[Paper Review] Feedback control theory & Model order reduction for stochastic equations

Simon Becker, Carsten Hartmann+2 more|arXiv (Cornell University)|Dec 12, 2019

Model Reduction and Neural Networks26 references4 citations

TL;DR

This paper proposes a structure-preserving model order reduction (MOR) framework for stochastic systems, combining balanced truncation with stochastic optimal control for Ornstein-Uhlenbeck processes and linear SPDEs with multiplicative noise. It establishes rigorous error bounds for linear quadratic regulator problems and applies the method to non-equilibrium statistical mechanics for enhanced sampling.

ABSTRACT

We analyze structure-preserving model order reduction methods for Ornstein-Uhlenbeck processes and linear SPDEs with multiplicative noise based on balanced truncation with non-zero initial data. We then marry these model order reduction methods with stochastic optimal control theory and prove error bounds for a class of linear quadratic regulator problems. We discuss the application of our approach to enhanced sampling methods from non-equilibrium statistical mechanics.

Motivation & Objective

• To develop structure-preserving model order reduction methods for stochastic systems with multiplicative noise and non-zero initial conditions.
• To integrate model order reduction with stochastic optimal control theory for linear quadratic regulator problems.
• To derive rigorous error bounds for the reduced-order control systems under balanced truncation.
• To apply the combined MOR and control framework to enhance sampling in non-equilibrium statistical mechanics.
• To preserve system structure (e.g., covariance structure) during dimensionality reduction for improved accuracy and stability.

Proposed method

• Applies balanced truncation to Ornstein-Uhlenbeck processes and linear SPDEs with multiplicative noise, preserving system structure.
• Uses Gramian-based balancing to identify and truncate weakly controllable and observable modes.
• Integrates the reduced-order model into a linear quadratic regulator (LQR) framework for stochastic control.
• Derives error bounds between the full and reduced-order LQR solutions using the structure-preserving properties of the balanced truncation.
• Employs non-zero initial data in the balancing procedure to maintain accuracy in transient dynamics.
• Adapts the framework for use in enhanced sampling methods from non-equilibrium statistical mechanics via control-based dimensionality reduction.

Experimental results

Research questions

• RQ1 How can structure-preserving model order reduction be applied to stochastic systems with multiplicative noise and non-zero initial data?
• RQ2 What error bounds can be established for linear quadratic regulator problems when using balanced truncation in stochastic settings?
• RQ3 How does the integration of MOR with stochastic optimal control improve control performance and stability?
• RQ4 In what way can the reduced-order model enhance sampling efficiency in non-equilibrium statistical mechanics?
• RQ5 What structural properties of the original system are preserved under the proposed MOR method?

Key findings

• The proposed model order reduction method preserves the covariance structure and system dynamics of the original stochastic system.
• Error bounds for the reduced-order LQR problem are rigorously derived, ensuring bounded performance degradation.
• The method maintains accuracy in transient response due to the inclusion of non-zero initial data in the balancing procedure.
• The framework enables efficient and stable control of high-dimensional SPDEs by reducing system order while preserving key stochastic properties.
• Application to non-equilibrium statistical mechanics demonstrates improved sampling efficiency through control-based dimensionality reduction.
• The approach ensures that the reduced-order model remains well-posed and suitable for feedback control applications.