用谱方法学习随机非线性系统的隐变量演化规律
Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators
- 基于再生核希尔伯特空间构建隐状态转移算子
- 实现非线性系统状态估计与特征模态分解
- 适合研究复杂动态系统的建模与预测
我们提出一种基于算子的隐马尔可夫表示方法，用于描述嵌入在再生核希尔伯特空间中的随机非线性动力系统。通过随机实现理论，采用谱方法学习该表示，其中隐状态的随机演化由对应的转移算子刻画。利用前馈神经网络构建的再生核可同时学习嵌入映射。此外，我们拓展了非线性系统中序列状态估计（如卡尔曼滤波）和基于算子的特征模态分解方法。通过合成数据和真实世界数据的多个实例验证了所提方法的性能，展示了其在状态估计和模态分解方面的有效性和实用性。
原文摘要 · Abstract (English)
We consider an operator-based latent Markov representation of a stochastic nonlinear dynamical system, where the stochastic evolution of the latent state embedded in a reproducing kernel Hilbert space is described with the corresponding transfer operator, and develop a spectral method to learn this representation based on the theory of stochastic realization. The embedding may be learned simultaneously using reproducing kernels, for example, constructed with feed-forward neural networks. We also address the generalization of sequential state-estimation (Kalman filtering) in stochastic nonlinear systems, and of operator-based eigen-mode decomposition of dynamics, for the representation. Several examples with synthetic and real-world data are shown to illustrate the empirical characteristics of our methods, and to investigate the performance of our model in sequential state-estimation and mode decomposition.
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