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.

翻译：当显式粗粒度方程不可得时，高维分布参数系统的控制仍具挑战性。经典的无方程方法依赖于被视为黑箱时间步进器的精细尺度模拟器。然而，为稳态计算、线性化及控制设计而进行的重复模拟通常在计算上不可行，甚至微观时间步进器可能根本不可用，仅留下数据作为唯一资源。我们提出一种数据驱动的替代方案，利用在时空微观/介观数据上训练的局部神经算子，以获取高效的短时解算子。这些替代模型被用于Krylov子空间方法中，以计算粗粒度稳态与非稳态，同时以无矩阵方式提供雅可比信息。随后，Krylov-Arnoldi迭代近似主导特征谱，产生能捕获开环慢速动力学而无需显式组装雅可比矩阵的降阶模型。离散时间线性二次调节器与极点配置控制器均基于此降阶系统构建，并提升回完整的非线性动力学，从而形成闭环反馈控制。