Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators
This work addresses the challenge of modeling complex stochastic dynamics for applications in fields like control or data analysis, but it appears incremental as it builds on existing operator-based and kernel methods.
The authors tackled the problem of learning stochastic nonlinear dynamical systems by developing a spectral method based on operator-based latent Markov representations and stochastic realization theory, with results demonstrated through examples on synthetic and real-world data for tasks like state-estimation and mode decomposition.
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.