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

• URL: http://arxiv.org/abs/2509.23975v1
• Date: Sun, 28 Sep 2025 17:01:53 GMT
• Title: Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operators
• Authors: Gianluca Fabiani, Constantinos Siettos, Ioannis G. Kevrekidis,
• Abstract summary: We propose a data-driven alternative that uses local neural operators, trained on microscopic/mesoscopic data, to obtain efficient short-time solution operators.<n> Krylov-Arnoldi then approximate the dominant eigenspectrum, yielding reduced models that capture the open-loop slow dynamics without explicit Jacobian assembly.
• Score: 1.5484595752241122
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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