linear Representations of Nonlinear Dynamics

摘要： 虽然线性系统已被很好地理解，但不存在一般非线性系统的显式解。 一种使线性系统的理解适用于非线性环境的经典方法是通过线性模型表示非线性系统。 尽管已经取得了将线性化技术扩展到更大域和更复杂吸引子几何结构的进展，但最近的研究强调了这些技术在应用于非线性动力学（如具有共存吸引子的动力学）时的局限性。 在这项工作中，我们展示了具有连续Koopman谱、极限环和共存解的非线性动力学，这些动力学可以全局线性化。 为此，我们明确构造了模仿这些非线性行为的线性系统。 随后，我们用深度神经网络近似线性和非线性系统之间的变换。 这种方法得到的有限维线性化最多比底层线性系统的相空间维度高一维。 显示英文摘要

摘要： While linear systems are well-understood, no explicit solution for general nonlinear systems exists. A classical approach to make the understanding of linear system available in the nonlinear setting is to represent a nonlinear system by a linear model. While progress has been made in extending linearization techniques to larger domains and more complex attractor geometries, recent work has highlighted the limitations of these techniques when applied to nonlinear dynamics, such as those with coexisting attractors. In this work, we show nonlinear dynamics with a continuous Koopman spectrum, a limit cycle, and coexisting solutions that can be globally linearized. To this end, we explicitly construct linear systems mimicking these nonlinear behaviors. Subsequently, we approximate transformations between linear and nonlinear systems with deep neural networks. This approach yields finite dimensional linearizations exceeding the phase space dimension of the underlying linear system by one at most.