Equation-Free Coarse Control of Distributed Parameter Systems via Local Neural Operators
This work addresses the computational burden in controlling distributed parameter systems for applications like fluid dynamics or materials science, offering a novel data-driven approach that is incremental in combining neural operators with Krylov methods.
The paper tackles the challenge of controlling high-dimensional distributed parameter systems without explicit coarse-grained equations by proposing a data-driven method using local neural operators to efficiently compute coarse states and Jacobian information, enabling the design of discrete-time LQR and pole-placement controllers that achieve closed-loop feedback control.
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.