Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operators

2025-09-28

AI Summary

Controlling high-dimensional distributed parameter systems (DPS) without explicit partial differential equation (PDE) descriptions remains challenging due to the absence of governing equations and prohibitive computational costs of traditional model-based approaches. Method: This paper proposes a fully data-driven closed-loop feedback control framework comprising: (i) learning short-time solution operators directly from spatiotemporal data via local neural operators—bypassing explicit PDE modeling and Jacobian assembly; (ii) integrating Krylov–Arnoldi iteration with matrix-free techniques for steady-state computation and dynamic model reduction; and (iii) designing controllers in the reduced space using discrete linear quadratic regulator (dLQR) and pole placement. Contribution/Results: To the best of our knowledge, this is the first work synergizing local neural operators with Krylov subspace methods for both open-loop slow-dynamics identification and closed-loop control of DPS. The approach achieves efficient, stable, and computationally lightweight control without requiring coarse-grained PDE models, significantly enhancing the feasibility and practicality of data-driven control for high-dimensional distributed systems.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Intelligent Robots: State Estimation

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The control of high-dimensional distributed parameter systems (DPS) remains a challenge when explicit coarse-grained equations are unavailable. Classical equation-free (EF) approaches rely on fine-scale simulators treated as black-box timesteppers. However, repeated simulations for steady-state computation, linearization, and control design are often computationally prohibitive, or the microscopic timestepper may not even be available, leaving us with data as the only resource. We propose a data-driven alternative that uses local neural operators, trained on spatiotemporal microscopic/mesoscopic data, to obtain efficient short-time solution operators. These surrogates are employed within Krylov subspace methods to compute coarse steady and unsteady-states, while also providing Jacobian information in a matrix-free manner. Krylov-Arnoldi iterations then approximate the dominant eigenspectrum, yielding reduced models that capture the open-loop slow dynamics without explicit Jacobian assembly. Both discrete-time Linear Quadratic Regulator (dLQR) and pole-placement (PP) controllers are based on this reduced system and lifted back to the full nonlinear dynamics, thereby closing the feedback loop.

Problem

• Control high-dimensional distributed systems without explicit coarse-grained equations
• Reduce computational cost of repeated simulations for control design
• Develop data-driven controllers when microscopic simulators are unavailable

Innovation

• Local neural operators replace fine-scale simulators
• Krylov subspace methods compute coarse steady states
• Reduced models enable dLQR and pole-placement controllers