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Meta-State-Space Learning: An Identification Approach for Stochastic Dynamical Systems

Gerben I. Beintema, Maarten Schoukens, Roland Tóth

TL;DR

This work introduces a meta-state-space (MSS) framework to identify nonlinear stochastic dynamical systems without restrictive noise assumptions. By representing the state distribution with a deterministic meta-state $z_t$ evolving as $z_{t+1}=f_z(z_t,u_t)$ and recovering $p^x_t=M^{\dagger}(z_t)$ and $p^y_t=h_z(z_t,u_t)$, the authors transform stochastic identification into a deterministic-like learning problem. They propose an ANN-based estimator that parameterizes $f_\theta$ and a Mixture Density Network for $p_\theta(y|z,u)$, and solve a MAP optimization with multiple shooting to efficiently learn from IO data. Simulation on a challenging nonlinear stochastic system shows mean log-likelihood close to a theoretical upper bound, indicating that MSS can capture complete output distributions with high fidelity. The approach offers a scalable, data-driven path to accurate probabilistic modeling of complex stochastic dynamics, with avenues for extending to multi-step predictions and filtering tasks.

Abstract

Available methods for identification of stochastic dynamical systems from input-output data generally impose restricting structural assumptions on either the noise structure in the data-generating system or the possible state probability distributions. In this paper, we introduce a novel identification method of such systems, which results in a dynamical model that is able to produce the time-varying output distribution accurately without taking restrictive assumptions on the data-generating process. The method is formulated by first deriving a novel and exact representation of a wide class of nonlinear stochastic systems in a so-called meta-state-space form, where the meta-state can be interpreted as a parameter vector of a state probability function space parameterization. As the resulting representation of the meta-state dynamics is deterministic, we can capture the stochastic system based on a deterministic model, which is highly attractive for identification. The meta-state-space representation often involves unknown and heavily nonlinear functions, hence, we propose an Artificial Neural Network (ANN)-based identification method capable of efficiently learning nonlinear meta-state-space models. We demonstrate that the proposed identification method can obtain models with a log-likelihood close to the theoretical limit even for highly nonlinear, highly stochastic systems.

Meta-State-Space Learning: An Identification Approach for Stochastic Dynamical Systems

TL;DR

This work introduces a meta-state-space (MSS) framework to identify nonlinear stochastic dynamical systems without restrictive noise assumptions. By representing the state distribution with a deterministic meta-state evolving as and recovering and , the authors transform stochastic identification into a deterministic-like learning problem. They propose an ANN-based estimator that parameterizes and a Mixture Density Network for , and solve a MAP optimization with multiple shooting to efficiently learn from IO data. Simulation on a challenging nonlinear stochastic system shows mean log-likelihood close to a theoretical upper bound, indicating that MSS can capture complete output distributions with high fidelity. The approach offers a scalable, data-driven path to accurate probabilistic modeling of complex stochastic dynamics, with avenues for extending to multi-step predictions and filtering tasks.

Abstract

Available methods for identification of stochastic dynamical systems from input-output data generally impose restricting structural assumptions on either the noise structure in the data-generating system or the possible state probability distributions. In this paper, we introduce a novel identification method of such systems, which results in a dynamical model that is able to produce the time-varying output distribution accurately without taking restrictive assumptions on the data-generating process. The method is formulated by first deriving a novel and exact representation of a wide class of nonlinear stochastic systems in a so-called meta-state-space form, where the meta-state can be interpreted as a parameter vector of a state probability function space parameterization. As the resulting representation of the meta-state dynamics is deterministic, we can capture the stochastic system based on a deterministic model, which is highly attractive for identification. The meta-state-space representation often involves unknown and heavily nonlinear functions, hence, we propose an Artificial Neural Network (ANN)-based identification method capable of efficiently learning nonlinear meta-state-space models. We demonstrate that the proposed identification method can obtain models with a log-likelihood close to the theoretical limit even for highly nonlinear, highly stochastic systems.
Paper Structure (10 sections, 27 equations, 3 figures, 2 tables)

This paper contains 10 sections, 27 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The meta-state-space representation
  3. Identification by meta-state-space learning
  4. The identification problem
  5. Neural meta-state-space estimator
  6. Simulation studies
  7. Conclusion
  8. Mixture Density Network parametrization
  9. Mean log-likelihood upper limit
  10. Estimation of the Gaussian baselines

Figures (3)

  • Figure 1: Both $\mathcal{S}^\mathbb{X}$ and the meta-state-space with mapping $M$ and inverse mapping $M^{\dagger}$ are visualized, showing that a transition from $z_t$ to $z_{t+1}$ can be computed in two ways by following either the blue or the red path and thus $z_{t+1} = f_\mathrm{z}(z_t,u_t) = M(F(M^{\dagger}(z_t),u_t))$.
  • Figure 2: A graphical representation of the evolution of a PDF of the state according to \ref{['eq:pdf-propagation-old']} and the evolution of the meta-state in terms of \ref{['eq:meta-state-space']}. This figure shows that meta-state vectors $z_t$ can represent the state distribution $p_t^\mathrm{x}$ through the mapping $M$.
  • Figure 3: Probability density histograms of the system output under different noise realizations compared with the predicted output distribution of the meta-state-space model. Since the model output distribution is parameterized as a weighted sum of normal distributions, these weighted components are also displayed.