Learning Theory for Dynamical Systems
Learning Theory for Dynamical Systems
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DOI：
10.1137/22m1516865
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发表时间：
2022-08
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影响因子：
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通讯作者：
Tyrus Berry;Suddhasattwa Das
中科院分区：
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作者：
Tyrus Berry;Suddhasattwa Das
动力系统建模和预测的任务是最古老的问题之一，并且仍然具有挑战性。概括地说，该任务有两个子任务：从部分观察中提取完整的动态信息；然后明确地从这些信息中学习动态。我们提出了一个数学框架，其中动态信息以嵌入的形式表示。该框架使用空间、地图和交换语言将两个子任务结合起来。该框架还统一了两种最常见的学习范式——延迟坐标和储层计算。我们使用这个框架作为重建系统的另外两项研究的平台 - 其动态稳定性；以及迭代下误差的增长。我们表明，这些问题与底层系统的更基本属性密切相关 - 矩阵余循环在基本动力学上的行为、其非均匀双曲行为及其相关性的衰减。因此，我们的框架弥补了普遍观察到的动力学建模行为之间的差距；以及动力学固有的谱、微分和遍历特性。
The task of modelling and forecasting a dynamical system is one of the oldest problems, and it remains challenging. Broadly, this task has two subtasks - extracting the full dynamical information from a partial observation; and then explicitly learning the dynamics from this information. We present a mathematical framework in which the dynamical information is represented in the form of an embedding. The framework combines the two subtasks using the language of spaces, maps, and commutations. The framework also unifies two of the most common learning paradigms - delay-coordinates and reservoir computing. We use this framework as a platform for two other investigations of the reconstructed system - its dynamical stability; and the growth of error under iterations. We show that these questions are deeply tied to more fundamental properties of the underlying system - the behavior of matrix cocycles over the base dynamics, its non-uniform hyperbolic behavior, and its decay of correlations. Thus, our framework bridges the gap between universally observed behavior of dynamics modelling; and the spectral, differential and ergodic properties intrinsic to the dynamics.