DySLIM：基于不变测度的混沌系统动力学稳定学习
DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems
AI总结：
针对混沌系统轨迹学习易发散的问题，提出DySLIM框架，通过同时学习不变测度与动力学，以可扩展正则化项实现稳定训练，提升逐点跟踪和长期统计精度。
摘要：
从耗散混沌系统中学习动力学因其固有的不稳定性而极为困难，这种不稳定性由其正的Lyapunov指数所形式化，该指数会以指数方式放大所学动力学中的误差。然而，许多此类系统表现出遍历性和吸引子：一个紧致且高度复杂的流形，轨迹在有限时间内收敛到该流形上，并且该流形支撑一个不变测度，即在动力学作用下保持不变的概率分布，它决定了系统的长期统计行为。在本工作中，我们利用这一结构提出了一个新的框架，目标是同时学习不变测度和动力学，这与仅针对轨迹之间失配的典型方法形成对比，后者在轨迹长度增加时往往会导致发散。我们利用该框架提出了一个易于处理且样本高效的目标函数，可以与任何现有的学习目标结合使用。我们提出的基于不变测度的动力学稳定学习（DySLIM）目标函数使得模型训练能够相对于其他学习目标实现更好的逐点跟踪和长期统计精度。通过以可扩展的正则化项来针对该分布，我们希望这种方法能够扩展到表现出缓慢变化分布的更复杂系统，例如天气和气候模型。
英文摘要：
Learning dynamics from dissipative chaotic systems is notoriously difficult due to their inherent instability, as formalized by their positive Lyapunov exponents, which exponentially amplify errors in the learned dynamics. However, many of these systems exhibit ergodicity and an attractor: a compact and highly complex manifold, to which trajectories converge in finite-time, that supports an invariant measure, i.e., a probability distribution that is invariant under the action of the dynamics, which dictates the long-term statistical behavior of the system. In this work, we leverage this structure to propose a new framework that targets learning the invariant measure as well as the dynamics, in contrast with typical methods that only target the misfit between trajectories, which often leads to divergence as the trajectories' length increases. We use our framework to propose a tractable and sample efficient objective that can be used with any existing learning objectives. Our Dynamics Stable Learning by Invariant Measure (DySLIM) objective enables model training that achieves better point-wise tracking and long-term statistical accuracy relative to other learning objectives. By targeting the distribution with a scalable regularization term, we hope that this approach can be extended to more complex systems exhibiting slowly-variant distributions, such as weather and climate models.
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。
