用柯尔莫哥洛夫算子理论分析随机系统中不完全观测的影响
Koopman operator-based discussion on partial observation in stochastic systems
- 基于柯尔莫哥洛夫算子理论研究随机系统的部分观测问题
- 发现延迟嵌入可有效缓解部分观测误差，误差随噪声呈幂律下降
- 揭示状态空间与函数空间区分的重要性，适合动力系统研究者
在某些情况下，难以获取完整可观测变量，部分观测成为必要。对于确定性系统，莫里-曾瓦齐格形式化提供了处理部分观测的理论框架。近年来，基于柯尔莫哥洛夫算子理论的数据驱动算法取得显著进展，有研究尝试将莫里-曾瓦扎格形式化与柯尔莫哥洛夫算子理论相连接。本文利用柯尔莫哥洛夫算子理论讨论随机系统中部分观测的影响。研究澄清了在随机系统中区分状态空间与函数空间的重要性。即使在随机系统中，延迟嵌入技术仍对部分观测有益。多个数值实验表明，误差随加性噪声幅值呈现幂律行为。同时讨论了幂律指数与部分观测效应之间的关系。
原文摘要 · Abstract (English)
It is sometimes difficult to achieve a complete observation for a full set of observables, and partial observations are necessary. For deterministic systems, the Mori-Zwanzig formalism provides a theoretical framework for handling partial observations. Recently, data-driven algorithms based on the Koopman operator theory have made significant progress, and there is a discussion to connect the Mori-Zwanzig formalism with the Koopman operator theory. In this work, we discuss the effects of partial observation in stochastic systems using the Koopman operator theory. The discussion clarifies the importance of distinguishing the state space and the function space in stochastic systems. Even in stochastic systems, the delay-embedding technique is beneficial for partial observation, and several numerical experiments show a power-law behavior of error with respect to the amplitude of the additive noise. We also discuss the relation between the exponent of the power-law behavior and the effects of partial observation.
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