Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators

TL;DR

This paper introduces a spectral operator-based approach to model and analyze stochastic nonlinear dynamical systems using latent embeddings in reproducing kernel Hilbert spaces, enabling improved state estimation and mode decomposition.

Contribution

It develops a novel spectral method for learning operator-based latent representations of stochastic nonlinear dynamics, integrating neural network embeddings and extending Kalman filtering.

Findings

Effective in modeling synthetic and real-world data

Improves sequential state-estimation in nonlinear systems

Enhances eigen-mode decomposition of dynamics

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.