Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators

2025-01-06

AI Summary

Identifying latent states and decoupling dynamic modes in stochastic nonlinear dynamical systems remains challenging due to inherent nonlinearity and randomness. Method: This paper proposes a hidden Markov operator representation framework embedded in a reproducing kernel Hilbert space (RKHS). It uniquely integrates stochastic realization theory with kernel embedding techniques to jointly learn latent-space structure and state-transition operators; extends the Kalman filter to nonlinear stochastic systems; and introduces operator-theoretic dynamic mode decomposition (O-DMD). The technical pipeline encompasses spectral learning, neural-network-based adaptive kernel construction, generalized sequential state estimation, and operator spectral analysis. Results: Evaluated on synthetic and real-world datasets, the method significantly improves latent-state estimation accuracy, enables interpretable and decoupled extraction of dynamical modes, and establishes a unified, robust, and interpretable paradigm for modeling and inference in stochastic nonlinear systems.

Technology Category

• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Complex Systems
• Hidden State Transition
• Continuous State Estimation

Innovation

• Stochastic Learning
• Hidden State Transitions
• Dynamic Pattern Decomposition