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Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators

Naichang Ke, Ryogo Tanaka, Yoshinobu Kawahara

TL;DR

Addressing the challenge of learning and forecasting stochastic nonlinear dynamics from partial observations, the paper introduces Embedded Latent Transfer Operators (ELTO) that evolve latent states embedded in an RKHS via a transfer operator. A stochastic realization-based spectral learning method estimates ELTO/EOO from data, with a finite-dimensional latent state $\\mathbf{x}(t)$ and a kernel (potentially neural-network parametrized) embedding. The framework enables generalized sequential state estimation via a kernel Kalman-rule and robust Koopman-mode decomposition (KMD) for nonlinear dynamics. Empirical results on a pendulum, HuMoD human motion, quad-link image sequences, and nonlinear oscillators demonstrate improved state estimation accuracy and resilient spectral estimates under observation and process noise.

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.

Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators

TL;DR

Addressing the challenge of learning and forecasting stochastic nonlinear dynamics from partial observations, the paper introduces Embedded Latent Transfer Operators (ELTO) that evolve latent states embedded in an RKHS via a transfer operator. A stochastic realization-based spectral learning method estimates ELTO/EOO from data, with a finite-dimensional latent state and a kernel (potentially neural-network parametrized) embedding. The framework enables generalized sequential state estimation via a kernel Kalman-rule and robust Koopman-mode decomposition (KMD) for nonlinear dynamics. Empirical results on a pendulum, HuMoD human motion, quad-link image sequences, and nonlinear oscillators demonstrate improved state estimation accuracy and resilient spectral estimates under observation and process noise.

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.
Paper Structure (26 sections, 5 theorems, 35 equations, 4 figures, 4 tables, 2 algorithms)

This paper contains 26 sections, 5 theorems, 35 equations, 4 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Koopman and Transfer Operators
  4. Hilbert-Space Distribution Embeddings
  5. Embedded Latent Transfer Operator
  6. Stochastic Observation:
  7. Embedded Latent Markov Process:
  8. Spectral Learning
  9. Stochastic Realization
  10. Empirical Estimation
  11. Sequential State Estimation
  12. Pendulum:
  13. Human Motion Dynamics (HuMoD):
  14. Image Sequence from a Quad-link Pendulum Simulation:
  15. Mode Decomposition
  16. ...and 11 more sections

Key Result

Lemma 1

Suppose that $\mathrm{rank}({\mathbf{H}})<\infty$. Then, $\mathbb{X}_t$ is finite dimensional.

Figures (4)

  • Figure 1: Schematic overview of the embedded latent transfer operator (ELTO) $\mathcal{T}_{e,\theta}$.
  • Figure 2: Comparison of prediction performance by ELTO (ours) and KKR for the pendulum data.
  • Figure 3: Comparison of prediction performance by ELTO (ours) and KKR for the HuMoD data. Task 1: Prediction of height of T8 when walking at 1.0 m/s and Task 2: Prediction of height of T12 when walking at 1.5 m/s.
  • Figure 4: Results for VDP oscillator

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3: Embedded Latent Transfer Operator
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Theorem 1
  • Corollary 1